New Trends in the Theory of Hyperbolic Equations

Operator Theory: Advances and Applications Vol. 159 Editor: I. Gohberg Editorial Office: School of Mathematical Scienc...

Author: Michael Reissig | Bert-Wolfgang Schulze

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Operator Theory: Advances and Applications Vol. 159 Editor: I. Gohberg

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) G. Heinig (Chemnitz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) T. Kailath (Stanford) P. D. Lax (New York) M. S. Livsic (Beer Sheva)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Institut für Mathematik Universität Potsdam 14415 Potsdam Germany

Sergio Albeverio Institut für Angewandte Mathematik Universität Bonn 53115 Bonn Germany

Michael Demuth Institut für Mathematik Technische Universität Clausthal 38678 Clausthal-Zellerfeld Germany

Elmar Schrohe Institut für Mathematik Universität Hannover 30060 Hannover Germany

New Trends in the Theory of Hyperbolic Equations

Michael Reissig Bert-Wolfgang Schulze Editors

Advances in Partial Differential Equations

Birkhäuser Verlag Basel . Boston . Berlin

Editors: Michael Reissig TU Bergakademie Freiberg Fakultät für Mathematik und Informatik Institut für Angewandte Analysis 09596 Freiberg Germany e-mail: [email protected]

Bert-Wolfgang Schulze Institut für Mathematik Universität Potsdam 14415 Potsdam Germany e-mail: [email protected]

2000 Mathematics Subject Classification 35L05, 35L10

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at

ISBN 3-7643-7283-4 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Cover design: Heinz Hiltbrunner, Basel Printed in Germany ISBN-10: 3-7643-7283-4 ISBN-13: 978-3-7643-7283-5 987654321

www.birkhauser.ch

Contents Editorial Preface

xi

Wave Maps and Ill-posedness of their Cauchy Problem Piero D’Ancona and Vladimir Georgiev

1

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Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational motivation of the wave maps equations . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Harmonic maps and special harmonic maps on the sphere . 2.3 Equivariant wave maps and construction of special solutions Local existence result for equivariant wave maps . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Localization in time . . . . . . . . . . . . . . . . . . . . . . 3.3 Estimates for the homogeneous problem . . . . . . . . . . . 3.4 Estimates for the non-homogeneous problem . . . . . . . . 3.5 Bilinear estimates for the homogeneous problem in H s,δ . . 3.6 Bilinear estimates in H s,δ for the inhomogeneous problem . Concentration of the local energy . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Construction of the solutions . . . . . . . . . . . . . . . . . 4.3 Higher regularity of the solution . . . . . . . . . . . . . . . 4.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-uniqueness result in the subcritical case . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Equivariant and self-similar solutions . . . . . . . . . . . . . 5.3 Low regularity self-similar solutions . . . . . . . . . . . . . 5.4 Appendix A: The self-similar ODE . . . . . . . . . . . . . . 5.5 Appendix B: Some technical lemmas . . . . . . . . . . . . . Ill-posedness in the critical case (Fourier analysis approach) . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Well-posedness of the Cauchy problem for semilinear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 4 6 12 17 17 20 24 26 31 36 40 40 42 46 52 55 55 57 60 66 72 77 77 79

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6.3 The wave map system in stereographic projection . . . . . 6.4 Conclusion of the proof of Theorem 6.1 . . . . . . . . . . 6.5 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . Ill-posedness in the critical case (fundamental solution approach) 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . 7.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 80 . 83 . 86 . 90 . 90 . 92 . 102 . 109

On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations Hideo Kubo and Masahito Ohta 1 2

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Introduction . . . . . . . . . . . . . Single wave equation . . . . . . . . 2.1 Blow-up . . . . . . . . . . . 2.2 Small data global existence 2.3 Almost global existence . . 2.4 Self-similar solution . . . . 2.5 Asymptotic behavior . . . . Semilinear system, I . . . . . . . . 3.1 Blow-up . . . . . . . . . . . 3.2 Small data global existence 3.3 Self-similar solution . . . . 3.4 Asymptotic behavior . . . . Semilinear system, II . . . . . . . . 4.1 Small data global existence 4.2 Self-similar solution . . . . 4.3 Generalization . . . . . . . Semilinear system, III . . . . . . . 5.1 Blow-up . . . . . . . . . . . Small data global existence 5.2 Nonlinear system . . . . . . . . . . 6.1 Blow-up . . . . . . . . . . . 6.2 Null condition . . . . . . . Appendix . . . . . . . . . . . . . . References . . . . . . . . . . . . . .

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Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains Mitsuhiro Nakao 1 2

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213

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

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Local energy decay . . . . . . . . . . . . . . . . . . . . . . . . . . 218 3.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 218 3.2 Proof of Theorem 3.1. . . . . . . . . . . . . . . . . . . . . . 219 3.3 Proof of Corollary 3.1. . . . . . . . . . . . . . . . . . . . . . 223 Total Energy decay for the wave equation with a localized dissipation226 4.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 226 4.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . 227 4.3 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . 232 Linear equations with variable coeﬃcients; Unique continuation property and a basic inequality . . . . . . . . . . . . . . . . . . . . . . . 233 5.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 233 5.2 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . 235 5.3 Proof of Theorems 5.1 and 5.2 . . . . . . . . . . . . . . . . 237 5.4 Proof of Proposition 5.2 . . . . . . . . . . . . . . . . . . . . 239 Lp estimates for the wave equation in exterior domains . . . . . . . 241 6.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 241 6.2 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . 243 Semilinear wave equations . . . . . . . . . . . . . . . . . . . . . . . 249 7.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 249 7.2 Proof of Theorems 7.1 and 7.2 . . . . . . . . . . . . . . . . 251 7.3 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . 253 Quasilinear wave equations . . . . . . . . . . . . . . . . . . . . . . 259 8.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 259 8.2 Energy decay for the quasilinear equation . . . . . . . . . . 261 8.3 Estimation of higher-order derivatives of solutions . . . . . 265 8.4 Proof of Theorems 8.2 and 8.3. . . . . . . . . . . . . . . . . 272 The wave equation with a half-linear dissipation . . . . . . . . . . 280 9.1 Problem and result . . . . . . . . . . . . . . . . . . . . . . . 280 9.2 A basic inequality . . . . . . . . . . . . . . . . . . . . . . . 283 9.3 Proof of Theorem 9.1 . . . . . . . . . . . . . . . . . . . . . 286 9.4 Proof of Theorems 9.2 and 9.3 . . . . . . . . . . . . . . . . 289 Some open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 293 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation Karen Yagdjian 301 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Counterexamples to the global existence . . . . . . . . Blow-up for the problem with large potential energy of Parametric resonance and wave map type equations .

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301 303 320 324

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6 7 8 9 10 11 12

Proof 5.1 5.2 5.3

of Theorem 4.1: Parametric resonance . . . . . . . . . . . . . Some properties of the Hill’s equation . . . . . . . . . . . . Borg’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . Construction of an exponentially increasing solution to Hill’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Construction of blow-up solutions . . . . . . . . . . . . . . Coeﬃcient stabilizing to a periodic one. Parametric resonance dominates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 6.1: Perturbation theory . . . . . . . . . . . . . . Nonexistence for equations with permanently restricted domain of inﬂuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global existence for a model equation with a polynomially growing coeﬃcient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example with an exponentially growing coeﬃcient . . . . . . . Fast oscillating coeﬃcients: no resonance ?! . . . . . . . . . . . . . Linear wave equations with oscillating coeﬃcients . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

On the Nonlinear Cauchy Problem Massimo Cicognani and Luisa Zanghirati 1 2

3

4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Well-posedness in C ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Function and symbol spaces . . . . . . . . . . . . . . . . . . 2.2 Levi conditions . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The linear problem . . . . . . . . . . . . . . . . . . . . . . . 2.5 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The equivalent quasilinear system . . . . . . . . . . . . . . 2.7 Local C ∞ solutions . . . . . . . . . . . . . . . . . . . . . . . 2.8 Analytic regularity . . . . . . . . . . . . . . . . . . . . . . . Well-posedness in Gevrey classes . . . . . . . . . . . . . . . . . . . 3.1 The linear problem . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gevrey-Levi conditions . . . . . . . . . . . . . . . . . . . . 3.3 Factorization under Gevrey-Levi conditions . . . . . . . . . 3.4 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The equivalent quasilinear system in Gevrey spaces . . . . . 3.6 Local Gevrey solutions and propagation of the analytic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strictly hyperbolic equations with non-Lipschitz coeﬃcients and C ∞ solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Log-Lipschitz coeﬃcients or unbounded derivatives . . . . . 4.2 The linear problem with non-regular coeﬃcients . . . . . . 4.3 The map u → v . . . . . . . . . . . . . . . . . . . . . . . . .

327 327 334 338 342 344 345 354 358 362 370 373 382

387 387 392 392 394 397 398 403 407 409 410 416 416 419 422 423 427 428 430 430 432 436

Contents 5

Holder ¨ coeﬃcients and Gevrey Solutions . . . . 5.1 Gevrey well-posedness . . . . . . . . . . 5.2 From the factorization to the quasilinear References . . . . . . . . . . . . . . . . . . . . .

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Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators Michael Dreher and Ingo Witt 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Well-posedness of the Cauchy problem . . . . . . . . . 1.2 Degenerate diﬀerential operators . . . . . . . . . . . . 1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Formulation of the results . . . . . . . . . . . . . . . . . . . . 2.1 Motivation and plan of the paper . . . . . . . . . . . . 2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . 3 A model case . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Taniguchi–Tozaki’s example . . . . . . . . . . . . . . . 3.2 Conversion into a 2 × 2 system . . . . . . . . . . . . . 3.3 Estimation of the fundamental matrix . . . . . . . . . 3.4 Function spaces: An approach via edge Sobolev spaces 3.5 Establishing energy estimates . . . . . . . . . . . . . . 3.6 Summary of Section 3 . . . . . . . . . . . . . . . . . . 4 Symbol classes and function spaces . . . . . . . . . . . . . . . 4.1 The symbol classes S m,η;λ . . . . . . . . . . . . . . . . 4.2 The symbol classes S˜m,η;λ . . . . . . . . . . . . . . . . m,η;λ 4.3 The symbol classes S+ for η ∈ Cb∞ (Rn ; R) . . . . . 4.4 Function spaces: An approach via weight functions . . 4.5 Summary of Section 4 . . . . . . . . . . . . . . . . . . 5 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . 5.1 Improvement of G˚ ˚ arding’s inequality . . . . . . . . . . 5.2 Symmetric-hyperbolic systems . . . . . . . . . . . . . 5.3 Symmetrizable-hyperbolic systems . . . . . . . . . . . 5.4 Higher-order scalar equations . . . . . . . . . . . . . . 5.5 Local uniqueness . . . . . . . . . . . . . . . . . . . . . 5.6 Sharpness of energy estimates . . . . . . . . . . . . . . A Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . B Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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449 449 450 451 454 455 455 457 462 463 464 464 465 470 470 471 471 472 474 475 478 478 480 481 483 487 489 491 498 506 507

Editorial Preface Hyperbolic partial diﬀerential equations describe phenomena of material or wave transport in the applied sciences. Despite of considerable progress in the past decades, the mathematical theory still faces fundamental questions concerning the inﬂuence of nonlinearities or multiple characteristics of the hyperbolic operators or geometric properties of the domain in which the evolution process is considered. The current volume is dedicated to modern topics of the theory of hyperbolic equations such as evolution equations – multiple characteristics – propagation phenomena – global existence – inﬂuence of nonlinearities. It is addressed both to specialists and to beginners in these ﬁelds. The contributions are to a large extent self-contained. The ﬁrst contribution is written by Piero D’Ancona and Vladimir Georgiev. Piero D’Ancona graduated in 1982 from Scuola Normale Superiore of Pisa. Since 1997 he is full professor at the University of Rome 1. Vladimir Georgiev graduated in 1981 from the University of Soﬁa. Since 2000 he is full professor at the University of Pisa. The ﬁrst part of the paper treats the existence of low regularity solutions to the local Cauchy problem associated with wave maps. This introductory part follows the classical approach developed by Bourgain, Klainerman, Machedon which yields local well-posedness results for supercritical regularity of the initial data. The nonuniqueness results are established by the authors under the assumption that the regularity of the initial data is subcritical. The approach is based on the use of self-similar solutions. The third part treats the ill-posedness results of the Cauchy problem for the critical Sobolev regularity. The approach is based on the eﬀective application of the properties of a special family of solutions associated with geodesics on the target manifold. The second contribution is written by Hideo Kubo and Masahito Ohta. Hideo Kubo graduated from Hokkaido University in 1996. Since 2003 he is associate professor at Osaka University. Masahito Ohta graduated from the University of Tokyo in 1996. Since 2003 he is associate professor at Saitama University. Initially they consider in their contribution wave equations with small nonlinear perturbation. The problems of interest are local well-posedness in time, blow-up, asymptotic behaviour and existence of self-similar solutions. The main topic of their contribution is how the theory of wave equations is transferred to systems of nonlinear wave equations with diﬀerent propagation speeds. They explain, in particular, how the fact of diﬀerent propagation speeds can be utilized to the advantage of a detailed analysis of initial value problems. The systematic study of such systems is based on pointwise decay estimates for solutions of the Cauchy problem for inhomogeneous wave equations in L∞ spaces with hyperbolic

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weights. The blow-up results are established by deriving lower estimates of this type. Further ﬁelds of interest of H. Kubo are associated with the study of the system of elastic equations and the Maxwell system. Among other things, M. Ohta is also interested in issues of stability and instability of standing wave solutions for nonlinear Klein-Gordon equations. The third contribution is written by Mitsuhiro Nakao. He received his doctoral degree (Doctor of Science) in 1977 from Kyushu University and is full professor at the same university since 1976. He has been strongly interested in decay problems for the wave equation with various types of dissipations in bounded domains. He developed his own techniques, which many authors use today (“Nakao’s inequality”). Since 1998 he focused his interest on the Cauchy problem and on initial-boundary value problems in exterior domains. His strategy is to combine the energy method with the geometry of the exterior domain. For non-trapping domains a restricted localized eﬀective dissipation is employed. He has derived decay results with algebraic rates for local and total energies. Moreover, he found the critical order of nonlinearities in his models. Here the property of stability comes in. In the future M. Nakao wants to apply his knowledge of decay properties for nonlinear damped wave equations to the problems concerning global attractors. He also plans to consider related problems for nonlinear degenerate parabolic equations which is another ﬁeld of his research interest. The fourth contribution is written by Karen Yagdjian. Karen Yagdjian has received his Doctor of Physical and Mathematical Sciences degree from Moscow State University in 1990. Since 2004 he is assistant professor at the University of Texas-Pan American. His main interests are microlocal analysis and its application to partial diﬀerential equations. The main goal of his contribution is to study the phenomenon of parametric resonance for wave map type equations. In particular, he is interested in the study of the inﬂuence of the oscillating behaviour of coeﬃcients in t on the global existence of small data solutions. A special transformation reduces the Cauchy problem for the wave map type equations to linear Cauchy problems for the wave equation with a special constraint. To attack these Cauchy problems Floquet’ theory, especially Borg’s theorem for Hill’s equation is used. The question for stability and instability is discussed in a systematic way. Other models with growing coeﬃcients or stabilizing coeﬃcients are treated in a similar fashion. The ﬁfth contribution is written by Massimo Cicognani and Luisa Zanghirati. Luisa Zanghirati graduated from Ferrara University in 1965 and is full professor there since 1985. Massiomo Cicognani graduated from Bologna University in 1983 and is full professor there since 1998. Their contribution is devoted to local in time existence of smooth solutions for nonlinear degenerate hyperbolic problems. Diﬀerent kinds of degeneracies are

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explored. On the one hand they consider weakly hyperbolic Cauchy problems (characteristics of constant multiplicity), thus Levi conditions of nonlinear type come into play. On the other hand degeneracies are produced by low regularity in time of the coeﬃcients. The goal of the authors is to present a uniﬁed approach for these problems. The main tools of the approach are an eﬀective diagonalization procedure with regularized characteristic roots, an eﬀective representation of commutators, a sharp G˚ ˚ arding’s inequality for systems and a suitable transformation containing the loss of derivatives. The loss of derivatives is characteristic for degenerate hyperbolic problems. Moreover, they describe the propagation of analytic regularity of solutions. Further interests of L. Zanghirati are questions concerning hypoellipticity, and M. Cicognani is interested in microlocal methods to describe the propagation of singularities for pseudodiﬀerential operators. The sixth contribution is written by Michael Dreher and Ingo Witt. Michael Dreher graduated from the University of Freiberg in 1999. Since 2004 he is assistant professor at the University of Konstanz. Ingo Witt graduated from the University of Bonn in 1995. Since 2004 he has a DFG fellowship at the Imperial College London. The goal of the authors is to derive sharp energy estimates for weakly hyperbolic Cauchy problems with ﬁnite time degeneracy of the coeﬃcients at t = 0. From such estimates they obtain the precise loss of regularity that depends on the spatial variables. In the case of time-dependent coeﬃcients they show an interesting relation to the theory of edge Sobolev spaces, a tool which is used for the study of diﬀerential operators on manifolds with singularities. In the general case (where the coeﬃcients’ depend on time–and spatial variables) the authors introduce Sobolev spaces of variable order. The main step is to ﬁnd the correct class of pseudodiﬀerential symbols and to establish a pseudodiﬀerential calculus which contains a symmetrizer. An aparding’s inequality gives rise to a sharp energy estimate. plication of a sharp G˚ Sharpness is proved by using the method of Lyapunov functionals, where suitable estimates lead to an instability result. We would like to thank all the referees for their valuable contribution in the evaluation process. We also wish to thank Dr. Jens Wirth (TU Bergakademie Freiberg) for the considerable eﬀort he put into producing the ﬁnal layout of this volume. Last not least, the editors would like to thank all the staﬀ of Birkh¨ auser Publishing Company, in particular, Dr. Thomas Hempﬂing, for the pleasant cooperation. Freiberg and Potsdam M. Reissig

B.-W. Schulze

April 2005

Operator Theory: Advances and Applications, Vol. 159, 1–111 c 2005 Birkhauser ¨ Verlag Basel/Switzerland

Wave Maps and Ill-posedness of their Cauchy Problem Piero D’Ancona and Vladimir Georgiev

Abstract. In this review article we present an introduction to the theory of wave maps, give a short overview of some recent methods used in this ﬁeld and ﬁnally we prove some recent results on ill–posedness of the corresponding Cauchy problem for the wave maps in critical Sobolev spaces. The low regularity solutions for the wave map problem are studied by the aid of appropriate bilinear estimates in the spirit of ones introduced by Klainerman and Bourgain. Our approach to obtain ill–posedness in critical Sobolev norms uses suitable family of wave maps constructed via geodesic ﬂow on the target manifold. We give two alternative proofs of the ill–posedness: the ﬁrst approach is based on the application of Fourier analysis tools, while the second proof is based on the application of the classical fundamental solution representation for the free wave equation. For the case of subcritical Sobolev norms we establish non–uniqueness of the corresponding Cauchy problem. Mathematics Subject Classiﬁcation (2000). 35L05, 35J10, 35P25, 35B40. Keywords. Equivariant wave maps, H s -spaces, blow-up of solution.

1. Preface Let (N, g) be a smooth n-dimensional Riemannian manifold with metric g, which with no loss of generality we can isometrically embed in Rn1 for some n1 > n. For the functions u : R × Rm → N deﬁned on the ﬂat Minkowski space Rt × Rm x with values in the target N , consider the functional J(u) = ∂α u, ∂ α ug(u) dtdx, R×Rm

The authors are partially supported by Research Training Network (RTN) HYKE, ﬁnanced by the European Union, contract number : HPRN-CT-2002-00282.

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Piero D’Ancona and Vladimir Georgiev

where summation over α = 0, 1, . . . , m is intended, with ∂t , ∂x1 , . . . , ∂xm ), (∂0 , . . . , ∂m ) = (∂

(∂ 0 , . . . , ∂ m ) = (∂ ∂t , −∂ ∂x1 , . . . , −∂ ∂xm )

as usual, while ·, ·g is the product in the metric g. The critical points of the functional J are called wave maps. If we choose a system of coordinates on N , then locally smooth wave maps satisfy the equation u + Γbc (u)∂α ub ∂ α uc = 0,

= 1, . . . , n,

(1.1)

where Γjk denote the Christoﬀel symbols on N in the chosen coordinates. The natural problem for this system of wave equations is clearly the Cauchy problem with data at t = 0 u(0, x) = u0 , ut (0, x) = u1 ; (1.2) the usual space for the data are Sobolev spaces (u0 , u1 ) ∈ H s (Rm , N ) × H s−1 (Rm , T N ) for suitable values of s ∈ R. Here we used the space H s (Rm ; N ) ≡ {v ∈ H s (Rm ; Rn1 ), v(Rm ) ⊆ N },

s∈R

(1.3)

with the induced norm; notice that H s (Rm ; N ) = ∅ if 0 ∈ N , but it causes no loss of generality to assume that 0 ∈ N after a translation in the ambient space. An alternative description of the wave map system, which usually gives a simpler expression in presence of symmetry of the target is the following: a wave map is a function u : R × Rm → Rn1 such that u(t, x) ∈ N,

u ⊥ N

for all (t, x).

A good introduction on this subject with comprehensive references f can be found in [30]. The Cauchy problem for wave maps has been extensively studied in recent years, starting with the work of Ginibre and Velo [15]. Not many general results for the Cauchy problem are known. One can unify the existing results in few groups. • Global existence of weak H 1 -solutions when the target is compact and has dimension 2 (several authors, see, e.g., [25], [29], [47]). • Local existence for data in H s , s > m/2. This is classical if s is large enough, but for s close to the critical value s = m/2 it is a much more diﬃcult result, due to Klainerman, Machedon, Selberg and obtained through careful bilinear estimates (see in particular [20] or Chapter 3 below). See also Tataru [41] for the case of Besov spaces. • Global existence for small data. Again, this result can be proved by “standard” methods in the smooth case (Y. Choquet-Bruhat), but the recent results of Tao (see in particular [40], see also [31] and [26]) show that it is suﬃcient to assume that the data are in H s for some s > m/2 and that they are small in the homogeneous H˙ m/2 -norm.

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• In the presence of symmetry and space dimension two one has sharper results; to this class belong the radial case, considered by Christodoulou, TahvildarZadeh and more recently by Struwe (see [8], [37], [38]), and the equivariant case, for which a fairly complete theory exists, due to Shatah, TahvildarZadeh, Struwe, and Grillakis (see, e.g., [13], [29], [32], [36]). However several outstanding problems remain open. In particular we mention the case of two-dimensional wave maps u : R × R2 → N, when the target N is the two-dimensional sphere in R3 . The wave map system for u can be written in this case u + u(|ut |2 − |∇x u|2 ) = 0. It is not known if the local smooth solutions (even with equivariant symmetric data) may develop singularities in a ﬁnite time; or, equivalently, if the global weak solutions (which exist) are unique. In this case the “null form” character of the nonlinear term is particularly evident, and it is easy to check that the critical space with respect to scaling is exactly H˙ m/2 (in the scale of Sobolev spaces H s ). For this we shall call this exponent s = m/2 critical (with respect to the scaling argument). For the deﬁnition and the properties of the quantity sc = m/2 as a critical (with respect to the scaling) exponent one can see Section 1.3 in [21]. In the present review we would like to give a detailed exposition of some recent results, obtained by the authors in a series of works, concerning the problem of the behavior of the wave map system for the cases s < m/2 and s = m/2. These results are closely connected with the general well-posed conjecture formulated in section 1.3 in [21], that asserts that for a large class of wave type equations (called initial value problems for basic ﬁeld theories) the scale critical exponent sc is such that • a) the corresponding Cauchy problem is (locally in time) well-posed for initial data in H s with s > sc ; • b) the Cauchy problem has a global solution provided the initial data have small H sc norm; • c) the Cauchy problem is ill-posed for s < sc . More precisely, we shall see that the third part c) of this conjecture is indeed true for the wave map problem and we can show non-uniqueness of suitable weak solutions to the wave map problem. For the case s = m/2 it is natural to ask if the Cauchy problem is well-posed. This situation is quite similar to the well-posedness problem studied by Bourgain [3], Kenig, Ponce, Vega [22] for the case of several dispersive equations, such as the nonlinear Schr¨ o¨dinger or the Korteweg-de Vries equations. The basic idea is to study the properties of the solution map “data → solution” and show that in the critical or subcritical case some change of behavior occurs. Indeed, for s > m/2 the bilinear estimates in [19], [20], [21] show that a contraction method works and yields local well-posedness for the corresponding Cauchy problem. Moreover, this

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contraction argument shows also that the solution map is Lipschitz continuous (and hence even smoother). A breakdown of this property may indicate an instability of the equation, and suggest that a blow up may occur in a ﬁnite time. As a minimal goal, if one can show that the solution map is not Lipschitz or even uniformly continuous, one obtains that the contraction method cannot be applied to prove well-posedness. This is indeed the case for the wave map system with data in the Sobolev space H˙ m/2 (see Section 6). The review is organized as follows. Section 2 is devoted to some background material, i.e., to show the eﬀect that suitable symmetry assumptions have on the form of the equations, and to give a self contained proof of the local well-posedness in the supercritical case. In Section 3 we consider some bilinear estimates in the spirit of the results in [20] and show how they can improve the local existence result for the Cauchy problem for wave maps. In Section 4 we show a ﬁrst instability example, by constructing smooth solutions for which the energy is concentrated in a cusp-like domain; notice that the energy cannot concentrate in a self-similar way for the 2-dimensional case as it is well known. Section 5 is devoted to the subcritical case; for the 2-D wave maps with values into the 2-D sphere one can prove (as expected) a quite strong ill-posedness result, namely the non uniqueness of weak solutions. In Section 6 we begin the study of the solution map in the critical space, which is completed in Section 7, where we prove that the solution map can never be locally uniformly continuous for any base space dimension and an arbitrary (non ﬂat) target. The authors are grateful to A. Ivanov, N. Visciglia for the careful reading and corrections of the manuscript of this work.

2. Variational motivation of the wave maps equations 2.1. Introduction Wave maps arise in several problems of mathematical physics (see, e.g., the Higgs ﬁeld model in [15], and the relativity models in [7]). To be more precise, let (N, g) be an n-dimensional manifold endowed with a Riemannian metric structure, i.e., a positive deﬁnite bilinear form g in every point of N . We call N the target manifold. Let M = Rm+1 be the Minkowski space time equipped with the ﬂat metric h = (−1, 1, . . . , 1). A wave map is a map that satisﬁes the equation Dα ∂α u = 0; here

(2.1)

∂ , (xα ) = (t, x) ∈ R1+m , α = 0, 1, . . . , m, ∂xα while Dα is the covariant pull-back derivative in the bundle u∗ T N. As usual, the Greek indices α, β run from 0 to m; we use the summation convention over repeated indices. By the Nash embedding theorem, we may assume that the target N is isometrically embedded in some Rd for d large enough. So, we can consider u as ∂α =

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an Rd valued function u = (u1 , . . . , ud ). Then the intrinsic equation (2.1) can be rewritten in extrinsic form utt − ∆u − B(u)(∂α u, ∂ α u) = 0,

(2.2)

where

B(p) : Tp N × Tp N → Tp N ⊥ is the second fundamental form of N ⊂ Rd . Given any function F : (t, x) ∈ Rm+1 → F (t, x) ∈ Tu(t,x) N we shall also consider the following inhomogeneous version of (2.2) utt − ∆u − B(u)(∂α u, ∂ α u) = F.

(2.3)

In this work we study the Cauchy problem for (2.3) subject to the initial conditions u(0, x) = u0 (x) ∈ H s (Rm ; N ),

∂t u(0, x) = u1 (x) ∈ H s−1 (Rm ; T N ).

(2.4)

More precisely, we identify H = H (R ; N ), s ≥ 0, with the space of funcs

s

m

tions u(x) ∈ H s (Rm ; Rd ), Rd is satisfying u(x) ∈ N for almost every x ∈ Rm (here the embedding N → m used in an essential way). Similarly, if U ⊂ R is an open set, then H s (U ) = H s (U ; N ), s ≥ 0, is the space of functions u(x) ∈ H s (U ; Rd ), satisfying u(x) ∈ N for almost every x ∈ U. The homogeneous case F = 0 is treated in [19], [21], [42]. From these results it follows that given any s > m/2, any data (u0 , u1 ) ∈ H s × H s−1 one can ﬁnd a ﬁnite time interval [0, T ], 0 < T < T0 , such that there exists a unique solution u ∈ C([0, T ]; H s ) to the Cauchy problem (2.2), (2.4). Our ﬁrst goal in this chapter is to verify that the Cauchy problem (2.3), (2.4) has a local solution u ∈ C([0, T ]; H s ) whenever the initial data satisfy (u0 , u1 ) ∈ H s × H s−1 with s > m/2 and the source term is such that F ∈ L1 ((0, T0 ); H s−1 ). After this we shall concentrate our attention to the speciﬁc case, when m = 2 and the target is one typical compact manifold namely N = S2 . The next step in

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this chapter is the study of special types of wave maps. More precisely, we consider equivariant wave maps and study their relation with the corresponding harmonic maps on S2 of arbitrary degree. The results in [32] guarantee the existence of a global solution, when the target is S2 , the degree of the harmonic map is 1 and the initial data are small in the H 1 -norm. We shall show in particular that this result can be extended to the case of arbitrary harmonic map degree. Our approach in this introductory chapter follows the argument in [32] and relies on suitable local (in time) a priori estimates. We can apply the general local existence result, which in dimension 2 implies the local well-posedness of the Cauchy problem in H s with s > 1. We shall precise the dependence on the index s > 1 of the a priori estimate, which is the essential tool in the proof of the well-posedness, by computing the constants appearing in the estimate as a function of s. Our main observation is that these constants are uniform in s > 1 as s tends to 1. 2.2. Harmonic maps and special harmonic maps on the sphere Harmonic maps can be regarded as a special case of the above construction, when the manifold M is a m-dimensional Riemannian manifold equipped with a metric σ. The target (N, g), as above, is a n-dimensional manifold endowed with a Riemannian metric g. Thus the harmonic map satisﬁes the equation m

Dj ∂j u = 0,

(2.5)

j=1

where ∂ ∂xj and xj , j = 1, . . . , m are the local coordinates on M. Moreover, Dj is the covariant pull-back derivative in the bundle u∗ T N. To introduce the energy functional we suppose that y 1 , . . . , y n are local coordinates on N provided y1, . . . , yn ∈ Y ∂j =

with Y being a small neighborhood of 0 ∈ Rn . Given any small neighborhood X of 0 ∈ Rm and any map U : x = (x1 , . . . , xm ) ∈ X → y = (y 1 , . . . , y n ) ∈ Y we can deﬁne locally the energy functional m n √ E(U ) = σ jk (x)gab (y)∂ ∂j y a (x)∂k y b (x) σdx. X a,b=1 j,k=1

To simplify the notations we use the summation convention for repeated indices, so that we can write √ σ jk (x)gab (y)∂ ∂j y a (x)∂k y b (x) σdx. (2.6) E(U ) = X

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The Euler–Lagrange equation associated with this functional has the form √ √ √ −2gab ∂k σ jk σ∂ ∂j y b − 2∂c gab ∂k y c σ jk σ∂ ∂j y b + (∂a gbc ∂j y c σ jk σ∂k y b ) = 0. (2.7) Since the Laplace–Beltrami operator ∆M has local representation ∆M =

m j,k=1

√ 1 √ ∂j σ σ jk ∂k , σ

(2.8)

where σ = det (σ σjk ) , we may write 1 gab ∆M y b + ∂c gab ∂k y c σ jk ∂j y b − (∂a gbc ∂j y c σ jk ∂k y b ) = 0. (2.9) 2 We recall the explicit expression of the Christoﬀel symbols: 1 γc;ab = (∂a gbc + ∂b gac − ∂c gab ). (2.10) 2 If we write 1 ∂c gab ∂k y c σ kj ∂j y b = (∂c gba ∂k y c σ kj ∂j y b + ∂b gac ∂k y c σ kj ∂j y b ) 2 and use the expression of Christoﬀel symbols, then we arrive at the following equation gab ∆M y b + γa;bc ∂k ub σ jk (x)∂ ∂j uc = 0. Raising the index a, we obtain a ∆M y a + γbc ∂k y b σ jk (x)∂ ∂j y c = 0.

(2.11)

By the Nash embedding theorem, we may assume that the target N is embedded in some Rd for d large enough. Hence u is a d-dimensional function u = (u1 , . . . , ud )l on the other hand, the local coordinates y 1 , . . . , y n on N enable us to parameterize the manifold N locally, and we can think of u as a function u = u(y),

u : Y ⊆ Rn → R d .

In the following we shall restrict to the simpler case d = n + 1, i.e., when N is a hypersurface in Rd . The Riemannian metric g on N is induced by the Euclidean metric on Rd which is simply gab = ∂ ∂ya u, ∂yb uRd ,

(2.12)

where ·, ·Rd is the scalar product in Rd . Then the intrinsic equation (2.11) can be rewritten in extrinsic form as ∆M u +

m

σ jk B(u)(∂ ∂j u, ∂k u) = 0,

j,k=1

where B(p) : Tp N × Tp N → Tp N ⊥

(2.13)

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Piero D’Ancona and Vladimir Georgiev

is the second fundamental form on N ⊂ Rd and ∆M is the Laplace–Beltrami operator on the manifold M. Recall that the second fundamental form is deﬁned by n B(u)(v, w) = bac (u)v a wc ν(u) (2.14) a,c=1

for any two vectors v, w ∈ Tp N , with coeﬃcients bac deﬁned as follows bac = −∂ ∂ya ∂yc u(y), ν(u(y))Rd , a, c = 1, . . . , n,

(2.15)

where ν(u) is the unit normal at u ∈ N. In the above local representation we have used a local basis dy 1 , . . . , dy n in Tp N, which is dual to the basis of vector ﬁelds ∂y1 , . . . , ∂yn so that n n v= vj dy j , w = wj dy j . j=1

j=1

To verify the above assertion it is suﬃcient to rewrite the energy functional in (2.6) as follows √ E(U ) = σ jk (x)∂ ∂j u(y(x))∂k u(y(x))Rd σdx. (2.16) X

Taking the variation of this integral over u ∈ H 1 , under the constraint u(y) ∈ N , we obtain the Euler–Lagrange equation with Lagrange multiplier ∆M u = −µν(u),

(2.17)

where the Lagrange multiplier µ can be obtained by scalar multiplication with u, i.e., √ 1 µ = − √ ∂ ∂j σ σ jk ∂k u, νRd . σ Using the property ∂k u ∈ Tu (N ), ν(u) ∈ Tu (N )⊥ , we get µ = − σ jk ∂ya ∂yc u, νRd ∂j y a ∂k y c . Now the deﬁnition (2.15) of the second fundamental form leads to (2.13). From now on we restrict our attention to the special case when M = Sm−1 = N. Then we can assume that the standard metric on M = N is induced by the embedding Sm−1 ⊂ Rm . If ω ∈ Sm−1 , then ω = (ω1 , . . . , ωm ) ∈ Rm and |ω| = 1. We shall denote by κ = (κ1 , . . . , κm−1 ) any local coordinates on Sm−1 and by x = (x1 , . . . , xm ) the coordinates on Rm ; then the standard metric on Sm−1 can be written m−1 j,k=1

σjk (κ)dκj dκk

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for suitable coeﬃcients σjk . Introducing spherical coordinates x ∈ Sm−1 , r = |x|, ω = |x| we have the following decomposition of the Laplace operator in Rm m−1 1 ∂r + 2 ∆Sm−1 . ∆x = ∂r2 + (2.18) r r Then the intrinsic form of harmonic map equation in (2.11) implies that a map κ = (κ1 , . . . , κm−1 ) −→ λ = λ(κ) = (λ1 , . . . , λm−1 ) is a (local) harmonic map if ∆M λa +

m−1 m−1

a γbc (λ)σ jk (κ)∂κk λb ∂κj λc = 0,

(2.19)

j,k=1 b,c=1

where a = 1, . . . , m − 1. The embedding Sm−1 ⊂ Rm enables us to consider the corresponding diﬀeomorphism λ ∈ Rm−1 −→ θ = θ(λ) ∈ Sm−1

(2.20)

onto a small neighborhood which takes a small neighborhood of the origin in R on the sphere Sm−1 . Then the equation (2.17) shows that a map m−1

κ = (κ1 , . . . , κm−1 ) −→ θ = θ(κ) = (θ1 , . . . , θm ) ∈ Sm−1 is a (local) harmonic map if ∆Sm−1 θ = −Kθ,

(2.21)

where K > 0 is a constant. Lemma 2.1. Let κ = (κ1 , . . . , κm−1 ) −→ θ = θ(κ) = (θ1 , . . . , θm ) ∈ Sm−1 be a local C 2 solution to ∆Sm−1 θ = −Kθ,

(2.22)

for some constant K > 0. Then K=

m−1

σ jk (κ)∂κj θ, ∂κk θRm

(2.23)

j,k=1

and K=

m−1 m−1

σbc (λ)σ jk (κ)∂κj λb ∂κk λc ,

(2.24)

b,c=1 j,k=1

where and θ

−1

λ(κ) = θ−1 θ(κ) is the inverse diﬀeomorphism to (2.20).

(2.25)

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Piero D’Ancona and Vladimir Georgiev

Remark 2.1. The map (2.25) in the above lemma can be extended as a map ω ∈ Sm−1 −→ λ(ω) = (λ1 (ω), . . . λm−1 (ω)), since κ = (κ1 , . . . , κm−1 ) are local coordinates on S

m−1

(2.26)

.

Proof. Multiplying the equation (2.22) by θ, we get K = −∆Sm θ, θRm , where (see (2.8)) ∆M =

m−1 j,k=1

√ 1 √ ∂κj σ σ jk ∂κk . σ

(2.27)

From the relation θ(κ), θ(κ)Rm = 1 we obtain ∂κk θ(κ), θ(κ)Rm = 0 so that K =−

m−1

σ jk ∂κj ∂κk θ, θRm =

j,k=1

m−1

σ jk ∂κj θ, ∂κk θRm

j,k=1

and this proves the ﬁrst relation (2.23). The second relation (2.24) follows from σbc (λ) = ∂λb θ(λ), ∂λc θ(λ)Rm and the chain rule ∂κj θ =

m−1

∂λb θ ∂κj λb .

b=1

This completes the proof of the lemma.

To construct the solutions of equation (2.21) we can follow the idea from [17] and look for polynomial functions x = (x1 , . . . , xm ) ∈ Rm −→ P (x) = y = (y1 , . . . , yn ) ∈ Rn ,

(2.28)

such that P (x) = (P P1 (x), . . . , Pn (x)) and Pj (x) are homogeneous polynomials of order L ≥ 1, harmonic in x, i.e., ∆Rm Pj (x) = 0

(2.29)

P1 (x))2 + · · · + (P Pn (x))2 = 1. (x1 )2 + · · · + (xm )2 = 1 ⇒ (P

(2.30)

and such that By homogeneity, (2.30) is a consequence of L Pn (x))2 = (x1 )2 + · · · + (xm )2 . (P P1 (x))2 + · · · + (P

(2.31)

Once the above problem (2.29) and (2.31) is solved, we can introduce polar coordinates x r = |x|, ω = |x|

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and set u(ω) = P (ω). Using the decomposition of the Laplace operator together with (2.29) and the relation Pj (x) = rL Pj (ω) we can rewrite (2.29) as rL−2 (L(L − 1) + (m − 1)L + ∆Sm−1 ) Pj (ω) = 0 so that u(ω) = P (ω) satisﬁes ∆Sm−1 u = −L(L + m − 2)u

(2.32)

and the equation (2.21) is satisﬁed with µ = L(L + m − 2). First, we consider the case n = m = 2. Then we can take κ ∈ [0, 2π) as a local coordinate on M = S1 while λ ∈ [0, 2π) is the local coordinate on N = S1 . We have simply ∆S1 = ∂κκ and setting u(κ) = (cos λ , sin λ), λ = λ(κ), the equation (2.32) becomes − sin λ ∂κκ λ − cos λ (∂κ λ)2 cos λ ∂κκ λ − sin λ (∂κ λ)2

= −L2 cos λ, = −L2 sin λ.

An obvious solution is λ = Lκ. An alternative approach to the solution of the system (2.29), (2.31) can be found using the embeddings S1 ⊂ R1+1 = C. If x1 , x2 are the coordinates on R2 and we can deﬁne the polynomial vector-valued function z = x1 + ix2 −→ P (z) = z L . Since ∆C = ∂z ∂z¯, we see that P (z) are harmonic polynomials of order L so that (2.29) is satisﬁed. Property (2.31) follows from the obvious relation |z L |2 = |z|2L . For L = 2 we obtain in particular P1 (x) = (x1 )2 − (x2 )2 , P2 (x) = 2x1 x2 .

(2.33)

Next, we consider the case m = n = 3. For L = 1 we can take Pj (x) = xj and see that (2.29) and (2.31) are satisﬁed. For L = 2 we use the argument of the previous case m = n = 2 and see that all polynomials (see (2.33)) (x2 )2 − (x3 )2 , (x3 )2 − (x1 )2 , (x1 )2 − (x2 )2 as well as x1 x2 , x2 x3 , x3 x1

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are harmonic. For this we choose P1 (x) = a (x2 )2 − (x3 )2 + b (x1 x2 + x2 x3 + x3 x1 ) , P2 (x) = a (x3 )2 − (x1 )2 + b (x1 x2 + x2 x3 + x3 x1 ) , P3 (x) = a (x1 )2 − (x2 )2 + b (x1 x2 + x2 x3 + x3 x1 ) , where a, b are suitable constants chosen so that (2.31) is fulﬁlled. Note that P2 (x))2 + (P P3 (x))2 (P P1 (x))2 + (P

= 2a2 ((x1 )4 + (x2 )4 + (x3 )4 ) + (3b2 − 2a2 ) (x1 x2 )2 + (x2 x3 )2 + (x3 x1 )2 . Comparing this relation with 2 (x1 )2 + (x2 )2 + (x3 )2

= ((x1 )4 + (x2 )4 + (x3 )4 ) + 2 (x1 x2 )2 + (x2 x3 )2 + (x3 x1 )2 ,

we see that it is suﬃcient to take 2a2 = 1,

3b2 − 2a2 = 2,

i.e., 1 a= √ , 2

b = 1.

With this choice we have

2 (P P1 (x))2 + (P P2 (x))2 + (P P3 (x))2 = (x1 )2 + (x2 )2 + (x3 )2 ,

so (2.31) is satisﬁed with L = 2. For higher-dimensional case L ≥ 3 or for n ≥ m ≥ 3 the existence of harmonic polynomial maps satisfying (2.29) and (2.31) is discussed in [17]. For our considerations concerning the concentration of local energy for two-dimensional wave maps the case alone n = m = 2 is suﬃcient. 2.3. Equivariant wave maps and construction of special solutions In this section we shall derive brieﬂy the wave map equation and shall construct a special class of equivariant wave maps that solve the inhomogeneous problem (2.3). The equation (2.1) is the Euler–Lagrange equation related to the density ∂α u, ∂ α ug(u) ,

(2.34)

which in a small neighborhood of a ﬁxed u0 ∈ N has the form ∂α u, ∂ α ug(u) = hαβ gab ∂α ua ∂β ub . Here and below the Greek indices α, β run from 0 to m, while the Latin indices a, b, c, d run from 1 to n. A summation convention for repeated indices is also used. The corresponding Lagrangian is given by: hαβ gab ∂α ua ∂β ub . (2.35) L[u] = M

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Since we assumed M to be the Minkowski space R1+m with the ﬂat metric h = diag(−1, 1, . . . , 1), we can simplify the Lagrangian to

L[u] = R1+m

gab ∂ α ua ∂α ub .

(2.36)

Then the Euler-Lagrange equations become −2∂α (gab ∂ α ub ) + ∂α uc ∂ α ub ∂a gbc = 0,

(2.37)

or equivalently 1 −gab ∂α ∂ α ub − ∂c gab ∂α uc ∂ α ub + (∂a gbc ∂α uc ∂ α ub ) = 0. 2 In terms of D’Alembertian we may write 1 gab 2ub + ∂c gab ∂α uc ∂ α ub − (∂a gbc ∂α uc ∂ α ub ) = 0, (2.38) 2 where 2 = −∂α ∂ α = ∂02 − ∂12 − · · · − ∂n2 . The Christoﬀel symbols are given by the expressions 1 (2.39) Γc;ab = (∂a gbc + ∂b gac − ∂c gab ). 2 If we write 1 ∂c gab ∂α uc ∂ α ub = (∂c gba ∂α uc ∂ α ub + ∂b gac ∂α uc ∂ α ub ) 2 and use the above expression of Christoﬀel symbols, then we arrive at the following equation gab 2ub + Γa;bc ∂α ub ∂ α uc = 0. Raising the index a, we obtain 2ua + Γabc ∂α ub ∂ α uc = 0.

(2.40)

In order to handle the inhomogeneous case, a minor modiﬁcation of the density (2.34) is suﬃcient: ∂α u, ∂ α ug(u) + F, ug(u) ,

(2.41)

where F : x = (x0 , x1 , . . . , xm ) ∈ R1+m → F (x) ∈ Tu(x) N is the given source term. The corresponding inhomogeneous problem has the form 2ua + Γabc ∂α ub ∂ α uc = F a .

(2.42)

As in the previous section we can rewrite these equations in extrinsic form. To this purpose assume that N is a n-dimensional surface in Rn+1 with metric induced by the Euclidean metric on Rn+1 . Thus u is a d = n + 1-dimensional vector u = (u1 , . . . , ud ); on the other hand, on N we can take local coordinates y 1 , . . . , y n so that N is described locally by a chart u = u(y),

y ∈ Y ⊂ Rn .

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Piero D’Ancona and Vladimir Georgiev

The Riemannian metric g on N is induced by the Euclidean metric on Rd (see (2.12) of the previous section) gab = ∂ ∂ya u, ∂yb uRd ,

(2.43)

where ·, ·Rd is the scalar product in R . Then the wave map (locally) is a function d

x = (x0 , x1 , . . . , xm ) ∈ X ⊂ Rm+1 −→ y = y(x) ∈ Rn deﬁned in a small neighborhood X of the origin in Rn+1 , satisfying the intrinsic equation (2.42), i.e., 2y a + Γabc ∂α y b ∂ α y c = F a . (2.44) It is easy to verify that the wave map v(x) := u(y(x)), x ∈ X satisﬁes the extrinsic equation m v + hαβ B(v)(∂α v, ∂β v) = 0,

(2.45)

α,β=0

where B(p) : Tp N × Tp N → Tp N ⊥ is the second fundamental form of N ⊂ Rd . We recall the explicit form (2.14) of the second fundamental form from the previous section: n bac (u)v a wc ν(u) (2.46) B(u)(v, w) = a,c=1

for any two vectors v, w ∈ Tp N , and with coeﬃcients bac deﬁned as follows (see (2.15)) bac = −∂ ∂ya ∂yc u(y), ν(u(y))Rd , a, c = 1, . . . , n; (2.47) ν(u) denotes as usual the unit normal at u ∈ N. To verify the above claim it is suﬃcient to rewrite the energy functional in (2.35) as follows L[u] = hαβ ∂α u(y(x))∂ ∂β u(y(x))Rd dx. (2.48) X

Taking the variation of this integral over u ∈ H 1 , under the constraint u(y) ∈ N we obtain the Euler-Lagrange equation with Lagrange multiplier 2u = −µν(u), (2.49) where the Lagrange multiplier µ can be obtained by scalar multiplication with u: µ = hαβ ∂α ∂β u, νRd . Using the property ∂yk u ∈ Tu (N ), ν(u) ∈ Tu (N )⊥ ,

Wave Maps

15

we get µ = hαβ ∂ya ∂yc u, νRd ∂α y a ∂β y c . Now the deﬁnition (2.15) of the second fundamental form leads to (2.13). In the case when N = Sn , we have ν(u) = u, and equation (2.45) simpliﬁes to m hαβ ∂α u, ∂β Rn+1 u = 0. (2.50) 2 u− α,β=0

We now recall the equivariant wave map ansatz. Assume that N is a smooth n-dimensional rotationally symmetric, wrapped product manifold deﬁned as N = {(φ, λ); φ ∈ (0, φ∗ ), λ ∈ Sn−1 } with metric dφ2 + g(φ)2 dσ 2 , where dσ is the standard metric on S 2

n−1

(2.51)

which we shall denote by

2

dσ = σjk (λ)dλj dλk , while (λ1 , . . . , λn−1 ) are the local coordinates on Sn−1 . In these coordinates we have gφλj = 0, gφφ = 1, gλi λj = g 2 (φ)σij (λ). (2.52) If at least two of the indices a, b, c are equal to φ, then (2.52) implies that Γa,bc = 0. If only one of indices a, b, c is φ, then Γφ,λi λj = −g (φ)g(φ)σij and Γλi ,λj φ = g (φ)g(φ)σij . Finally, Γλi ,λj λk = g 2 (φ)γi,jk where 1 γi,jk = ∂λj σik + ∂λk σij − ∂λi σj k , 2 are the Christoﬀel symbols for the metric σ. The equivariant ansatz is the following: a wave map u, expressed in the coordinates of N as u = (φ, λ), is equivariant if φ(t, x) = φ(t, r),

λj = λj (ω),

(2.53)

where j = 1, . . . , n − 1 ω ∈ Sm−1 → λj (ω) ∈ R, is the map of (2.25). Recall that this map in the local coordinates κ = (κ1 , . . . , κm−1 ) on S

m−1

deﬁnes a solution to the equation ∆Sm−1 λj + Kλj = 0,

where K = L(L + m − 2), L ≥ 1 is an integer, and K = σbc (λ)σ jk (κ)∂κj λb ∂κk λc , due to (2.24) in Lemma 2.1. Choosing a = φ in (2.40) we obtain 2φ + Γφλb λc (u)∂α λb ∂ α λc = 0, where

Γφλb λc (u) = −g (φ)g(φ)σbc (λ).

(2.54)

16

Piero D’Ancona and Vladimir Georgiev Note that ∂α λb ∂ α λc = σ jk (κ)

∂κj λb ∂κk λc r2

so from (2.54) we ﬁnd Kg (φ)g(φ) = 0. r2 The corresponding inhomogeneous problem is of course 2φ +

(2.55)

Kg (φ)g(φ) = f. (2.56) r2 In the special case when the target is the two-dimensional sphere S2 , the metric on S2 has the form dφ2 + sin2 φ dλ2 . Let u : R × R2 → S2 be an equivariant wave map. Then u = (u1 , u2 , u3 ) with 2φ +

u1 = cos(φ) cos(λ), u2 = cos(φ) sin(λ), u3 = sin(φ).

(2.57)

Introducing polar coordinates (r, κ) in R2 , we have x1 = r cos κ, x2 = r sin κ; so the equivariant ansatz (2.53) shows that φ = φ(t, r) satisﬁes (2.55) and λ = λ(κ) is a harmonic map between S1 and S1 . The simplest possible choice of λ is clearly the identity map λ(κ) = κ, and the equation (2.56) becomes then sin(2φ) = f, (2.58) 2r2 where 2φ = (∂ ∂t2 − ∂r2 − 1r ∂r )φ. The vector-valued function u in (2.57) solves the equation utt − ∆u + |ut |2 − |∇x u|2 u = F, (2.59) provided φ solves the inhomogeneous equation (2.58). Indeed, we have the relations 2φ +

ut = ∂t φ ∂φ u, utt =

−φ2t u

ur = ∂r φ ∂φ u, ∂r2 u

+ φtt ∂φ u,

(2.60) 2

= − (∂ ∂r φ) u + ∂rr φ∂φ u

and the representation formula 1 1 2 = ∂t2 − ∂r2 − ∂r − 2 ∂κ2 . r r From

(2.61)

|∇x u|2 = |∂ ∂r u|2 + r−2 |∂κ u|2

and (2.57) we get |∇x u|2 = |∂ ∂r φ|2 +

cos2 φ , r2

|∂ ∂t u|2 = |∂ ∂t φ|2

whence |∂ ∂t u|2 − |∇x u|2 = φ2t − φ2r −

cos2 φ . r2

(2.62)

Further, from (2.60) and (2.61) we ﬁnd 2u = −φ2t u + φtt ∂φ u + φ2r u − φrr ∂φ u −

φr 1 ∂φ u − 2 ∂κκ u. r r

(2.63)

Wave Maps

17

To conclude our computation, we need the simple Lemma 2.2. The function u satisﬁes the identity sin(2φ) ∂κκ u = − (cos φ)2 u + ∂φ u. 2 Proof. Consider the vectors e = (cos λ, sin λ, 0) and e3 = (0, 0, 1). Then

(2.64)

u = e cos φ + e3 sin φ, ∂φ u = −e sin φ + e3 cos φ and from these relations we get immediately e = u cos φ − ∂φ u sin φ. This relation and the identity ∂κκ u = −e cos φ imply (2.64). This completes the proof. Combining the above Lemma and (2.62), we obtain Corollary 2.1. If u is deﬁned by (2.57), then the following relation holds: 2 sin(2φ) 2 2u + |ut | − |∇x u| u = 2φ + ∂φ u. 2r2

(2.65)

We conclude this section by a ﬁnal remark. If φ solves the inhomogeneous Cauchy problem for (2.56), i.e., sin(2φ) = f, 2r2 φ(0, x) = φ0 (|x|, ∂t φ0 (0, x) = φ1 (|x|) 2φ +

(2.66)

then we get immediately a solution of the corresponding extrinsic problem (2.67) utt − ∆u + |ut |2 − |∇x u|2 u = F, u(0, x) = u0 (x), ∂t u(0, x) = u1 (x), where u0 = u(φ0 ), u1 = u(φ1 ), F = f ∂φ u (2.68) with u = u(φ), deﬁned according to (2.57). An analogous connection exists between the Sobolev spaces associated with these two problems (see Lemmas 5.1 and 5.2 in the Appendix to Chapter 5).

3. Local existence result for equivariant wave maps 3.1. Introduction The wave map system can be written (locally) as a semilinear problem for the wave equation utt − ∆u = Q(u, ∇u) + F. (3.1) Here n u = u(t, x), F = F (t, x) : Rt × Rm x −→ R , ∇u = (∂ ∂t u, ∂x1 u, . . . , ∂xm u),

18

Piero D’Ancona and Vladimir Georgiev

while Q(u, v) is the nonlinear term, i.e., a smooth function Q : u × v ∈ Rn × Rn(m+1) −→ Rn , satisfying the condition Q(0, 0) = 0, ∇u Q(0, 0) = 0, ∇v Q(0, 0) = 0.

(3.2)

In this chapter we study the existence of a local solution to the Cauchy problem for (3.1) with initial data u(0, x) = u0 (x), ∂t u(0, x) = u1 (x).

(3.3)

Clearly, by rewriting the wave map equation in this local form, we are dropping the geometric constraint that u should take its values on the target. This simpliﬁcation is crucial for the study of the local existence for the Cauchy problem with data u0 ∈ H s , u1 ∈ H s−1 .

(3.4)

Here and below H s = H s (Rm ) is the classical Sobolev space with norm f H s = (1 + | · |)s fˆL2 for any s ∈ R, while fˆ(ξ) denotes the Fourier transform of f (x) ˆ e−ixξ f (x)dx. f (ξ) = Rm

The linear analogue of the Cauchy problem (3.1), (3.3) is the wave equation utt − ∆u = F, u(0, x) = u0 (x), ut (0, x) = u1 (x).

(3.5)

The standard method to solve this problem in Sobolev spaces is an application of the Fourier transform with respect to space variables e−ixξ u(t, x)dx (3.6) u˜(t, ξ) = Rm

which gives immediately the standard a priori energy estimate uC([0,T ];H s) ≤ C u0 H s + u1 H s−1 + F L1 ((0,T );H s−1 ) .

(3.7)

This leads to the following existence result: Lemma 3.1. If s ∈ R, T > 0, and the data satisfy u0 ∈ H s , u1 ∈ H s−1

(3.8)

F ∈ L ((0, T ); H

(3.9)

1

s−1

),

then there exists a unique solution u ∈ C([0, T ]; H s ) to the Cauchy problem (3.5).

(3.10)

Wave Maps

19

Turning back to the nonlinear problem (3.1), we see that a nonlinear analogue of Lemma 3.1 should be a consequence of the following property: if R > 0, then there exists C = C(R) > 0, such that for any T, 0 < T ≤ 1, the conditions uC([0,T ];H s ) ≤ R, wC([0,T ];H s ) ≤ R

(3.11)

imply that Q(u, ∇u) − Q(w, ∇w)L1 ((0,T );H s−1 ) ≤ C(R)T δ u − wC([0,T ];H s ) ,

(3.12)

δ

for some δ > 0. Indeed, the factor T enables one to apply a contraction argument, when T > 0 is suﬃciently small. Note that for the typical example, when Q(u, ∇u) = Q(∇u) is a quadratic form in ∇u, we have the following estimate Q(u, ∇u) − Q(w, ∇w)L1 ((0,T );H s−1 ) ≤ C(R)T u − wC([0,T ];H s ) provided s>1+

n 2

(3.13) (3.14)

and uC([0,T ];H s) ≤ R, wC([0,T ];H s ) ≤ R. However, in some cases a special structure of the nonlinear term allows to improve the condition (3.14) and to construct local solutions with a lower regularity. To this end, we shall follow the approach developed by Bourgain [2], [3], for the Schrodinger ¨ type equations and by Klainerman, Machedon and Selberg [19], [20], [21] for the wave type equations. This approach is based on the use of the full space-time Fourier transform u ˆ(τ, ξ) = e−itτ −ixξ u(t, x)dxdt. R

Rm

The crucial tool of this method is the modiﬁed Sobolev space H s,θ = H s,θ (Rm+1 ) with norm u2H s,θ = |τ | + |ξ|2s |τ | − |ξ|2θ | u(τ, ξ)|2 dξdτ, (3.15) R

where

Rm

1/2 r = 1 + r2 .

It is clear that this norm is invariant for translations with respect to time Ta u(t, x) = u(t + a, x), or translations in space. Indeed, we can write −iτ a u (τ, ξ), T a u(τ, ξ) = e

so that

u(τ, ξ)|. Ta u(τ, ξ) = |

20

Piero D’Ancona and Vladimir Georgiev

3.2. Localization in time In this section we follow the approach in [21]. We begin by recalling some basic properties of the spaces H s,θ . It is not diﬃcult to see that the embedding 1 H s,θ → C([0, T ]; H s (Rn )), s ∈ R, θ > , T > 0 2 is continuous. To verify this property we recall the relation ∞ 1 u ˜(t, ξ) = eitτ u ˆ(τ, ξ)dτ 2π −∞ so that, applying the Cauchy inequality, we ﬁnd for any θ > 1/2 ∞ 2 2 u(t, ξ)| ≤ C |τ | + |ξ|2s |τ | − |ξ|2θ |ˆ u(τ, ξ)| dτ ξ2s |˜

(3.16)

(3.17)

(3.18)

−∞

and this relation implies (3.16) after an integration in ξ. An essential property is that localization in time is a bounded operation. Indeed, given any function in the Schwartz class ϕ(t) ∈ S(Rt ), the multiplication operator u ∈ H s,θ → ϕu ∈ H s,θ is continuous. But also a stronger property holds: frequently, the localization in time is realized by means of a function t ϕT (t) = ϕ , T where the parameter T is a positive number and ϕ(s) is a smooth compactly supported non-negative function such that

1, if |s| ≤ 1; ϕ(s) = (3.19) 0, if |s| ≥ 2. Then we have the following uniform estimate ϕ t u s,θ ≤ CuH s,θ , T

(3.20)

H

where C > 0 is a constant independent of the parameters T > 1, s, θ. Indeed, we have the relation ∞ ϕ u(τ, ξ) = C ϕ T (τ − τ1 ) u(ττ1 , ξ)dτ1 . T −∞

Now ϕ T (τ ) = T ϕ(τ ˆ T) and by our assumption on ϕ we have |ϕ T (τ )| ≤ T

CN τ T N

(3.21)

Wave Maps

21

for any real N ≥ 0. Then, estimate (3.20) is a consequence of the following estimate |τ | + |ξ|s |τ | − |ξ|θ τ1 | + |ξ|s |ττ1 | − |ξ|θ N1 −N |τ ≤ C(1 + T ) |τ − τ1 |T N |τ − τ1 |T N1

(3.22)

which is valid for all positive integers N, N1 and all reals s > 0 and θ > 0 such that |s + θ| ≤ N − N1 . N ≥ N1 , We shall prove inequality (3.22) by means of the the following technical Lemmas. Lemma 3.2. If a, b are real numbers with a ≥ b and a ≥ 0, then there exists a constant C = C(a, b) > 0 so that x ± yb ≤ Cxa yb

(3.23)

for any real numbers x, y. If in addition we have the inequality |b| ≤ a, then we can choose C = C(a) independent of b. Proof. We shall establish only the estimate with sign + in (3.23), since the case of opposite sign is similar. The assertion is clear when |x| ≤ 2 and |y| ≤ 2. If |x| ≤ 1 and |y| ≥ 2, then x ∼ 1, x + y ∼ y and (3.23) takes the form yb ≤ Cyb which is obvious. If |x| ≥ 2 and |y| ≤ 1, then y ∼ 1, x + y ∼ x and (3.23) takes the form xb ≤ Cxa and recalling the assumption a ≥ b we see that also in this case the inequality is true. Finally, for |x| ≥ 2 and |y| ≥ 2, we can use the inequality |x + y| ≤ C|x| |y| whence x + yb ≤ Cxb yb ≤ Cxb yb

(3.24)

and this completes the proof of the Lemma for the case 0 ≤ b ≤ a. If −a ≤ b ≤ 0, then we can use the following modiﬁcation of the inequality (3.24) x + yb ≤ Cx−b yb ≤ Cxa yb (3.25) and this completes the proof of the Lemma.

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Piero D’Ancona and Vladimir Georgiev

Lemma 3.3. If the real numbers s, s1 , N, N1 satisfy the conditions N ≥ N1

s1 ≥ N1 − N,

s ≤ N − N1 ,

s − s1 ≤ 0,

(3.26)

then there exists a constant C0 = C0 (N, N1 ) > 0, such that |τ | − As τ1 | − As1 N1 −N |τ ≤ C (1 + T ) 0 |τ − τ1 |T N |τ − τ1 |T N1

(3.27)

for any T > 0 and any real numbers τ, τ1 , A. Proof. The proof is divided in a few cases. If |τ | ≤ 2 and |ττ1 | ≤ 2, then |τ | − A ∼ |ττ1 | − A ∼ A and the conditions N ≥ N1 , s1 ≥ s imply the estimate As C0 As1 ≤ N |τ − τ1 |T |τ − τ1 |T N1 which in turn implies (3.27). If |τ | ≤ 1 and |ττ1 | ≥ 2, then |τ | − A ∼ A, |τ − τ1 | ∼ |ττ1 | ∼ ττ1 and using the chain of inequalities |τ | − As |τ − τ1 |T N

≤ ≤

1 C|τ | − As N 1 |τ − τ1 |T |τ − τ1 |T N −N1 T N1 −N C|τ | − As , N 1 |τ − τ1 |T ττ1 N −N1

we see that in order to prove (3.27) it is suﬃcient to show that (recall that s1 ≥ s) As ≤ C|ττ1 | − As . ττ1 N −N1 This last inequality follows from Lemma 3.2 and the assumptions N ≥ N1 , N − N1 ≥ s. If |τ | ≥ 2 and |ττ1 | ≤ 1, then |ττ1 | − A ∼ A, |τ − τ1 | ∼ |τ | ∼ τ and using the inequalities |τ | − As |τ − τ1 |T N

≤ ≤

1 C|τ | − As |τ − τ1 |T N1 |τ − τ1 |T N −N1 T N1 −N C|τ | − As , N |τ − τ1 |T 1 τ N −N1

we see that (3.27) follows from (recall that s1 ≥ s) |τ | − As ≤ CAs ; τ N −N1

Wave Maps

23

now, the last inequality follows from Lemma 3.2 and the assumptions N ≥ N1 , N − N1 ≥ s. It remains to consider the case |τ | ≥ 2 and |ττ1 | ≥ 2. Here we have two additional subcases. When |τ − τ1 | ≤ 1, then |τ | − A ∼ |ττ1 | − A and from the conditions s1 ≥ s and N ≥ N1 we obtain |τ | − As |τ | − As1 ≤ C0 N |τ − τ1 |T |τ − τ1 |T N1 so (3.27) is fulﬁlled. On the other hand, when |τ − τ1 | ≥ 1, then we take advantage of the inequalities |τ | − As |τ − τ1 |T N

≤ ≤

1 C|τ | − As |τ − τ1 |T N1 |τ − τ1 |T N −N1 T N1 −N C|τ | − As . |τ − τ1 |T N1 τ − τ1 N −N1

As before, (3.27) follows from |τ | − As ≤ C|ττ1 | − As , τ − τ1 N −N1 and the last inequality is a consequence of Lemma 3.2 and the assumptions N ≥ N1 , N − N1 ≥ s. The proof of the lemma is concluded. Now we are in the position to verify the inequality (3.22). Proposition 3.1. If s, θ, N, N1 are real numbers such that N ≥ N1 ,

|s + θ| ≤ N − N1 ,

(3.28)

then there exists a positive constant C = C(N, N1 ), such that τ1 | + |ξ|s |ττ1 | − |ξ|θ |τ | + |ξ|s |τ | − |ξ|θ N1 −N |τ ≤ C(1 + T ) |τ − τ1 |T N |τ − τ1 |T N1

(3.29)

for any T > 0 and any τ, τ1 ∈ R, ξ ∈ Rn . Proof. Estimate (3.29) is an immediate consequence of estimate (3.27) of the preceding lemma.

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Piero D’Ancona and Vladimir Georgiev

3.3. Estimates for the homogeneous problem The spaces H s,θ can be regarded as Sobolev spaces “adapted” to the wave equation, as the following properties show. Consider the linear homogeneous wave equation utt − ∆u = 0, t ∈ R, x ∈ Rn u(0, x) = u0 , ut (0, x) = u1 (x).

(3.30)

The classical energy estimates, namely u(t, ·)H s ≤ C (u0 H s + u1 H s−1 ) ,

(3.31)

where 0 ≤ t ≤ T0 and C > 0, imply for any s ∈ R and T0 > 0 the existence of a solution u0 ∈ H s , u1 ∈ H s−1 ⇒ u ∈ C([0, T0 ]; H s ). (3.32) This solution, properly localized, belongs to the space H s,θ : t u0 ∈ H s , u1 ∈ H s−1 ⇒ ϕ u ∈ H s,θ , θ ∈ (1/2, n − 1/2), T

(3.33)

provided ϕ(s) is a smooth compactly supported non-negative function satisfying (3.19), and T > 0. Indeed, the solution to the Cauchy problem (3.30) can be expressed using Dirac’s delta in the following way: u ˆ(τ, ξ) = c1 δ(τ 2 − |ξ|2 ) |ξ| u 0 (ξ) + c2 δ(τ 2 − |ξ|2 ) u 1 (ξ)

(3.34)

and this relation shows that we need the cut-oﬀ function ϕ in (3.33). More precisely, we have: Lemma 3.4. Assume s ≥ 0,

θ > 0.

Then for any T > 0 the condition u0 ∈ H s , u1 ∈ H s−1 implies that ϕ t Dt,x u T

≤ CT 1−N (1 + T s ) (u0 H s + u1 H s−1 ) ,

(3.35)

H s−1,θ

for all 1 N >s+θ+ . 2 Proof. Write for brevity

t ϕT (t) = ϕ . T

Then ϕ T (τ ) = T ϕ(τ ˆ T) and we can apply estimate (3.21) for any real N ≥ 0.

(3.36)

Wave Maps

25

Consider now the case u1 = 0 (the complementary case is similar). Then the relation (3.34) (with u1 = 0) takes the form ϕ T (τ − |ξ|) + ϕ T (τ + |ξ|)) u 0 (ξ); T u(τ, ξ) = C (ϕ

(3.37)

using (3.21), we obtain immediately the claimed inequality (3.35). Combining the above estimate and (3.31) we can derive the following: Corollary 3.1. For any s≥0

θ > 0, C0∞ (R)

and for any ϕ(t) ∈ one can ﬁnd a constant C = C(ϕ, s, θ), such that the solution u of the free wave equation with data u0 ∈ H s ,

u1 ∈ H s−1

satisﬁes ϕuH s,θ ≤ C (u0 H s + u1 H s−1 ) .

(3.38)

Proof. We can reduce the proof to the case when suppξ u 0 (ξ) ∪ suppξ u 1 (ξ) ∪ suppξ u (τ, ξ) ⊆ {|ξ| ≤ 1}. Then we have

L2 L2 . ϕuH s,θ ∼ τ s+θ ϕu τ

Since

(3.39)

|τ |≥1

ξ

s+θ 2 2 τ ϕu(τ, ·)L2 dτ ≤ C ϕ∂ ∂t uH s−1,δ , ξ

an application of Lemma 3.4 implies that 2 s+θ τ ϕu(τ, ·)L2 dτ ≤ C (u0 H s + u1 H s−1 ) . |τ |≥1

ξ

Thus, it only remains to show that 2 ϕu(τ, ·)L2 dτ ≤ C u0 2H s + u1 2H s−1 , ξ

|τ |≤1

and this follows immediately from the Plancherel identity 2 2 ϕu(τ, ·)L2 dτ = C ϕ(t)˜ u(t, ·)L2 dt ξ

R

R

ξ

and from estimate (3.31), which under the assumption (3.39) takes the form u(t, ·)L2 ≤ C (u0 L2 + u1 L2 ) . This completes the proof.

(3.40)

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Piero D’Ancona and Vladimir Georgiev

3.4. Estimates for the non-homogeneous problem The relation established in the preceding section between the homogeneous wave equation and the H s,θ spaces has a partial extension to the non-homogeneous case. Indeed, consider the non-homogeneous problem utt − ∆u = F,

u(0, x) = ut (0, x) = 0

(3.41)

under the assumptions F ∈ H s−1,θ−1 ,

suppt F ⊆ [−1, 1],

s > 1, θ > 1/2.

(3.42)

A ﬁrst useful property is the following: Theorem 3.1. Let n ≥ 2. If F satisﬁes the assumptions (3.42) and u is the solution to (3.41), then for any t ≥ 0 we have u(t, ·)H s ≤ CF H s−1,θ−1 .

(3.43)

Proof. Following the idea of the proof of Corollary 3.1, we see that it is suﬃcient to verify the following inequality ∇u(t, ·)H s−1 ≤ CF H s−1,θ−1 . Consider the function v(t, x) deﬁned as t sin((t − t1 )|ξ|) ˜ F (t1 , ξ)dt1 , v˜(t, ξ) = |ξ| −∞

(3.44)

(3.45)

where v˜(t, ξ) denotes the Fourier transform with respect to space variables (see (3.6)). Then v solves the equation 2v = F. If we can verify the following inequality similar to (3.44) ∇v(0, ·)H s−1 ≤ CF H s−1,θ−1 ,

(3.46)

then we are in the position to conclude that v(0, ·) ∈ H s , ∂t v(0, ·) ∈ H s−1 . Now an application of the energy inequality will complete the proof, since the diﬀerence u − v satisﬁes the homogeneous wave equation 2(u − v) = 0 with initial data ∇x (u(0, ·) − v(0, ·)) ∈ H s−1 , ∂t (u0 (t, ·) − v(0, ·)) ∈ H s−1 . To prove (3.46) consider the operators t −ixξ e e±i(t1 −t)|ξ| F˜ (t1 , ξ)dt1 dξ. I(F )(t, x) = Rn

−∞

(3.47)

Wave Maps

27

From (3.45) it is suﬃcient to show that the operators 0 −ixξ I(F )(0, x) = e e±it1 |ξ| F˜ (t1 , ξ)dt1 dξ,

(3.48)

−∞

Rn

maps H s−1,θ−1 into H s−1 . Taking, e.g., the sign + (the other case is identical), we obtain 0 )(0, ξ) = c I(F eit1 |ξ| F˜ (t1 , ξ)dt1 . (3.49) −∞

1 F˜ (t1 , ξ) = eit1 τ F (τ, ξ)dτ 2π R and the well-known identity (in a suitable distributional sense) 0 1 + πδ(τ + |ξ|) eit1 (τ +|ξ|) dt1 = −iPv τ + |ξ| −∞ From the relation

we see that it is suﬃcient to prove that the operators Jj (F )(ξ) = ξs−1 vj (τ + |ξ|)F (τ, ξ)dτ = ξs−1 vj (τ )F (τ − |ξ|, ξ)dτ, j = 1, 2, R

R

(3.50)

1 , τ are bounded operators from H s−1,θ−1 into L2 provided F satisﬁes the properties (3.42). We also recall that the action of the distribution Pv(1/τ ) may be written as ∞ 1 ϕ(τ ) − ϕ(−τ ) Pv dτ. , ϕ = τ τ 0 Now, take a function ψ(t) ∈ C0∞ (R) such that ψ(t) = 1 for |t| ≤ 1. Then F (t, x) = ψ(t)F (t, x) and − τ1 )F (ττ1 , ξ)dτ1 . F (τ, ξ) = ψ(τ

where

v1 (τ ) = δ(τ ),

v2 (τ ) = P v

R

Consider ﬁrst the operator J1 , deﬁned by s−1 J1 (F )(ξ) = ξ ψ(−|ξ| − τ1 )F (ττ1 , ξ)dτ1 , j = 1, 2.

(3.51)

R

Since ψˆ is a rapidly decreasing function, we can use the inequality | − |ξ| − τ1 | ≥ ||ττ1 | − |ξ|| to ﬁnd |ψ(−|ξ| − τ1 )| ≤

CN , ∀N ∈ N. |ττ1 | − |ξ|N

Hence, by the Cauchy inequality we obtain |J J1 (F )(ξ)|2 ≤ Cξ2s−2 |ττ1 | − |ξ|2θ−2 |F(ττ1 , ξ)|2 dττ1 . R

(3.52)

(3.53)

28

Piero D’Ancona and Vladimir Georgiev

Integrating over ξ, we arrive at the required estimate: J J1 (F )(ξ)L2 ≤ CF H s−1,θ−1 . Next, consider the operator J2 deﬁned by (3.50), i.e., − |ξ| − τ1 )F (ττ1 , ξ)dττ1 dτ. J2 (F )(ξ) = ξs−1 v2 (τ ) ψ(τ R

(3.54)

(3.55)

R

To estimate this double integral it is suﬃcient to verify the inequality − a)dτ ≤ C . v2 (τ )ψ(τ a

(3.56)

R

Indeed, if this inequality is true, we can take a = τ1 + |ξ| and we ﬁnd C − |ξ| − τ1 )dτ ≤ v2 (τ )ψ(τ |ττ1 | − |ξ| .

(3.57)

R

From this inequality and (3.55) we easily derive |J J2 (F )(ξ)| = ξs−1 |ττ1 | − |ξ|−1 F (ττ1 , ξ)dττ1 ,

(3.58)

so that, by the Cauchy inequality, we obtain as in (3.53) |J J2 (F )(ξ)|2 ≤ Cξ2s−2 |ττ1 | − |ξ|2θ−2 |F(ττ1 , ξ)|2 dττ1 ,

(3.59)

R

R

for all θ > 1/2. As before, this inequality implies J J2 (F )(ξ)L2 ≤ CF H s−1,θ−1 .

(3.60)

Thus it remains to prove the estimate (3.56). To this end, we start from the following two simple properties of the distribution v2 (τ ) = P v(1/τ ): v2 (ξ) = iπ sgn ξ, C |v2 (τ )χ(τ )| ≤ τ for any smooth function χ(τ ) ∈ L∞ such that χ(τ ) = 0 near τ = 0. Now, the proof of (3.57) can be reduced to the proof of − a)dτ ≤ C , v2 (τ )χ(τ )ψ(τ a

(3.61) (3.62)

(3.63)

R

where a is a real number and χ satisﬁes one of the following two conditions: either χ ∈ C0∞ (R), or

(3.64)

χ ∈ C ∞ (R), χ(τ ) = 0 for |τ | ≤ 1, χ(τ ) = 1 for |τ | ≥ 2. (3.65) In the case (3.64) we use the property (3.61) and see that the needed inequality (3.63) follows from the estimate C − a)e−iτ ξ dτ ≤ χ(τ )ψ(τ (3.66) ξN aN R

Wave Maps

29

for any integer N ≥ 1. This inequality in turn follows easily from the fact that χ and ψ are rapidly decreasing functions in the Schwartz class S(R). In the case (3.65) we use (3.62) and see that it is suﬃcient to apply the obvious estimate τ −1 τ − a−N dτ ≤ C , N ≥ 2. (3.67) a R This completes the proof of (3.63) and of the theorem.

Theorem 3.2. Assume that n ≥ 2, F satisﬁes the assumptions (3.42) and u is a solution to utt − ∆u = F, with initial data u(0, x) = ut (0, x) = 0. Then for any ϕ(t) ∈ C0∞ (R) we have ϕu ∈ H s,θ ,

(3.68)

for all s > n/2 and 1/2 < θ < 1. Proof. As in the proof of the previous theorem we consider the function v(t, x) deﬁned by (3.45). Further, we can localize the space Fourier transform in x so that suppξ F (t, ξ) ⊂ {|ξ| ∼ 2j }, j = 0, 1, 2, . . .

(3.69)

Then we take two smooth compactly supported functions ϕj (t), j = 1, 2, and see that it is suﬃcient to replace F (t, x) by ϕ2 (t)F (t, x) and show that the function deﬁned by t sin((t − t1 )|ξ|) v˜(t, ξ) = ϕ2 (t1 )F (t1 , ξ)dt1 , (3.70) |ξ| −∞ satisﬁes ϕ1 v ∈ H s,θ . (3.71) It is not diﬃcult to show that ϕ 1 v(τ, ξ) = C

R

K(τ, τ1 , ξ)F (ττ1 , ξ)dττ1 ,

(3.72)

where (the integrals are in classical sense and ξ = 0) sin(t − t1 )|ξ| H(t − t1 ) ϕ1 (t)ϕ2 (t1 ) e−i(tτ −t1 τ1 ) dtdt1 . (3.73) K(τ, τ1 , ξ) = |ξ| R R First, we consider the case when ξ is bounded, i.e., j is bounded in (3.69). Then we can integrate by parts in (3.73) and in this way we ﬁnd 1 |K(τ, τ1 , ξ)| ≤ C ττ1 τ τ − τ1 N for any integer N ≥ 1; this estimate implies τ M |K(τ, τ1 , ξ)| ≤ Cττ1 M−2 f (τ − τ1 ),

(3.74)

30

Piero D’Ancona and Vladimir Georgiev

for any integer M ≥ 1 and any non-negative function f ∈ L1 (R). In the general case, when j ≥ 2, we shall prove the following variant: |τ | + |ξ|s |τ | − |ξ|θ |K(τ, τ1 , ξ)| ≤ C|ττ1 | + |ξ|s−1 |ττ1 | − |ξ|θ−1 f (τ − τ1 ). (3.75) By an integration by parts argument similar to the one used in the proof of (3.74), we can easily reduce the proof of (3.75) to the case when |τ | ∼ |ξ| ∼ 2j and |ττ1 | ∼ |ξ| ∼ 2j . Then we can rewrite (3.75) as |τ | − |ξ|θ |K(τ, τ1 , ξ)| ≤ C|ξ|−1 |ττ1 | − |ξ|θ−1 f (τ − τ1 ). Using (3.73), we can represent the kernel K(τ, τ1 , ξ) in the form K(τ, τ1 , ξ) = U (t1 − t)V (t)W (t1 ) dtdt1 , R

(3.76)

(3.77)

R

where U (t) = −

sin t|ξ| H(−t), |ξ|

(3.78)

V (t) = ϕ1 (t)e−itτ ,

(3.79)

W (t) = ϕ2 (t)e−itτ1 .

(3.80)

Now the relation (ττ2 )V (ττ2 )W (ττ2 )dττ2 U U (t1 − t)V (t)W (t1 ) dtdt1 = C R

R

R

as well as the identities ˆ (ττ2 ) = − U

1 , (ττ2 + i0)2 − |ξ|2

(3.81)

Vˆ (ττ2 ) = ϕ 1 (ττ2 + τ ),

(3.82)

ˆ (ττ2 ) = ϕ W 2 (ττ2 + τ1 ),

(3.83)

show that it is suﬃcient to establish the inequality (3.76), with the kernel K replaced by ˆ (ττ2 )χ(ττ2 ) Kχ (τ, τ1 ) = U ϕ1 (ττ2 + τ ) ϕ2 (ττ2 + τ1 ) dττ2 , (3.84) R

where χ(ττ2 ) is a smooth function. Take ﬁrst χ(ττ2 ) with support close to ±|ξ|; we may assume for determinacy that the cut-oﬀ function χ has support near τ = |ξ|. Then we can take χ(ττ2 ) = χ0 (ττ2 − |ξ|) with χ0 supported near the origin and we can use the following equality (in the sense of distributions) −

1 1 Φ(ττ2 − |ξ|, ξ), χ(ττ2 ) = 2 2 (ττ2 + i0) − |ξ| τ2 + i0 − |ξ|

where Φ(τ, ξ) = −

1 χ0 (τ ) τ + 2|ξ|

(3.85)

Wave Maps

31

is a smooth compactly supported function (with support near τ = 0), such that for any integer N ≥ 0 we have C |∂ ∂τN Φ(τ, ξ)| ≤ N |ξ| with some constant C = C(N ) independent of τ, ξ. Using the change of variables τ2 → σ = τ2 − |ξ|, we get

Kχ (τ, τ1 ) =

R

1 Φ(σ, ξ) ϕ1 (σ + |ξ| + τ ) ϕ2 (σ + |ξ| + τ1 ) dσ. σ + i0

(3.86)

Now we can use a suitable variant of the estimate (3.63). More precisely, given any two smooth rapidly decreasing functions f1 , f2 and any family Φα (σ), α ∈ A, of smooth functions supported in |σ| ≤ 1, which satisfy the estimates |∂ ∂τN Φα (τ )| ≤ CN with a constant C(N ) independent of α ∈ A, we have the estimate 1 C Φα (σ)f1 (σ − a)ff2 (σ − b)dσ ≤ , N σ + i0 a bN

(3.87)

R

for any integer N ≥ 1. It remains to consider the case when χ(ττ2 ) is identically 0 near τ2 = ±|ξ|. Then we have the obvious inequality C ˆ , U (ττ2 )χ(ττ2 ) ≤ |ττ2 | − |ξ||ττ2 | + |ξ| hence (3.84) implies |Kχ (τ, τ1 )| ≤

R

1 C dττ2 |ττ2 | − |ξ||ττ2 | + |ξ| ττ2 + τ N ττ2 + τ1 N

(3.88)

and this estimate leads to the needed estimate (3.76). This completes the proof of the theorem. 3.5. Bilinear estimates for the homogeneous problem in H s,δ The ﬁrst step in the iteration scheme for the non-linear problem is an estimate for the homogeneous problem utt − ∆u = 0,

u(0, x) = f,

ut (0, x) = 0.

(3.89)

More precisely, given two solutions uj , j = 1, 2, to the free wave equation having initial data uj (0, x) = fj (x), ∂t uj (0, x) = 0, the main result of this section concerns an estimate for the bilinear form n Q(∇u1 , ∇u2 ) = ∂t u1 (t, x)∂ ∂t u2 (t, x) − ∂xk u1 (t, x)∂ ∂xk u2 (t, x). (3.90) k=1

32

Piero D’Ancona and Vladimir Georgiev

Theorem 3.3. If f1 , f2 ∈ H s , s > n/2, then Q(∇u1 , ∇u2 ) ∈ H s1 −1, θ−1 , whenever s1 + θ < s + 1/2 and θ > 1/2. Proof. f Our starting point is the following representation Q(∇u 1 , ∇u2 )(τ, ξ)

=C

(3.91)

[(τ − τ1 )ττ1 − (ξ − ξ1 )ξ1 ] u1 (τ − τ1 , ξ − ξ1 ) u2 (ττ1 , ξ1 )dξ1 dττ1 .

Since uj is a solution to (3.89), we have the relation δ(τ − σ|ξ|)fj (ξ). (3.92) uj (τ, ξ) = c δ(τ − |ξ|)fj (ξ) + δ(τ + |ξ|)fj (ξ) = c σ=±1

Given any ξ, ξ1 ∈ Rn , we can consider δ(τ − τ1 − σ1 |ξ − ξ1 |)δ(ττ1 − σ2 |ξ|),

σ1 , σ2 = ±1,

as a distribution on Rτ × Rτ1 deﬁned as follows: δ(τ − τ1 − σ1 |ξ − ξ1 |)δ(ττ1 − σ2 |ξ|), ψ(τ, τ1 ) = ψ(σ2 |ξ1 | + σ1 |ξ − ξ1 |, σ2 |ξ1 |) for any test function ψ(τ, τ1 ) ∈ C0∞ (R2 ). Having in mind this property as well as the obvious relation 1 2 (τ − τ1 )ττ1 − (ξ − ξ1 )ξ1 = τ − ξ 2 − (τ − τ1 )2 + |ξ − ξ1 |2 − τ12 + |ξ1 |2 , (3.93) 2 we see that it is suﬃcient to verify the inequality u1 u2 H s1 , θ ≤ Cf1 H s ff2 H s . The representation formula u 1 u2 (τ, ξ) = c

σ1 ,σ2 =±1

σ1 |ξ−ξ1 |+σ2 |ξ1 |=τ

f1 (ξ − ξ1 )f2 (ξ1 )dξ1

reduces the proof of (3.94) to the following estimate ∞ |τ | + |ξ|2s1 |τ | − |ξ|2θ |II± (τ, ξ)|2 dξdτ ≤ Cf1 2H s ff2 2H s ,

(3.94)

(3.95)

(3.96)

0

where I± (τ, ξ) =

|ξ −ξ1 |±|ξ1 |=τ

Sξ1 . f1 (ξ − ξ1 )f2 (ξ1 )dS

(3.97)

Here and below dS Sξ1 is the surface element on the surfaces S± (τ, ξ) = {ξ1 ; |ξ − ξ1 | ± |ξ1 | = τ }. Note that S+ (τ, ξ) is an ellipsoid for τ > |ξ|, and ∞ P dξ1 = P dS Sξ1 dτ Rn

for any function P ∈ L1 (Rn ).

0

S+ (τ,ξ)

(3.98)

Wave Maps The surface S− (τ, ξ) is a hyperboloid for τ < |ξ|, and ∞ P dξ1 = P dS Sξ1 dτ ξ1 ,ξ<|ξ|2 /2

0

33

(3.99)

S− (τ,ξ)

for any function P ∈ L1 (Rn ). Setting (3.100) F1 (ξ) = ξs f1 (ξ), F2 (ξ) = ξs f2 (ξ), we see that after a Paley-Littlewood partition of unity in the variables ξ we can assume supp F1 (ξ) = {|ξ| ∼ 2j }, supp F2 (ξ) = {|ξ| ∼ 2k } for some j, k ≥ 0 and the proof of (3.96) will follow from the estimate ∞ F1 2L2 F F2 2L2 . (3.101) |τ | + |ξ|2s1 |τ | − |ξ|2θ |II± (τ, ξ)|2 dξdτ ≤ CF 0

We have two cases for the localized frequencies |ξ − ξ1 | ∼ 2j , |ξ1 | ∼ 2k in the integral (3.97). If j ≥ k + 2, then and |II± (τ, ξ)| ≤

ξ − ξ1 ≥ Cξ

C ξs

|ξ −ξ1 |±|ξ1 |=τ

If j ≤ k + 2, then

ξ1 −s |F F1 (ξ − ξ1 )||F F2 (ξ1 )|dS Sξ1 .

ξ1 ≥ Cξ

and we have |II± (τ, ξ)| ≤

C ξs

|ξ −ξ1 |±|ξ1 |=τ

ξ − ξ1 −s |F F1 (ξ − ξ1 )||F F2 (ξ1 )|dξ1 .

An application of the Cauchy inequality implies 2s 2 |F F1 (ξ − ξ1 )|2 |F F2 (ξ1 )|2 dξ1 J± (τ, ξ), ξ |II± (τ, ξ)| ≤ C

(3.102)

|ξ −ξ1 |±|ξ1 |=τ

where J± (τ, ξ) =

J± (τ, ξ) =

(3.103)

ξ − ξ1 −2s dξ1 , if j ≤ k + 2.

(3.104)

|ξ −ξ1 |±|ξ1 |=τ

and

ξ1 −2s dξ1 , if j ≥ k + 2;

|ξ −ξ1 |±|ξ1 |=τ

The inequality (3.102) implies that ∞ ξ2s |II± (τ, ξ)|2 dτ ≤ C |F F1 (ξ − ξ1 )|2 |F F2 (ξ1 )|2 dξ1 , J± (τ, ξ) 0 Rn

(3.105)

34

Piero D’Ancona and Vladimir Georgiev

so the proof shall be ﬁnished if we integrate the last inequality in ξ and verify the inequality C J± (τ, ξ) ≤ (3.106) |τ | − |ξ|2θ for some θ > 1/2. Thus, the ﬁnal step of the proof is a consequence of the following lemma: Lemma 3.5. If n ≥ 2 and s > n/2, then for |ξ| < τ we have the inequality C ξ − ξ1 −2s dS Sξ1 + ξ1 −2s dS Sξ1 ≤ , |τ | − |ξ|2s−n+1 |ξ −ξ1 |+|ξ1 |=τ |ξ −ξ1 |+|ξ1 |=τ (3.107) while for |ξ| > τ > 0 we have C ξ − ξ1 −2s dS Sξ1 + ξ1 −2s dS Sξ1 ≤ . 2s−n+1 |τ | − |ξ| |ξ −ξ1 |−|ξ1 |=τ |ξ −ξ1 |−|ξ1 |=τ (3.108) Proof. It is easy to show the identities ξ − ξ1 −2s dS Sξ1 = S± (τ,ξ)

S± (±τ,ξ)

ξ1 −2s dS Sξ1 ,

where S± (τ, ξ) = {ξ1 ; |ξ − ξ1 | ± |ξ1 | = τ }. Therefore, it is suﬃcient to show that for τ > |ξ| we have C L+ (τ, ξ) ≡ ξ1 −2s dS Sξ1 ≤ , 2s−n+1 |τ | − |ξ| |ξ −ξ1 |+|ξ1 |=τ while for |ξ| > τ > 0 we have L− (τ, ξ) ≡

|ξ −ξ1 |−|ξ1 |=τ

ξ1 −2s dS Sξ1 ≤

C . |τ | − |ξ|2s−n+1

(3.109)

(3.110)

We now introduce polar coordinates ρ = |ξ1 |, ϕ ∈ (0, π), ω ∈ Sn−2 , such that ξ1 = (ρ cos ϕ, ρ sin ϕω). Then the Euclidean metric in Rn can be written (if n ≥ 3) dρ2 + ρ2 (dϕ2 + sin2 ϕdω 2 ), where dω 2 is the standard metric on Sn−2 . A trivial modiﬁcation in the above relations is necessary for the special case n = 2, when ξ1 = (ρ cos ϕ, ρ sin ϕ) and the metric in R simpliﬁes to 2

dρ2 + ρ2 dϕ2 .

Wave Maps

35

The ellipsoid S+ (τ, ξ) = {|ξ − ξ1 | + |ξ1 | = τ } in the new coordinates will have the equation (recall that S+ (τ, ξ) is an ellipsoid if and only if τ > |ξ|) ρ=

τ 2 − |ξ|2 , 2(τ − |ξ| cos ϕ)

ϕ ∈ (0, π).

(3.111)

We have further

(τ 2 − |ξ|2 )|ξ| sin ϕ . 2(τ − |ξ| cos ϕ)2 Thus ρ(ϕ) is a decreasing function and ρ (ϕ) = −

(3.112)

|τ | − |ξ| |τ | + |ξ| ≥ ρ(ϕ) ≥ ρ(π) = . 2 2 The surface element dS Sξ1 can be written dS Sξ1 = ρ2 + |ρ |2 ρn−2 sinn−2 ϕdϕdω, ρ(0) =

so that we have

L+ (τ, ξ) = C ≤C

π

ρ−2s

ρ2 + |ρ |2 ρn−2 sinn−2 ϕdϕ

0 π

(3.113)

ρ−2s ρn−1 sinn−2 ϕdϕ + C

0

π

ρ−2s |ρ |ρn−2 sinn−2 ϕdϕ.

0

The inequalities (3.113) imply that π ρ−2s+n−1 sinn−3 ϕdϕ ≤ 0

By a simple change of variables we get π − ρ−2s ρ ρn−2 sinn−2 ϕdϕ ≤ 0

ρ(0)

C . |τ | − |ξ|2s−n+1

ρ−2s ρn−2 dρ ≤

ρ(π)

C |τ | − |ξ|2s−n+1

and hence

C . |τ | − |ξ|2s−n+1 This argument completes the proof of (3.107). The corresponding equation for the hyperboloid L+ (τ, ξ) ≤

S− (τ, ξ) = {|ξ − ξ1 | − |ξ1 | = τ } is ρ= where

|ξ|2 − τ 2 , ϕ ∈ (0, ϕ0 ) 2(|ξ| cos ϕ + τ ) τ ϕ0 = arccos − . |ξ|

(3.114)

(3.115)

Obviously, ρ ≥ C(|ξ| − τ ).

(3.116)

36

Piero D’Ancona and Vladimir Georgiev

For 0 < τ < |ξ| the denominator τ + |ξ| cos ϕ vanishes if and only if ϕ = ϕ0 . We have the following relation ρ (ϕ) =

(|ξ|2 − τ 2 )|ξ| sin ϕ . 2(τ + |ξ| cos ϕ)2

Thus, ρ(ϕ) is an increasing function and ϕ0 −2s n−1 ρ ρ dϕ + C L− (τ, ξ) ≤ C 0

0

ϕ0

ρ−2s ρn−2 ρ dϕ.

(3.118)

0

Combining (3.116) and (3.117), we ﬁnd ϕ0 ρ−2s ρn−1 dϕ ≤ C and

ϕ0

(3.117)

ρ−2s ρn−2 ρ dϕ ≤

∞

C |τ | − |ξ|2s−n+1

ρ−2s ρn−2 dρ ≤ C

ρ(0)

0

C . |τ | − |ξ|2s−n+1

This completes the proof of (3.110) and of the lemma.

3.6. Bilinear estimates in H s,δ for the inhomogeneous problem We now consider the inhomogeneous problem utt − ∆u = F,

u(0, x) = 0,

ut (0, x) = 0,

(3.119)

where F ∈ H with s > n/2, θ > 1/2. In this section we shall consider the bilinear form (3.90), namely s,θ

Q(∇u1 , ∇u2 ) = ∂t u1 (t, x)∂ ∂t u2 (t, x) −

n

∂xk u1 (t, x)∂ ∂xk u2 (t, x)

(3.120)

k=1

for two functions u1 , u2 ∈ H s,δ (Rt ×Rm x ). Our main goal is to establish the property Q(H s,δ , H s,δ ) → H s−1,δ−1 .

(3.121)

The starting point is the following representation of the Fourier transform of Q(∇v, ∇w): ∇w)(τ, ξ) Q(∇v, (3.122) =c ((τ − τ1 )ττ1 − (ξ − ξ1 )ξ1 ) v(τ − τ1 , ξ − ξ1 )w(τ τ1 , ξ1 )dξ1 dττ1 . R

Rm

Using the identity (ττ1 + τ2 )2 − (ξ1 + ξ2 )2 − (ττ12 − ξ12 ) − (ττ22 − ξ22 ) , 2 we see that (3.121) follows from the embeddings τ1 τ2 − ξ1 ξ2 =

H s,θ · H s,θ → H s,θ

(3.123)

H s−1,θ−1 · H s,θ → H s−1,θ−1 .

(3.124)

and

Wave Maps

37

In the following we shall give a proof of (3.123) only; the proof of property (3.124) is similar. Theorem 3.4. If n ≥ 2, s > n/2 and θ > 1/2, then H s,θ . H s,θ · H s,θ →

(3.125)

Proof. We start again with the following modiﬁcation of (3.122): u 1 u2 (τ, ξ) = c

u 1 (ττ2 , ξ2 ) u2 (ττ1 , ξ1 )dξ1 dτ1 ,

(3.126)

τ1 +τ2 =τ, ξ1 +ξ2 =ξ

where here and below F (ττ1 , ξ1 , τ2 , ξ2 )dξ1 dττ1 = τ1 +τ2 =τ, ξ1 +ξ2 =ξ

Rm

F (ττ1 , ξ1 , τ − τ1 , ξ − ξ1 )dξ1 dτ1 .

We begin by proving the following elementary inequality. Lemma 3.6. If τ1 + τ2 = τ and ξ1 + ξ2 = ξ, then we have the inequality ||ττ1 + τ2 | − |ξ1 + ξ2 || ≤ ||ττ1 | − |ξ1 || + ||ττ2 | − |ξ2 || + r(ξ1 , ξ2 ), where

⎧ ⎨ ||ξ1 | + |ξ2 | − |ξ||, if τ1 τ2 ≥ 0; ||ξ1 | − |ξ2 | − |ξ||, if τ1 τ2 < 0, |ττ1 | > |ττ2 |; r(ξ1 , ξ2 ) = ⎩ ||ξ2 | − |ξ1 | − |ξ||, if τ1 τ2 < 0, |ττ1 | ≤ |ττ2 |.

(3.127)

(3.128)

Proof. If τ1 τ2 ≥ 0, then |ττ1 + τ2 | = |ττ1 | + |ττ2 |, so that |ττ1 + τ2 | − |ξ1 + ξ2 | = |ττ1 | − |ξ1 | + |ττ2 | − |ξ2 | + |ξ1 | + |ξ2 | − |ξ1 + ξ2 | and applying the triangle inequality, we deduce (3.127). In the case when τ1 τ2 < 0, |ττ1 | > |ττ2 | we have |ττ1 + τ2 | = |ττ1 | − |ττ2 |, so that |ττ1 + τ2 | − |ξ1 + ξ2 | = |ττ1 | − |ξ1 | − |ττ2 | + |ξ2 | + |ξ1 | − |ξ2 | − |ξ1 + ξ2 | and (3.127) follows again. In a similar way we treat the case τ1 τ2 < 0, |ττ1 | ≤ |ττ2 | and this completes the proof of the lemma. Turning back to the proof of the theorem, we set Fj (τ, ξ) = |τ | + |ξ|s |τ | − |ξ|θ u j (τ, ξ),

j = 1, 2.

(3.129)

Since the integrals in (3.126) are invariant when exchanging u1 and u2 , there is no loss of generality in assuming that |ττ1 | + |ξ1 | ≥ |ττ2 | + |ξ2 |.

(3.130)

38

Piero D’Ancona and Vladimir Georgiev

Then we can use Lemma 3.6 and we ﬁnd |τ | + |ξ|s |τ | − |ξ|θ u 1 u2 (τ, ξ) ≤ C(I(τ, ξ) + II(τ, ξ) + III(τ, ξ)), where

(3.131)

|ττ1 | + |ξ1 |s |ττ1 | − |ξ1 |θ F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 )dξ1 dτ1 , h(ττ1 , ξ1 )h(ττ2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ (3.132) s θ |ττ1 | + |ξ1 | |ττ2 | − |ξ2 | F1 (ττ1 , ξ1 )F II(τ, ξ) = F2 (ττ2 , ξ2 )dξ1 dττ1 , h(ττ1 , ξ1 )h(ττ2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ (3.133) s θ |ττ1 | + |ξ1 | r(ξ1 , ξ2 ) F1 (ττ1 , ξ1 )F III(τ, ξ) = F2 (ττ2 , ξ2 )dξ1 dττ1 , h(ττ1 , ξ1 )h(ττ2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ (3.134) and h(τ, ξ) = |τ | + |ξ|s |τ | − |ξ|θ . I(τ, ξ) =

The property (3.125) will be established if we can verify the inequality F1 L2τ,ξ F F2 L2τ,ξ . |τ | + |ξ|s |τ | − |ξ|θ u 1 u2 (τ, ξ)L2τ,ξ ≤ CF To estimate the term I we use the inequality F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 ) dξ1 dτ1 , I(τ, ξ) ≤ h(τ τ 2 , ξ2 ) τ1 +τ2 =τ, ξ1 +ξ2 =ξ and obtain

(3.135)

(3.136)

I

L2τ,ξ

≤ ≤

|F F2 (ττ2 , ξ2 )| dξ2 dττ2 h(ττ2 , ξ2 ) 1 CF F1 L2τ ,ξ F F2 L2τ ,ξ . h2 2 1 1 2 2 L CF F1

L2τ ,ξ 1 1

τ1 ,ξ1

The conditions s > n/2, θ > 1/2 imply that 1 < ∞, h2 2 L τ1 ,ξ1

so that IL2τ,ξ ≤ CF F1 L2 F F2 L2 . As to the term II we have IIL2τ,ξ

≤ ≤

C

F1 θ |ττ1 | − |ξ1 | L2

1 ξ1 Lτ1

CF F1 L2τ

1 ,ξ1

F F2 L2τ

2 ,ξ2

.

(3.137)

F2 |ξ2 |s

L1ξ L2τ2 2

(3.138)

Wave Maps

39

Finally, we consider the term III. Note that (3.134) implies |III(τ, ξ)| ≤

r(ξ1 , ξ2 )θ F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 )dξ1 dττ1 . τ1 | − |ξ1 |θ |ξ2 |s |ττ2 | − |ξ2 |θ τ1 +τ2 =τ, ξ1 +ξ2 =ξ |τ (3.139) It will be suﬃcient to study only the cases F2 ⊆ {ττ2 > 0}, Case A: suppF F1 ⊆ {ττ1 > 0}, suppF Case B: suppF F1 (ττ1 , ξ1 )F F2 (ττ2 , ξ2 ) ⊆ {ττ1 > 0, τ2 < 0, τ1 + τ2 > 0}, since the other cases are similar. In Case A we make the change of variables τ1 → κ = τ1 − |ξ1 |

(3.140)

and introduce elliptic coordinates λ = |ξ1 | + |ξ − ξ1 |

(3.141)

so that an application of Lemma 3.6 gives r(ξ1 , ξ2 ) = |λ − |ξ|| and |ττ2 | − |ξ2 | = τ2 − |ξ2 | = τ − τ1 − |ξ2 | = τ − κ − λ. From (3.139) we deduce ∞ λ − |ξ|θ |III(τ, ξ)| ≤ F ∗ F ∗ dS Sξ1 dλdκ, θ s θ 1 2 0 |ξ1 |+|ξ−ξ1 |=λ κ |ξ − ξ1 | τ − κ − λ (3.142) where F1∗ = F1 (κ + |ξ1 |, ξ1 ), F2∗ = F2 (τ − κ − λ + |ξ − ξ1 |, ξ − ξ1 ). Applying the Cauchy inequality together with Lemma 3.5, we ﬁnd ∞ 2 |III(τ, ξ)| ≤ C |F F1∗ F2∗ |2 dS Sξ1 dλdκ (3.143) 0 |ξ1 |+|ξ−ξ1 |=λ = C |F F1 (ττ1 , ξ1 )|2 |F F2 (ττ2 , ξ2 )|2 dξ1 dττ1 , τ1 +τ2 =τ, ξ1 +ξ2 =ξ

so integrating in τ, ξ, we obtain IIIL2τ,ξ ≤ CF F1 L2 F F2 L2 .

(3.144)

In Case B we make the change of variables τ1 → κ = τ1 − |ξ1 |

(3.145)

and introduce hyperbolic coordinates λ = |ξ1 | − |ξ − ξ1 | so that an application of Lemma 3.6 gives r(ξ1 , ξ2 ) = |λ − |ξ||

(3.146)

40

Piero D’Ancona and Vladimir Georgiev

and |ττ2 | − |ξ2 | = −ττ2 − |ξ2 | = −τ + τ1 − |ξ2 | = −τ + κ + λ. From (3.139) we deduce ∞ λ − |ξ|θ F ∗ F ∗ dS Sξ1 dλdκ, |III(τ, ξ)| ≤ θ s τ − κ − λθ 1 2 κ |ξ − ξ | 1 0 |ξ1 |−|ξ−ξ1 |=λ (3.147) and the above argument used in Case A implies again (3.144). Using the inequalities (3.137), (3.138) and (3.144), we complete the proof of the theorem.

4. Concentration of the local energy 4.1. Introduction In this chapter we shall study a special property of the wave maps with inhomogeneous right-hand side Dα ∂α u = F ; (4.1) we recall that ∂ ∂α = , (xα ) = (t, x) ∈ R1+m , α = 0, 1, . . . , m ∂xα denote the partial derivatives with respect to the variables xα , while Dα are the covariant pull-back derivatives in the bundle u∗ T N. As usual, the Greek indices α, β run from 0 to m. We use summation convention over repeated indices. The corresponding initial data are u(0, x) = u0 (x) ∈ H s (Rm ; N ),

ut (0, x) = u1 (x) ∈ H s−1 (Rm ; T N ).

(4.2)

If the target is a hyperboloid (or a manifold with negative curvature and suitable symmetry) results due to Shatah and Tahvildar-Zadeh [32], Christodoulou and Tahvildar-Zadeh [8], Grillakis [13], show that for any C ∞ initial data a global unique C ∞ solution exists, at least in the class of radial or equivariant wave maps. The key point in this approach is the following property, called the nonconcentration of energy: |∇t,x u(t, x)|2 dx → 0, (4.3) Ω(t)

as t → 0 and Ω(t) = {x ∈ R : |x| < t}. In this chapter we study the property (4.3) for more general domains of the form Ωα (t) = {x ∈ R2 : |x|α < t}, where α ∈ (0, 1] and the target is S 2 . We shall consider the inhomogeneous problem (4.1), assuming u is an equivariant wave map. Our goal is to compare the following local norms: t and F (s, ·)H ε (Ωα (s)) ds, u(t, ·)H 1+ε (Ωα (t)) 2

0

Wave Maps

41

for ε ≥ 0. When ε = 0 and α = 1, these are exactly the norms appearing in the local energy estimate T 1 k ∂t u(T, ·) 1−k +C F (s, ·)H 0 (Ω(s)) ds, (4.4) u(t, ·)H 1 (Ω(t)) ≤ C H (Ω(T )) t

k=0

where 0 < t < T. This energy estimate, combined with a suitable control of L∞ norm, leads to the property (4.3) (see, e.g., [32] for details ). If one tries to improve the regularity in (4.4) by a small additional ε > 0, then a natural question arises: does an estimate like u(t, .)H 1+ε (Ωα (t)) ≤C

(4.5)

1 k ∂t u(T, .) 1−k+ε +C H (Ωα (T )) k=0

T

t

F (s, .)H ε (Ωα (s)) ds

hold for t ∈ [0, T ]? This question of course is particularly interesting in the case of dimension n = 2, when the energy space H 1 coincides with the critical space with respect to scaling and local existence; the behavior of (4.5) as ε → 0 can be regarded as a measure of the instability of H 1 from the point of view of local existence. Our goal is to show that (4.5) fails in general in n = 2, and actually an even stronger statement holds: we shall construct wave maps for which lim u(t, ·)H 1+ε (Ωα (t)) = ∞,

t→0+

while the expression 1 k ∂t u(T, ·) 1+ε + H (Ωα (T ))

0

k=0

T

F (s, ·)H ε (Ωα (s)) ds

is bounded for 0 < α < 1 and ε < min 12 , 1 − α . More precisely, we have the following. Theorem 4.1. Consider the inhomogeneous wave maps equation (4.1) deﬁned on R2+1 with target N = S2 . Then one can ﬁnd positive numbers ε > 0, T > 0, initial data (u0 , u1 ) ∈ H 1+ε × H ε and a source term F ∈ L1 ((0, T ); H ε (Ωα (t))) such that the Cauchy problem for (4.1) with initial data at t = T u(T, x) = u0 (x),

ut (T, x) = u1 (x)

has a solution u deﬁned (at least) on the domain Kα (T ) = {(t, x); t ∈ (0, T ], x ∈ Ωα (t)}, with α ∈ (0, 1 − ε), satisfying u ∈ C((0, T ]; H 1+ε (Ωα (t)))

(4.6)

42

Piero D’Ancona and Vladimir Georgiev

and lim u(t)H 1+ε (Ωα (t)) = ∞.

t→0+

Remark 4.1. The condition u ∈ C((0, T ]; H 1+ε (Ωα (t))) means that the function deﬁned as v(t) = u(t, ·)H 1+ε (Ωα (t)) belongs to C((0, T ]). This theorem shows that the naive intuitive argument based on approximation of (weak) H 1 solutions in the light cone K(T ) = {(t, x); t ∈ (0, T ], x ∈ Ω(t)} by means of sequences of smoother H 1+ε solutions in the slightly distorted “cones” Kα (T ) = {(t, x); t ∈ (0, T ], x ∈ Ωα (t)}, might have a concentration of the local energy as shown by the property lim u(t)H 1+ε (Ωα (t)) = ∞.

t→0+

The plan of the chapter is the following. In Section 4.2 we construct the wave maps used in the proof of the theorem by a rescaling in time of solutions of a suitable elliptic problem. This method does not yield an exact solution of the wave maps equation, but produces instead an error which is exactly the source term in (4.1); we then estimate the solution and the source term in the energy norms. In Section 4.3 we compute the higher-order norms H 1+ε of the solution and the C([0, T ]; H ε )-norm of the source term, and conclude the proof of the theorem. The Appendix is devoted to the proof of some technical lemmas. 4.2. Construction of the solutions From now on we will use the following notations: if f and g are two positive, real-valued functions, we write f g if there exists a constant C > 0 such that f ≤ Cg; we shall write f ∼ g, if there exist constants A > 0 and B > 0, such that Ag ≤ f ≤ Bg. Consider the equation (2.66) for equivariant wave maps with initial data φ(1, r) = φ0 (r),

φt (1, r) = φ1 (r),

(4.7)

where φ = φ(t, r) depends only on t and r. We shall construct a solution of the following special form: v(r) φ(t, r) = Q ; (4.8) t the function Q must satisfy a suitable ordinary diﬀerential equation, which we derive now. By the deﬁnition of ϕ we have: v(r) v(r) (v(r))2 v(r) 2v(r) v(r) Q Q , ∂t2 φ = + , ∂t φ = − 2 Q t t t4 t t3 t v (r) v(r) (v (r))2 v(r) v (r) v(r) 2 Q Q Q ∂r φ = , ∂r φ = + . t t t2 t t t

Wave Maps Plugging these quantities into (2.66) we see that ϕ satisﬁes the identity 2 sin 2φ ∂t − ∆ φ(t, x) + 2 2r 2 (v(r)) v(r) 2v(r) v(r) (v (r))2 v(r) = Q Q + 3 Q − t4 t t t t2 t v (r) v(r) sin 2Q v (r) v(r) Q Q . − + − t t rt t 2r2

43

(4.9)

Our main idea is to regard all the terms involving the time derivatives as a source term, i.e., to choose (v(r))2 v(r) 2v(r) v(r) 2 f = ∂t u = Q + 3 Q ; (4.10) t4 t t t then the equation (4.9) (∂ ∂t2 − ∆)φ(t, x) +

sin 2φ =f 2r2

simpliﬁes to sin 2φ = 0, 2r2 and, recalling our choice of ϕ, this leads to the following equation for Q −∆φ(t, x) +

(v (r))2 v(r) v (r) + rv (r) sin 2Q Q − Q + = 0. t2 t rv(r) 2r2

(4.11)

As v = v(r) we may choose a solution of the following ordinary diﬀerential equation: v (r) + rv (r) c = 2, (4.12) rv(r) r where the positive constant c is a parameter to be chosen. This is an equation of Euler-type: r2 v (r) + rv (r) − cv(r) = 0. A special solution to (4.12) is v = rα , provided we take c = α2 . With these choices, the equation for Q becomes α2

α 1 r2α sin 2Q 2 1 r Q − Q + α = 0. r 2 t2 r2 t 2r2

Setting for brevity z=

(4.13)

rα , t

we can rewrite (4.13) as follows: α2 z 2 Q + α2 zQ −

sin 2Q = 0. 2

Now, making change of the variable s=

1 ln z, α

(4.14)

44

Piero D’Ancona and Vladimir Georgiev

we get sin 2Q(s) = 0. (4.15) 2 We have not yet chosen the initial data for Q. Multiplying the equation by Q we obtain 1 + cos 2Q = const (4.16) (Q )2 + 2 and this means that the quantity Q (s) −

1 + cos 2Q ≡ (Q )2 + (cos Q)2 2 is constant on the integral curves of (4.15) or, in other words, is a ﬁrst integral of the equation. Now we may choose the initial data for Q such that I(s) is equal to 1: indeed, it is suﬃcient to take π Q(0) = , Q (0) = 1 = =⇒ I(s) = (Q )2 + (cos Q)2 = 1. 2 The last equation has the two solutions Q tan = c0 e±s , 2 I(s) = (Q )2 +

and our choice of the initial data for Q implies c0 = 1. Discarding the solution with sign − we ﬁnally obtain 1 Q(z) = 2 arctan z α . (4.17) Our next step is to study the regularity properties of the remainder f (t, r) deﬁned as above by r2α rα rα rα f (t, r) = 4 Q +2 3Q (4.18) t t t t in the region rα < t. In fact, we shall prove the following: Lemma 4.1. For any α ∈ (0, 1) and T > 0 we have f ∈ L1 ((0, T ); H 0 rad (Ωα (t))). Proof. As before we use the notation rα . t From the equations (4.18) and (4.14) for Q we have: 1 sin 2Q f = 2 zQ + . t 2α2 z=

So, the norm of f in H 0 rad (Ωα (t)) = L2 rad (Ωα (t)) for ﬁxed t ∈ (0, T ) is: 2 α1 1 t sin 2Q 2 r dr, zQ (z) + f (t, ·)L2 rad (Ωα (t)) = 4 t 0 2α2

(4.19)

(4.20)

Wave Maps

45

where in terms of z we have 2 z α −1 , α 1 + z α2 1

Q (z) = so we get 2 f (t, ·)L2 rad (Ωα (t))

≤ ≤

≤

C t4

1

tα

2

zα

2

+ (sin 2Q)

2

(1 + z α )2

0

⎛

1

2 α

tα

r dr ⎞

1

tα

C⎝ z r dr + (sin 2Q)2 r dr⎠ 2 2 t4 α (1 + z ) 0 0 ⎛ 1 ⎞ α 2 C C ⎝ t r3 α⎠ = 2 dr + t 2 , t4 tα t4− α 0

where we have used the inequality

1 2

1+z α

≤ 1 and C = C(α, T ) > 0 is a constant

independent of t. Finally, we have the following inequality: C (4.21) f (t, ·)L2 rad (Ωα (t)) ≤ 2− 1 . t α Now it is obvious that the function t−2+1/α is in L1 (0, T ) if α < 1, and this completes the proof. The next lemma shows that the solution ϕ(t, ·) belongs to the energy space H 1 : Lemma 4.2. The solution φ(t, r) of (2.66), deﬁned according to (4.8), belongs to H 1 rad (Ωα (t)) for every ﬁxed t > 0. Proof. For any ﬁxed t > 0 we have t α1 2 2 = |∂ ∂r φ| r dr ≤ C φ(t, ·)H˙ 1 rad

0

0

≤

1 tα

C

1

tα

2

zα (1 + z

r

2 α

)2

1 dr r

2 dr = const. tα Note that the reverse inequality is also true. Indeed, we have t α1 2 zα 1 2 2 dr φ(t, .)H˙ 1 |∂ ∂r φ| r dr = C 2 (Ωα (t)) = 2 rad(loc) r (1 + z α ) Ωα (t) 0

(4.22)

0

≥C

1

tα

r 2

0

tα

dr = const ,

where we used inequalities 1 1 ≤ ∀z ∈ [0, 1]. 2 ≤ 1, 2 1 + zα

(4.23)

46

Piero D’Ancona and Vladimir Georgiev

The solution of the equation has the form 1 1 φ = 2 arctan z α z α =: w, and this gives the bound 2

2

u(t, ·)L2 rad (Ωα (t)) w(t, ·)L2 rad (Ωα (t)) . Then we get φ(t, ·)L2 rad (Ωα (t))

≤ C

2 w(t, ·)L2 rad (Ωα (t))

1

tα

= 0

r2 t

2 α

1

r dr = Ct α .

This concludes the proof of the lemma. 4.3. Higher regularity of the solution

We now estimate higher-order norms of the solution constructed in the previous section and of the corresponding source term. In order to carry an explicit computation of the fractional Sobolev norms, the following well-known alternative deﬁnition is useful: Deﬁnition 4.1. (see, e.g., [Triebel 2.5.1]) Let s > 0. The fractional H s (R2 ) norm can be deﬁned as follows: 1/2 2 −(1+2s) [s]+1 f H s (R2 ) = f L2 (R2 ) + |h| dh . ∆ h f (x) R2

L2 (R2 )

(4.24) The Sobolev space H s consists of all functions f such that the above norm is ﬁnite. Here ∆k h f (x) is the diﬀerence of order k deﬁned as follows: the diﬀerence of order 0 and 1 of the function f are simply ∆0 h f (x) = f (x) and ∆1 h f (x) = f (x + h) − f (x); then the diﬀerence of order k may be deﬁned inductively as ∆k h f (x) = ∆1 h (∆k−1 h f (x)). In order to extend the deﬁnition of fractional Sobolev spaces to subdomains Ω of R2 , it is suﬃcient to consider, for each function f on Ω, the set of all its extensions f˜ to R2 (i.e., f˜Ω = f ): Deﬁnition 4.2. Let s > 0. We say that the function f ∈ H s (Ω) if the following norm is ﬁnite: f H s (Ω) = inf f˜H s (R2 ) , f˜

(4.25)

where the inf is taken over all extensions f˜ ∈ H s (R2 ) of f . We now specialize the above deﬁnitions to our (radial) situation. We consider ﬁrst the case s ∈ (0, 1):

Wave Maps

47

s Deﬁnition 4.3. Let 0 < s < 1. We say that f = f (r) belongs to the space Hrad (R2 ) if the following norm is ﬁnite: 1/2 1 ∞ 1 −(1+2s) ∆ h f (r) 2 r dr dh f H s rad (R2 ) = f L2 rad (R2 ) + |h| . 0

0

(4.26) Now we consider the case s ∈ (1, 2): Deﬁnition 4.4. Let s = 1 + ε ∈ (1, 2). We say that f = f (r) belongs to the space 1+ε 1 Hrad (R2 ), if f ∈ Hrad (R2 ) and the following norm is ﬁnite: 1/2 1 ∞ 1 −(1+2ε) ∆ h ∂r f (r) 2 r dr dh f H 1+ε rad (R2 ) = f H 1 rad (R2 ) + |h| . 0

0

(4.27) The deﬁnition of the fractional Sobolev spaces on a ball Ω of radius R centered at 0 is identical to the above one (4.25). Moreover, for any radially symmetric function f (x) = f (|x|), we have the following obvious inequality 1/2 R/2 2 [s]+1 −(1+2s) f H s (|x|
0

(4.28) On the other hand, for R = 1 and for any integer k the norm f H k (|x|<1) is equivalent to the norm ∂ ∂xα f˜L2 (|x|<2) . |α|≤k

Here

f˜(r) =

f (r), if 0 ≤ r < 1; f (2 − r)ϕ(r − 1), if 1 < r < 2

(4.29)

where ϕ(s) is a cut-oﬀ function such that ϕ(s) = 1 for |s| ≤ 1/2 and ϕ(s) = 0 for |s| ≥ 1. Using this fact and an interpolation argument, we see that for any real s > 0 we have (4.30) f H s (|x|<1) ∼ f˜H s (R2 ) . To prove Theorem 4.1 we can use the equivalence of the H s -norms of the solutions u and φ obtained in Lemma 4.3 (and the analogous relation for the source terms as given in Lemma 4.4); this reduces the proof to the analysis of the solution φ = φ(t, r) of the Cauchy problem (2.66). Moreover, making a shift in the time variable we can impose data of the form (4.7) (i.e., with initial time t = 1). Then the key point of the proof is an improvement of the regularity result of the previous section. Indeed, we shall show that the source term f = f (t, r), deﬁned ε (Ωα (t))), while the solution φ = φ(t, r) in (4.18), is in the class L1 ((0, T ); Hrad deﬁned in (4.8) belongs to 1+ε (Ωα (t))) C((0, T ]; Hrad

48

Piero D’Ancona and Vladimir Georgiev

for 0 < ε < 1 − α. In this way the proof of Theorem 4.1 can be reduced to the proof of the following: Theorem 4.2. Let α ∈ (0, 1) and T > 0. For any ε > 0 such that 1 ,1 − α ε < min 2 we have the properties: ε f ∈ L1 ((0, T ); Hrad (Ωα (t))),

(4.31)

φ∈

(4.32)

1+ε C((0, T ]; Hrad (Ωα (t))),

lim φ(t, ·)H 1+ε (Ωα (t)) = ∞.

t→0+

(4.33)

rad

Proof. There is no loss of generality if we assume T = 1. First of all, we will ε estimate the Hrad (Ωα (t))-norm of f = f (t, r) for ﬁxed t ∈ (0, 1); recall that in the previous section we have estimated the L2rad (Ωα (t))-norm of f = f (t, r). Applying Deﬁnition (4.25), we see that it is suﬃcient to estimate the quantity 2θ 1 1 |∆1h f˜(t, r)|2 r dr dh, (4.34) 1+2ε 0 h 0 where θ = t1/α , the extension f˜ is constructed as follows:

f (t, r), if 0 < r < θ; ˜ f (t, r) = (4.35) f (t, r)ϕ((r − θ)/θ)), if r > θ, and ϕ(s) is a standard cut-oﬀ function, such that ϕ(s) = 1 for |s| ≤ 1/2 and ϕ(s) = 0 for |s| ≥ 1. Note that h (4.36) θ with ϕθ (r) = ϕ((r − θ)/θ)) and C is a constant, independent of θ ∈ (0, 1). In the quantity (4.34) we can split the integral in r in the two pieces 0 < r < θ and θ < r < 2θ; we shall estimate only the ﬁrst piece, i.e., 1 θ 1 I(t) = |∆1h f˜(t, r)|2 rdrdh, (4.37) 1+2ε 0 h 0 |∆1h ϕθ (r)| ≤ C

since the estimate of the second piece 1 2θ 1 |∆1h f˜(t, r)|2 r dr dh, 1+2ε 0 h θ is completely similar and uses only (4.36) in addition to the argument presented below. Recalling (4.19), we know that the function f is given by the following expression: 2rθ 1 1 + sin 2Q f (t, r) = 2 t α(r2 + θ2 ) 2α2

Wave Maps

49

where θ = t1/α . Now change the order of integration in I(t), ﬁrst with respect to h, then with respect to r, and divide the integral in two parts:

θ

θ−r

I1 = 0

0

θ

I2 =

1

1

θ −r

0

1 h1+2ε

h1+2ε

1 ∆ f (t, r) 2 r dh dr, h

(4.38)

2 1˜ ∆h f (t, r) r dh dr.

(4.39)

This allows to simplify the estimate for the function f = f (t, r). Indeed, writing explicitly the integral I1 and using the trivial estimate (a + b)2 ≤ 2(a2 + b2 ), we arrive at the following expression: I1

≤

C 0

θ

r t4

θ−r 0

1 h(1+2ε)

2 (r + h)θ rθ − (r + h)2 + θ2 r2 + θ 2

1 2 + (1+2ε) |(sin 2Q(r + h) − sin 2Q(r))| dh dr h 2 θ θ−r h(θ2 − r2 − rh) r θ2 4 2 2 2 2 (1+2ε) ((r + h) + θ )(r + θ ) h 0 t 0 1 1 2 + (1+2ε) 2 |sin 2Q(r + h) − sin 2Q(r)| dh dr. 2α h

(4.40)

The function θ2 − (r2 + rh) ≥ 0 for h ∈ [0, θ − r] has a maximum at h = 0, so we have 2 θ − (r2 + rh) ≤ (θ2 − r2 ). On the other hand, we know that 1 1 ≤ 2 ((r + h)2 + θ2 )(r2 + θ2 ) (r + θ2 )2 and |sin 2Q(r + h) − sin 2Q(r)| ≤ |2Q(r + h) − 2Q(r)| ≤ 2 |Q (r + ωh)h| for some 0 < ω < 1. Note that Q (r + ωh) =

2θ , (r + ωh)2 + θ2

so we see that the following inequality is true: 2θ 2θ . ≤ 2 2 2 (r + ωh) + θ (r + θ2 )

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Piero D’Ancona and Vladimir Georgiev

Hence, the integral I1 can be estimated by the following chain of inequalities: θ θ−r 2 2 2 r 4θ2 h(1−2ε) (1−2ε) (θ − r ) 2 I1 h θ + dh dr 4 (r2 + θ2 )4 2α2 (r2 + θ2 )2 0 t 0 θ2 θ θ4 4θ2 1 2θ (2−2ε) r(θ − r) dr + r(θ − r)(2−2ε) 2 dr 4 2 2 4 4 t 0 (r + θ ) t 0 (r + θ2 )2 θ θ 1 1 (2−2ε) r(θ − r) dr + 2 4 r(θ − r)(2−2ε) dr. (4.41) θ 2 t4 0 θ t 0 The function (θ − r)2−2ε is decreasing in the interval [0, θ], so we have that (θ − r)2−2ε ≤ θ2−2ε . This remark gives the estimate I1

θ2−2ε t4

(4.42)

and recalling that θ = t1/α we obtain I1

1 2(1−ε) t4− α

.

Consider now the second integral: θ 1 2 1 1 ˜ I2 = f (t, r) ∆ r dh dr. h 1+2ε 0 θ −r h The obvious estimate |a − b|2 ≤ 2(a2 + b2 ) implies that

2 2 2 1˜ ∆h f (t, r) ≤ 2 f˜(t, r) + 2 f˜(t, r + h) .

Further, we have the estimates

rθ + 1 , r2 + θ 2 C C rθ hθ ˜ |f (t, r + h)| ≤ 2 +1 + 2 , t (r + h)2 + θ2 t (r + h)2 + θ2 and using the inequalities |f˜(t, r)| ≤

C t2

rθ rθ 1 ≤ 2 ≤ , (r + h)2 + θ2 r + θ2 2 rh rh 1 ≤ 2 ≤ 2 2 2 (r + h) + θ h +θ 2 we obtain

2 C 1 ∆h f (t, r) ≤ 2 . t

(4.43)

(4.44)

Wave Maps

51

Then I2 can be estimated as follows: 1 θ 1 1 r C θ I2 ≤ 4 r dh dr ≤ C − 1 dr 1+2ε 4 t 0 (θ − r)2ε θ −r h 0 t θ 1 r 1 1 4 dr 4 θ2−2ε 2−2ε . 2ε t t t4− α 0 (θ − r) Now, Deﬁnition 4.3 shows that 2 2 f H ε rad (Ωα (t)) ≤ f˜ 2 + rad (|x|<2θ)

L

1

h−(1+2ε)

0

2θ

(4.45)

1 ˜ 2 ∆ h f (r) r dr dh. (4.46)

0

The estimate for the ﬁrst term on the right-hand side (see(4.21)) is C f (t, ·)L2 rad (|x|<2θ) ≤ 2− 1 . (4.47) t α On the other hand, we have just established the following estimate of the second term: 2θ 1 1 1 |∆1h f (t, r)|2 r dr dh (4.48) 2−2ε . 1+2ε 4− h α t 0 0 Summing up we obtain 1 f H ε rad (Ωα (t)) (4.49) 1−ε . 2− α t With this estimate we have that if ε < 1 − α, then the function f = f (t, r) lies in ε the desired space, i.e., f ∈ L1 ((0, 1); Hrad (Ωα (t))). Our next step is to show that the solution φ = φ(t, r), deﬁned in (4.8), belongs 1+ε (Ωα (t)). To this purpose, we need an estimate for the function φ = φ(t, r) to Hrad similar to the one just proved for f . We proceed in a similar way: ﬁrst of all, we ˜ The extend φ as it was done in (4.35) and consider the corresponding extension φ. 1 (Ωα (t))-norm of φ = φ(t, r) at a ﬁxed time t > 0 was obtained estimate of the Hrad in Lemma 4.2. Now we will estimate from above and from below the integral 2θ 1 2 1

r) r dr dh. h−(1+2ε) J= ∆ h ∂r φ(t, 0

0

Thus, in particular, we shall prove that the solution is in the desired space for strictly positive t. As above, it is suﬃcient to consider the integral 1 θ 2 1

r) r dr dh, J0 = h−(1+2ε) ∆ h ∂r φ(t, 0

0

and again, we can split the integral J0 as follows: J0 = J1 + J2 , where

J1 = 0

θ

h−(1+2ε)

0

θ−r

2 1

r) r dr dh, ∆ h ∂r φ(t,

52

Piero D’Ancona and Vladimir Georgiev

while

J2 =

θ

h−(1+2ε)

1

θ −r

0

2 1

r) r dr dh. ∆ h ∂r φ(t,

We have the following upper bound for J1 : 2 θ θ−r 1 1 1 J1 = 2θ2 r − dh dr r2 + θ 2 h(1+2ε) (r + h)2 + θ2 0 0 θ θ−r (2r + h)2 h2 2 2θ r 4 dh dr h(1+2ε) (r2 + θ2 ) 0 0 θ−r θ (r + θ)2 h1−2ε 2θ2 r dh dr θ8 0 0 θ 1 1 r(θ − r)2−2ε (θ + r)2 dr 2ε . θ6 0 θ

(4.50)

In a similar way we obtain the lower bound for J1 θ θ−r θ θ−r (2r + h)2 h2 h4 2 J1 θ2 r dr dh θ r dh dr. 4 h(1+2ε) θ8 h(1+2ε) (r2 + θ2 ) 0 0 0 0 Taking the smaller domain of integration {r ∈ [0, θ/2], h ∈ [θ/4, θ/2]} we get J1 θ−3−2ε

θ/2

θ/2

dh dr ∼

r 0

θ/4

C . θ2ε

(4.51)

Concerning the second integral J2 , we shall establish only an upper bound, since 0 is a suﬃcient lower bound. We proceed in a way similar to the study of I2 and ﬁnd 2 θ 1 1 1 1 2 θ r + 2 dh dr (4.52) J2 (1+2ε) (r + h)2 + θ 2 r + θ2 0 θ −r h θ 1 1 1 1 1 θ 1 θ2 r dh dr r − 1 dr 2ε 2 2ε (1+2ε) (r2 + θ 2 )2 θ (θ − r) θ h 0 θ −r 0 provided 2ε < 1. In conclusion, the estimates from above proved for J1 , J2 yield (4.32), while property (4.33) follows from the lower bound for J1 obtained in (4.51). This completes the proof of the theorem. 4.4. Appendix In this section we will prove the two technical lemmas needed in the proof of Theorem 4.1. We begin with a deﬁnition: 1+ε 1 Deﬁnition 4.5. The space H(ε) := H˙ rad ∩ H˙ rad is the Hilbert space obtained by ∞ 2 completing C0 (R ) with respect to the norm vH := vH˙ 1 + vH˙ 1+ε .

Wave Maps

53

The deﬁnition of the space H 1+ε (Ω) for Ω ⊂ R2 is the following (see, e.g., [44], Deﬁnition 4.2.1.1): Deﬁnition 4.6. We say that the distribution u belongs to the space H 1+ε (Ω) if there exists an extension u ˜ ∈ H 1+ε (R2 ) and in this case the H 1+ε (Ω)-norm of u is given by: uH 1+ε (Ω) = inf ˜ uH 1+ε (R2 ) , u ˜

where inf is taken over all extensions u ˜ of u. Our ﬁrst result applies to the function u(φ) deﬁned as in (2.57) as follows u1 = cos(φ)cos(λ), u2 = cos(φ) sin(λ), u3 = sin(φ) with φ = φ(t, r) = 2 arctan

(4.53)

r . t1/α

Then we have: 1+ε 1 Lemma 4.3. For every function φ = φ(r) ∈ H(ε) = H˙ rad ∩ H˙ rad such that φ (r) ≥ 0 (almost everywhere in r > 0) we have:

u(φ)H(ε) ≤ CφH(ε) , ∀ ε ∈ [0, 1],

(4.54)

u(φ)H(ε) ≥ C1 φH(ε) , ∀ ε ∈ [0, 1].

(4.55)

Proof. The proof relies on the fact that the function u takes its values on the unit sphere, i.e., the vector-valued function u has norm 1 as a vector in R3 . The ﬁrst inequality then is just a special case of Theorem 1 in Section 5.3.6 of the book of Runst and Sickel ([28]), and we shall not reproduce it here. Consider now the second estimate from below. We will study separately the cases ε = 0 and ε = 1 ﬁrst. In the case ε = 0 we have ∞ 2 = |∂ ∂r (u(φ(r)))|2 r dr u(φ)H˙ 1 rad 0 ∞ ∞ = |u (φ(r))|2 |φ (r)|2 r dr = |φ (r)|2 r dr = φ2H˙ 1 . 0

0

rad

For the case ε = 1 we have analogously: ∞ ∞ |∂ ∂r2 (u(φ(r)))|2 r dr = |u (φ(r))(φ (r))2 + u (φ)φ (r)|2 r dr u(φ)2H˙ 2 = rad 0 0 ∞ (|u (φ(r))|2 (φ (r))2 + 2 (u (φ(r)), u (φ(r))) φ (r)φ (r) = 0

= 0

+|u (φ(r))|2 (φ (r))2 )r dr ∞ (φ (r))2 + (φ (r))2 r dr ≥ φ2H˙ 2 , rad

since the vectors u (φ(r)) and u (φ(r)) are orthogonal.

54

Piero D’Ancona and Vladimir Georgiev

We now consider the fractional case 0 < ε < 1. We have 1 ∞ 1 u(φ)2H˙ 1+ε = |h ∂r (u(φ(r)))|2 r dr dh 1+2ε rad 0 h 0 1 ∞ 1 = |u (φ(r + h))(φ (r + h)) − u (φ(r))φ (r)|2 r dr dh 1+2ε 0 h 0 1 ∞ 1 = r|u (φ(r + h))|2 (φ (r + h))2 1+2ε h 0 0 −2r (u (φ(r + h)), u (φ(r))) φ (r + h)φ (r) + |u (φ(r))|2 (φ (r))2 r dr dh ∞ 1 1 (φ (r + h))2 − 2φ (r + h)φ (r) + (φ (r))2 r dr dh = φ2H˙ 1+ε , ≥ 1+2ε rad h 0 0 where we used the Cauchy-Schwartz inequality and the positivity of the ﬁrst derivative of the function φ. We now extend the above result to bounded domains Ω ⊂ R2 . To this end we need to prove the following proposition: Proposition 4.1. For a bounded domain Ω and under the conditions of the previous lemma we have the following estimates: u(φ)H 1+ε (Ω) ≤ CφH 1+ε (Ω) , ∀ ε ∈ [0, 1],

(4.56)

u(φ)H 1+ε (Ω) ≥ C1 φH 1+ε (Ω) , ∀ ε ∈ [0, 1], with C, C1 independent of ε.

(4.57)

Proof. We have 1+ε u(φ) ˜ H(ε) + φ ˜ L2 (Ω ) u(φ)H 1+ε (Ω) = inf u(φ) H g u(φ)

˜ H(ε) + φ ˜ L2 (Ω ) = φ ˜ H 1+ε φH 1+ε (Ω) + δ, φ (4.58) ˜ where we used the fact that u(φ) is an extension of u(φ); notice that we are allowed to choose the extension of φ in a slightly larger domain Ω such that φ˜ = 0 ˜ H 1+ε − φH 1+ε (Ω) < δ, for any ﬁxed δ > 0. Since δ is arbitrary, in R2 Ω and φ this concludes the proof of the ﬁrst inequality. To prove the opposite inequality, we use the fact that |u| = 1 as a vector in R3 . Then, taking the same extension φ˜ on a larger domain Ω ⊆ Ω as above, we have ˜ L2 (Ω ) . ˜ L2 (Ω ) u(φ) (4.59) φ Moreover, recalling the proof of the previous lemma, we have also H(ε) + δ. ˜ H(ε) u(φ) ˜ H(ε) u(φ) φ Taking the sum between these two inequalities we conclude the proof. Lemma 4.4. Let φ ∈ C([0, T ]; H ) and f ∈ L ((0, T ); H s

1

s−1

F = f ∂φ u(φ) ∈ L1 ((0, T ); H s−1 ).

(4.60)

) for some s > 1. Then

Wave Maps

55

Proof. We ﬁrst estimate some norms with respect to space variables at a ﬁxed time t > 0. The L2 -norm of the term F = f ∂φ u(φ) can be computed immediately as follows: f ∂φ u(φ)L2 = f L2 .

(4.61)

µ

Next we use the fact that for µ > 1 the space H is an algebra, and this gives the estimate f ∂φ u(φ)H µ f H µ ∂φ u(φ)H µ . 2

(4.62)

2

µ

Thus, if we consider f as an operator from L into L and from H into H µ , an interpolation argument implies that f is bounded on H s for each 0 < s < µ, with a norm bounded by f H s . Hence we have f ∂φ u(φ)H s f H s ∂φ u(φ)H s . s

(4.63)

But from the previous lemma we may control the H norm of u (φ) with the H s norm of φ. In conclusion, the result follows by integrating the above estimate and applying H¨ ¨older’s inequality with respect to time. An analogous result holds on the restricted cones: Lemma 4.5. Let φ ∈ C([0, T ]; H s (Ωα (t))) and f ∈ L1 ((0, T ); H s−1 (Ωα (t))) for some s > 1. Then F = f ∂φ u(φ) ∈ L1 ((0, T ); H s−1 (Ωα (t))). Proof. The argument is identical to the above one (use Proposition 4.1).

5. Non-uniqueness result in the subcritical case 5.1. Introduction The general wave map problem in local coordinates has the form (see equation (2.40)) (5.1) u + Γbc (u)∂α ub ∂ α uc = 0. The natural problem for (5.1) is the Cauchy problem with initial data at t = 0 u(0, x) = u0 (x),

ut (0, x) = u1 (x).

(5.2)

When N is the unit sphere in R with the metric induced by the Euclidean metric, the equations become very symmetrical using the full set of functions (u1 , . . . , ud ), indeed we obtain (see equation (2.50)) d

u + (|ut |2 − |∇x u|2 )u = 0

(5.3)

subject to the constraint |u| = 1. It is easy to see that if a smooth function u(t, x) solves (5.3) and moreover satisﬁes the constraint at t = 0, i.e., |u(0, x)| = 1, u(0, x) · ut (0, x) = 0, then we have |u| = 1 for all t. It is also possible to consider system (5.3) without geometric constraints, from a purely analytical point of view.

56

Piero D’Ancona and Vladimir Georgiev

The minimal regularity required of u in order to give a meaning to the nonlinear term (in distributional sense) is Dt,x u ∈ L2loc , provided |u| = 1. This can be further relaxed if we remark that u is parallel to u by the equation, hence we have also u ∧ u = 0, |u| = 1 or equivalently ∂t (ut ∧ u) =

n

∂xj (uxj ∧ u),

(5.4)

j=1

since u · ut = u · uxj = 0 by the condition |u| = 1. For smooth functions with norm 1, (5.4) and (5.3) are equivalent; moreover, equation (5.4) has a distributional sense for Du ∈ L1loc only, provided |u| = 1. Thus in the following we shall say that u is a weak solution of equation (5.3) if Du ∈ L1loc , |u| = 1 a.e. and u satisﬁes (5.4) in the sense of distributions. One of the basic open questions of the theory is the well-posedness of the Cauchy problem for (5.3) in two dimensions, i.e. with u : R × R2 → S2 ⊆ R3 . In this case the local existence of smooth solutions follows by classical arguments, while global existence meets essential diﬃculties. The critical space for equation (5.4) in two dimensions is H 1 , which is also the energy space; thus an important question is the well-posedness of (5.4) in H 1 . The aim of this chapter is to investigate the behavior of (5.3) for solutions of low regularity, i.e., below the energy space H 1 × L2 , and indeed to show that in general the problem is not well-posed in this situation. Our main result is the following: Theorem 5.1 (Nonuniqueness). (See Theorems 5.2 and 5.3 below.) It is possible to construct two weak solutions u, v : R × R2 → S2 to the equation (5.4), continuous in time with values in H 1−ε (Ω) for all ε > 0 and all bounded Ω, and also L∞ with 1 values in the Besov space B2,∞ , such that u ≡ v for t < 0 and u ≡ v for t > 0. We can construct u, v such that their Besov norm is arbitrarily small. 1 1 , recalling Tataru’s local well-posedness result in B2,1 , we may say Since H 1 = B2,2 1 that the question of local well-posedness in H is conﬁned in the gap between the 1 1 and B2,1 . In this regard it is necessary to mention Tao’s two Besov spaces B2,∞ result [40] showing that the well-posedness holds provided the data are slightly smoother (H 1+ε × H ε ) and small in the H˙ 1 norm. See also the paper [27], where a model (scalar) equation with the same type of nonlinearity is considered. The plan of this chapter is the following: in Section 5.2 we recall the deﬁnition of equivariant and self-similar solutions, which are necessary for the following constructions. The self-similar ansatz leads to an ODE which is studied in detail in Appendix A; in particular all the solutions are computed explicitly. Section 5.3 is devoted to the non-uniqueness result Appendix B collects some technical lemmas.

Wave Maps

57

5.2. Equivariant and self-similar solutions When the manifolds have rotational symmetry, two interesting classes of special solutions arise, the equivariant and the self-similar solutions. The general construction is standard and can be found in [5], [30], [32]. We recall brieﬂy the equivariant ansatz. Assume that N is a smooth dimensional rotationally symmetric manifold deﬁned as N = {(φ, χ); φ ∈ [0, φ∗ ), χ ∈ S−1 } (φ∗ may be +∞), with metric dφ2 + g(φ)2 dχ2 ,

(5.5)

where dχ is the standard metric on S . In the coordinates (φ; χ1 , . . . , χ−1 ), denoting by hij the coeﬃcients of the metric dχ2 , the only nonzero Christoﬀel symbols for (5.5) are 2

−1

Γφχi χj = −g (φ)g(φ)hij ,

Γχχij φ =

g (φ) δij , g(φ)

i Γχχij χs = γjs ,

(5.6)

i where γjs are the Christoﬀel symbols for the metric hij . The equivariant wave maps are the maps u satisfying the ansatz

u(t, x) = (φ, χ),

φ = φ(t, r),

χ = χ(ω),

(5.7)

where (r, ω) are the spherical coordinates on Rn . Under this assumption, the equations for χ decouple and in fact we obtain that χ : Sn−1 → S−1 must be a harmonic map. For suitable choices of the dimensions n, (and of χ) further simpliﬁcations occur. In the special case = n = 2 under consideration here, it is easy to see that χ must be a rotation of degree k = 1, 2, 3, . . . of S1 into itself. With this choice the equation for φ decouples and we obtain 1 k2 φtt − φrr − φr + 2 g (φ)g(φ) = 0. (5.8) r r This is the equivariant wave map equation. When N is the sphere S2 , the above framework corresponds to the standard choice of coordinates (φ, χ) with φ ∈ [0, π] and χ ∈ S1 . Then the metric can be written dφ2 + sin2 φ dχ2 , and equation (5.8) becomes 1 k2 φtt − φrr − φr + 2 sin(2φ) = 0. (5.9) r 2r Actually it is more convenient to embed N = S2 in R3 ; with the usual coordinate system on N ⎛ ⎞ sin φ sin χ ⎝sin φ cos χ⎠ , cos φ

58

Piero D’Ancona and Vladimir Georgiev

where φ ∈ [0, π], χ ∈ [0, 2π], φ = 0 corresponding to the north pole and φ = π to the south pole of the unit sphere. Then we can express the solution u(t, x) to (5.3) as the vector (u1 , u2 , u3 ) with u1 = sin φ(t, |x|) · |x|−k Re (x1 + ix2 )k u2 = sin φ(t, |x|) · |x|−k Im (x1 + ix2 )k u3 = cos φ(t, |x|); since cos χ = cos(kω) = |x|−k Re (x1 + ix2 )k , sin χ = sin(kω) = |x|−k Im (x1 + ix2 )k . In order to introduce the self-similar solutions, we recast the equation in the hyperbolic coordinates r (5.10) ρ = t2 − r 2 , τ= ; t notice that the inverse transformations are given by τρ ρ , r= √ . (5.11) t= √ 2 1−τ 1 − τ2 Then we obtain 1 2 ∆H (5.12) ∂t2 − ∂r2 − ∂r = ∂ρ2 + ∂ρ − 2 , r ρ ρ where ∆H is the Laplace operator on the hyperboloid ρ = 1, that is, (1 − τ 2 )(2τ 2 − 1) ∂τ . τ In the new coordinates ρ, τ the equation (5.8) becomes ∆H = (1 − τ 2 )2 ∂τ2 −

(1 − τ 2 ) 2 1 sin 2φ = 0. ∂ρ2 φ + ∂ρ φ − 2 ∆H φ + k 2 ρ ρ 2τ 2 ρ2

(5.13)

(5.14)

We can now deﬁne the self-similar solutions as the solutions which are independent of ρ, i.e., r φ(t, r) = ψ =: ψ(τ ). t Under this assumption the ﬁrst two terms in (5.14) drop, and we obtain immediately the following equation for ψ = ψ(τ ): τ 2 (τ 2 − 1)ψ + τ (2τ 2 − 1)ψ + k 2 sin ψ cos ψ = 0.

(5.15)

Notice that if one could ﬁnd global smooth solutions to this equation, an immediate consequence would be a blow-up result for the wave map equation (5.3); but this is not possible, as shown in [30] (while in higher dimensions this idea is correct and was exploited in [32]). Nevertheless it is possible to utilize the singular (i.e., not in H 1 ) solutions thus obtained, as we shall do in the following section. It is not diﬃcult to see that all solutions to (5.15) are analytic and deﬁned for τ = 0, 1; in Appendix A we give a complete study of the equation, and we represent all its solutions using Jacobi’s elliptic functions. In the next sections, in particular, we shall use the following special solutions:

Wave Maps

59

(i) The function deﬁned as ψ(τ ) =

arcsin τ π/2

for 0 ≤ τ ≤ 1 for τ > 1

(5.16)

is a solution to the equation (for τ = 1) in the case k = 1. Notice that the only constant solutions are the integer multiples of π/2, and that the value of the constant chosen here for τ > 1 ensures (H¨ o¨lder) continuity. The fact that (5.16) is a solution can be veriﬁed directly, or can be obtained by setting λ = π/2, k = 1 in the general expression (5.72). (ii) A more general class of solutions in the case k = 1 is given by the expressions (δ ∈ (0, 1)) ⎧ ⎪ ⎪2 arctan tan λ · τ for 0 ≤ τ ≤ 1, ⎪ ⎨ 2 1 ± 1 − τ2 (5.17) ψ(τ ) = ⎪ 1 1 ⎪ am δ arctan for τ > 1. ⎪ ⎩ τ2 − 1 δ Here the two-parameter function am (τ |m) is the Jacobi amplitude in the case m > 1; a precise deﬁnition is given in Appendix A, here we shall only need to know that it is an analytic periodic function of τ , with the property am (0|m) = 0, hence (5.17) tends to 0 as τ → ∞. Again, we can ensure H¨¨older continuity at τ = 1 by imposing a suitable condition on the constants λ, δ. In the following section we shall also need to compute sin ψ and cos ψ for ψ given by (5.17); writing γ = tan we have

sin ψ(τ ) =

and

λ 2

⎧ 4γτ ⎪ ⎪ ⎪ ⎨ (1 ± 1 − τ 2 ) + γ 2 (1 ∓ 1 − τ 2 ) ⎪ ⎪ ⎪ ⎩

δ · sn arctan √τ 12 −1 δ

⎧ 2 ) − γ 2 (1 ∓ 2 ⎪ (1 ± 1 − τ ⎪ ⎪ 1 − τ ) ⎪ 2 ⎨ (1 ± 1 − τ 2 ) + γ (1 ∓ 1 − τ 2 ) cos ψ(τ ) = ⎪ ⎪ ⎪ ⎪ ⎩dn arctan √ 12 δ , τ −1

for 0 ≤ τ ≤ 1, (5.18) for τ > 1,

for 0 ≤ τ ≤ 1, (5.19) for τ > 1

(the signs ± are the same as in (5.17), the ∓ are opposite). For the deﬁnition and properties of the elliptic functions sn (τ |δ), dn (τ |δ) see Appendix A. These formulas are proved in Appendix A, see Remark 5.4 and (5.75), (5.76), (5.77).

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5.3. Low regularity self-similar solutions Our purpose here is to construct weak solutions to the wave map equation (5.3) from R × R2 to S2 below critical regularity. As mentioned in the Introduction, it is convenient to transform the equation in the form of a conservation law ∂t (∂ ∂t u ∧ u) =

2

∂j (∂ ∂j u ∧ u)

(5.20)

j=1

in order to handle weak solutions of very low regularity; indeed if u is a locally bounded function such that ∂u ∈ L1loc (R × R2 ), all the terms in (5.20) have a well-deﬁned meaning in the sense of distributions. Starting from a solution ψ(τ ) of (5.15), we can construct a self-similar solution to the wave map equation by setting ⎛ ⎞ u1 (t, x) u(t, x) = ⎝ u2 (t, x) ⎠ , (5.21) u9 (t, x) with u1 (t, x) = |x|−k Re (x1 + ix2 )k sin (ψ (|x|/t)) , u2 (t, x) = |x|−k Im (x1 + ix2 )k sin (ψ (|x|/t)) , u3 (t, x) = cos (ψ (|x|/t)) . We shall restrict ourselves to the case k = 1 and the special solutions examined in the preceding section. Our ﬁrst result is based on the solution (5.17); this means simply ⎛ ⎞ x1 1⎝ ⎠ x2 (5.22) u(t, x) = t 2 2 t − |x| inside the cone, and

⎞ x1 /|x| u(t, x) = ⎝x2 /|x|⎠ 0 ⎛

(5.23)

outside the cone. Our aim is to prove Theorem 5.2. The function u(t, x) deﬁned in (5.22), (5.23) is a solution in the distributional sense of the wave map equation (5.20). Moreover, given any bounded open set Ω ⊆ R2 , we have u ∈ C(R, H s (Ω)),

∀s < 1

and

1 u ∈ L∞ (R, B2,∞ (Ω)).

(5.24)

A second solution is the function v(t, x), independent of t, deﬁned as (5.23) every1 (Ω)) for any Ω, and v ≡ u for t ≤ 0, thus the weak where; we have v ∈ C(R, B2,∞ solution to (5.20) is not unique in the spaces (5.24).

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Remark 5.1. Actually the solution u is continuous with values in Besov spaces for t ≥ 0 and t ≤ 0, and has a jump at t = 0 only. We also notice that, thanks to the 1 scaling properties of Besov spaces, the B˙ 2,∞ norm of u(t, x) on the ball B(0, t) is constant as t ↓ 0, i.e., it concentrates at 0. Proof. We remark that the ﬁrst derivatives of u in the sense of distributions coincide with the derivatives a.e., which are locally integrable functions on R × R2 . This follows at once by Lemma 5.3 in the Appendix (with obvious choices); indeed, the singularities of u are concentrated along the positive cone t = |x|, where u is Holder ¨ continuous, and along the line x = 0 for negative t, where it is bounded. Thus we can compute the derivatives ut , uxj directly by diﬀerentiating the above formulas, obtaining locally integrable functions. A similar remark holds for v. Outside the positive light cone, i.e., for t ≤ |x| (including negative t) we have u ≡ v, and an explicit computation gives ∂t u ∧ u = ∂t v ∧ v = 0, ⎞ ⎛ 0 ∂x1 u ∧ u = ∂x1 v ∧ v = ⎝ 0 ⎠ , x2 /|x|2 ⎞ ⎛ 0 ⎠. 0 ∂x2 u ∧ u = ∂x2 v ∧ v = ⎝ 2 −x1 /|x|

(5.25) (5.26)

(5.27)

These formulas hold on the whole space R × R2 for the second solution v. To check that v is a solution of (5.20) we may proceed directly; indeed, we have to check that x1 x2 ∂x1 − 2 + ∂x2 =0 |x| |x|2 in the sense of distributions, and this follows from the identities xj ∂xj |x| = |x| (notice that these functions are locally integrable). To check that u is a solution, it is simpler to express the equation in spherical coordinates: 1 1 ∂t (ut ∧ u) − ∂r (ur ∧ u) − ur ∧ u − 2 ∂ω (∂ ∂ω u ∧ u) = 0. (5.28) r r Outside the light cone we have u = (cos ω, sin ω, 0), while inside it holds u=

1 (r cos ω, r sin ω, t2 − r2 ). t

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We can write the wedge products in a global form valid everywhere: writing for brevity ⎛ ⎞ − sin ω Z(ω) = ⎝ cos ω ⎠ 0 and √ 1 if r ≤ t, t2 −r 2 (5.29) w(t, r) = 0 otherwise, we ﬁnd after some computations r ut ∧ u = w(t, r) · Z, t ur ∧ u = −w(t, r) · Z,

(5.30) (5.31)

while

r2 t − r2 · Z. (5.32) t2 Thus, to check that u is a solution we must show that √ 1 rw t2 − r 2 ∂t χK = 0 (5.33) + ∂r w + w − t r rt2 in distribution sense, where χK is the characteristic function of the future light cone K. Given any test function φ, we must show that √ rw 1 t2 − r 2 wφ − ∂t φ − w∂ ∂r φ − φ rdrdt = 0. r t rt2 K ∂ω (∂ ∂ω u ∧ u) = ∆ω u ∧ u =

This amounts to say that, writing K ε = {(t, r) : t ≥ r + ε}, the limit of the same integral over K ε tends to zero when ε → 0. We integrate by parts, the integrals over K ε cancel since the solution is smooth in K ε and the terms on the boundary t = r + ε give (keeping into account the fact that the components νt and νr of the normal unit vector are opposite) ∞ r ε rwφdS = rw(r + ε, r)φ(r + ε, r)dr → 0. 1− t r+ε 0 ∂K ε It remains now to show that the solutions belong to the stated spaces. For the solution v this follows directly from Lemma 5.1 in Appendix B, and the remark 1 that the H s -norms for s < 1 are controlled by the Besov norm of B2,∞ . Consider now the solution u; for t ≤ 0 it coincides with v, hence the same argument is applicable and we obtain the Besov-valued continuity on (−∞, 0]. We now consider the case t > 0. We know by Lemma 5.1 in Appendix B that x1 /|x| 1 is locally in B = B2,∞ , and this implies that also the function x1 x1 − |x|

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63

is locally in B since x1 is smooth. Then also the function x1 x1 − |x| for |x| ≤ 1, g(x) = 0 for |x| ≥ 1, is in B locally, since g(x) is Lipschitz continuous at |x| = 1. But g(x) is compactly 1 (R2 ) globally. Recalling that the homogeneous norm supported, hence g ∈ B2,∞ 1 2 ˙ B2,∞ (R ) is invariant for the scaling x → x/t, we obtain that the function g(x/t) 1 (R2 ). This implies immediately that is continuous on [0, +∞) with values in B2,∞ x1 + g(x/t) (5.34) u1 = |x| 1 is continuous on [0, +∞) with values in B2,∞ (Ω) for any bounded open Ω; u2 is identical. As to u3 , we only have to remark that

u3 (t, x) = λ(x/t),

(5.35)

where λ(x) is the function deﬁned in Lemma 5.2 in Appendix B, which belongs to 1 B2,∞ (R2 ), and argue as before. 1 Continuity with values in H s for t = 0 follows, since the norm of B2,∞ is stronger. Continuity also at t = 0 follows from (5.34) and (5.35) since the scaling properties of H s for s < 1 imply that g(·/t)H s (R2 ) → 0 as t → 0.

The situation does not improve if we consider small solutions. Indeed, we can construct two diﬀerent weak solutions depending on a parameter δ which are small and coincide for t < 0. The ﬁrst solution is the self-similar solution obtained using the functions (5.17) with the plus sign; recalling (5.18), (5.19), we have inside the light cone, i.e., for |x| < |t|, 2γxj uj (t, x) = , (5.36) 2 (1 + γ )t + (1 − γ 2 ) t2 − |x|2 for j = 1, 2, and

(1 − γ 2 )t + (1 + γ 2 ) t2 − |x|2 u3 (t, x) = ; (1 + γ 2 )t + (1 − γ 2 ) t2 − |x|2

(5.37)

here γ = tan λ2 is a small parameter. On the other hand, outside the light cone, i.e., for |x| > |t|, we have t xj (5.38) δ sn arctan uj = δ |x| |x|2 − t2 for j = 1, 2, and

u3 = dn

t

arctan |x|2 − t

δ 2

.

(5.39)

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For a small ﬁxed δ we choose γ such that 2γ = δ · sn (π/2|δ) (5.40) 1 + γ2 to ensure continuity; recall that for δ < 1 it holds d sn (s|δ) ≤ 1, |sn (s|δ)| ≤ 1, ds while d 2 1 − δ ≤ dn (s|δ) ≤ 1, ds dn (s|δ) ≤ δ. So the function u(t, x) thus constructed takes its values in a δ-neighborhood of the North pole (0, 0, 1). The second solution v(t, x) coincides with u(t, x) for negative t and outside the forward light cone; inside it, i.e., for t > |x|, we deﬁne it using again (5.17) but we choose now the minus sign; in a more explicit form we obtain (see again (5.18), (5.19)) 2γxj vj (t, x) = , (5.41) 2 (1 + γ )t − (1 − γ 2 ) t2 − |x|2 for j = 1, 2, and (1 − γ 2 )t − (1 + γ 2 ) t2 − |x|2 (5.42) v3 (t, x) = (1 + γ 2 )t − (1 − γ 2 ) t2 − |x|2 (compare with (5.36), (5.37)). Remark 5.2. Notice that the solution u(t, x) takes its values in a δ-neighborhood of the North pole (0, 0, 1); on the other hand, the second solution v(t, x) covers the entire sphere, and for t > 0, x = 0 we have v(t, 0) = (0, 0, −1), the South pole. Then we have: Theorem 5.3. Consider the function u(t, x) deﬁned by (5.36), (5.37) for |x| ≤ |t| and by (5.38), (5.39) for |x| > |t|. Moreover, consider the function v(t, x) deﬁned as u(t, x) for t < |x| and by (5.41), (5.42) inside the future light cone t > |x|. Then both u(t, x) and v(t, x) are weak solutions of the wave map equation (5.20), and have the same regularity properties (5.24). Moreover, for any bounded Ω ⊆ R2 and t < 0 ﬁxed we have 1 1 u(t, ·) − N B2,∞ (Ω) ≡ v(t, ·) − N B2,∞ (Ω) = O(γ) as γ → 0,

(5.43)

where N denotes the North pole N = (0, 0, 1). Proof. The regularity of u, v is proved as before, and we obtain as above the continuity with values in a Besov space for t > 0 and for t < 0, with a jump in t = 0 (u is actually continuous with values in a Besov space also at t = 0). Also the estimate (5.43) follows by a simple argument. To prove that u, v solve (5.20) we begin as above by applying Lemma 5.3 in the Appendix. The singularity of u and v is concentrated on the light cones

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65

K = {|t| = |x|}. Choosing a ball Ω near a point of the cone diﬀerent from the origin, and setting f = u, h(t, x) = u(|x|, x), we see that assumption (5.92) of this lemma is satisﬁed, using the H¨¨older continuity of u near the point; thus the distributional derivative of u on (R × R2 ) \ (0, 0) coincides with the derivative a.e. Now we can apply again this lemma choosing K = the origin, and in this case assumption (5.92) follows from the fact that u is bounded near the origin. In conclusion, the ﬁrst distributional derivative of u on R × R2 coincides with its derivative a.e. The argument for v is identical. Thus it is suﬃcient to prove that for any test function φ the following identity holds: 1 1 φt ut ∧ u − φr ur ∧ u + φ ur ∧ u + 2 φ uωω ∧ u r dr dω dt = 0 r r and an analogous one for v(t, x). Now u is a function of the form ⎛ ⎞ cos ω sin ψ(r/t) u(t, x) = u(t, r cos ω, r sin ω) = ⎝ sin ω sin ψ(r/t) ⎠ , cos ψ(r/t) and this gives, writing

the identities ut ∧ u = Z(ω)

r r , ψ t2 t

⎞ ⎛ − sin ω Z(ω) = ⎝ cos ω ⎠ 0 1 r ur ∧ u = −Z(ω) ψ , t t

uωω ∧ u = Z(ω) sin ψ cos ψ.

ω). We are It is not restrictive to consider test functions of the form φ(t, r)φ(t, thus reduced to prove that r r 1 1 r 1 φ ψ · φ φ + 2 φ sin(2ψ) r dr dt = 0; (5.44) ψ + + t r t2 t t t r 2r recall that ψ(s) is smooth for s = ±1 and ψ (s) means the derivative a.e. Introduce now the sets Aε = {|t| ≥ |x| + ε}, whose boundary is made of the two cones ∂ ± Aε = {±t = |x| + ε}, and Bε = {|x| ≤ |t| + ε}, whose boundary is made of the two sets ∂ + Bε = {t = |x| − ε, t ≥ 0},

∂ − Bε = {−t = |x| − ε, t ≤ 0}.

Identity (5.44) will follow if we prove that the integral restricted to Aε ∪ Bε converges to 0 as ε → 0. On Aε ∪ Bε the functions are smooth, hence we can integrate by parts and the integrals on the interior cancel (since ψ solves the self-similar

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ODE away from the singularity). Only the boundary terms remain, i.e., we must prove that ∞ r r r −εr νt + νr ψ φ dS = C ψ φ(r+ε, r)dr → 0, (5.45) (r + ε)2 r+ε 0 ∂ + Aε t t with a similar relation on ∂ − Aε , and ∞ r r t+ε t+ε νt + νr ψ φ dS = εψ φ(t, t + ε)dt → 0 t2 t 0 ∂ + Bε t t

(5.46)

with a similar relation on ∂ − Bε . To prove (5.45), (5.46) it is suﬃcient to recall that ψ satisﬁes in all cases the condition τ 2 (1 − τ 2 )ψ (τ )2 − sin ψ 2 = const , whence the estimate, valid for any τ = ±1, C . |ψ (τ )| ≤ τ |1 − τ 2 | This implies

3/2 r ψ ≤ C (r +√ε) r+ε r ε

which gives (5.45), and

t+ε t2 ψ ≤C √ t (t + ε)3/2 ε

which gives (5.46). The proof for v is identical.

5.4. Appendix A: The self-similar ODE This section is devoted to a complete study of the equation τ 2 (τ 2 − 1)ψ + τ (2τ 2 − 1)ψ + k 2 sin ψ cos ψ = 0

(5.47)

which governs the proﬁle of self-similar solutions φ(t, r) = ψ(r/t) to equation (5.9). Here k ≥ 1 is any integer. We can express all solutions to (5.47) in an explicit form using Jacobi elliptic functions. These functions are usually introduced as doubly periodic meromorphic functions with suitable additional properties; but the standard deﬁnition is only given for restricted values of the parameters, hence from our standpoint it is both too general and too restrictive. For convenience of the reader, we construct them from scratch in a very short but complete way. Remark 5.3 (Jacobi elliptic functions on R). Consider the system of ODEs for the functions f, g, h : R → R f = gh,

(5.48)

g = −hf,

h = −m f g, 2

(5.49) (5.50)

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67

where m is a ﬁxed real number, subject to the initial conditions f (0) = 0,

g(0) = 1,

h(0) = 1.

(5.51)

It is clear that the system admits a unique C 1 solution (f, g, h) which is in fact real analytic and can be prolonged on the whole R using the ﬁrst integrals of the system (5.52) f 2 + g 2 = 1, m2 f 2 + h2 = 1, m2 g 2 − h2 = 1 which follow at once from the equations. The standard notation for these functions is f = sn (t|m), g = cn (t|m), h = dn (t|m). Notice that in the literature the parameter m is usually restricted to the range [0, 1[, which is not suﬃcient for our purposes. It is clear by the deﬁnition that sn (t|m) = sn (t| − m),

cn (t|m) = cn (t| − m),

dn (t|m) = dn (t| − m).

Squaring the equations and using the conservation laws (5.52), we obtain immediately 2 (5.53) f = (1 − f 2 )(1 − m2 f 2 ), g = (1 − g 2 )(1 − m2 + m2 g 2 ), 2

h

2

= −(1 − h2 )(1 − m2 − h2 ).

(5.54) (5.55)

From (5.53)–(5.55) it is immediate to obtain several properties of the elliptic functions. Consider ﬁrst the case |m| < 1. Then sn (t|m) and cn (t|m) are periodic with period 4K(m), where π/2 ds K(m) = (5.56) 0 1 − m2 sin2 s is called the complete integral of the ﬁrst species. The couple (sn (t|m), cn (t|m)) has a behavior similar to the couple (sin, cos), i.e., they oscillate between ±1, and indeed we have in the special case m = 0 sn (t|0) = sin t,

cn (t|0) = cos t.

Moreover, the zeros of sn (t|m) are t = 2jK(m) and those of cn (t|m) are (2j + 1)K(m), j ∈ Z. On the other hand, the √ function dn (t|m) has period 2K(m) and oscillates between the values 1 and 1 − m2 , thus it is strictly positive (and is identically 1 when m = 0). A fundamental property connecting elliptic functions with elliptic integrals is the following: for ﬁxed |m| < 1 it holds α ds √ = β, then sin α = sn (β|m), cos α = cn (β|m). (5.57) if 2 sins s 1 − m 0 This follows from (5.53), (5.54) through the change of variables f → sin f , g → cos g.

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When |m| > 1 the behavior changes. The function sn (t|m) oscillates between ±1/m, and indeed we have the formula 1 (5.58) sn (t|m) = sn (mt|m−1 ) m from which we see that the period is 4K(m−1 )m−1 and the zeros 2jK(m−1 )m−1 . The functions cn (t|m) and dn (t|m) exchange with each other according to the formulas cn (t|m) = dn (mt|m−1 ), dn (t|m) = cn (mt|m−1 ), (5.59) √ so that, for |m| > 1, cn (t|m) is strictly positive, oscillates between 1 and 1 − m−2 and has period 2K(m−1 )m−1 , while dn (t|m) oscillates between ±1 and has period 4K(m−1 )m−1 . When |m| = 1 we have simply et − e−t 2 1 = t , cn (t|1) = dn (t|1) = . (5.60) t −t e +e cosh t e + e−t We ﬁnally introduce the fourth Jacobi function called the amplitude and connected to the above through the relations sn (t|1) = tanh t =

sn (t|m) = sin(am (t|m)),

cn (t|m) = cos(am (t|m)).

(5.61)

When |m| < 1 we can compute for |t| < K(m) (i.e., between the ﬁrst zeros of cn) am (t|m) = arctan

sn (t|m) . cn (t|m)

(5.62)

Actually am (t|m) extends as an analytic function for all t ∈ R; indeed, by (5.57) we have immediately α ds √ = β, then α = am (β|m) (|m| < 1), (5.63) if 2 sins s 1 − m 0 which means that am (t|m) is the inverse function of the integral to the right, regarded as a function of α. This equivalent deﬁnition of am (t|m) is meaningful for any t, provided |m| < 1. When |m| > 1 we can use for all t deﬁnition (5.62), since cn (t|m) is strictly positive in this case; thus am (t|m) is 4K(m−1 )/m−1 periodic and oscillates between the values ± arctan[(m2 − 1)−1/2 ]. Notice that also in this case am (t|m) inverts the elliptic integral as in (5.63), but only on a ﬁnite interval: α ds √ if = β, then α = am (β|m) (|m| > 1, |α| < arcsin(|m|−1 )). 1 − m2 sins s 0 (5.64) The zeros of am (t|m) are the same of sn (t|m), that is to say 2jK(m−1 )m−1 . When |m| = 1 we have am (t|1) = 2 arctan(et ) − We are now ready to prove the

et − 1 π = 2 arctan t . 2 e +1

(5.65)

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Proposition 5.1. Consider the equation τ 2 (τ 2 − 1)ψ + τ (2τ 2 − 1)ψ + k 2 sin ψ cos ψ = 0.

(5.66)

(i) If ψ is a solution, also jπ ± ψ are solutions, j ∈ Z. (Thus it is suﬃcient to study the solutions in the range [0, π/2].) The only constant solutions are given by ψ(τ ) = jπ/2. (ii) If ψ is a C 1 solution near a point τ0 ∈]0, 1[, then ψ can be extended to an analytic function on the whole interval ]0, 1[. Moreover, the limits µ = lim ψ (τ ) 1 − τ 2 λ = lim ψ(τ ), τ ↑1

τ ↑1

exist and characterize uniquely the solution ψ. Indeed, for λ ∈ (0, π/2) and any µ, or for λ = 0, π/2 and µ = 0 (the excluded cases correspond to the constant solutions), we can represent ψ(τ ) as follows: π ψ(τ ) = − am (1 + q0 )1/2 q1 + sgn µ · k · arctanh 1 − τ 2 (1 + q0 )−1/2 , 2 (5.67) where π/2 ds µ2 2 q1 = (5.68) q0 = 2 − sin λ, k λ q0 + sin2 s and sgn µ must be replaced by −1 when µ = 0. (iii) If ψ is a C 1 solution near a point τ0 > 1, then ψ can be extended to an analytic function on the whole interval ]1, ∞[. Moreover, the limits µ = lim ψ (τ ) τ 2 − 1 λ = lim ψ(τ ), τ ↓1

τ ↓1

exist and characterize uniquely the solution ψ. Indeed, for λ ∈]0, π/2[ and any µ, or for λ = 0, π/2 and µ = 0 (the excluded cases correspond to the constant solutions), we can represent ψ(τ ) as follows: 1/2 −1/2 q1 + sgn µ · k · arctan τ 2 − 1 q0 , (5.69) ψ(τ ) = am q0 where q0 =

µ2 + sin2 λ, k2

q1 = 0

λ

ds q0 − sin2 s

(5.70)

and sgn µ must be replaced by −1 when µ = 0. Remark 5.4. Before sketching the proof, we single out a few solutions with special properties that are used in the paper. (a) For τ ∈ (0, 1), most of the solutions given by (5.67) have a nasty behavior near 0, indeed arctanh is unbounded near 1 and am (t|m) either has a linear growth

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(|m| < 1) or oscillates (|m| > 1). The only good solutions are obtained when q0 = 0, i.e., with the choice π/2 λ ds = − log tan ; λ ∈ (0, π/2], µ = ±k sin λ ⇒ q0 = 0, q1 = sin s 2 λ (5.71) recalling (5.65) and the identities √ π 1 1 + 1 − τ2 2 , − arctan x = arctan exp(arctanh 1 − τ ) = τ 2 x we obtain ! k " τ λ √ ψ(τ ) = 2 arctan tan · , (5.72) 2 1 ± 1 − τ2 where ± = sgn µ. It is useful to compute also 2γτ k √ √ , (1 ± 1 − τ 2 )k + γ 2 (1 ∓ 1 − τ 2 )k

λ γ = tan , 2

(5.73)

√ √ (1 ± 1 − τ 2 )k − γ 2 (1 ∓ 1 − τ 2 )k √ √ cos ψ(τ ) = , (1 ± 1 − τ 2 )k + γ 2 (1 ∓ 1 − τ 2 )k

λ γ = tan , 2

(5.74)

sin ψ(τ ) = and

where ± = sgn µ (and ∓ = − sgn µ). (b) The solutions for τ > 1 have a nice behavior, and in particular they are monotone; we shall be interested in small solutions with the property ψ → 0 as τ → ∞. Actually we have: for any ﬁxed k ≥ 1 and small δ > 0 we can ﬁnd (unique) λ, µ such that q0 = δ 2 and the solution given by (5.69) tends to 0 as τ → +∞. The solution is monotone decreasing for odd k and monotone increasing for even k. In particular, for k = 1 we can choose q1 = π/2 and the solution is given by 1 1 , (5.75) ψ(τ ) = am δ arctan √ τ2 − 1 δ so that, recalling formulas (5.59), 1 sin ψ(τ ) = δ · sn arctan √ (5.76) δ , 2 τ −1 1 (5.77) cos ψ(τ ) = dn arctan √ δ . 2 τ −1 Indeed, since q0 < 1, the amplitude function in (5.69) has a periodic behavior, 1/2 1/2 with zeros in the points 2jK(q0 )q0 . Recall that π/2 arcsin √q0 ds ds 1/2 = , K(q0 ) = 2 0 1 − q0 sin s 0 q0 − sin2 s

Wave Maps while

q1 = 0

Notice that q1 ≤

λ

71

ds . q0 − sin2 s

1/2 K(q0 )

for small λ, µ since # √ λ ≤ arcsin q0 = arcsin sin2 λ + µ2 /k 2 ;

more precisely, if we keep q0 ﬁxed and change the values of λ, µ, we see that q1 1/2 takes all the values from 0 (when λ = 0) to K(q0 ) (when µ = 0). Moreover, K(m) is a strictly increasing function for 0 < m < 1, with K(m) ↓ π/2 as m ↓ 0 and K(m) ↑ ∞ as m ↑ 1. Now the solution (5.69) tends to 0 as τ → ∞ provided −1/2 ) approaches one of the zeros, i.e., provided we can the argument s of am (s|q0 ﬁnd j ∈ Z such that π 1/2 (5.78) q1 + sgn µ · k = 2jK(q0 ). 2 If k > 0 is odd, we can write k = −2j + 1 for a negative integer j and choosing µ < 0 condition (5.78) becomes π π 1/2 − K(q0 ) . q1 = + |2j| 2 2 The right-hand side is slightly less than π/2 for small q0 , and keeping the value of q0 ﬁxed we can ﬁnd λ, µ such that the condition is satisﬁed (since q1 ranges from 0 1/2 to K(q0 ) > π/2). Thus our claim is proved for odd k. A similar argument holds for even k (we choose now 2j = k and µ > 0). Proof. The claims of part (i) are self-evident. We now prove (ii). It is clear that ψ deﬁned near τ0 ∈]0, 1[ is analytic; if we multiply the equation by ψ we obtain the identity $ 2 % τ (1 − τ 2 )ψ (τ )2 − k 2 sin2 ψ = 0, i.e., τ 2 (1 − τ 2 )ψ (τ )2 − k 2 sin2 ψ = const. A ﬁrst consequence is that we can extend the solution ψ of (5.66) on the whole interval ](0, √ 1) since ψ must be bounded on any compact subinterval. Moreover, 2 setting g( 1 − τ ) = ψ(τ ), we see that the function g(s) satisﬁes the diﬀerential equation τ 4 g ( 1 − τ 2 )2 − k 2 sin2 g( 1 − τ 2 ) = const , i.e., (1 − s2 )2 g (s)2 − k 2 sin2 g(s) = const. Thus we see that analytic function near√s = 0, and this implies that √ g(s) is an √ both ψ(τ ) = g( 1 − τ 2 ) and 1 − τ 2 ψ (τ ) = −τ g ( 1 − τ 2 ) have a limit as τ ↑ 1, as claimed. Deﬁning λ, µ, q0 , q1 as in the statement, we can write τ 2 (1 − τ 2 )ψ (τ )2 − k 2 sin2 ψ = q0 k 2 ,

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or equivalently,

ψ (τ )2 k2 , = 2 2 τ (1 − τ 2 ) q0 + sin ψ

and hence

ψ (τ ) k =± √ , (q0 + sin2 ψ)1/2 τ 1 − τ2 where ± = sgn µ. To solve this we set χ = −ψ + π/2 and obtain χ (τ ) k =∓ √ . (q0 + 1 − sin2 χ)1/2 τ 1 − τ2 We can now integrate between τ and 1; for τ close enough to 1, recalling (5.63), we obtain immediately that (5.67) holds in a left neighborhood of 1 and hence on ]0, 1[ by analyticity. The proof of part (iii) is analogous. 5.5. Appendix B: Some technical lemmas We recall the standard deﬁnitions j 1 uB2,∞ uL2 (2j−1 ≤|ξ|≤2j+1 ) , (R2 ) = uL2 + sup 2 j∈Z

and, for any open set Ω, 1 2 1 1 uB2,∞ (Ω) = inf{u1 B2,∞ (R2 ) : u1 ∈ B2,∞ (R ), u1 | Ω = u}. 1 (Ω) can be deﬁned as the space of restrictions to Ω of functions Of course B2,∞ 1 (R2 ), which in turn are the temperate distributions for which the above from B2,∞ 1 norm uB2,∞ (R2 ) is deﬁned and ﬁnite. For details see [45], Section 4.2.1.

Lemma 5.1. The function

x |x| 1 1 , i.e., it belongs to B2,∞ (Ω) for any bounded belongs locally to the Besov space B2,∞ 2 open set Ω ⊆ R . θ(x) =

1 (Ω), it is suﬃcient to show that ψ(x)θ(x) is in Proof. By the deﬁnition of B2,∞ 1 2 B2,∞ (R ) for any cut-oﬀ function ψ ∈ C0∞ (Rn ). First of all we recall that the Fourier transform of the function θ from L∞ (R2 ) can be expressed as

= P.V. ξ = lim χε ξ , θ(ξ) ε↓0 |ξ|3 |ξ|3

χε (ξ) = 1 for |ξ| ≥ ε, 0 elsewhere,

where P.V. (meaning principal value) is exactly deﬁned as the limit in the distributional sense at the right-hand side (see, e.g., [34], p. 164 ﬀ.). Thus we are led to estimate the quantities η − η) dη ψ(ξ |ξ|≥ε |η|3 2 j−1 j+1 L (2

≤|ξ|≤2

)

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uniformly with respect to ε. We split the integral as follows: − η) η dη = ψ(ξ + =: Iε (ξ) + II(ξ). |η|3 |η |≥ε 1≥|η|≥ε |η |≥1 The ﬁrst part can be handled by the standard trick η − η) − ψ(ξ)] − η) η dη = [ψ(ξ dη, ψ(ξ Iε = 3 |η| |η|3 1≥|η|≥ε 1≥|η|≥ε since η has average 0 on the sphere. We recall now that ψ is rapidly decreasing and hence satisﬁes η2 = 1 + |η|2 , (5.79) + |ψ(η)| ≤ CN η−N , |∇η ψ(η)| for any N ; thus by Taylor’s formula, if |η| ≤ 1, − η) − ψ(ξ)| |ψ(ξ ≤ sup ∇η ψ(ξ − θη) · |η|, ≤ CN ξ−N |η|, 0≤θ≤1

where CN is independent of ε. This gives −N |IIε (ξ)| ≤ CN ξ |η|−1 dη ≤ CN ξ−N , |η |≤1

whence, choosing N = 3, IIε L2 (2j−1 ≤|ξ|≤2j+1 ) ≤ C0 2−j

(5.80)

with C0 independent of ε. We now estimate II(ξ), which corresponds to the integration on {|η| ≥ 1}. When |ξ| ≤ 1 it is suﬃcient to remark that for any N it holds − η)| ≤ Cη−N |ψ(ξ and this gives immediately IIL2 (|ξ|≤1) ≤ C.

(5.81)

It remains to consider the L2 norm of II(ξ) when |ξ| ∼ 2j , j ≥ 0. We can split II as − η) η dη + − η) η dη =: II1 + III2 . ψ(ξ ψ(ξ II(ξ) = 3 |η| |η|3 |η |≤|ξ|/2 |η |≥|ξ|/2 When |η| ≤ |ξ|/2 we have − η)| ≤ Cξ − η−N ≤ Cξ−N |ψ(ξ which implies

II1 L2 (2j−1 ≤|ξ|≤2j+1 ) ≤ C2−j ; on the other hand, when |η| ≥ |ξ|/2, we have directly − η)|dη ≤ C|ξ|−2 , |III2 (ξ)| ≤ |ξ|−2 |ψ(ξ and this gives

III2 L2 (2j−1 ≤|ξ|≤2j+1 ) ≤ C2−j .

(5.82)

(5.83)

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In conclusion, by (5.80), (5.82), (5.83) we obtain η ψ(ξ − η) 3 dη |ξ|≥ε 2 j−1 |η| L (2

≤ C2−j ,

≤|ξ|≤2j+1 )

whence the result follows.

For our second lemma we need an alternative expression of the Besov norm (see, e.g., [34], Section 2.5.1): given δ > 0, which may be chosen arbitrarily, −1 1 uB2,∞ ∆2h uL2 (R2 ) . (R2 ) uL2 + sup |h| |h|≤δ

Here

∆2h

is the second diﬀerence operator ∆2h f (x) = f (x + 2h) − 2f (x + h) + f (x),

while the ﬁrst diﬀerence operator is simply ∆1h f (x) = f (x + h) − f (x). Lemma 5.2. Consider the function λ = λ(x) on R2 which is deﬁned by λ(x) = 1 − |x|2 for |x| ≤ 1, 0 elsewhere. Then, given any f (s, x) ∈ C 2 (R+ × R2 ) with bounded derivatives, the composition 1 1 f (λ(x), x) belongs locally to the Besov space B2,∞ , i.e., it belongs to B2,∞ (Ω) for 2 1 any bounded open set Ω ⊆ R . In particular, λ(x) itself belongs to B2,∞ (R2 ). Proof. Choose any cut-oﬀ function ψ(x), and deﬁne g(x) = ψ(x)f ( 1 − |x|2 , x). We need a suitable estimate for ∆2h g(x). Notice that, given any C 2 function Φ(y) on RN and any locally bounded function γ : Rn → RN , the following formulas hold: ∆1h Φ(γ(x)) = a, ∆1h γ(x) where a = a(γ(x + h), γ(x)), for a suitable C 1 vector-valued function a(y, z), and ∆2h Φ(γ(x)) = a, ∆2h γ(x) + A1 ∆1h γ(x + h), ∆1h γ(x) + A2 ∆1h γ(x), ∆1h γ(x), where a = a(γ(x + h), γ(x)), Aj = Aj (γ(x + 2h), γ(x + h), γ(x)), for a(y, z), Aj (y, z, p) continuous functions of their arguments (vector- and matrix-valued respectively). If we apply these formulas to g(x), since x, h run on a compact set we get |∆1h g(x)| ≤ C|∆1h λ(x)| (5.84) and |∆2h g(x)| ≤ C|∆2h λ(x)| + C|∆1h λ(x)|2 + C|∆1h λ(x + h)|2 (5.85) with C independent of x, h. We are thus reduced to estimate the diﬀerences of λ(x). For a ﬁxed |h| ≤ 1/4, we split R2 in the three domains |x| > 1 + 3|h|, 1 + 3|h| > |x| > 1 − 3|h| and |x| < 1 − 3|h|. In the ﬁrst one we have simply ∆1h λ(x) = ∆1h λ(x + h) = ∆2h λ(x) = 0.

(5.86)

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In the region 1 + 3|h| > |x| > 1 − 3|h| we have directly |∆1h λ(x)| ≤ 6|h|1/2 ,

|∆1h λ(x + h)| ≤ 7|h|1/2 ,

|∆2h λ(x)| ≤ 16|h|1/2 ,

and since the measure of the region is 12π|h| we obtain for some constant independent of h 1 |∆h λ(x)| + |∆1h λ(x + h)| 2 ≤ C|h|, (5.87) L (1+3|h|>|x|>1−3|h|) ∆2h λL2 (1+3|h|>|x|>1−3|h|) ≤ C|h|.

(5.88)

In the last region |x| < 1 − 3|h| some more computation is needed; notice that here λ(x) = 1 − |x|2 . We have ∆1h

|h|2 + 2x · h 1 − |x|2 = − 1 − |x|2 + 1 − |x + h|2

and this implies |∆1h the inequality |∆1h

1 − |x|2 | ≤ 2|h|1/2 ;

1 − |x + h|2 | ≤ 2|h|1/2

is analogous. Thus we have 1 ∆h 1 − |x|2 + ∆1h 1 − |x + h|2

L2 (|x|<1−3|h|)

≤ C|h|

(5.89)

with C independent of h. In a similar way, some elementary algebra gives |h|2 |h|2 |h|2 + 2x · h ∆2h 1 − |x|2 = −3 − 2x · h , (5.90) m1 m2 m1 · m2 · m3 where we have introduced the quantities m1 = 1 − |x|2 + 1 − |x + h|2 , m2 = 1 − |x|2 + 1 − |x + 2h|2 , m3 = 1 − |x + 2h|2 + 1 − |x + h|2 . Since |x| < 1 − 3|h|, we have m1 ≥ 2|h|1/2 , m2 ≥ 2|h|1/2 , and this allows to estimate the ﬁrst two terms in (5.90) as follows: 2 |h| |h|2 3/2 m1 − 3 m2 ≤ 2|h| . On the other hand, |x| ≤ 1 − 3|h| implies 2|x + h|2 ≤ |x|2 + |x|2 + 2|h|2 + 4|x| · |h| ≤ |x|2 + 1, whence 2 − 2|x + h|2 ≥ 1 − |x|2 , and this implies m3 ≥

1 1 − |x|2 , 2

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while it is obvious that m1 ≥

1 − |x|2 ,

m2 ≥

1 − |x|2 .

These inequalities give for the third term in (5.90) 2 12|h|2 2x · h |h| + 2x · h ≤ . m1 · m2 · m3 (1 − |x|2 )3/2 Summing up, we have obtained on the region |x| < 1 − 3|h| the inequality 12|h|2 |∆2h 1 − |x|2 |2|h|3/2 ≤ (1 − |x|2 )3/2 and integrating with respect to x we obtain 2 ≤ C|h| (5.91) ∆h 1 − |x|2 2 L (|x|<1−3|h|)

with C independent of h. In conclusion, (5.86), (5.87), (5.88), (5.89), (5.91) combined with (5.85) give ∆2h gL2 (R2 ) ≤ C|h| and this implies the thesis. The last lemma is useful to check whether the distributional derivative of a piecewise smooth function coincides with its derivative a.e.: Lemma 5.3. Let Ω ⊆ Rn be an open set and K ⊆ Rn be a closed set, and denote by Kε = {x : d(x, K) < ε} the ε-neighborhood of K. Let f ∈ L1loc (Ω) be a function, diﬀerentiable a.e. on Ω \ K with diﬀerential ∇f . Assume that (i) ∇f is in L1loc (Ω \ K) and coincides with the distributional derivative of f on Ω \ K; (ii) there exists a smooth function h deﬁned on a neighborhood of K such that for any compact set B ⊆ Ω 1 |f − h|dx = 0. (5.92) lim ε↓0 ε Kε ∩B Then the distributional derivative of f coincides with the function g ∈ L1loc (Ω) deﬁned as ∇f on Ω \ K and ∇h on K. If in addition K has measure 0, then the distributional derivative and the derivative a.e. of f on Ω coincide. Proof. Let ρε be a sequence of standard molliﬁers with support in B(0, ε) and set φε = ρε ∗ χ2ε , where χ2ε is the characteristic function of the set K2ε . Notice that φε = 1 on Kε ,

supp φε ⊆ K4ε ;

moreover, on any compact set B we have the pointwise estimate |∇φε | = |(∇ρε ) ∗ χ2ε | ≤

C(B) . ε

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Now ﬁx any test function φ with support in Ω; we can write − f ∇φdx = − f ∇[φ(1 − φε )]dx − (f − h)∇[φφε ]dx − h∇[φφε ]dx =: Iε + IIIε + IIIIε . By (i) we have immediately Iε = ∇f [φ(1 − φε )]dx →

gφdx Ω\K

as ε → 0. Moreover, we can write C |IIIε | ≤ |f − h|dx → 0, ε K4ε ∩B where B is the support of φ; we have used the pointwise estimate and assumption (ii). Finally, we have IIIIε = ∇hφφε dx → ∇hφdx = gφdx K

K

and summing up we obtain the thesis. The last claim follows immediately.

6. Ill-posedness in the critical case (Fourier analysis approach) 6.1. Introduction In this chapter and the following one we shall study the behavior of the Cauchy problem for the wave map system in the critical space H n/2 × H n/2−1 (or, more precisely, H˙ n/2 × H˙ n/2−1 ). From the results of the ﬁrst chapters it is natural to expect some diﬃculty, and indeed our main purpose here will be to show that the solution map (u0 , u1 ) → u(t, x) which takes the Cauchy data into the solution has an unstable behavior in the critical case. Here we shall focus on the special case of 2D wave maps from R × R2 , to the unit sphere Sn (embedded in Rn+1 ), n ≥ 2 as target. Thus u : Rt × R2x → Sn , and the system takes the form utt − ∆u + (|ut |2 − |∇x u|2 )u = 0, ut (0, x) = u1 (x), u(0, x) = u0 (x), with the additional geometric constraint |u| = 1. In this case the critical space for the data is of course H˙ 1 × L2 , which is also the energy space, this is one of the reasons that make the two-dimensional case especially interesting. For this problem several results of global well-posedness are available under suitable smallness assumptions on the initial data (see [20], [40], [41], [30]). Moreover, the existence of a global weak solution in H 1 is known, at least in the

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two-dimensional case ([29], [47], [25]); see also the closely connected results of existence and uniqueness of smooth solutions under suitable assumptions of symmetry for geodesically convex 2-dimensional targets ([32],[13]). In the case when the target is S2 the existence of smooth radial classical solutions was recently proved by Struwe [37]. However the well-posedness, even local in time, in the energy space, i.e., for (large) data in H 1 × L2 is still an open problem. Clearly, this is in close relation with the properties of continuity and regularity of the solution map. Indeed, the classical deﬁnition of well-posedness implies in particular the continuity of this map; even in a modern sense, we may remark that the standard proofs of existence and uniqueness, which resort to some contraction method, have as a natural consequence the Lipschitz continuity of that map. To quantify this property, denote by E(t, u) the energy of a solution u at the time t: E(t, u) = ∂ ∂t u(t, ·)2L2 (R2 ) + ∇x u(t, ·)2L2 (R2 ) . Then we may say that the solution map is Lipschitz continuous if we may ﬁnd a constant C such that for any two solutions u, v the following inequality holds: E(t, u − v) ≤ CE(0, u − v),

∀t ∈ [0, 1].

(6.1)

Note that in this deﬁnition the existence of the solution map is not assumed. The solution map is locally Lipschitz continuous if for any solution u one can ﬁnd positive constants δ, C such that for any solution v with E(0, u − v) ≤ δ inequality (6.1) holds. Our ﬁrst goal here is to show, by a suitable counterexample, that the solution map is not locally Lipschitz continuous. More precisely, we shall prove the following: Theorem 6.1. There exists a smooth solution u : R × R2 → Sn to the wave map system, such that for any C > 0, δ > 0, we can construct a smooth solution v : R × R2 → Sn to the wave map system so that E(0, u − v) ≤ δ while the Lipschitz condition (6.1) is not satisﬁed at t = 1. We remark that the solutions constructed in the counterexample are radially symmetric, hence the symmetry assumption does not improve the regularity of the solution map. By a diﬀerent method we can prove a more precise result concerning the uniform continuity of the solution map. Indeed, assume a solution map Φ : H 1 × L2 → C([−T, T ], H˙ 1) ∩ C 1 ([−T, T ], L2) possibly non unique, is deﬁned on some neighborhood U of 0 in H 1 × L2 . Then we can prove that Φ is not uniformly continuous in these spaces. More precisely we have:

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Theorem 6.2. Let U be a neighborhood of zero in H 1 (R2 ) × L2(R2 ), and let T > 0. Then for any δ > 0 we can construct two smooth solutions u, v : R × R2 → Sn to the wave map system, belonging to the ﬁxed neighborhood U , such that ∂ ∂t (u − v)(0, ·)L2 (R2 ) + ∇x (u − v)(0, ·)L2 (R2 ) ≤ δ while at t = T ∂ ∂t (u − v)(T, ·)L2 (R2 ) ≥ 1,

∇x (u − v)(T, ·)L2 (R2 ) ≥ 1.

In the last section of this chapter we shall show that this result is not conﬁned to the setting considered here but it is a quite general phenomenon due to the properties of the nonlinearity in the wave map equation; indeed, the uniform continuity in critical spaces is violated for arbitrary dimension and (non ﬂat) targets. 6.2. Well-posedness of the Cauchy problem for semilinear wave equations The linear wave equation ∂t2 u − ∆u = 0

(6.2)

with initial data u(0, x) = u0 (x),

ut (0, x) = u1 (x)

(6.3)

satisﬁes the energy estimate ∇x u(t, ·)L2 + ∂ ∂t u(t, ·)L2 ≤ C (∇x u0 (·)L2 + u1 (·)L2 ) , provided the initial data u0 , u1 belong to the Hilbert space H = H˙ 1 (Rn ) × L2 (Rn ).

(6.4)

(6.5)

In this case, for any T > 0 the solution map Φ0 is a well-deﬁned linear bounded (i.e., continuous) operator Φ0 : H → C(I; H),

I = [−T, T ],

which takes the data (u0 , u1 ) into a solution u(t, x) = Φ0 (u0 , u1 ) of (6.2) in distribution sense on [−T, T ] × Rn , satisfying the initial conditions (6.3) in a strong sense. The above setting can be slightly extended if we consider a Banach space X = X(I) ⊆ C(I; H) such that Φ0 is a continuous linear operator Φ0 : (u0 , u1 ) ∈ H → X. Now consider the nonlinear Cauchy problem utt − ∆u = F (u),

t ∈ [−T, T ],

x ∈ Rn ,

(6.6)

with initial data (6.3). Here F is a continuous map F : u ∈ X → F (u) ∈ Y

(6.7)

for some subset Y of the space of distributions D ((−T, T ) × R ). Classical wellposedness is usually formulated in terms of the continuity of the mapping datasolution, (u0 , u1 ) → u(t). More precisely, ﬁxed T > 0 and a suitable Banach space n

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X = X(T ) ⊂ C(I, H) we shall say that the Cauchy problem (6.6) is well posed in H if there exists a positive r > 0 and a continuous operator Φ : {(u0 , u1 ) ∈ H; (u0 , u1 )H ≤ r} → X = X(T ), so that u(t) = Φ(u0 , u1 )(t) is a solution in distribution sense of (6.6) and satisﬁes the initial condition (6.3). For an early example of this type of study see [39] where the well-posedness of the Cauchy problem for the wave maps in (t, x) ∈ R × R is considered, and a similar question is investigated by Kenig, Ponce and Vega in a series of papers (see in particular [22]) where, for several classes of equations, an even weaker regularity condition on Φ is assumed, namely uniform continuity. (This type of approach will be considered in the following f Section 7). In the situations when a standard contraction argument is suﬃcient to prove well-posedness and to construct the solution map Φ (see [33]), one can easily show that Φ is locally Lipschitz continuous. More precisely, one can ﬁnd a positive r > 0 and a constant C such that u0 , u ˜1 )H u − u ˜X ≤ C(u0 , u1 ) − (˜

(6.8)

u0 , u˜1 ) ∈ H, satisfying for all (u0 , u1 ), (˜ u0 , u ˜1 )H ≤ r. (u0 , u1 )H + (˜

(6.9)

Our aim is to show that (6.8), (6.9) cannot hold in the present case. A preliminary step will be a reduction of the problem to a simpler form using stereographic projection. 6.3. The wave map system in stereographic projection In the following it will be suﬃcient to consider the case when the target is the twodimensional sphere S2 , since the general case will follow from the trivial embedding S2 ⊂ Sn . Thus we consider the Cauchy problem (utt − ∆u) + Q(∂u)u = 0,

(6.10)

with initial data u(0, x) = u0 (x) ∈ H˙ 1 (R2 ),

ut (0, x) = u1 (x) ∈ L2 (R2 ),

(6.11)

where the quadratic form Q is given by Q(∂u) = |∂ ∂t u|2 − |∇x u|2 .

(6.12)

In order to reduce the vector-valued wave system (6.10) to a scalar one, we compose the map u : (t, x) ∈ R × R2 −→ u = u(t, x) ∈ S2 ⊂ R3 with the stereographic projection u = (u1 , u2 , u3 ) ∈ S2 −→ z ∈ C ∪ ∞, given by z=

u1 + iu2 ; 1 + u3

(6.13)

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the South pole S = (0, 0, −1) is mapped in ∞. The inverse map is u1 =

2 Re z 2 Im z 1 − |z|2 , u = , u = . 2 3 1 + |z|2 1 + |z|2 1 + |z|2

(6.14)

The metric induced by the projection is (1 + |z|2 )2 |dz|2 . The lines through the origin are geodesics on C. Hence, we can take a geodesic γ(s) given by γ(s) = (1 + Ai)s = s + h(s)i, h(s) = As, in C, where A is a real constant. The composition of a geodesic with any realvalued solution of the free wave equation generates a wave map u = uγ (see [30]). To see the form of the resulting equation, we shall write it in terms of the scalar function u1 X(t, x) = Re z(t, x) = 1 + u3 and a generic curve s + h(s)i; from (6.14) we get u1 =

2X , 1 + X 2 + h2 (X)

u2 =

2h(X) , 1 + X 2 + h2 (X)

u3 =

1 − X 2 − h2 (X) . (6.15) 1 + X 2 + h2 (X)

Substitution of this ansatz into the wave map equation gives the scalar equation M (X)2X − L(X)Q(∂X) = 0,

(6.16)

where L(X) = 4h(X)h (X)(−3X 2 +h2 (X)+1) − (1 − (h (X))2 )(2X 3 −6Xh2(X)−2X), (6.17) 4 2 2 3 2 M (X) = −X +(1+h (X)) −2X h(X)h (X)−2X(1+h (X))h(X)h (X). (6.18) Indeed, to prove (6.16), we start with the relations 1 − X 2 + h2 (X) − 2Xh(X)h (X) 2X ∂xj = 2∂ ∂ X , x j 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))2 −2Xh(X) + (1 + X 2 − h2 (X))h (X) 2h(X) = 2∂ ∂xj X ∂xj , 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))2 and X + h(X))h (X) 1 − X 2 − h2 (X) = −4∂ ∂ X ∂xj , xj 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))2 which imply 1 + (h (X))2 . Q(∂u) = 4Q(∂X) (1 + X 2 + h2 (X))2

(6.19)

(6.20)

(6.21)

(6.22)

Computing the second-order derivative (note that h = 0) gives ∂x2j xj

2(∂ ∂x2j xj X)M1 (X) − 4(∂ ∂xj X)2 L1 (X) 2X = , 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))3

(6.23)

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where L1 (X) = X(1 + X 2 + h2 (X))(1 + (h (X))2 ) + 2(X + h(X)h (X))(1 − X 2 + h2 (X) − 2Xh(X)h(X)), M1 (X) = (1 − X 2 + h2 (X) − 2Xh(X)h(X))(1 + X 2 + h2 (X)). From these identities we get 2X 2(2X)M1 (X) − 4Q(∂X)L1(X) 2 , = 1 + X 2 + h2 (X) (1 + X 2 + h2 (X))3

(6.24) (6.25)

(6.26)

and combining (6.26) and (6.22), we ﬁnally obtain (6.16). In the special case h(X) = AX, where A is a real constant, we obtain L(X) = 4A2 X(−3X 2 + A2 X 2 + 1) − 2X(1 − A2 )(X 2 − 3A2 X 2 − 1) (6.27) = 2X(1 + A2 )(1 − X 2 (1 + A2 )), M (X) = −X 4 + (1 + A2 X 2 )2 − 2A2 X 4 − 2A2 X 2 (1 + A2 X 2 ) = (1 − X 2 (1 + A2 ))(1 + X 2 (1 + A2 )). The equation (6.16) suggests us to take X so that the equation 2X + f (X)Q(∂X) = 0

(6.28)

is satisﬁed. Here

2X(1 + A2 ) . (6.29) 1 + X 2 (1 + A2 ) It is clear that (6.28) implies (6.16). This scalar nonlinear wave equation can be transformed into a linear wave equation (see [27]) by the aid of the transform X s Y = G(X) ≡ eF (s) ds, F (s) = f (σ)dσ. f (X) = −

0

0

√ So using (6.29), we ﬁnd F (s) = − ln(1 + B s ), where B = 1 + A2 and Y = B −1 arctan(BX). In conclusion, given any solution of the linear wave equation 2 2

2Y = 0

(6.30)

the function (6.31) X = B −1 tan(BY ) is a solution of the scalar nonlinear wave equation (6.28) and from (6.15) we see that the function u = uA (t, x) deﬁned by sin(2BY ) A sin(2BY ) , u2 = , u3 = cos(2BY ), B = 1 + A2 , (6.32) u1 = B B is a wave map. Now we can choose the special solutions of the free wave equation (6.30) that we shall use in the following: we take dξ Y (t, x) = Re (6.33) sin(t|ξ|)eix·ξ ϕ(ξ) , |ξ| R2

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which corresponds to the choice of initial data Y (0, x) = 0,

ˆ Yt (0, x) = Re ϕ.

This implies in particular ∂ ∂t Y (t, ·)L2 + ∇x Y (t, ·)L2 ≤ CϕL2 for any t ≥ 0. Thus for the resulting wave map uA , by (6.32) we have ∂ ∂t uA (0, ·)L2 + ∇x uA (0, ·)L2 ≤ CϕL2 ,

(6.34)

and noticing that ∂t uA (0) = ∂t Y (0, x)(2, 2A, 0),

∇x uA (0) = ∇x Y (0, x)(2, 2A, 0).

we have also C −1 |A1 − A2 |ϕL2 ≤ ∂ ∂t (uA1 (0, ·) − uA2 (0, ·))L2 + ∇x (uA1 (0, ·) − uA2 (0, ·))L2 ≤ C|A1 − A2 |ϕL2

(6.35)

for some constant C independent of ϕ, A, A1 , A2 . 6.4. Conclusion of the proof of Theorem 6.1 Assume by contradiction that the solution map is (locally) Lipschitz continuous. Then for any real numbers A, A˜ such that ˜ 0 < A < A, we can consider the wave maps u and uA˜ constructed in (6.32), and estimate (6.35) implies that ˜ ∂ ∂t (u(t, ·) − uA˜ (t, ·))L2 + ∇x (u(t, ·) − uA˜ (t, ·))L2 ≤ C|A − A|ϕ L2 . ˜ and taking the limit A˜ → A, we get Dividing by |A − A| ∂ ∂t ∂A u(t, ·)L2 + ∇x ∂A u(t, ·)L2 ≤ CϕL2 . The A-derivatives can be computed explicitly; from (6.32) we obtain ∂A u1 = ∂A u2 =

2A B2 Y 2A2 B2 Y

cos(2BY ) − cos(2BY )

A B 3 sin(2BY ), + B13 sin(2BY ),

∂A u3 = − 2A B Y sin(2BY ). Then taking the time derivative, we ﬁnd the following pointwise estimate: |∂ ∂t ∂A uA (t, x)| ≥ C0 (A)|Y (t, x)||∂ ∂t Y (t, x)| − C1 (A)|∂ ∂t Y (t, x)|, where C0 (A) > 0 provided A > 0. For space derivatives we have an analogous estimate: |∇x ∂A uA (t, x)| ≥ C0 (A)|Y (t, x)||∇x Y (t, x)| − C1 (A)|∇x Y (t, x)|.

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Therefore, if we ﬁx an A > 0, the assumption that the solution map is Lipschitz continuous implies the inequality ∂t Y (t, ·)L2 + C0 (A)Y (t, ·)∇x Y (t, ·)L2 − C1 (A)∂ ∂t Y (t, ·)L2 C0 (A)Y (t, ·)∂ − C1 (A)∇x Y (t, ·)L2 ≤ ∂ ∂t ∂A u(t, ·)L2 + ∇x ∂A u(t, ·)L2 ≤ CϕL2 . On the other hand, the classical energy identity gives ∂ ∂t Y (t, ·)L2 + ∇x Y (t, ·)L2 ≤ CϕL2 and we arrive at Y (t, ·)∂ ∂t Y (t, ·)L2 + Y (t, ·)∇x Y (t, ·)L2 ≤ CϕL2 .

(6.36)

Recall that this estimate is valid locally, i.e., only for ϕL2 ≤ r according to (6.9); but by a standard rescaling argument we obtain that the estimate (6.36) with ϕL2 ≤ r implies the scale invariant estimate Y (t, ·)∂ ∂t Y (t, ·)L2 + Y (t, ·)∇x Y (t, ·)L2 ≤ Cϕ2L2

(6.37)

also for large values of ϕL2 . In conclusion we have proved that for any solution Y (t, x) of the linear wave equation of the form (6.33), an estimate like (6.37) holds, with a constant C independent of the norm of the arbitrary initial datum ϕ. Clearly this implies that (6.37) is valid also for all solutions of the form dξ Y (t, x) = (6.38) sin(t|ξ|)eixξ ϕ(ξ) |ξ| R3 for arbitrary ϕ ∈ L2 . The proof will be concluded if we show that such an estimate is impossible. Indeed, (6.37) implies Ψ(x)Y (t, x)∂ ≤ CΨL2 ϕ2 2 ∂ Y (t, x)dx (6.39) t L for any Ψ ∈ L2 . Using the Plancherel identity and (6.38) we see that this inequality yields dη Ψ(ξ − η) cos(t|ξ|) sin(t|η|)ϕ(ξ)ϕ(η)dξ ≤ CΨL2 ϕ2L2 . (6.40) |η| Now for any even integer M > 2 we deﬁne the function (compare with [27]) 1 ϕM (ξ) = H(AM ) , (6.41) |ξ| ln5/8 |ξ| where (6.42) AM = {ξ ∈ R2 : 2 ≤ |ξ| ≤ M, dist(|ξ|, 8Z + 1) < 1/2} while H(A) denotes the characteristic function of the set A. The condition dist(|ξ|, 8Z + 1) < 1/2 ensures that the following inequalities are satisﬁed: ϕM (ξ) sin(t0 |ξ|) ≥ CϕM (ξ) ≥ 0, ϕM (ξ) cos(t0 |ξ|) ≥ CϕM (ξ) ≥ 0

(6.43)

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with C > 0 and t0 = π/4 (by the non-negativity of sin). The function Ψ will be chosen as follows: Ψ M (ξ) = H(2 ≤ |ξ| ≤ M )

1 |ξ| ln9/16 |ξ|

.

(6.44)

Notice that for any M > 3 we have the estimates ϕM L2 (R2 ) ≤ C, ΨM L2 (R2 ) ≤ C

(6.45)

with some constant C independent of M > 3. Further, we take N ∈ 8Z + 1 and M ∈ 16Z + 1 so that M . 3
|ξ |>2N

dη ϕM (ξ)ϕM (η)dξ |η| .

When |ξ| ≥ 3, |η| ≥ 3 and |η| < |ξ|/2 we can write |ξ − η| ∼ |ξ|,

ln |ξ − η| ∼ ln |ξ|,

so that by (6.40) and deﬁnition (6.44) of Ψ M we arrive at the estimate ϕM (ξ) dη ≤C ϕM (η)dξ 9/16 |η| |ξ| 3<|η|2N |ξ| ln

(6.46)

valid for a constant C > 0 independent of M, N. Now the deﬁnition (6.41) of ϕM implies 1 dη ∼ ϕM (η) ∼ ln3/8 N. 5/8 |η| j 3<|η|2N

|ξ| ln

9/16

|ξ|

dξ ∼

2N ≤j≤M,j∈8Z+1

1 9/16+5/8

j ln

j

∼

1 3/16

ln

N

provided M ≥ N 2 . In conclusion, estimate (6.46) leads to ln3/16 N ≤ C for some constant C > 0 independent of N , which is clearly absurd. This concludes the proof of the theorem.

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6.5. Proof of Theorem 6.2 Instead of the stereographic projection used above, we may use the standard polar coordinates θ ∈ [0, 2π) and φ ∈ [0, π) on the sphere. Then for ﬁxed θ the meridian ⎛ ⎞ cos θ sin φ γθ (φ) = ⎝ sin θ sin φ ⎠ cos φ is a geodesic on the sphere S2 . Thus, for any θ ∈ [0, 2π[, a function of the form ⎛ ⎞ cos θ sin v(t, x) uθ (t, x) = γθ (v(t, x)) = ⎝ sin θ sin v(t, x) ⎠ cos v(t, x) is a wave map, provided v(t, x) is a real-valued solution of the free wave equation v = 0. We shall now compare the ﬁrst-order derivatives of two meridian geodesic solutions uθ for two values θ = A and θ = B of the parameter and two diﬀerent choices of the free wave solution: u(t, x) = γA (v(t, x)),

u ˜(t, x) = γB (w(t, x)),

v = w = 0.

˜), Then we easily have for the ﬁrst component of ∂t (u − u [∂ ∂t (u − u ˜)]1 = cos A cos v · vt − cos B cos w · wt which can be written = cos A(cos v − cos w)vt + (cos A − cos B) cos wvt + cos B sin w(vt − wt ), and thus we obtain the inequality |[∂ ∂t (u − u˜)]1 | + | cos A − cos B| · |vt | + | cos B| · |vt − wt | ≥ | cos A| · | cos v − cos w| · |vt |. In a similar way, we have for the second and third components |[∂ ∂t (u − u˜)]2 | + | sin A − sin B| · |vt | + | sin B| · |vt − wt | ≥ | sin A| · | cos v − cos w| · |vt | and |[∂ ∂t (u − u ˜)]3 | + |vt − wt | ≥ | sin v − sin w| · |vt |. Squaring and summing these three inequalities, and using the identity (cos A − cos B)2 + (sin A − sin B)2 = 2(1 − cos(A − B)) = 4 sin2

A−B 2

we ﬁnally obtain A−B v−w 4 |vt |2 + |vt − wt |2 ≥ sin2 |vt |2 (6.47) 2 3 2 (we have used the trivial inequality a2 + b2 + c2 ≥ (a + b + c)2 /3). An identical pointwise inequality holds for the space derivatives: ˜)|2 + 4 sin2 |∂ ∂t (u − u

|∇x (u − u˜)|2 + 4 sin2

A−B v−w 4 |∇x v|2 + |∇x (v − w)|2 ≥ sin2 |∇x v|2 , (6.48) 2 3 2

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so in the following we shall focus on the behavior of ∂t u, but completely analogous results are obtained also for ∇x u. We now choose A = B and write w(t, x) = v(t, x) + z(t, x) (notice that z is also a solution of the free wave equation) so that (6.47) becomes |∂ ∂t (u − u˜)|2 + |zt |2 ≥

z 4 |vt |2 sin2 ; 3 2

(6.49)

recall that u(t, x) = γA (v(t, x)),

u˜(t, x) = γA (v(t, x) + z(t, x)).

Let now v(t, x) be a solution of the Cauchy problem v = 0,

v(T, x) = 0,

vt (T, x) = µρ−1 χB(0,ρ) ,

where χB(0,ρ) is the characteristic function of the disc B(0, ρ) and T > 0 is ﬁxed; on the other hand, let z(t, x) be the solution of the Cauchy problem z = 0,

z(0, x) = 0,

zt (0, x) = φ(x)

for a smooth function φ to be precise. Since z is smooth we can write 1 z(T, x) z(T, 0) ≥ sin2 2 2 2 thus (6.49) gives at t = T sin2

for all x ∈ B(0, ρ), ρ depending on φ,

˜(T ))|2 + |zt (T, x)|2 ≥ |∂ ∂t (u(T ) − u

z(T, 0) 2 µ2 χB(0,ρ) sin2 3 ρ2 2

(6.50)

and integrating on R2 and using the energy identity Dt,x z(T, ·)2L2 = Dt,x z(0, x)2L2 ≡ φ2L2 we obtain ∂ ∂t (u(T ) − u ˜(T ))2L2 + φ2L2 ≥

2 2 2 z(T, 0) µ sin . 3 2

Notice that the data of u at t = 0 are u(0, x) = γA (v(0, x)),

ut (0, x) = γA (v(0, x))vt (0, x)

which implies, using the energy identity for v and the fact that |γA | = 1,

Dt,x u|t=0 L2 ≤ µ; analogously we have for u ˜ u ˜(0, x) = γA (v(0, x)),

u ˜t (0, x) = γA (v(0, x))(vt (0, x) + φ(x))

which gives ˜|t=0 L2 ≤ µ + φL2 . Dt,x u

(6.51)

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Let now U be the neighborhood of 0 in H˙ 1 × L2 chosen in the statement of the theorem, which we can assume to be a ball with center 0 and radius r0 ; the assumption that the data of u, u ˜ lie in U will be ensured by the condition µ + φL2 < r0 ,

(6.52)

and in the following we can choose µ = r0 /2. Now, assume by contradiction that the solution map Φ is uniformly continuous. Then for any ε > 0 we can ﬁnd δ > 0 such that the condition ˜)|t=0 L2 ≡ φL2 < δ Dt,x (u − u implies, for any ﬁxed T > 0, Dt,x (u − u ˜)|t=T L2 < ε. In view of inequality (6.51), this means that for any ε > 0 we can ﬁnd δ > 0 such that 2 2 2 z(T, 0) φL2 < δ =⇒ µ sin < δ + ε; (6.53) = 3 2 recall that z(t, x) is a generic solution of the free wave equation, with data z(0, x) = 0, zt (0, x) = φ(x), subject to the only constraint φL2 < r0 − µ = r0 /2. Thus the proof is reduced to a property of the linear wave equation, and it is not diﬃcult to show that (6.53) is absurd. Here we shall give a proof based on harmonic analysis methods; a diﬀerent proof based on the explicit representation of the fundamental solution can be found in Section 7. More precisely, we shall construct a sequence of data φj (x) which converges to 0 in the norm L2 , such that the corresponding solution z(t, x) satisﬁes z(T, 0) = C > 0, and this is a contradiction with (6.53) and concludes the proof of the theorem. Proposition 6.1. Fix T > 0. There exists a sequence φj (x) of real-valued, smooth functions (actually in the Schwartz class) such that φj → 0 in L2 and the solution to the Cauchy problem zj = 0,

zj (0, x) = 0,

∂t zj (0, x) = φj (x)

satisﬁes zj (T, 0) = 1 for all j. Proof. To prove this statement we observe ﬁrst of all that for any function g(ξ) with the properties g(−ξ) = g(ξ), the function

g(ξ) is real-valued

z(t, x) = R2

sin(t|ξ|) cos(x · ξ)g(ξ)

dξ |ξ|

is a real-valued solution of the Cauchy problem z = 0,

z(0, x) = 0,

zt (0, x) = gˆ(x)

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(since gˆ(x) is also real-valued). Thus to achieve the proof it will be suﬃcient to construct a sequence of functions φN (ξ) with the properties φN (ξ) = φN (−ξ) ≥ 0,

φ(ξ) is real-valued,

φN → 0 in L2 ,

and such that, at a ﬁxed point x0 which we may choose, e.g., equal to x0 = (1, 0), dξ zN (T, x0 ) = ≥ 1. sin(t|ξ|) cos(ξ1 )φN (ξ) |ξ| 2 R To this end we deﬁne

Dj,m = ξ :

the sets Dj,m in R2 , m ∈ Z, j ≥ 1, & ' ( |ξ| π − (2j + 1) ≤ π ∩ ξ : |ξ1 − 2mπ| ≤ π ; T 2 4 4

the ﬁrst condition ensures that ξ ∈ Dj,m

sin(T |ξ|) ≥ C0 > 0,

= =⇒

while the second one implies that ξ ∈ Dj,m

cos(ξ1 ) ≥ C0 > 0

= =⇒

for some constant C0 independent of j, m. Moreover, denoting by ) Dj = Dj,m m∈Z

we shall choose the functions φN of the following form: φN (ξ) =

bN

cj χDj ,

cj ≥ 0,

j=aN

where χDj are the characteristic functions of the sets Dj . Notice that this implies immediately that φN are real-valued, non-negative, and φN (−ξ) = φN (ξ) since the sets Dj are symmetric for reﬂections. If we notice that, by the ﬁrst condition, ξ ∈ Dj,m

|ξ| ≥ T πj,

=⇒ =

we can write zN (T, x0 ) ≥ C02 T π

bN

cj j −1 µ(Dj ),

j=aN

where µ is the Lebesgue measure; on the other hand we have φN 2L2 =

bN

c2j µ(Dj ).

j=aN

It is an elementary exercise to obtain the estimates T 2 π 3 (j + 1/4) ≥ µ(Dj ) ≥

π2 j, 16

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and hence we obtain zN (T, x0 ) ≥ C02 T

bN π3 cj , 16 j=a

φN 2L2 ≤ 2T 2π 3

bN

jc2j .

j=aN

N

We need a sequence cj ≥ 0 and indices aN ≤ bN such that the ﬁrst sum is bounded from below by some positive constant independent of N , while the second sum tends to 0 as* N → ∞; this can be * obtained immediately if we choose a sequence ∞ ∞ cj such that 1 cj diverges while 1 jc2j converges, such as cj =

1 . j log j

The proof of the proposition is concluded.

7. Ill-posedness in the critical case (fundamental solution approach) 7.1. Introduction This ﬁnal section is devoted to a more detailed study of the properties of the solution map in the critical spaces Φ : H˙ n/2 × H˙ n/2−1 → C([0, T ]; H˙ n/2 ) ∩ C 1 ([0, T ]; H˙ n/2−1 ) in the case of a general target and arbitrary base space dimension. Notice that it is not known if such a map is deﬁned at all, or is unique if it exists; however, our counterexamples are always based on explicit smooth solutions, for which uniqueness holds, hence any possible solution map must contain our counterexamples in its image. The behavior of the wave map system in the critical case s = n/2 is largely an open problem (while in the subcritical case s < n/2 one has in general illposedness in a sharp sense, i.e., non-uniqueness, see, e.g., [39] and [10]). A possible line of attack was suggested by Bourgain (see [2], [3] and also Tzvetkov [46]) who proved that the map data → solution to some nonlinear evolution equations is not C 2 in the subcritical Sobolev spaces. This holds for the cubic NLS, for KdV and mKdV with diﬀerent critical indices. The result was sharpened by Kenig, Ponce and Vega [22] who proved that the solution map actually is not (locally) uniformly continuous in the subcritical spaces. We also mention [4] and [24] where the case of the supercritical nonlinear wave equation and of the Benjamin-Ono equation are considered. Our aim here is to prove a similar result for the wave map system in the critical case s = n/2. Our assumption on N will be quite general; essentially we only require that N is not ﬂat. More precisely, we assume that there exists a geodesic curve γ :] − s0 , s0 [→ N with γ(0) = 0, γ (0) = 0.

(7.1)

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91

From such generality it will be clear that the ill-posedness in the sense of uniform continuity is a general property of nonlinear systems like (1.1) more than a geometric property. Our result is the following: Theorem 7.1. Let N be a smooth Riemannian manifold, isometrically embedded in Rm , such that there exists a geodesic curve γ :] − s0 , s0 [→ N with γ (0) = 0.

γ(0) = 0 ∈ N,

(7.2)

Assume a solution map Φ : (u0 , u1 ) → u for system (1.1) with data (1.2) is deﬁned on some neighbourhood U of 0 in X × Y = H n/2 (Rn ; N ) × H n/2−1 (Rn ; N ). Then, for any T > 0, Φ is not uniformly continuous between the spaces Φ : U ⊆ H n/2 × H n/2−1 → C([0, T ]; H˙ n/2 (Rn , Rm )) or Φ : U ⊆ H n/2 × H n/2−1 → C 1 ([0, T ]; H˙ n/2−1 (Rn , Rm )). As usual, we say that Φ : U ⊆ X × Y → L is uniformly continuous on U if: (1)

(1)

(2)

(2)

for any ε > 0 there exists δ > 0 such that, for any (u0 , u1 ) and (u0 , u1 ) in U it holds (1)

(1)

(2)

(2)

(u0 , u1 ) − (u0 , u1 )X×Y ≤ δ (1)

(1)

(2)

⇒

u(1) − u(2) L ≤ ε,

(7.3)

(2)

where u(1) = Φ(u0 , u1 ), u(2) = Φ(u0 , u1 ). Thus the above result excludes in particular that Φ is (locally) Lipschitz or H¨ o¨lder continuous. Remark 7.1. It is not diﬃcult to prove by similar arguments that the solution map is not uniformly continuous also in the subcritical case, i.e., from H s × H s−1 , 1 ≤ s < n/2 with values in C([0, T ], H s ) or C 1 ([0, T ], H s−1 ). However, it is already known, at least in the case of a rotationally symmetric target, that a much stronger ill-posedness result holds, namely the local non-uniqueness can be proved. This was obtained for n = 3 in [32], for n ≥ 4 in [5] and for n = 2 in [10]. Since the arguments in these results have a local nature, it is reasonable to argue that non-uniqueness may hold also in the general nonsymmetric case. Remark 7.2. The proof of the theorem is based on an explicit construction of sequences of data such that the corresponding solutions violate (7.3); such solutions are of geodesic type, i.e., of the form γ◦v(t, x), where v(t, x) is a real-valued solution of the homogeneous wave equation. We recall that if γ(s) = (γ1 (s), . . . , γm (s)) is an arbitrary curve in Rm with values in N , and v(t, x) an arbitrary real-valued function, for the composition u(t, x) = γ(v(t, x)) we can write u + Γbc (u)∂α ub ∂ α uc ≡ γ · v + γ + Γbc (γ)γb γc · ∂α v∂ α v and this is identically zero as soon as v = 0 and γ(s) is a geodesic curve.

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Remark 7.3. The ill-posedness for the wave map problem in the case n = 1, s = 1/2 is proved in [39]. It is interesting to mention also the paper [27], where a scalar wave equation of the form u + f (u)∂α u∂ α u = 0 is studied in the critical spaces H n/2 × H n/2−1 . Of course in the scalar case it is possible to prove a much stronger ill-posedness result (actually, a blow-up result). Remark 7.4. It is important to notice that in the proof of Theorem 7.1 the fact that Γjk are Christoﬀel symbols of some Riemannian manifold is not essential. In other words, the result holds for any system of the form (1.1), provided the curves locally deﬁned by the system of equations γ + Γbc (γ)γb γc = 0 satisfy an assumption like (7.2) near some point. This means that the ill-posedness in the sense of uniform continuity is a general property of systems of the wave map type. 7.2. Proof of Theorem 7.1 It is not restrictive to assume that γ is parameterized by arc length; moreover, in the following we shall take T = 1 for simplicity of notations but the proof is unchanged in general. Assumption (7.2) implies that for some component γj of γ one has |γ γj (s)| ≥ c1 (N ) for |s| ≤ c0 (N ) (7.4) for suitable constants c0 , c1 depending only on the manifold N . Let v, w be two C ∞ real-valued solutions of the homogeneous wave equation v = w = 0 with data v(0, x) = w(0, x) = v0 (x),

vt (0, x) = v1 (x),

wt (0, x) = w1 (x).

Notice that v(0, x) ≡ w(0, x) and only the second datum is diﬀerent. Moreover, we shall always work with data of compact support, so that v(t, ·), w(t, ·) will have support in a ﬁxed ball (say B(0, 10)) for all t ∈ [−1, 1]. Then the functions u(1) = γ ◦ v, u(2) = γ ◦ w are solutions of the wave map equation (see Remark 7.2), provided v, w take their values in the domain of γ(s); more precisely we shall assume that |v| ≤ c0 (N ), |w| ≤ c0 (N ), (7.5) and these conditions will be veriﬁed in the explicit construction of v and w. The corresponding Cauchy data are given by u(1) (0, x) = u(2) (0, x) = γ(v0 )(x),

ut (0, x) = γ (v0 )v1 (x), (1)

ut (0, x) = γ (v0 )w1 (x). (2)

(7.6)

Assume now that the solution map is deﬁned and uniformly continuous on some neighbourhood U of 0 in H n/2 (Rn ; N ) × H n/2−1 (Rn ; N ) with values in the

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space C 1 ([0, 1]; H˙ n/2−1 ) (the case of C([0, 1]; H˙ n/2 ) is completely analogous). If we apply this to the data (7.6), we obtain that: for any ε > 0 there exists δ > 0 such that ∂t u(2) (0, ·)H n/2−1 ≤ δ ⇒ sup ∂ ∂t u(1) (t, ·)−∂ ∂t u(2) (t, ·)H˙ n/2−1 ≤ ε ∂ ∂t u(1) (0, ·)−∂ t∈[0,1]

(7.7) (1) (2) for all data u(1) (0, x) = u(2) (0, x) and ut (0, x), ut (0, x) in U . We can express this condition in terms of the data for v, w. Indeed, we have ∂t u(1) (0, x) − ∂t u(2) (0, x) = γ (v0 )(v1 − w1 )(x), where γ (v0 ) is a smooth function, equal to a constant outside some compact set in Rn . Applying Lemma 7.3 from the Appendix for s = n/2 − 1, we have γ(f )gH n/2−1 ≤ cn γ(f )L∞ ∩H n/2 · gH n/2−1 , where we are using the notation uX∩Y = uX + uY . Since γ(0) = 0 we can apply the standard Moser type estimate γ(f )H n/2 ≤ ρ0 (f L∞ ) · f H n/2

(7.8)

for a suitable continuous increasing function ρ0 (s) (see e.g. [43], Vol.III, Chapter 13, Proposition 10.2), we obtain an inequality like γ(f )gH n/2−1 ≤ ρ1 (f L∞ ∩H n/2 ) · gH n/2−1

(7.9)

for some continuous increasing ρ1 (s), which is valid provided the range of the realvalued function f is contained in a compact subset of the domain of the smooth function γ(s). Then we have ∂ ∂t u(1) (0) − ∂t u(2) (0)H n/2−1 ≤ ρ1 (v0 L∞ ∩H n/2 )v1 − w1 H n/2−1 ,

(7.10)

hence property (7.7) implies the following: for all ε > 0 there exists δ > 0 such that v1 − w1 H n/2−1 ≤ δ

⇒ sup ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 ≤ ε

(7.11)

t∈[0,1]

for all data (v0 , v1 ) and (w0 , w1 ) belonging to a suitable neighbourhood V of 0 in H n/2 (Rn ) × H n/2−1 (Rn ) and such that v0 = w0 and |v0 | ≤ c0 , where c0 = c0 (N ) is deﬁned in (7.4). We now estimate from below the second term in (7.11) ∂t v − γ (w)∂ ∂t wH˙ n/2−1 . ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 = γ (v)∂ We have ∂t v−γ (w)∂ ∂t wH˙ n/2−1 ≥ (γ (v)−γ (w))∂ ∂t vH˙ n/2−1 −γ (w)∂ ∂t (v−w)H˙ n/2−1 . γ (v)∂ (7.12) We apply (7.9) to the last term and obtain γ (v)∂ ∂t (v − w)H˙ n/2−1 ≤ ρ1 (vL∞ ∩H n/2 )∂ ∂t (v − w)H n/2−1 ;

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by the energy identity for (v − w) = 0, v0 = w0 , we know that ∂ ∂t (v − w)H n/2−1 ≤ v1 − w1 H n/2−1 and in conclusion γ (v)∂ ∂t (v − w)H˙ n/2−1 ≤ ρ1 (vL∞ ∩H n/2 )v1 − w1 H n/2−1 .

(7.13)

To estimate from below the ﬁrst term in the right-hand side of (7.12) we use Taylor’s formula and get γ (b) − γ (a) = γ (a)(b − a) + F (a, b)(b − a)2 ,

γ (a) = γ (0) + G(a) · a,

where F (a, b), G(a) are smooth functions of their arguments whose explicit expression is not relevant. Then γ (v) − γ (w) = γ (0) · (v − w) + R(v, w) · (v − w), where we have written for short R(u, v) = G(v) · v + F (v, w)(v − w).

Recalling that |γ (0)| ≥ c1 (see (7.4)), we have γ (v)(v − w)∂ ∂t vH˙ n/2−1 ≥ c1 (v − w)∂ ∂t vH˙ n/2−1 − R(v, w)(v − w)∂ ∂t vH˙ n/2−1 . Now we can apply (7.59) of Lemma 7.3 from the Appendix to obtain ∂t vH n/2−1 , R(v, w)(v − w)∂ ∂t vH˙ n/2−1 ≤ R(v, w)L∞ ∩H n/2 (v − w)∂ while using (7.8) it is standard to obtain R(v, w)L∞ ∩H n/2 ≤ ρ2 (v, wL∞ ∩H n/2 ) · v, wL∞ ∩H n/2 (v, w = v+w) with some continuous increasing function ρ2 (s) whose precise form is not relevant. In conclusion, recalling also (7.12) and (7.13), we have proved the inequality ∂t vH˙ n/2−1 ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 ≥ c1 (v − w)∂ − ρ2 (v, wL∞ ∩H n/2 ) · v, wL∞ ∩H n/2 (v − w)∂ ∂t vH n/2−1 − ρ1 (vL∞ ∩H n/2 )v1 − w1 H n/2−1 . (7.14) To proceed, we must construct explicitly the functions v and w. This is done with the help of a few lemmas. Lemma 7.1. Let n ≥ 2. There exists a sequence of real-valued functions φj ∈ C0∞ (Rn ) supported in the ball {|x| ≤ 2}, with φj → 0

in

H n/2−1 (Rn )

as

j→∞

(7.15)

such that, denoting by zj (t, x) : R × Rn → R the solution of the linear problem z = 0,

z(0, x) = 0,

zt (0, x) = φj (x)

(7.16)

one has zj (1, 0) = 1

for any

j.

(7.17)

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The functions φj and hence zj (t, x) can be chosen as radial functions in x, i.e., depending only on |x|. Proof. We begin with the case n = 2. For 0 < p < q < 1, we deﬁne ψp,q (y) on R2 as follows: I{p≤|y|≤q} (y) , (7.18) ψp,q (y) = − 1 − |y|2 log(1 − |y|2 ) where IA (y) denotes the characteristic function of the set A. An elementary computation gives 1 1 − . (7.19) 2ψp,q 2L2 (R2 ) = 2 log (1 − q ) log (1 − p2 ) Notice that taking any 0 < p1 < p < q < q1 < 1, and an arbitrary smooth radial cut-oﬀ function χp,q with I{p≤|y|≤q} (y) ≤ χp,q (y) ≤ I{p ≤|y|≤q } (y), we can modify deﬁnition (7.18) as follows: χ{p≤|y|≤q} (y) ψ p,q (y) = − 1 − |y|2 log(1 − |y|2 )

(7.20)

in order to obtain a smooth initial datum with similar norm: 1 1 1 1 − ≤ 2ψ p,q 2L2 (R2 ) ≤ − . (7.21) log (1 − q 2 ) log (1 − p2 ) log (1 − q12 ) log (1 − p21 ) On the other hand, the solution zp,q (t, x) of the problem z = 0,

z(0, x) = 0,

is explicitly given by zp,q (t, x) =

t 2π

|x−y|≤t

zt (0, x) = ψp,q ψp,q (y) dy, t2 − |x − y|2

and in particular at (t, x) = (1, 0) one has log(1 − q 2 ) 1 1 1 zp,q (1, 0) = − dy = log log(1 − p2 ) . 2π p≤|y|≤q (1 − |y|2 ) log(1 − |y|2 ) 4π By the positivity of the kernel we have immediately, for the solution z p,q obtained by replacing ψp,q with ψ p,q , 2 log(1 − q 2 ) 1 ≤ z p,q (1, 0) ≤ 1 log log(1 − q1 ) . log log(1 − p2 ) 4π log(1 − p2 ) 4π 1

If we now choose for δ ∈ (0, 1) 1 − p21 = δ,

1 − p2 = δ 2 ,

and write ψδ = ψ p,q , we obtain 1 √ | log δ|−1/2 ≤ ψ δ L2 ≤ 12

1 − q2 = δ3 , +

3 | log δ|−1/2 → 0 8

1 − q12 = δ 4

as δ → 0

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while zδ = z p,q satisﬁes 3 1 1 log ≤ zδ (1, 0) ≤ log 4. 4π 2 4π Deﬁning φj = zδj (1, 0)−1 ψδj for any δj ↓ 0 we obtain the thesis. The general case for even n ≥ 2 follows easily by modifying the above example, using the fact that the solution of (7.16) can be represented as |α|+1 aα t y α Dα φ(x + ty)(1 − |y|2 )−1/2 dy, z(t, x) = |y |≤1

0≤|α|≤(n−2)/2

which, for a radial function φ, gives

(n−2)/2

z(1, 0) =

1

cν

ν=0

∂rν φ(r)(1 − r2 )−1/2 rν+n−1 dr.

0

Here of course we shall choose a datum φ such that its radial derivative of order n/2 − 1 is of the form ψ p,q as seen above. Let us now consider the case of odd n, starting from n = 3. In this case it is suﬃcient to use the well-known fact (see, e.g., Theorem 11.1 in Volume I of [23]) that, for any bounded Ω ⊂ Rn with C ∞ boundary, C0∞ (Ω)

is dense in H 1/2 (Ω)

and also in H s for s ≤ 1/2. Since we shall need a special version of this result for radial functions, we shall give here a self-contained proof adapted to our situation. Indeed, consider the space Z = {φ ∈ C0∞ (R) : φ(x) = φ(−x), φ ≡ 0 near 1 and − 1}

(7.22)

(where “near ±1” means “on some neighbourhood of these two points, depending on φ”). It is easy to see that Z is a dense subset of the space of even functions from H 1/2 (R), that is, 1/2 Heven (R) = {u ∈ H 1/2 (R) : u(x) = u(−x)}

(7.23)

1/2

by the following argument: in the Hilbert space Heven (R) we can certainly choose a u0 orthogonal to Z, and we must only prove that u0 = 0. The tempered distribution T whose Fourier transform is given by T = ξ u0 belongs to H −1/2 (R) and by the identity 2 = (u , φ) 1/2 = 0 T (φ) = (ξ u0 , φ) 0 L H for any test function in Z, we see that the support of T is contained in the set {±1}, i.e., T is a linear combination of a ﬁnite number of derivatives of δ1 , δ−1 .

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97

Hence T(ξ) is a function of the form T(ξ) =

N

(c eiξ + d e−iξ )ξ

=0

for a suitable N ≥ 0 and complex numbers c , d , and at the same time ξ−1/2 T(ξ) must belong to L2 . It is trivial to see that the only such function T is 0, and this implies u0 ≡ 0 too. 1/2 Thus we have proved that C0∞ (]1, 1[) is dense in Heven (] − 1, 1[) since this last 1/2 space coincides with the space of restrictions of functions in Heven (R) to ] − 1, 1[, with the restriction norm (the norm of u is the inﬁmum of the norms of its possible extensions). It would be not diﬃcult to prove the same result for higher dimensions, but actually here we only need to construct a sequence of radial functions ψj ∈ C0∞ (B1 ) which converges to 1 in H 1/2 (B1 ), where B1 is the unit ball B1 = {x ∈ Rn : |x| < 1}. To this end, it is suﬃcient to remark that the operator s ((−1, 1)) → H s (B1 ) A : Heven

deﬁned as A(f )(x) = f (|x|) is bounded for all 0 ≤ s ≤ 1: this is proved directly for s = 0, 1 and follows, e.g., by interpolation for the intermediate values of s. Hence taken any sequence fj (x) in C0∞ ((−1, 1)), with f (x) = f (−x), converging to 1 in the H 1/2 ((−1, 1))-norm, we only need to deﬁne ψj (x) = fj (|x|) to obtain the desired result. Now, setting φj = 1 − ψj we obtain a sequence of radial smooth functions on B1 , converging to 0 in the norm of H 1/2 (B1 ), and identically equal to 1 on some neighbourhood of ∂B1 (depending on j). Then by Kirchhoﬀ’s formula we obtain 1 z(1, 0) = φj (y)dS = 1 4π ∂B1 as needed. In the general case n ≥ 3 odd, we proceed in a similar way using the general representation of the solution; notice that for radial φ the following formula holds

(n−3)/2

z(1, 0) =

bν ∂rν φ(1)

ν=0

with suitable constants bν .

After the constructions of the lemma we have no control on the L∞ -norm of the functions zj ; if we give up the requirement that the zj be radial, however, it is easy to obtain the following result:

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Corollary 7.1. Let n ≥ 2. There exists a sequence of real-valued functions φj ∈ C0∞ (Rn ) supported in the ball {|x| ≤ 5} with φj → 0

in

H n/2−1 (Rn )

as

j→∞

(7.24)

such that, denoting by zj (t, x) : R × R → R the solution of the linear problem n

z = 0,

z(0, x) = 0,

zt (0, x) = φj (x)

(7.25)

one has zj (tj , 0) = 1

for some sequence

tj ∈ (0, 1]

(7.26)

and |zj (t, x)| ≤ 1

for all (t, x, j) ∈ [0, 1] × Rn × N.

(7.27)

Proof. The functions zj constructed in the previous lemma are smooth and compactly supported, let (tj , xj ) be a point where |zj | attains its maximum value mj on the strip [0, 1] × Rn , and deﬁne z j (t, x) = m−1 j zj (t, x − xj ) (and possibly multiply by the sign of zj (tj , xj )). Notice that tj > 0 since z(0, x) ≡ 0. This concludes the proof. Before passing to the main part of the proof, a last elementary rescaling lemma is necessary. Lemma 7.2. Let χ(x) with χH˙ n/2−1 = 0 be a smooth compactly supported (radial) function, vanishing for |x| ≥ 2, and with the property χ(x)dx = 0. (7.28) Rn

Let R ≥ 1, M ≥ 0 be positive numbers, 0 ≤ T ≤ 1, and denote by vR,M,T (t, x) the (radial) solution of the homogeneous wave equation v = 0,

v(T, x) = 0,

vt (T, x) = χR,M (x) := M χ(Rx)

(7.29)

with data at t = T > 0. Denote by v0 , v1 the traces v0 = vR,M,T (0, x),

v1 = ∂t vR,M,T (0, x)

(7.30)

so that (7.29) is equivalent to a Cauchy problem for the homogeneous wave equation with data v0 , v1 at t = 0. Then the following estimates hold, for a constant cn depending only on the space dimension n and on the function χ(x): M (7.31) v0 H n/2 + v1 H n/2−1 ≤ cn , R and, for all (t, x) ∈ [0, 1] × Rn , M (7.32) |vR,M,T (t, x)| ≤ cn . R Finally, for all 0 ≤ s ≤ n/2 and all t ∈ R M v(t, ·)H˙ s + ∂ · Rs−n/2 . ∂t v(t, ·)H˙ s−1 ≤ cn (7.33) R

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99

Proof. Rescale v(t, x) as v(t, x) = w(Rt, Rx) so that M χ(x). R By the energy estimates we have for all real s and all t ∈ R w = 0,

w(RT, x) = 0,

wt (RT, x) =

w(t, ·)H˙ s + ∂ ∂t w(t, ·)H˙ s−1 ≤ 2

M χH˙ s−1 R

(7.34)

which scaling back to v gives M s−n/2 R χH˙ s−1 . R Notice that (7.34) gives a ﬁnite bound also for s = 0; indeed, by assumption (7.28) we have χ (0) = 0 and hence χ/ |ξ| ∈ L2 , i.e., χ ∈ H˙ −1 . Thus for all 0 ≤ s ≤ n/2 we obtain (R ≥ 1) ∂t v(t, ·)H˙ s−1 ≤ 2 v(t, ·)H˙ s + ∂

M s−n/2 R (χH˙ n/2−1 + χH˙ −1 ). R This proves (7.33); inequality (7.31) is just the special case s = n/2 computed at t = 0. To prove (7.32) we use (7.34) again for s = n/2 + 1, which gives v(t, ·)H˙ s + ∂ ∂t v(t, ·)H˙ s−1 ≤ 2

sup w(t, ·)H˙ n/2+1 ≤ cn t∈R

M χH˙ n/2 , R

while for s = 0 it gives sup w(t, ·)L2 ≤ cn t∈R

M χH˙ −1 , R

and this is bounded by (7.28) as already remarked. Thus, by Sobolev embedding, we have M wL∞ (R×Rn ) ≤ cn sup w(t, ·)H n/2+1 ≤ cn (χH˙ n/2 + χH˙ −1 ). R t∈R Since vL∞ = wL∞ , this concludes the proof; the constant cn depends only on n and the quantity χH˙ n/2 + χH˙ −1 . We revert now to the main part of the proof. The next step is the explicit construction of sequences of functions v, w appearing in (7.14). As v we shall choose the function vR,M,T constructed in the preceding lemma, with a suitable choice of the parameters. Notice that by (7.31) we can assume that the initial data v0 , v1 belong to the neighbourhood V of 0 in H n/2 × H n/2−1 on which property (7.11) holds, as soon as M/R is small enough; e.g., if V contains a ball of radius r0 (V ) centered in 0, we may assume that 4cn χH n/2−1

M < r0 . R

(7.35)

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Notice also that, in order to deﬁne the composition γ ◦ v, we must ensure that |v| < s0 (at least on the strip [0, 1] × Rn ) since the geodesic curve is only deﬁned on the interval (−s0 , s0 ), or even better, that |v| < c0 given by (7.4). Using (7.32), we see that therefore it is suﬃcient to decrease M/R, e.g., to impose the condition M < c0 /2. (7.36) R In connection with Remark 7.2, we observe that condition (7.36) is not necessary when we assume that γ(s) is deﬁned for all s ∈ R. Then we deﬁne vj = vR,M,T with the following choices: the parameter T will be chosen as ⇒ vR,M,T (tj , x) = M χ(Rx), (7.37) T = tj cn χH n/2

where tj are given by Corollary 7.1; the parameter R = Rj will be chosen such that 1 on the ball {|x| ≤ 2Rj −1 }; (7.38) zj (tj , x) ≥ 2 this is possible in view of (7.26) and of the continuity of zj ; it is not restrictive to assume that Rj ↑ +∞. On the parameter M = Mj , besides (7.35), (7.36) further smallness conditions will be imposed in the following. We now deﬁne wj ; let µ > 0 be a small parameter, and set (w0 := v0 and) wj := vj + µzj ,

(7.39)

where vj was deﬁned above and zj is given by Corollary 7.1. Thus the data for wj are w0 := v0 , w1 := v1 + µφj , where v0 , v1 are the traces of vj at t = 0, studied in Lemma 7.2. Again, in order to deﬁne the composition γ ◦ w, we must ensure that |w| < c0 , at least for 0 ≤ t ≤ 1. Using (7.27) and recalling (7.32), (7.36), we see that it is suﬃcient to impose the condition (7.40) 0 < µ < c0 /2. Notice that the data w0 , w1 belong to the given neighbourhood V as soon as j is large enough, since φj → 0 in H n/2−1 . Consider inequality (7.14); our aim is to estimate its right-hand side from below. The ﬁrst term at t = tj gives (vj − wj )∂ ∂t vj H˙ n/2−1 = µzj (tj , ·)χRj ,M ˙ n/2−1 ; Mj H

(7.41)

we can apply (7.62) from the Appendix with s = n/2 − 1; since zj ≥ 1/2 on the support of χR,M , we have c zj (tj , ·)χRj ,M ˙ n/2−1 ≥ χRj ,M ˙ n/2−1 − c zj H n/2 χRj ,M Mj H Mj H Mj H n/2−1 . 2 Now we have for R large enough χR,M H˙ n/2−1 =

M κ, R

χR,M H n/2−1 ≤ 2

M κ, R

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101

where by assumption κ = χH˙ n/2−1 = 0. Moreover, by the energy identity we have for all t the relations zj (t, ·)H˙ n/2 ≤ cφj H˙ n/2−1 → 0 as j → ∞ and also for all |t| ≤ 1 sin(t|ξ|) j ≤ φj L2 → 0 zj (t, ·)L2 = φ 2 |ξ| L

⇒

(7.42)

zj (t, ·)H n/2 → 0. (7.43)

Hence we have proved that ∂t vj H˙ n/2−1 ≥ c(n, κ) (vj − wj )∂

Mj . Rj

(7.44)

In view of the second term in (7.14) we need also a bound from above for the quantity (v − w)∂ ∂t v; by (7.59) from the Appendix with s = n/2 − 1, we have zj (tj , ·)χRj ,M Mj H n/2−1 ≤ Czj L∞ ∩H n/2 χRj ,M Mj H n/2−1 ≤ C

Mj zj L∞ ∩H n/2 , Rj

for j large enough, and recalling that zj → 0 in H n/2 uniformly in |t| ≤ 1 as remarked above, and |zj | ≤ 1 by construction, we ﬁnally obtain ∂t vj H n/2−1 ≥ c (n, κ) (vj − wj )∂

Mj Rj

(7.45)

provided j is large enough. We notice that, by (7.32), (7.33), vj L∞ ∩H n/2 ≤ c

Mj , Rj

while, recalling that |zj | ≤ 1 and that zj H n/2 ≤ 1 for j large enough, we have wj L∞ ∩H n/2 = vj + µzj L∞ ∩H n/2 ≤ cµ + c

Mj . Rj

Together with (7.45) this gives us the following estimate for the second term in (7.14): ∂t vj H n/2−1 ρ2 (vj , wj L∞ ∩H n/2 )vj , wj L∞ ∩H n/2 (vj − wj )∂ Mj Mj ≤ ρ3 (µ + Mj /Rj ) · µ + . (7.46) Rj Rj We can impose now the last smallness condition on µ and Mj (recall that Mj /Rj is bounded): Mj 1 (7.47) ρ3 (µ + Mj /Rj ) · µ + ≤ c(n, κ), Rj 2

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where c(n, κ) is the constant appearing in (7.44). Thus we get Mj 1 c(n, κ) . 2 Rj (7.48) The last term in (7.14) is quite easy to estimate: we have for j → ∞

ρ2 (vj , wj L∞ ∩H n/2 )vj , wj L∞ ∩H n/2 (vj − wj )∂ ∂t vj H n/2−1 ≤

ρ1 (vL∞ ∩H n/2 )v1 − w1 H n/2−1 ≤ ρ4 (µ + Mj /Rj ) · µφj H n/2−1 → 0.

(7.49)

We can ﬁnally choose Mj and µ; we deﬁne Mj = λ · Rj , and λ, µ are two positive constants so small that conditions (7.35), (7.36), (7.47) are satisﬁed. Summing up, by (7.44), (7.48), (7.49), we obtain Mj 1 1 ∂ ∂t u(1) (tj , ·) − ∂t u(2) (tj , ·)H˙ n/2−1 ≥ c(n, κ) (7.50) = c(n, κ) · λ 4 Rj 4 provided j is large enough. We can now conclude the proof of the theorem. Recalling (7.30), we can choose as data for v the sequences (j)

(j)

v0 := vRj ,M Mj ,tj (0, x),

v1 := ∂t vRj ,M Mj ,tj (0, x)

while the data for w are chosen as (j)

(j)

(j)

w0 := v0 ,

(j)

(j)

w1 := v1 + µφj = v1 + c0 φj /2.

By (7.35) the data for v belong to V ; as a consequence, the data for w belong to V provided j is large enough, since φj → 0 in H n/2−1 . Thus we are in the position (j) (j) to apply the uniform continuity property (7.11); since w1 − v1 = µφj /2 we have that for all ε > 0 there exists δ > 0 such that φj H n/2−1 < δ

⇒

sup ∂ ∂t u(1) (t, ·) − ∂t u(2) (t, ·)H˙ n/2−1 ≤ ε;

t∈[0,1]

hence in particular at t = tj we must have φj H n/2−1 < δ

⇒

∂ ∂t u(1) (tj , ·) − ∂t u(2) (tj , ·)H˙ n/2−1 ≤ ε

and this is in clear a contradiction with (7.50).

7.3. Appendix The aim of this Appendix is to prove two multiplicative estimates needed in the proof of Theorem 7.1. The ﬁrst one has the following form: f gH s ≤ Cf H s gL∞∩H n/2 , s < n/2.

(7.51)

Notice that this estimate is asymmetric in f, g. We can obtain this estimate from the Kato-Ponce estimate (see Lemma 2.2 in [18]) f gH s ≤ Cf Lp1 J s gLp2 + CJ s f Lp3 gLp4 −1 −1 −1 which is valid for all s ≥ 0, for all p2 , p3 ∈ (1, ∞), and 1/2 = p−1 1 +p2 = p3 +p4 ; s s/2 here J = (1 − ∆) . Then (7.51) follows taking p3 = 2, p4 = ∞, 2n n , p2 = p1 = n − 2s s

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103

and using the Sobolev embeddings f Lp1 ≤ Cf H s ,

J s gLp2 ≤ CgH n/2 .

Also the second commutator estimate we need, i.e., J s (f g) − gJ s f L2 ≤ Cf H s gH n/2 ,

s < n/2,

n ≥ 3,

(7.52)

can be proved by a similar argument based on the the Kato-Ponce commutator estimate (see Lemma 2.2 in [18]) J s (f g) − gJ s f L2 ≤ C∇gLp1 J s−1 f Lp2 + CJ s gLp3 f Lp4 −1 −1 −1 which is valid for all s ≥ 0, for all p2 , p3 ∈]1, ∞[, and 1/2 = p−1 1 + p2 = p3 + p4 . Now (7.52) follows taking p1 = n, p3 = s/n,

2n , n−2 and using the Sobolev embeddings

p4 =

p2 =

∇gLp1 ≤ CgH n/2 ,

2n n − 2s

J s−1 f Lp2 ≤ Cf H s ,

J s gLp3 ≤ CgH n/2 ,

f Lp4 ≤ Cf H s .

For completeness, we give a self-contained proof of (7.51), (7.52) and a reﬁned version of (7.52) involving homogeneous Sobolev norms; we hope that our method is of independent interest. To this end, we must introduce some basic tools from the theory of Sobolev and Besov spaces. 1) Diﬀerence operators. Given h ∈ Rn and a function f : Rn → C, we denote by fj (x) the j-th translate of f in the direction h: fj (x) = f (x + j · h),

j ∈ Z,

and the diﬀerence operator ∆h = ∆ deﬁned as ∆f = f1 − f,

i.e.,

∆f (x) = f (x + h) − f (x).

We denote by ∆ the iterates of ∆. Trivial properties are f0 ≡ f , (ffi )j = fi+j , ∆i (∆j f ) = ∆i+j f , ∆(ffj ) = (∆f )j ≡ ∆ffj . Here of special interest will be the behavior of the diﬀerence operator with respect to products. We have immediately

∆(f g) = f1 g1 − f g = f1 (g1 − g) + (f1 − f )g which can be written shortly ∆(f g) = ∆f · g + f1 · ∆g. By induction one proves easily the Leibnitz rule k k ∆ (f g) = ∆ fm ∆m g. +m=k

(7.53)

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2) Sobolev spaces with fractional index. All the functions (and the spaces) considered here are deﬁned on the whole Rn . The homogeneous Sobolev seminorms ˙ k,p with k ≥ 0 integer, 1 < p < ∞, are deﬁned as W Dα uLp ; uW˙ k,p = α=k

˙ k,2 . Thus the standard Sobolev norms can be written as we write H˙ k for W uW k,p = uLp + uW˙ k,p . For our purposes it is not necessary to enter into the topological details of the deﬁnition of the corresponding spaces; only the norms are suﬃcient, and we shall ˙ s,p , W s,p (semi)norms with noninalways apply them to smooth functions. The W teger s > 0 are more troublesome; the usual deﬁnition by interpolation is not well suited to prove multiplicative estimates. A handier equivalent characterization can be given using the fractional integrals 1/p [s]+1 |∆h u(x)|p dxdh , Is,p (u) = |h|n+sp where [s] is the integer part of the noninteger s > 0, 1 < p < ∞, and integration is performed over R2n ; we shall write Is,2 = Is . Then we have uW k,p uLp + Is,p (u)

(7.54)

(see, e.g., 2.3.1 and Theorem 2.5.1 in [44]). The integral Is,p (u) plays the role of the homogeneous norm; this can be seen by a simple rescaling argument. For the following application it will be suﬃcient to consider the L2 case, in which we have a simple deﬁnition using the Fourier transform uH˙ s = |ξ|s u L2 ,

uH s ≡ (1 + |ξ|2 )s/2 u L2 uL2 + uH˙ s .

Indeed, let Sλ be the scaling operators for λ > 0 which are deﬁned by (Sλ u)(x) = u(λx); it is easy to check the scaling properties Sλ uL2 = λ−n/2 uL2 ,

Sλ uH˙ s = λs−n/2 uH˙ s ,

Is (Sλ u) = λ2s−n Is (u).

Thus, for ﬁxed u ∈ H s , if we apply the two equivalent deﬁnitions for Sλ u we obtain λs−n/2 uH˙ s + λ−n/2 uL2 λs−n/2 Is (u) + λ−n/2 uL2 . Letting λ → ∞, we obtain immediately uH˙ s Is (u).

(7.55)

3) Besov spaces. With the same type of norms it is possible to deﬁne the Besov s as follows (see Theorem 2.5.1 in [44]): for any s > 0, 1 < p < ∞, spaces Bp,q 1 < q < ∞ set 1/q [s]+1 ∆h f qLp s f Bp,q = f Lp + dh (7.56) |h|n+sq

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and deﬁne the spaces accordingly. From this deﬁnition in particular it is evident s = W s,p for non-integer s. We shall use the fact that that Bp,p s B2,2 ≡ Hs

for all values of s (including integers). We ﬁnally recall the continuous embedding (see, e.g., Theorem 7.58 in [1] and Theorem 2.8.1 in [44]): for s, t ≥ 0 n n ⇒ W s,p ⊆ W t,q (7.57) 1 < p ≤ q < ∞, s − = t − p q and, more generally, the Besov version r ∈ [1, ∞],

1 < p ≤ q < ∞,

s−

n n =t− p q

⇒

s t Bp,r ⊆ Bq,r .

(7.58)

We are ready to prove our lemma. We use the notation uX∩Y := uX + uY for any two Banach spaces X, Y and u ∈ X ∩ Y . We state the following lemma for smooth functions, the extension to f, g belonging to the appropriate spaces is obvious. Lemma 7.3. For all real 0 ≤ s < n/2 and any smooth functions f, g, the following inequalities hold: (7.59) f gH s (Rn ) ≤ Cf H s gL∞ ∩H n/2 and, for all λ with s < λ < n/2, f gH s (Rn ) ≤ Cf H n/2+s−λ gH λ .

(7.60)

|g(x)| ≥ C1 > 0 on the support of f ;

(7.61)

f gH˙ s ≥ cC1 f H˙ s − c f H s gH n/2

(7.62)

Moreover, assume that

then we have also for some constants c, c > 0 depending only on s, n. Remark 7.5. Estimate (7.59) can be regarded as the limit case of (7.60) as λ → n/2; when λ → s we obtain (7.59) with f and g exchanged. Proof. Notice that in order to prove (7.59), (7.60) it is suﬃcient to prove them with the H s -norm on the left-hand side replaced by the homogeneous H˙ s -norm, since the estimates are trivially true for the term f gL2 . We need two diﬀerent (but parallel) proofs in the cases s integer or non-integer, since we have two diﬀerent representations of the norm in these cases. The proof for integer s is simple. Indeed, by the Sobolev embedding 2n n H s (Rn ) ⊆ L n−2s (Rn ), ∀0≤s< 2

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(see (7.57)), and by H¨ o¨lder’s inequality we have uvL2 ≤ uLn/µ vL2n/(n−2µ) ≤ CuH n/2−µ vH µ

(7.63)

for any real number n . 2 We can apply (7.63) to the product of two derivatives (here and in the following we shall use the shorthand notation D to denote any derivative of order ): 0<µ<

D f Dm gL2 ≤ Cf H n/2+−µ gH m+µ . For any integer s < n/2 we can write D f Dm gL2 . f gH˙ s

(7.64)

(7.65)

+m=s

Now, (7.60) follows directly by applying (7.64) to each term 0 ≤ < n/2 with the choice µ = + λ − s, since in this case we have 0 < µ < n/2 for all = 0, . . . , s. To prove the limit case (7.59), i.e., with λ = n/2, the same methods works if we choose for = 0, . . . , s − 1 µ = + n/2 − s and (7.64) gives D f Dm gL2 ≤ f H s gH n/2 ; but we must consider the term with = s separately since µ = n/2 in that case, and we have Ds f gL2 ≤ gL∞ f H s and this concludes the proof. Consider now (7.62) for s < n/2 integer; by (7.65) we have f gH˙ s ≥ gDs f L2 − c D f Dm gL2

(7.66)

+m=s m≥1

and applying (7.64) to each term in the sum, with µ = n/2 − + s as above (so that 0 < µ < n/2), we obtain f gH˙ s ≥ gDs f L2 − cgH n/2 f H s . Recalling (7.61), we obtain (7.62). From now on, assume 0 < s < n/2 is not an integer. To estimate from above f gH˙ s we use the characterization (7.55) and the Leibnitz rule (7.53): 1/2 |∆ fm ∆m g|2 f gH˙ s Is (f g) ≤ C dxdh . |h|n+2s +m=[s]+1

Thus we need an analogue of (7.64) for fractional integrals. Consider ﬁrst the terms with both ≥ 1 and m ≥ 1. By H¨ o¨lder’s inequality we can write for any

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p−1 + q −1 = 1 and any ρ + σ = s 1/p 1/q |∆ f |2p |∆ fm ∆m g|2 |∆m g|2q dxdh ≤ dxdh dxdh , |h|n+2s |h|n+2pρ |h|n+2qσ (7.67) where we replaced fm with f after a translation in the variable x. The parameters p, q, ρ, σ must be chosen in an appropriate way. First of all we can set (since , m ≥ 1) 1 {s} 1 {s} , σ =m− + , (7.68) ρ=− + 2 2 2 2 where {s} = s − [s] is the fractional part of s, so that ρ + σ = s,

= [ρ] + 1,

m = [σ] + 1.

Recalling the deﬁnition of Is,p , from (7.67) we obtain 1/2 |∆ fm ∆m g|2 dxdh ≤ Iρ,2p (f ) Iσ,2q (g) ≤ cf W ρ,2p gW σ,2q . |h|n+2s

(7.69)

Now let λ be such that s ≤ λ ≤ n/2 (extreme cases included) and choose p, q ∈ (1, ∞[) as follows: n 2n 2p = , 2q = ; (7.70) λ−σ n − 2(λ − σ) notice that λ − σ ≥ s − σ > 0 and n − 2(λ − σ) ≥ 2σ > 0. Thus by (7.57) we have the embeddings H n/2+s−λ ⊆ W ρ,2p , H λ ⊆ W σ,2q . (7.71) In conclusion we have proved for all , m ≥ 1 with + m = [s] + 1, and any s ≤ λ ≤ n/2, the inequality 1/2 |∆ fm ∆m g|2 dxdh ≤ cf H n/2−λ+s · gH λ . (7.72) |h|n+2s Two terms are left. The term with m = 0, = [s] + 1 is bounded simply by writing 1/2 |g∆[s]+1 f |2 dxdh ≤ gL∞ · Is (f ) ≤ cgL∞ f H s . (7.73) |h|n+2s On the other hand, the term with = 0 and m = [s] + 1 is more delicate since we o¨lder’s inequality can not use the L∞ norm of f . We proceed as follows: we apply H¨ in dx to obtain 1/2 1/2 ffm 2Ln/s ∆[s]+1 g2L2n/(n−2s) |ffm ∆[s]+1 g|2 dxdh ≤ dh , |h|n+2s |h|n+2s and we notice that the norm ffm Ln/s = f Ln/s is independent of h and can be drawn out of the integral. What remains is exactly a Besov-norm (see (7.56)) and

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we conclude

|ffm ∆[s]+1 g|2 dxdh |h|n+2s

1/2 ≤ f L2n/(n−2s) gB sn ,2 ≤ cf H s gH n/2

(7.74)

s

by the continuous embeddings n/2

H s ⊆ L2n/(n−2s) ,

B sns ,2 ⊆ B2,2 ≡ H n/2

(see (7.57), (7.58)). By (7.72) for λ = n/2, (7.73) and (7.74) we obtain (7.59). By the same method we can write

|ffm ∆[s]+1 g|2 dxdh |h|n+2s

1/2

ffm 2Lp ∆[s]+1 g2Lq dh |h|n+2s

≤

1/2 ,

where, for an arbitrary λ with s < λ < n/2, p and q are chosen as 2p =

n , λ−s

2n ; n − 2(λ − s)

2q =

proceeding exactly as in the proof of (7.74) we obtain

|ffm ∆[s]+1 g|2 dxdh |h|n+2s

1/2 ≤ f Ln/(λ−s) gB s

2n ,2 n−2(λ−s)

≤ cf H n/2+s−λ gH λ .

(7.75) By (7.72) and (7.75) we obtain immediately (7.60) for non-integer s. The proof of (7.62) for non-integer s proceeds in a similar way. Using again the Leibnitz rule (7.53) we can write Is (f g) ≥

|g∆[s]+1 f |2 dxdh |h|n+2s

1/2 −c

+m=[s]+1 m≥1

|∆ fm ∆m g|2 dxdh |h|n+2s

1/2

which by (7.72) for λ = n/2 and (7.74) implies Is (f g) ≥

|g∆[s]+1 f |2 dxdh |h|n+2s

1/2 − cf H s gH n/2 .

Using now assumption (7.61) we have Is (f g) ≥ C1 Is (f ) − cf H s gH n/2 and recalling that Is (u) uH˙ s , we conclude the proof.

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References [1] R. Adams, Sobolev Spaces. Academic Press, New York, 1975. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, II, I Geom. Funct. Anal. 3 (1993) 107–156, 209–262. [3] J. Bourgain, Periodic Korteweg–de Vries equation with measures as initial data, Selecta Math. 3 (1997) 115–159. [4] Ph. Brenner and P. Kumlin, On wave equations with supercritical nonlinearities, Arch. Math. 74 (2000) 129–147. [5] T. Cazenave, J. Shatah, and A. S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills ﬁelds, Ann. Inst. H. Poincare´ Phys. Th´eor. 68, 3 (1998), 315–349. [6] Y. Choquet-Bruhat, Global existence for non-linear σ-models, Rend. Sem. Mat. Univ. Pol. Torino, Special Issue (1988), 65–86. [7] Y. Choquet-Bruhat, Global wave maps on curved space times, Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, 1998), 1–29, Lecture Notes in Phys., 537, Springer, Berlin, 2000. [8] D. Christodoulou and A. S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46, 7 (1993), 1041–1091. [9] P. D’Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system, accepted in CPAM. [10] P. D’Ancona and V. Georgiev, Low regularity solutions for the wave map equation into the 2-D sphere, accepted in Math. Zeitschrift. [11] A. Freire, S. M¨ u ¨ller and M. Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. Henri Poincare´ 15 No.6 (1998) 425–759. [12] V. Georgiev and A. Ivanov, Concentration of local energy for two-dimensional wave maps, preprint 2003. [13] M. G. Grillakis, Classical solution for the equivariant wave maps in 1+2 dimensions, preprint, 1991. [14] M. G. Grillakis, The wave map problem, In Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 227–230. [15] J. Ginibre and G. Velo, The Cauchy problem for the O(N ), CP(N −1), and GC (N, p) models, Ann. Physics 142 (1982), no. 2, 393–415. [16] C. H. Gu, On the Cauchy problem for harmonic maps deﬁned on two-dimensional Minkowski space, Comm. Pure Appl. Math. 33, 6 (1980), 727–737. [17] H. Karcher and J. C. Wood , Non-existence results and growth properties of harmonic maps and forms, J. Reine Angew. Math. 353 (1984) 165–180. [18] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907. [19] S. Klainerman and M. Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. J. 87 (1997), no. 3, 553–589. [20] S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Diﬀerential Equations 22, 5-6 (1997), 901–918.

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[21] S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations (English. English summary), Commun. Contemp. Math. 4 (2002), no. 2, 223–295. [22] C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. Journal 106 (2001) 617–632. [23] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol.I, Springer Verlag, Berlin 1972. [24] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations., SIAM J. Math. Anal. 33 (2001) 982–988. [25] S. M¨ u ¨ller and M. Struwe, Global existence of wave maps in 1 + 2 dimensions with ﬁnite energy data, Topol. Methods Nonlinear Anal. 7, 2 (1996), 245–259. [26] A. Nahmod, A. Stefanov and K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom. 11, Number 1, 49–83, 2003. [27] K. Nakanishi and M. Ohta, On global existence of solutions to nonlinear wave equations of wave map type, Nonlinear Anal. TMA 42, (2000), 1231–1252. [28] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Diﬀerential Equations, Walter de Gruyter, Berlin 1996. [29] J. Shatah, Weak solutions and development of singularities of the su(2) σ-model, Comm. Pure Appl. Math. 41, 4 (1988), 459–469. [30] J. Shatah and M. Struwe, Geometric wave equations, New York University Courant Institute of Mathematical Sciences, New York, 1998. [31] J. Shatah and M. Struwe, The Cauchy problem for wave maps, Preprint; to appear on International Math. Research Notices. [32] J. Shatah and A. Sh. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754. [33] I. Sigal, Nonlinear semi-groups, Ann. of Math. 78 No. 2 (1963) 339–364. [34] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32. [35] M. Struwe, Wave maps, In Nonlinear partial diﬀerential equations in geometry and physics (Knoxville, TN, 1995). Birkh¨ ¨ auser, Basel, 1997, pp. 113–153. [36] M. Struwe, Equivariant wave maps in two space dimensions, preprint, to appear in Comm. Pure and Appl. Math. [37] M. Struwe, Radially symmetric wave maps from 1 + 2–dimensional Minkowski space to the sphere, Preprint; to appear on Math. Zeitschrift. [38] M. Struwe, Radially symmetric wave maps from 1 + 2–dimensional Minkowski space to general targets, Preprint. [39] T. Tao, Ill-posedness for one-dimensional wave maps at the critical regularity, Amer. J. Math. 122 (2000), no. 3, 451–463. [40] T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys. 224, (2001), 443–544. [41] D. Tataru, Local and global results for wave maps I , Comm. Part. Diﬀ. Eq. 23 (1998) 1781–1793.

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[42] D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77. [43] M. Taylor, Partial diﬀerential equations, Vol.III, I Springer Verlag, New York, 1997. [44] H. Triebel, Interpolation Theory, Function Spaces, Diﬀerential Operators, North Holland Co., Amsterdam 1978. [45] H. Triebel, Interpolation theory, function spaces, diﬀerential operators, second ed., Johann Ambrosius Barth, Heidelberg, 1995. [46] N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Ser. ` I Math. 329 (1999) 1043–1047. [47] Y. Zhou, Global weak solutions for (1 + 2)-dimensional wave maps into homogeneous spaces, Ann. Inst. H. Poincar´ ´e Anal. Non Lin´eaire 16, 4 (1999), 411–422. Piero D’Ancona Universita ` di Roma “La Sapienza” Dipartimento di Matematica Piazzale A. Moro 2 I-00185 Roma Italy e-mail: [email protected] Vladimir Georgiev Universit` ` a di Pisa Dipartimento di Matematica Via Buonarroti No. 2 56127 Pisa Italy e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 113–211 c 2005 Birkhauser ¨ Verlag Basel/Switzerland

On the Global Behavior of Classical Solutions to Coupled Systems of Semilinear Wave Equations Hideo Kubo and Masahito Ohta Abstract. The aim of this work is twofold. One is to develop an approach for dealing with semilinear wave equations adopted by John [38]. In Section 2, the basis of the argument will be explained in a self-contained way. The other is an application of the approach to systems of wave equations. We shall make use of it to handle the semilinear case in Sections 3,4 and 5, and to consider the quasilinear case in Section 6. In these argument we bring such systems that the single wave components obey diﬀerent propagation speeds into focus. Mathematics Subject Classiﬁcation (2000). Primary 35L70; Secondary 35B20. Keywords. Nonlinear hyperbolic systems, global solutions, blow-up, lifespan, self-similar solutions, asymptotic behavior.

1. Introduction This article is concerned with systems of nonlinear wave equations (∂ ∂t2 − c2i ∆)ui = Fi (u, ∂u, ∂∇u),

(t, x) ∈ [0, ∞) × Rn (1 ≤ i ≤ N ),

(1.1)

where ∂ = (∂0 , ∂1 , . . . , ∂n ) = (∂ ∂t , ∇), ∇ = (∂ ∂x1 , . . . , ∂xn ), ci is a positive constant, and u = (u1 , . . . , uN ) is an RN -valued unknown function of (t, x). We study the global behavior of solutions of the initial value problem to (1.1). The equation has a long history. Indeed, the case where N = 1, c1 = c, and F1 (u, ∂u, ∂∇u) = |u|p with p > 1, that is (∂ ∂t2 − c2 ∆)u = |u|p ,

(t, x) ∈ [0, ∞) × Rn ,

(1.2)

The ﬁrst author is partially supported by: Grant-in-Aid for Science Research #14740114, JSPS. The second author is partially supported by: Grant-in-Aid for Science Research #14740099, JSPS.

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was considered by Strauss [100] in 1968. Since we know well the feature of the homogeneous wave equation (∂ ∂t2 − c2 ∆)u = 0,

(t, x) ∈ [0, ∞) × Rn ,

(1.3)

one of the basic question is if it is possible to regard (1.2) as a small perturbation of (1.3). When n = 3, John [38] proved in 1979 that √ there exists a unique classical solution to (1.2) for small initial data if p > 1 + 2,√and that the solution may develop a singularity in a ﬁnite time if 1 < p < 1 + 2 however small the initial data is. (For the other spatial dimensional case, see the introductive part of Section 2 below.) This result means that in the former case, the eﬀect of the nonlinearity in (1.2) is not so strong that one can think of it just as a small perturbation. On the other hand, we see that (1.2) has a diﬀerent structure from (1.3) in the latter case. Roughly speaking, the behavior of the blowing-up solution is governed by the ordinary diﬀerential equation v (y) + v (y) = |v(y)|p ,

y ≥ 0,

(1.4)

or “almost” equivalently to (2.33) below. Having these points in mind, our ﬁrst problem for (1.1) is formulated as follows. Problem: Find a sharp condition about the small data global existence and blowup for the system (1.1). Here small data global existence means that the initial value problem to (1.1) admits a unique global (mild) solution for all “small” initial data. On the contrary, we say blow-up occurs if small data global existence does NOT hold. In other words, it means that one can choose a pair of initial data such that the lifespan of the corresponding solution is ﬁnite. On the one hand, the equation (1.2) is invariant with respect to the scale transform u −→ uλ , where 2

uλ (t, x) = λ p−1 u(λt, λx)

(λ > 0).

(1.5)

This will give us a possibility to construct a self-similar solution to (1.2). When n = 3, the existence and non-existence of the self-similar solution is again characterized √ by the exponent of p = 1 + 2. Our next question is concerned with the case where the √ nonlinearity of (1.2) is regarded as a small perturbation of (1.3) (i.e., p > 1 + 2 when n = 3). What we wish to know is whether the eﬀect of the nonlinearity to the solution of (1.2) remains for large time or not. For the equation (1.2), the answer is the following. The global solution of (1.2) tends to a solution of (1.3) in the sense of the energy for large values of t. In other words, we can say that (1.2) is a nonlinear perturbation from (1.3), or the nonlinearity in (1.2) is of “short-range” type. In Section 2, we give a complete proof of all the results mentioned in the above for n = 2, 3, in the spirit of the work of John [38]. The core of the argument is the point-wise estimate which is involved by the following weight t + |x|−

n−1 2

ct − |x|−ν

(ν > 0).

(1.6)

Coupled Systems of Semilinear Wave Equations

115

In analyzing the case where the exponent p is close to the critical value, the factor ct − |x|−ν plays an essential role. It reﬂects the fact that the wave propagates along the characteristic cone ct = |x|. Most of the argument presented in Section 2 below contains modiﬁcation and simpliﬁcation of earlier works, so that it can be applicable to the system (1.1). We underline that Subsection 2.3 is devoted to a new proof for establishing optimal lower bounds of the lifespan with respect to the size of the initial data. One of the advantages of this approach is that we ﬁnd point-wise behaviors of the solutions by product. While, we need to pose higher regularity of the initial data than that of the solution itself. So, for example, to consider the well-posedness of the problem in Sobolev spaces, Strichartz estimates come into play. For the estimates, we refer to [24, 52, 78, 84] and the references cited therein. We also refer to Strauss [102] concerning the other aspect of the equation (1.2) and do not go further in these direction. Based on the preparation in Section 2, we next consider the system of semilinear wave equations

2 (t, x) ∈ [0, ∞) × Rn , (∂ ∂t − c21 ∆)u1 = λ1 |u1 |p1 |u2 |p2 , (1.7) 2 2 q1 q2 (∂ ∂t − c2 ∆)u2 = λ2 |u1 | |u2 | , (t, x) ∈ [0, ∞) × Rn where λ1 , λ2 ∈ R and n = 2, 3, in three diﬀerent cases. In Section 3, we treat the case where p1 = q2 = 0. Then the feature of the system (1.7) becomes similar to that of the single wave equation (1.2) in the sense that we can treat it without paying attention to the propagation speeds c1 and c2 . We underline that the proof of the blow-up part given in Subsection 3.1 improves the treatment of the reduced integral equations in the previous works [65, 66]. Actually, we reduce the problem to the single integral equation (3.21) below. In Section 4, we deal with the case where p1 , p2 , q1 , q2 ≥ 1.

(1.8)

In this case, the discrepancy of the propagation speeds comes into play through the interaction between u1 and u2 in the nonlinearity. Indeed, if c1 = c2 , then c1 t − |x| is equivalent to t + |x| when (t, x) is close to the light cone c2 t = |x|, and c2 t − |x| is so when (t, x) is close to c1 t = |x| vice versa. Therefore it is possible to extract an additional decay from the nonlinearity, because the solution n−1 ui (t, x) is supposed to behave like t + |x|− 2 ci t − |x|−ν with some ν > 0. On the other hand, if c1 = c2 , then we don’t have such an extra decay, so that we can simply adopt the argument in the preceding section. In Section 5, we consider a mixed case of the preceding two sections. In particular, we study the case where n = 3 and p1 = p2 = 1, q1 = 3, q2 = 0.

(1.9)

Then we have the following remarkable result: When 0 < c1 < c2 , small data global existence holds good. On the contrary, if c1 > c2 > 0, then blow-up occurs. In order to show the existence part, we need a diﬀerent type of weight from the

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standard one given in (1.6). While, in proving the blow-up part, we modify the reduction of the problem to a single integral equation. As for the viewpoint of mathematical physics, we refer to Menzala and Ebihara [81] and the references cited therein. Now we turn our attention to the general case where the nonlinearity of the system (1.1) depends also on the derivatives of the unknown functions. This case is treated in Section 6. In order to avoid the loss of derivatives, we suppose the solution to be suﬃciently smooth. This in turn requires that the nonlinear term is also smooth in its arguments. Then the quadratic nonlinearity is of special interest, as long as the small amplitude solutions are concerned. When n = 3, it was shown by John [39] that blow-up occurs for the system (1.1) with some quadratic nonlinearity. Therefore we need some structure of the nonlinearity for establishing small data global existence for that case. Christodoulou [12] and Klainerman [54] independently introduced a condition on the nonlinearity called null condition. When the propagation speeds are common, it ensures small data global existence for (1.1). We underline that if the propagation speeds are distinct, then we can relax the condtion on the nonlinearity, and there are many contributions which aim at ﬁnding a wider class of nonlinearities for which (1.1) admits a global solution. In Subsection 6.1, we present a blow-up result due to Ohta [85]. To our knowledge, only few results on blow-up for (1.1) with multiple speeds are known. An interpretation of the null condition will be given in Subsection 6.2, together with the key identity (6.57) below. They are not only useful to show small data global existence for the system (1.1) with multiple speeds, but also to examine asymptotic behaviors of the global solution to that. As we shall see, the system can be regarded as a nonlinear perturbation from the system of homogeneous wave equations (∂ ∂t2 − c2i ∆)ui = 0,

(t, x) ∈ [0, ∞) × Rn

(1 ≤ i ≤ N ),

(1.10)

in most of the cases. In the appendix, we give an elementary proof for the uniqueness of classical solutions for the system (1.1). We conclude this section by collecting several notations which will be used in the sequel. – For any x ∈ Rn , the symbol x denotes 1 + |x|2 . – For A ≥ 1 we denote A[a]+ = Aa

if

a > 0 ; A[a]+ = 1

if a < 0 ; A[0]+ = 1 + log A.

– We put a ∨ b := max{a, b} and a ∧ b := min{a, b} for a, b ∈ R. Besides, [a]+ = max{a, 0}. – We set R∗ := R \ {0} and denote by S n−1 the unit sphere in Rn . – For a set A, χA denotes the characteristic function of A.

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In what follows, the letter C stands for various constants which may change from line to line. Especially C(. . . ) denotes a constant depending essentially only on the quantities indicated in the parentheses.

2. Single wave equation This section is concerned with the initial value problem to (∂ ∂t2 − c2 ∆)u = |u|p ,

(t, x) ∈ [0, ∞) × Rn ,

(2.1)

where c > 0 and p > 1, with u(0, x) = ϕ(x),

∂t u(0, x) = ψ(x),

x ∈ Rn ,

(2.2)

where ϕ ∈ C0∞ (Rn ) and ψ ∈ C0∞ (Rn ). For this problem Strauss [101] introduced a number p0 (n) which is the positive root of the following quadratic equation: n−1 n+1 p− Φ(n, p) ≡ p − 1 = 0. (2.3) 2 2 Note that p0 (n) is strictly decreasing with respect to n and for instance √ √ p0 (2) = (3 + 17)/2, p0 (3) = 1 + 2, p0 (4) = 2. It was conjectured by [101] that p0 (n) plays a role as the critical exponent for the problem (2.1)–(2.2). More precisely, small data global existence holds if p > p0 (n), while blow-up occurs if 1 < p < p0 (n). Though the number seems to be strange at ﬁrst glance, one can understand it based on the scaling invariance of the semilinear equation. The scaling invariance means that if u(t, x) is a solution of (2.1), then Dλ,p u(t, x) also satisﬁes the same equation for all λ > 0, where we denoted by Dλ,p u(t, x) the dilation of u(t, x) deﬁned by 2

Dλ,p u(t, x) = λ p−1 u(λt, λx)

(λ > 0).

(2.4)

Then the quadratic equation (2.3) follows from the self-similarity of the function w(r, ˙ t) = (t + r)−

n−1 2

|ct − r|−(

n−1 n+1 2 p− 2 )

for

r, t ∈ [0, ∞).

Namely, if p = p0 (n), then we have the dilation invariance Dλ,p0 (n) w(|x|, ˙ t) = w(|x|, ˙ t) for all λ > 0. Now we recall the historical background about the initial value problem (2.1)– (2.2). When n = 3, John [38] ﬁrstly established that W. Strauss’ conjecture is true, provided that the initial data is of compact support. This result was extended by Glassey [25, 26] to the case n = 2. Moreover, it was shown by Schaeﬀer [92] that blow-up occurs for the critical case p = p0 (n) when n = 2, 3. In the case where 1 < p ≤ p0 (n), the lifespan of the solution was studied by Lindblad [76] and by Zhou [116, 117], independently. For the case n ≥ 4, Sideris [95] proved that blow-up occurs if 1 < p < p0 (n). Recently Yordanov and Zhang [115] extends the result to the critical case. On the other hand, the existence part was considered by so many authors. In fact, Zhou [118] treated the case n = 4 for p > p0 (4), and Christodoulou [12], Li

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and Yun [74] and Choquet-Bruhat [11] handled the general case for suﬃciently large p (see also the references cited therein). Moreover, when n ≥ 5, the case where p0 (n) < p < (n + 3)/(n − 1) was treated by Lindblad and Sogge [79] and by Georgiev [21], independently. In the sequel, it was shown by Georgiev, Lindblad and Sogge [23] that a weak global solution of the problem (2.1)–(2.2) exists for p0 (n) < p < (n + 3)/(n − 1) and n ≥ 2. The proof is based on the weighted Strichartz estimate. It was simpliﬁed by Georgiev [22] and by Tataru [109] independently, by using the Fourier transform on the hyperboloid. Coming back to the work [38], we discuss about the assumption on the initial data. When we are interested in the asymptotic behavior of the global solution, this kind of consideration is important. For the case n = 3, Asakura [10] showed that small data global existence holds without assuming the compactness of the support of the initial data. More precisely, if the initial data (ϕ, ψ) behaves like (x−κ , x−κ−1 ) as |x| → ∞ with κ > 2/(p − 1), then small data global existence holds for p > p0 (n). Kubota [71] and Tsutaya [112] independently extended this result to the case κ = 2/(p − 1). On the contrary, if we take the initial data in such a way that (2.5) ϕ(x) ≡ 0, ψ(x) ≥ εx−κ−1 for x ∈ Rn with κ < 2/(p − 1), then blow-up occurs even though p > p0 (n) and ε is arbitrary small. (Notice that 2/(p − 1) is the number related to the exponent of the scaling invariance for (2.1).) For the case n = 2, the existence part of the above result was proven by [71] and Tsutaya [110], independently. While the blow-up part was shown by Agemi and Takamura [4] and by [110], independently. When n ≥ 4, Takamura [108] proved the blow-up part, while Kubo [59], Kubo and Kubota [61] showed the existence part under the assumption that the initial data is radially symmetric. Finally, D’Ancona, Georgiev and Kubo [14] established small data global existence for initial data being in some weighted L2 space. Next we turn our attention to the asymptotic behavior of the global solution of (2.1)–(2.2). Then the condition p > p0 (n) is necessary for this purpose. It was shown by Strauss [101], Mochizuki and Motai [82, 83] that the scattering operator exists on a dense set of a neighborhood of zero in the energy space, provided p1 (n) p0 (n), since Φ(n, p0 (n)) = 0 and −(n − 1)p0 (n) + (n + 1) = −2/p0(n) < 0. Moreover, when 2 ≤ n ≤ 4 and p > p0 (n), analogous results were obtained by Pecher [87] for n = 3, by Tsutaya [111], Kubota and Mochizuki [72] for n = 2 independently, and by Hidano [28] for n = 4. As for the case p ≥ (n + 3)/(n − 1), Lindblad and Sogge obtained similar results in [78] for n ≥ 3.

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Next we consider the self-similar solution of the initial value problem to (2.1). The existence of it was proved by Pecher [88, 89] for p > p0 (3), and by Hidano [29] for p0 (n) < p < (n + 3)/(n − 1) and n = 2, 3, independently. It was also proved by [89] that even for small data no self-similar solution exists in general if p ≤ p0 (3). On the one hand, Ribaud and Youssﬁ [91] treated the problem for general n(≥ 2) and for p > p1 (n) with p1 (n) is the number in (2.6). This result was extended by Kato and Ozawa [47, 48, 49] for p0 (n) < p < (n + 3)/(n − 1), provided that the initial data is radially symmetric. Moreover, without assuming the radial symmetricity, Kato, Nakamura and Ozawa [50] obtained the same result for 2 ≤ n ≤ 5. In particular, the argument in [50] gives us a new proof based on the weighted Strichartz estimate for the earlier results of [89] and [29]. As we have seen, there are so many contributions that it is diﬃcult to cover all issues in this article. For this reason, we shall concentrate on the approach initiated by John [38]. It is based on the point-wise estimation of the fundamental solution from above and below. To realize this, we consider only the case where n = 2, 3. In the following subsections, we shall explain the details in a selfcontained and simpliﬁed way, so that they provide basic ideas to handle the hyperbolic systems later on. This section is organized as follows. In the next subsection, we establish the blow-up result for the problem (2.1)–(2.2). The point is to consider the quantity uc,p∗ deﬁned by (2.30) and to reduce the problem to an integral inequality (2.33) with (2.32) concerning the quantity. Subsection 2.2 is devoted to the existence part. The main step is to establish the basic estimate (2.49). In Subsection 2.3 we extend the argument in the preceding subsection to the critical case (see (2.65)) and obtain the “almost” global existence result. Subsection 2.4 is concerned with the selfsimilar solution whose existence is shown by the application of (2.92). Finally, we consider asymptotic behaviors in Subsection 2.5. In the super-critical case where p > p0 (n), the eﬀect of the nonlinearity is not so strong that the global solution tends to some free solution as t → +∞ in the sense of the energy. Moreover, the densely deﬁned scattering operator related to (2.1) exists in a neighborhood of the origin of the energy space. These results essentially follow from (2.100) and (2.132). 2.1. Blow-up In this subsection we sketch the proof of the blow-up result. Suppose that u(t, x) is a classical solution of the problem (2.1)–(2.2). Then it satisﬁes the following integral equation: u = Kc [ϕ, ψ] + Lc [|u|p ] in [0, ∞) × Rn ,

(2.7)

where we put Kc [ϕ, ψ](t, x) = Jc [ψ](t, x) + ∂t Jc [ϕ](t, x), t Jc [F (s, ·)](t − s, x) ds. Lc [F ](t, x) = 0

(2.8) (2.9)

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H. Kubo and M. Ohta (n)

Here Jc [ψ](t, x) ≡ Jc [ψ](t, x) (n = 2, 3) is deﬁned by t Jc(3) [ψ](t, x) = ψ(x + ctω) dS Sω , (t, x) ∈ [0, ∞) × R3 , 4π |ω|=1 Jc(2) [ψ](t, x) =

t 2π

|ξ |<1

ψ(x + ctξ) dξ, 1 − |ξ|2

(t, x) ∈ [0, ∞) × R2 .

(2.10)

(2.11)

We take the initial data in such a way that ϕ(x) = 0,

ψ(x) = εg(x),

(2.12)

where ε > 0 and g ∈ C(Rn ) satisﬁes g(x) ≥ 0 for all x ∈ Rn ,

g(0) > 0.

(2.13)

Theorem 2.1. Let n = 2, 3 and 1 < p ≤ p0 (n). Suppose that ε ∈ (0, 1] and g ∈ C(Rn ) satisﬁes (2.13). Then the solution of (2.7) with (2.12) blows up in a ﬁnite time T ∗ (ε). Moreover, there exists a positive constant C ∗ independent of ε such that

exp(C ∗ ε−p(p−1) ) if p = p0 (n), ∗ ∗ (2.14) T (ε) ≤ C ∗ ε−p(p−1)/(1−pp ) if 1 < p < p0 (n). In order to prove Theorem 2.1, we prepare a couple of estimates, Lemma 2.1 and Proposition 2.1 below. By (2.7), (2.12) and (2.13), we have u(t, x) ≥ εJ Jc [g](t, x), u(t, x) ≥ Lc [|u|p ](t, x),

(t, x) ∈ [0, ∞) × Rn , (t, x) ∈ [0, ∞) × Rn .

(2.15) (2.16)

Moreover, by (2.13), there exist δ > 0 and φδ ∈ C([0, ∞)) such that g(x) ≥ φδ (|x|) ≥ 0 for x ∈ Rn ,

φδ (ρ) > 0 for ρ ∈ [0, δ].

(2.17)

Note that we can assume that δ is suﬃciently small. In the sequel we shall make use of the following identity. Lemma 2.1. Let n ≥ 2 and let g ∈ C([0, ∞)). Then we have n−3 23−n ωn−1 ρ+r g(|x + ρω|)dS Sω = λg(λ)[h(λ, ρ, r)] 2 dλ n−2 (rρ) |ρ−r| |ω |=1

(2.18)

for ρ > 0 and x ∈ Rn with r = |x| > 0, where ωn−1 = 2π (n−1)/2 /Γ((n − 1)/2) is the area of the unit sphere in Rn−1 , and h(λ, ρ, r) is deﬁned by h(λ, ρ, r) = {λ2 − (ρ − r)2 }{(ρ + r)2 − λ2 }. Proof. We put λ = |x + ρω|,

x · ω = r cos θ

(0 ≤ θ ≤ π).

Then we have λ2 = r2 + 2rρ cos θ + ρ2 ,

sin θ =

[h(λ, ρ, r)]1/2 , 2rρ

(2.19)

Coupled Systems of Semilinear Wave Equations and

|ω |=1

π

g(λ)ωn−1 [sin θ]n−2 dθ

g(|x + ρω|)dS Sω =

= ωn−1

121

0 ρ+r

g(λ)[sin θ]n−2 |ρ−r|

λ dλ. rρ sin θ

Thus we obtain (2.18).

Next we prepare basic estimates for Jc [G](t, x) and Lc [F ](t, x) from below. Proposition 2.1. Let n = 2, 3 and let G ∈ C(Rn ), g ∈ C([0, ∞)). If G(x) ≥ g(|x|) ≥ 0 for all x ∈ Rn , then we have r+ct n−1 1 Jc [G](t, x) ≥ λ 2 g(λ) dλ (2.20) n−1 2cr 2 |r −ct| for all (t, x) ∈ [0, ∞) × Rn , where r = |x|. Moreover, let F ∈ C([0, T ) × Rn ), f ∈ C([0, ∞) × [0, T )) with T > 0 and suppose that F (t, x) ≥ f (|x|, t) ≥ 0 for all (t, x) ∈ [0, T ) × Rn . Then we have n−1 1 Lc [F ](t, x) ≥ λ 2 f (λ, s) dλ ds (2.21) n−1 2cr 2 Dc (r,t) for all (t, x) ∈ [0, T ) × Rn , where we put Dc (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ s ≤ t,

(2.22)

|r − c(t − s)| ≤ λ ≤ r + c(t − s)}. Proof. First we prove (2.20). By G(x) ≥ g(|x|) for x ∈ Rn , (2.10) and (2.11) imply Jc [G](t, x) ≥ Jc [g(| · |)](t, x) for (t, x) ∈ [0, ∞) × Rn . Therefore it is easy to see from (2.10) and Lemma 2.1 that (2.20) holds for n = 3. When n = 2, by (2.11) and Lemma 2.1 we have ct ρ dρ 1 Jc(2) [g(| · |)](t, x) = g(|x + ρω|) dS Sω 2πc 0 (ct)2 − ρ2 |ω|=1 ρ+r ct ρ dρ λg(λ) 2 = dλ. 2 2 cπ 0 (ct) − ρ |ρ−r| h(λ, ρ, r) Since h(λ, ρ, r) = h(ρ, λ, r) by (2.19), we can write r+ct ct ρ dρ 2 (2) (2.23) Jc [g(| · |)](t, x) = λg(λ) dλ 2 cπ |r−ct| (ct) − ρ2 h(ρ, λ, r) |r −λ| [ct−r]+ r+λ 2 ρ dρ + . λg(λ) dλ 2 − ρ2 h(ρ, λ, r) cπ 0 (ct) |r −λ| Recalling g(λ) ≥ 0 for λ > 0 and noting that for ρ ≥ |r − λ| h(ρ, λ, r) = {(r + λ)2 − ρ2 }{ρ2 − (λ − r)2 } ≤ 4rλ{ρ2 − (λ − r)2 }

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we get Jc(2) [g(| · |)](t, x)

≥

1 √ cπ r

=

1 √ 2c r

r+ct

√ λg(λ) dλ

|r −ct| r+ct √

ct |r −λ|

ρ dρ (ct)2 − ρ2 ρ2 − (r − λ)2

λg(λ) dλ.

|r −ct|

Here we have used the following identity: For 0 ≤ a < b, we have b ρ dρ π = . 2 2 2 2 2 b −ρ ρ −a a

(2.24)

Thus we obtain (2.20) for n = 2. Moreover, (2.21) follows from (2.9) and (2.20). This completes the proof. Now we shall give the proof of Theorem 2.1. In what follows, we put n−1 p∗ = mp − m − 1, m = . (2.25) 2 Step 1. We see from (2.17) and Proposition 2.1 that r+ct 1 λm φδ (λ) dλ. Jc [g](t, x) ≥ 2crm |r−ct| Therefore, if |ct − r| ≤ δ/2 and ct + r ≥ δ, then by (2.15) we have

where we put C0 = (2c)−1

u(t, x) ≥ C0 εr−m ,

(2.26)

δ

λm φδ (λ) dλ( > 0). δ/2

Step 2. We shall show that there is a positive constant C1 = C1 (g, δ, c, n, p) such that C1 εp r1−m (2.27) u(t, x) ≥ (ct + r)(ct − r)p∗ holds for c(t − δ) ≥ r = |x|. Note that if c(t − δ) ≥ r, then we have cs + λ ≥ cδ for (λ, s) ∈ Dc (r, t). By (2.16), (2.26) and Proposition 2.1, for c(t − δ) ≥ r we have Cεp Cεp u(t, x) ≥ m λm−mp dλ ds ≥ m (cs + λ)−m(p−1) dλ ds, r r E E where we put E = {(λ, s) ∈ [0, ∞)2 : |cs − λ| ≤ δ/2, ct − r ≤ cs + λ ≤ ct + r}. Changing the variables by cs − λ , (2.28) ξ = cs + λ, η = c we have ct+r Cεp δ/(2c) dξ dξ Cεp ct+r u(t, x) ≥ m dη = m . m(p−1) m(p−1) r r ξ ξ −δ/(2c) ct−r ct−r

Coupled Systems of Semilinear Wave Equations

123

Thus, using (2.29) below, we arrive at (2.27). Lemma 2.2. For any κ ∈ R, t+r t−r

dρ Cr ≥ ρκ+1 (t + r)(t − r)κ

(2.29)

holds for 0 < r < t, where C = 2/ max{κ, 1}. Proof. Since 1 − sκ ≥ min{κ, 1}(1 − s) for 0 ≤ s ≤ 1, when κ ≥ 1, we have

κ & t+r t−r dρ 1 2r = . 1− ≥ κ+1 κ κ(t − r) t+r κ(t + r)(t − r)κ t−r ρ While, when κ < 1, we have t+r t+r dρ dρ 2r 1−κ ≥ (t − r) = . κ+1 2 (t + r)(t − r)κ t−r ρ t−r ρ

This completes the proof.

Step 3. In view of (2.27), for c, y > 0 and κ ∈ R, we introduce the following quantity: ˜ y)}, (2.30) uc,κ (y) = inf{|x|−(1−m) (ct + |x|)(ct − |x|)κ |u(t, x)| : (t, x) ∈ Σ(c, ˜ y) = {(t, x) ∈ [0, ∞) × Rn : (|x|, t) ∈ Σ(c, y)}, Σ(c, Σ(c, y) = {(r, t) ∈ [0, ∞)2 : r ≤ c(t − y)}.

(2.31)

Since we may assume 0 < δ ≤ 1, (2.27) yields uc,p∗ (y) ≥ C1 εp

for y ≥ 1.

(2.32)

Next we shall show that there exists a constant C2 > 0 such that b y η [uc,p∗ (η)]p 1− uc,p∗ (y) ≥ C2 dη for y ≥ 1, y η pp∗ 1

(2.33)

˜ y), we where b = m + (1 − m)p. Let y ≥ 1. By (2.16) and (2.21), for (t, x) ∈ Σ(c, have u(t, x) ≥ Lc [|u|p ](t, x) cs − λ λb 1 ∗ ≥ u c,p p pp∗ 2crm c Dc (r,t)∩Σ(c,1) (cs + λ) (cs − λ)

p

dλ ds.

Changing the variables by (2.28), we have (ct−r)/c ct+r C (ξ − cη)b [uc,p∗ (η)]p u(t, x) ≥ dξ dη rm 1 ξ p η pp∗ ct−r ct+r C dξ (ct−r)/c (ct − r − cη)b [uc,p∗ (η)]p ≥ dη, rm ct−r ξ p 1 η pp∗

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and by (2.29), we have

(ct−r)/c (ct − r − cη)b [uc,p∗ (η)]p Cr1−m u(t, x) ≥ dη (ct + r)(ct − r)p−1 1 η pp∗ b (ct−r)/c [uc,p∗ (η)]p Cr1−m cη = dη. 1 − ∗ p (ct + r)(ct − r) ct − r η pp∗ 1 Since the function b y η [uc,p∗ (η)]p y → 1− dη y η pp∗ 1 ˜ y), we have is non-decreasing, for any (t, x) ∈ Σ(c, b y [uc,p∗ (η)]p η −(1−m) p∗ r (ct + r)(ct − r) u(t, x) ≥ C dη, 1− y η pp∗ 1 which implies (2.33).

Step 4. Now we are in a position to employ Lemma 2.3 below. Then we see that uc,p∗ (y) blows up in a ﬁnite time y = T∗ (ε), provided pp∗ ≤ 1. The last condition is equivalent to 1 < p ≤ p0 (n) according to (2.3). Therefore the solution of (2.7) with (2.12) blows up in a ﬁnite time T ∗ (ε) ≤ T∗ (ε), if 1 0, α, β ≥ 0, b > 0, κ ≤ 1, ε ∈ (0, 1], and p > 1. Suppose that f (y) satisﬁes b y f (η)p η α β dη, y ≥ 1. 1− f (y) ≥ C1 ε , f (y) ≥ C2 ε y ηκ 1 Then, f (y) blows up in a ﬁnite time T∗ (ε). Moreover, there exists a constant C ∗ = C ∗ (C1 , C2 , b, p, κ) > 0 such that

exp(C ∗ ε−{(p−1)α+β} ) if κ = 1, T∗ (ε) ≤ C ∗ ε−{(p−1)α+β}/(1−κ) if κ < 1. Proof. First, we consider the case κ = 1. We put F (z) = (C1 εα )−1 f (exp(ε−µ z)), Since the function z → (1 − e F (z) ≥ 1,

µ = (p − 1)α + β.

−z b

) is increasing on [0, ∞) and 0 < ε ≤ 1, we have z b 1 − e−(z−ζ) F (ζ)p dζ, z ≥ 0. F (z) ≥ C1p−1 C2 (2.34) 0

Since F (z) blows up in a ﬁnite time (see Lemma 3.3 in Subsection 3.1), we obtain the desired estimate for the case κ = 1. Next, we consider the case κ < 1. We put (p − 1)α + β G(z) = (C1 εα )−1 f (ε−ν ez ), ν = . 1−κ Then we see that G(z) satisﬁes (2.34). Thus we obtain the desired estimate for the case κ < 1. This completes the proof.

Coupled Systems of Semilinear Wave Equations

125

Remark 2.4. The upper bound (2.14) for T ∗ (ε), including the critical case p = p0 (n), was ﬁrst obtained by Zhou [116] for n = 3 and [117] for n = 2. Then Takamura [107] gave a uniﬁed proof for both n = 2 and 3 whose argument is applicable to the system (3.1) below if the propagation speeds c1 and c2 are the same (see [65]). However it seems to be diﬃcult to apply the argument from [107] directly to the system (3.1) for the case c1 = c2 . On the other hand, the proof in the present subsection is applicable to the case c1 = c2 as well. One of the essential diﬀerences to the argument from [107] is to consider the quantity (2.30). In [107], instead of (2.30), the quantity inf{|x|m (t − |x|)κ |u(t, x)| : (t, x) ∈ Σ (y)}, Σ (y) = {(t, x) ∈ [0, ∞) × Rn : t/2 ≤ |x| ≤ t − y} is employed in the case c = 1. Notice that |x|m is equivalent to |x|−(1−m) (t + |x|) for (t, x) ∈ Σ (y), since Σ (y) is away from the t-axis. However the quantity does not give any information about the behavior of the solution close to the t-axis. ˜ y) On the contrary, we studied the behavior of the solution for all (t, x) ∈ Σ(1, including the region that is close to the t-axis. Eventually this will give us some advantage especially for the case n = 2, because we do not have sharp Huygens’ principle in that case. Moreover, the restriction t/2 ≤ |x| in Σ (y) makes it diﬃcult to treat systems with diﬀerent propagation speeds. 2.2. Small data global existence Following John [38], we look for a solution of the integral equation (2.7) in the weighted L∞ -space X(c, κ) deﬁned by X(c, κ) = {u ∈ C([0, ∞) × Rn ) : uX(c,κ) < ∞}, vX(c,κ) =

sup (t,x)∈[0,∞)×Rn

t + |x|

n−1 2

ct − |x|κ |v(t, x)|,

where c > 0, κ ≥ 0. Besides, we introduce a class of initial data Y (ν) deﬁned by Y (ν) = {(ϕ, ψ) ∈ C 1 (Rn ) × C(Rn ) : (ϕ, ψ)Y (ν) < ∞}, ν+ n−1 2

(ϕ, ψ)Y (ν) = sup {x x∈Rn

ν+ n+1 2

|ϕ(x)| + x

(2.35)

(|∇ϕ(x)| + |ψ(x)|)}.

Then we have the following theorem. Theorem 2.2. Let n = 2, 3 and let p > p0 (n). When n = 3, we take ν = p − 2. When n = 2, we take ν so that ν = (p − 3)/2 if p < 4, and 1/4 < ν < 1/2 if p ≥ 4. Suppose (ϕ, ψ) ∈ Y (ν). Then there are constants ε0 = ε0 (p, ν, c, n) > 0 and C0 = C0 (ν, c, n) > 0 such that if (ϕ, ψ)Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there is a unique solution of (2.7) in C0 ε}. {v ∈ C([0, ∞) × Rn ) : vX(c,ν) ≤ 2C First we prepare basic estimates for Jc [G](t, x) and Lc [F ](t, x) from above.

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Proposition 2.2. Let n = 2, 3 and let G ∈ C(Rn ), g ∈ C([0, ∞)). If |G(x)| ≤ g(|x|) for all x ∈ Rn , then we have r+ct 1 (3) λg(λ) dλ (2.36) |J Jc [G](t, x)| ≤ 2cr |r−ct| for all (t, x) ∈ [0, ∞) × R3 , and |J Jc(2) [G](t, x)|

≤

1 c

r+ct

|r −ct|

+

1 c

0

λg(λ) dλ (r + λ)2 − (ct)2 [ct−r]+

(2.37)

λg(λ) dλ (ct)2 − (r + λ)2

for all (t, x) ∈ [0, ∞) × R . Moreover, let F ∈ C([0, T ) × Rn ), f ∈ C([0, ∞) × [0, T )) with T > 0 and suppose that |F (t, x)| ≤ f (|x|, t) for all (t, x) ∈ [0, T ) × Rn . Then we have 1 λf (λ, s) dλ ds (2.38) |Lc [F ](t, x)| ≤ 2cr Dc (r,t) 2

for all (t, x) ∈ [0, T ) × R3 , where Dc (r, t) is deﬁned by (2.22), and 1 λf (λ, s) |Lc [F ](t, x)| ≤ dλ ds 2 − c2 (t − s)2 c (r + λ) Dc (r,t) 1 λf (λ, s) + dλ ds 2 c f c (t − s)2 − (r + λ)2 Dc (r,t)

(2.39)

for all (t, x) ∈ [0, T ) × R2 , where we put ,c (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ λ ≤ c(t − s) − r}. D

(2.40)

Proof. First we prove (2.36) and (2.37). By |G(x)| ≤ g(|x|) for x ∈ Rn , (2.10) and (2.11) imply |J Jc [G](t, x)| ≤ Jc [g(| · |)](t, x) for (t, x) ∈ [0, ∞) × Rn . Therefore it is easy to see from (2.10) and Lemma 2.1 that (2.36) holds. Besides, (2.37) follows from (2.23) and (2.24). Moreover, (2.38) and (2.39) follow from (2.9), (2.36) and (2.37), respectively. This completes the proof. We shall carry out the proof of Theorem 2.2 by dividing the argument into three steps. Step 1. We shall show that for (ϕ, ψ) ∈ Y (ν) with ν > 0 there is a positive constant C0 = C0 (ν, c, n) such that Kc [ϕ, ψ]X(c,ν) ≤ C0 (ϕ, ψ)Y (ν) ,

(2.41)

provided 0 < ν < 1/2 when n = 2. By (2.8) it suﬃces to show r + t

n−1 2

r − ctν |J Jc [ψ](t, x)| ≤ C·ν+

n+1 2

ψL∞ (Rn )

(2.42)

Coupled Systems of Semilinear Wave Equations

127

and r + t ≤

n−1 2

r − ctν |∂ ∂t Jc [ϕ](t, x)|

ν+ n−1 2

C(·

ν+ n+1 2

ϕL∞ (Rn ) + ·

(2.43) ∇ϕL∞ (Rn ) )

for (t, x) ∈ [0, ∞) × Rn , provided 0 < ν < 1/2 when n = 2. To this end, we prepare the following elementary inequalities whose proof will be given at the end of this step. Lemma 2.5. For κ > 0, there exists a constant C = C(κ) > 0 such that r+t ρ−1−κ dρ ≤ C min{t, r} r + t−1 r − t−κ , |r −t| r+t |r −t|

ρ− 2 −κ (ρ + r − t)− 2 dρ ≤ Cr − t−κ ,

[t−r]+

1

1

(2.45)

ρ− 2 −κ (t − r − ρ)− 2 dρ ≤ Cr − t− 2 r − t[ 2 −κ]+ 1

1

(2.44)

1

1

(2.46)

0

for any (r, t) ∈ [0, ∞)2 . Here we used the notations [a]+ = max{a, 0} for a ∈ R, and A[a]+ = Aa if a > 0, A[a]+ = 1 if a < 0, and A[0]+ = 1 + log A, where A ≥ 1. Let us continue the proof of (2.41). First we prove (2.42). When n = 3, we have from (2.36) ·ν+2 ψL∞ (R3 ) r+ct −ν−1 (3) |J Jc [ψ](t, x)| ≤ λ dλ. 2cr |r −ct| This estimate yields (2.42) with n = 3, by virtue of (2.44). When n = 2, we have from (2.37) 3 3 λλ−ν− 2 1 r+ct (2) ν+ 2 dλ |J Jc [ψ](t, x)| ≤ · ψL∞ (R2 ) c |r−ct| (λ + r + ct)(λ + r − ct) 3 1 [ct−r]+ λλ−ν− 2 + dλ . c 0 (λ + r + ct)(ct − r − λ) Since λ(λ + r + ct)−1 ≤ Cλr + t−1 for λ ≥ 0, we get r + t

1 2

|J Jc(2) [ψ](t, x)|

≤

ν+ 32

C·

+ 0

r+ct

ψL∞ (R2 )

[ct−r]+

|r −ct|

λ−ν− 2 √ dλ λ + r − ct 1

λ−ν− 2 √ dλ . ct − r − λ 1

Therefore (2.42) for n = 2 follows from (2.45) and (2.46), since 0 < ν < 1/2. Next we prove (2.43). It follows from (2.10) and (2.11) that |∂ ∂t Jc [ϕ](t, x)| ≤ t−1 Jc [|ϕ|](t, x) + cJ Jc [|∇ϕ|](t, x).

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In view of (2.42), it suﬃces to show that r + t

n−1 2

r − ctν Jc [|ϕ|](t, x) ≤ Ct·ν+

n−1 2

ϕL∞ (Rn ) ,

(2.47)

provided 0 < ν < 1/2 when n = 2. When n = 3, we have from (2.36) r + ct r+ct −ν−1 (3) ν+1 Jc [|ϕ|](t, x) ≤ · ϕL∞ (R3 ) × λ dλ. 2cr |r−ct| By (2.44) we get (2.47). When n = 2, we have from (2.37) 1

cJ Jc(2) [|ϕ|](t, x) ≤ ·ν+ 2 ϕL∞ (R2 ) (I1 + I2 ), where we have set

I1 =

r+ct

|r −ct|

λλ−ν− 2 dλ, (λ + r + ct)(λ + r − ct)

[ct−r]+

I2 = 0

1

λλ−ν− 2 dλ. (λ + r + ct)(ct − r − λ) 1

We evaluate Ij by dividing the argument into three cases. First suppose r ≥ ct ≥ 0. Then I2 = 0 and r+ct r+ct 1 1 I1 ≤ λ−ν− 2 dλ ≤ Cr + t 2 λ−ν−1 dλ. |r −ct|

|r −ct|

By (2.44) we get 1

r + t 2 r − ctν (I1 + I2 ) ≤ Ct.

(2.48)

Next suppose 0 ≤ r ≤ ct and r + ct ≥ 1. Then we have

r+ct 1 1 − 12 λ−ν− 2 (λ + r − ct)− 2 dλ I1 + I2 ≤ Ctr + t +

ct−r

ct−r

& 1 1 λ−ν− 2 (ct − r − λ)− 2 dλ ,

0

which yields (2.48) by (2.45) and (2.46), because of 0 < ν < 1/2. Finally, suppose 0 ≤ r ≤ ct and r + ct ≤ 1. Then we have

& r+ct ct−r √ − 12 − 12 (λ + r − ct) dλ + (ct − r − λ) dλ ≤ Ct. I1 + I2 ≤ C t ct−r

0

Therefore we ﬁnd (2.48), so that (2.47) is valid for n = 2. Hence (2.43) holds. Thus we conclude the proof of (2.41) from (2.42) and (2.43), assuming that Lemma 2.5 is valid. Here we give a proof of the lemma. √ Proof of Lemma 2.5. First, we note that ρ ≤ 1 + ρ ≤ 2ρ for ρ ≥ 0. To prove (2.44), without loss of generality, we can assume that r ≥ t. Then, we have

κ & r+t r+t −1 1+r−t dρ 1 −κ (1 + ρ) = = 1− . 1+κ κ κ(1 + r − t)κ 1+r+t r −t (1 + ρ) r−t

Coupled Systems of Semilinear Wave Equations

129

Since 1 − sκ ≤ max{1, κ}(1 − s) for 0 ≤ s ≤ 1, we have κ 1+r−t 1+r−t 2 max{1, κ}t . ≤ max{1, κ} 1 − 1− = 1+r+t 1+r+t 1+r+t Since min{r, t} = t, we obtain (2.44). Next we prove (2.45). We put a = |r − t|. Then we have r+t ∞ 1 1 1 1 (1 + ρ)− 2 −κ (ρ + r − t)− 2 dρ ≤ (1 + ρ)− 2 −κ (ρ − a)− 2 dρ |r −t|

− 12 −κ

a

2a+1

≤ (1 + a)

− 12

(ρ − a)

∞

dρ +

(ρ − a)−1−κ dρ ≤ C(1 + a)−κ ,

2a+1

a

which implies (2.45). Finally, we prove (2.46). We can assume t > r. We put a = [t − r]+ = t − r, and a 1 1 I(a) = (1 + ρ)− 2 −κ (a − ρ)− 2 dρ. 0

Since I(a) is bounded for 0 ≤ a ≤ 1, we have only to consider the case a > 1. Then, we have a − 12 a/2 1 1 a − 12 −κ a (1 + ρ)− 2 −κ dρ + 1 + (a − ρ)− 2 dρ I(a) ≤ 2 2 0 a/2 ≤

Ca− 2 (1 + a)[ 2 −κ]+ + C(1 + a)− 2 −κ a 2 , 1

1

1

1

which implies (2.46). This completes the proof.

Step 2. We need the following weighted decay estimate for Lc [F ](t, x) to handle the nonlinearity. Proposition 2.3. Let n = 2, 3 and let a, c, µ and κ > 0. Suppose F ∈ C([0, T )×Rn ) and κ = 1. Then there exists a constant C1 = C1 (a, c, µ, κ) > 0 such that sup (t,x)∈[0,T )×Rn

≤ C1

t + |x|m ct − |x|κ0 |Lc [F ](t, x)|

sup (t,x)∈[0,T )×Rn

(2.49)

t + |x|µ at − |x|κ |F (t, x)|,

where we put m = (n − 1)/2 and

µ−m−1 κ0 = µ+κ−m−2

if κ > 1, µ > m + 1 , if κ < 1, µ + κ > m + 2

provided in the case n = 2 that µ < 2 when κ > 1, and that µ + κ < 3 when κ < 1. Fc )(x, ct) with Fc (x, t) = c−2 F (x, t/c), it suﬃces to Proof. Since Lc (F )(x, t) = L1 (F show (2.49) for c = 1. When n = 3, we have from (2.38) |L1 [F ](t, x)| ≤ I ×

sup (t,x)∈[0,T )×Rn

t + |x|µ at − |x|κ |F (t, x)|,

(2.50)

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where we put

1 I= λ + s−µ+1 λ − as−κ dλ ds. 2r D1 (r,t) Changing the variables by β = λ − s,

α = λ + s, we have I

= ≤ ≤

1 4r

t+r

|t−r|

(2.51)

1 + a −κ 1−a α+ β dβ 2 2 r −t α t+r α−µ+1 dα σ−κ dσ −µ+1

α

α

dα

1 2(1 + a)r |t−r| −aα C t+r α−µ+1 α[1−κ]+ dα, r |t−r|

since r − t > −α. Therefore, if κ > 1 and µ > 2, then we have I ≤ Ct + r−1 r − t−µ+2 , by (2.44). While, if κ < 1 and µ + κ > 3, then we have I ≤ Ct + r−1 r − t−µ−κ+3 . Thus we get I ≤ Ct + r−1 r − t−κ0 , hence (2.49) for n = 3 follows from (2.50). When n = 2, we have from (2.39) |L1 [F ](t, x)| ≤ (I1 + I2 ) where we put I1

=

(t,x)∈[0,T )×Rn

t + |x|µ at − |x|κ |F (t, x)|,

D1 (r,t)

λ + s−µ+1 λ − as−κ dλ ds, (λ − s + t + r)(λ + s + r − t)

f1 (r,t) D

λ + s−µ+1 λ − as−κ dλ ds. (λ − s + t + r)(t − r − λ − s)

=

I2

sup

Changing the variables by (2.51), we have α 1−a −κ 2 α + 1+a 1 t+r α−µ+1 2 β √ √ dα dβ, I1 = 2 |t−r| α + r − t β+t+r r −t α 1−a −κ 2 α + 1+a 1 [t−r]+ α−µ+1 2 β √ √ I2 = dβ. dα 2 0 β+t+r t−r−α −α

(2.52)

(2.53) (2.54)

We remark that µ > 3/2 by the assumption and that we may assume µ < 2 without loss of generality. We divide the argument into three cases. First suppose 2r ≥ t ≥ 0 and t + r ≥ 1. Since β + t + r ≥ 2r if either β > −α and 0 < α < t − r or β > r − t, we

Coupled Systems of Semilinear Wave Equations

131

have β + t + r ≥ Ct + r in this case. Therefore we can handle the β–integrals as before and get t+r 1 α−µ+1 α[1−κ]+ 2 √ dα (2.55) t + r (I1 + I2 ) ≤ C α+r−t |t−r| [t−r]+ α−µ+1 α[1−κ]+ √ + dα . t−r−α 0 By (2.45) and (2.46) we can conclude I1 + I2 ≤ Ct + r− 2 r − t−κ0 . 1

(2.56)

Next suppose 2r ≥ t ≥ 0 and t + r ≤ 1. Then both β–integrals in I1 and I2 are bounded. Therefore, I1 + I2 is estimated by the right-hand side of (2.55). Hence (2.56) also holds in this case. Finally suppose t ≥ 2r ≥ 0. We begin with proving α 1−a −κ 2 α + 1+a 1 2 β √ dβ ≤ Ct + r− 2 t + r[1−κ]+ (2.57) β + t + r −α a−1 for 0 ≤ α ≤ t+r. Denoting the left-hand side by I and setting d = 1+a 2 (t+r)+ 2 α, we get 12 α 12 d/a 2 2 σ−κ σ−κ √ √ dσ ≤ dσ, I= 1+a 1+a σ+d σ+d −aα −d

since d ≥ aα for 0 ≤ α ≤ t + r. By the fact that d ≥ min{1, a}(t + r) for 0 ≤ α ≤ t + r, the application of (2.58) below gives (2.57). Now it follows from (2.53), (2.57) and (2.45) that I1 ≤ Ct + r− 2 t + r[1−κ]+ r − t−µ+ 2 , 1

3

since µ > 3/2. Moreover, (2.54), (2.57) and (2.46) yield I2 ≤ Ct + r− 2 t + r[1−κ]+ r − t−µ+ 2 , 1

3

since 3/2 < µ < 2. For t ≥ 2r ≥ 0, we see that these estimates imply (2.56). Therefore we obtain (2.49) for n = 2 via (2.52). This completes the proof. The following lemma has been used in the proof of (2.57). Lemma 2.6. Let κ, a, d > 0. Then there exists a positive constant C = C(κ, a) such that d/a 1 1 σ−κ (d + σ)− 2 dσ ≤ Cd− 2 d[1−κ]+ . (2.58) −d

Proof. We denote by I the left-hand side of (2.58). First suppose 0 < a < 1. Then, splitting the integral at σ = d and putting d d/a 1 −κ − 12 σ (d + σ) dσ, P2 = σ−κ (d + σ)− 2 dσ, P1 = −d

d

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we have I = P1 + P2 . It is easy to see that d/a 1 1 (d + σ)− 2 dσ ≤ Cd−κ+ 2 . P2 ≤ d−κ d

To treat P1 , we consider two diﬀerent cases. Let 0 < d < 1. Then we easily see 1 that P1 is estimated by Cd−κ+ 2 , since d 1 P1 = (d + σ)− 2 dσ ≤ C. −d

On the other hand, when d ≥ 1, we further split the integral as −d/2 d 1 1 Q1 = σ−κ (d + σ)− 2 dσ, Q2 = σ−κ (d + σ)− 2 dσ. −d

−d/2

It follows that Q2 ≤

− 12 d d 1 σ−κ dσ ≤ Cd− 2 d[1−κ]+ . 2 −d/2

While, we have −κ

Q1 ≤ Cd

d

−d

(d + σ)− 2 dσ ≤ Cd−κ+ 2 . 1

1

Thus we have shown (2.58) for the case 0 < a < 1. Finally suppose a ≥ 1. Then I ≤ P1 . Since the argument concerning P1 in the above still works for the present case, we obtain the desired estimate. This completes the proof. Step 3. For ε > 0 we put C0 ε}, Xε := {v ∈ C([0, ∞) × Rn ) : vX(c,ν) ≤ 2C

(2.59)

where ν is the number from the theorem. Deﬁne a map from Xε to C([0, ∞) × Rn ) by Φ[v] := Kc [ϕ, ψ] + Lc [|v|p ]. (2.60) Let C0 and C1 are the constants in (2.41) and (2.49), respectively. We shall show C0 ε0 )p−1 ≤ 1, then for 0 < ε ≤ ε0 we have that if ε0 satisﬁes p2p+1 C1 (C 2C C0 ε for v ∈ Xε , (2.61) 1 Φ[v1 ] − Φ[v2 ]X(c,ν) ≤ v1 − v2 X(c,ν) for v1 , v2 ∈ Xε . (2.62) 2 Once we establish (2.61) and (2.62), we ﬁnd that Φ[v] is a contraction on Xε , hence (2.7) admits a unique solution in Xε for 0 < ε ≤ ε0 . Let a = c, µ = mp and κ = pν. Then µ > m + 1 is equivalent to p∗ > 0 by (2.25). Since p > p0 (n) implies pp∗ > 1 by (2.3), we see from (2.49) that Φ[v]X(c,ν)

≤

Lc [|v|p ]X(c,p∗ ) ≤ C1 [vX(c,ν)]p .

Coupled Systems of Semilinear Wave Equations

133

Recalling that ν < 1/2 ≤ p∗ for n = 2 and p ≥ 4, we can deduce Lc [|v|p ]X(c,ν) ≤ C1 [vX(c,ν) ]p . This estimate, together with (2.41) and the choice of ε0 , implies (2.61). Since ||s|p − |t|p | ≤ p(|s|p−1 + |t|p−1 )|s − t| for p > 1 and s, t ∈ R, we have the following estimate by (2.49) with the same choice of the parameters as in the above: p−1 Lc [|v1 |p ] − Lc [|v2 |p ]X(c,p∗ ) ≤ pC1 (v1 p−1 X(c,ν) + v2 X(c,ν) )v1 − v2 X(c,ν) ,

which yields (2.62). This completes the proof of Theorem 2.2. Remark 2.7. In Theorem 2.2 we can relax the condition on the initial data. We put n−1 2 − m, m = . (2.63) p∗ = p−1 2 Let p satisfy p < (n + 3)/(n − 1), so that p∗ > 0. If (ϕ, ψ) ∈ Y (p∗ ), then there exists a global solution u ∈ X(c, p∗ ) of (2.7), provided (ϕ, ψ) is suﬃciently small in Y (p∗ ). Indeed, because of the fact that pp∗ < 1 for p > p0 (n), mp+pp∗ −m−2 = p∗ and that mp + pp∗ < 3 for p > 3 and n = 2, we can apply Proposition 2.3 as κ < 1. Therefore we can prove such an existence theorem analogously to the proof of Theorem 2.2. While, we see from (2.25), (2.63) and (2.3) that p∗ > p∗ for p > p0 (n), hence ∗ Y (p ) ⊂ Y (p∗ ). Therefore the above result improves Theorem 2.2 concerning the assumption on the initial data when p0 (n) < p < (n + 3)/(n − 1). Remark 2.8. If the initial data (ϕ, ψ) is more regular, then the solution u(t, x) of (2.7) also gain the corresponding regularity. For instance, let (ϕ, ψ) ∈ C 3 (Rn ) × C 2 (Rn ) and (∂ α ϕ, ∂ α ψ) (|α| ≤ 2) are suﬃciently small in Y (ν) with ν is the number from Theorem 2.2. Then u ∈ C 2 ([0, ∞) × Rn ), hence it becomes a classical solution of (2.1)–(2.2). 2.3. Almost global existence In this subsection we consider (2.7) for the case of p = p0 (n). As we have seen in the Subsection 2.1, the lifespan of the solution of (2.7) is estimated from above in terms of the size of the initial data. Here we shall investigate the lower bound of the lifespan. The main result in this subsection reads as follows. Theorem 2.3. Let n = 2, 3 and let p = p0 (n). Suppose (ϕ, ψ) ∈ Y (ν) with ν > 1/p. Then there is a constant ε0 = ε0 (p, c, n) > 0 such that if (ϕ, ψ)Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution u ∈ C([0, T ∗ (ε)) × Rn ) of (2.7). Moreover, there is a constant C = C(p, c, n) > 0 such that T ∗ (ε) ≥ exp(Cε−p(p−1) ).

(2.64)

Remark 2.9. In view of (2.14), the above estimate is sharp with respect to ε. In order to obtain such an optimal order in ε, we adapt the argument used in [45]. The idea is to consider the diﬀerence between the solution of (2.7) and that of the

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H. Kubo and M. Ohta

homogeneous wave equation. This consideration simpliﬁes both arguments in [3] and [67]. To prove the theorem, we need the following extension of Proposition 2.3 to the case κ = 1, since pp∗ = pp∗ = 1 for p = p0 (n). Proposition 2.4. Let n = 2, 3 and let a, c, µ > 0. Suppose F ∈ C([0, T ) × Rn ) and µ > m + 1. Then there exists a positive constant C1 = C1 (a, c, n, µ) such that sup (t,x)∈[0,T )×Rn

t + |x|m ct − |x|κ0 |Lc [F ](t, x)|

≤ C1 log(2 + T )

sup (t,x)∈[0,T )×Rn

(2.65)

t + |x|µ at − |x||F (t, x)|,

where κ0 = µ − m − 1, provided µ < 2 when n = 2. Proof. Let b = max(a, c). First we shall prove that if 2bt ≤ r := |x|, then we have t + |x|m ct − |x|κ0 |Lc [F ](t, x)| ≤C

(2.66)

t + |x| at − |x||F (t, x)|. µ

sup (t,x)∈[0,T )×Rn

We shall use Proposition 2.2. Since c(t − s) − r ≤ 0 for 2bt ≤ r and s ≥ 0, we see ,c (r, t) = ∅. Moreover, if (λ, s) ∈ Dc (r, t), then that D λ ≥ r − c(t − s) ≥ r − 2b(t − s) ≥ 2bs ≥ 2as. This means that as − λ is equivalent to s + λ when (λ, s) ∈ Dc (r, t). In the following, we assume c = 1. When n = 3, we have from (2.38) |L1 [F ](t, x)| ≤ CI ×

sup (t,x)∈[0,T )×Rn

where we put 1 I= 2r

t + |x|µ at − |x||F (t, x)|,

(2.67)

λ + s−µ dλ ds.

D1 (r,t)

Changing the variables by (2.51), we have α t+r 1 I = α−µ dα dβ 4r |t−r| r −t t+r 1 ≤ α−µ+1 dα, 2r |t−r| since r − t > −α. Recalling κ0 = µ − 2 > 0, we have I ≤ Ct + r−1 r − t−κ0 . Hence (2.66) with n = 3 follows from (2.67). When n = 2, we have from (2.39) |L1 [F ](t, x)| ≤ CI

sup (t,x)∈[0,T )×Rn

t + |x|µ at − |x||F (t, x)|,

(2.68)

Coupled Systems of Semilinear Wave Equations where we put

I= D1 (r,t)

135

λ + s−µ dλ ds, (λ − s + t + r)(λ + s + r − t)

Changing the variables by (2.51), we have α 1 1 t+r α−µ √ √ dβ. I= dα 2 |t−r| α + r − t β+t+r r −t

(2.69)

We divide the argument into two cases. First suppose 2r ≥ t ≥ 0 and r ≥ 1. Since β +t+r ≥ 2r if β > r −t, we have β + t + r ≥ Ct+r in this case. Therefore we get t+r 1 α−µ+1 √ t + r 2 I ≤ C dα α+r−t |t−r| ≤ Cr − t−κ0 , by (2.45), since κ0 = µ − (3/2) > 0. Next suppose t ≥ 2r ≥ 0 or 0 ≤ r ≤ 1. It is easy to see that the β–integral is 1 estimated by Ct + r 2 for α < t + r. Therefore we get t+r 1 α−µ 2 √ dα I ≤ Ct + r α+r−t |t−r| ≤ Ct + r 2 r − t−(µ− 2 ) , 1

1

by (2.45). Since t + r is equivalent to r − t in this case, we have I ≤ Ct + r− 2 r − t−κ0 . 1

Thus we have shown (2.66) with n = 2. Next we shall prove that if 2bt ≥ r, then we have (2.70) t + |x|m ct − |x|κ0 |Lc [F ](t, x)| µ t + |x| at − |x||F (t, x)|. ≤ C log(1 + r − ct) sup (t,x)∈[0,T )×Rn

In this case, we cannot avoid the logarithmic tail in the β–integral. We can show that (2.70) holds as well, by proceeding as in the proof of Proposition 2.3 and using the lemma below instead of Lemma 2.5. In conclusion we obtain (2.65) from (2.66) and (2.70). We ﬁnish the proof, assuming the following lemma is valid. Lemma 2.10. For κ > 0 and > 0, there exists a constant C = C(κ, ) > 0 such that r+t C min{t, r} [log(1 + r − t)] ρ−1−κ [log(1 + ρ)] dρ ≤ , (2.71) t + rr − tκ |r −t| r+t 1 1 C[log(1 + r − t)] ρ− 2 −κ (ρ + r − t)− 2 [log(1 + ρ)] dρ ≤ (2.72) r − tκ |r −t|

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for (r, t) ∈ [0, ∞)2 . Besides, assuming κ < 1/2, we have t−r 1 1 C[log(1 + r − t)] ρ− 2 −κ (t − r − ρ)− 2 [log(1 + ρ)] dρ ≤ r − tκ 0

(2.73)

for (r, t) ∈ [0, ∞)2 with t > r. Proof. It is easy to see from (2.46) that (2.73) is valid. Since (2.72) can be proved similarly to (2.71), we shall only deal with (2.71). Putting f (ρ) =

[log(2 + ρ)] , (2 + ρ)κ/2

we ﬁnd that the left-hand side of (2.71) is estimated by some constant times t+r f (ρ) I := dρ. 1+κ/2 ρ |t−r| Note that f (ρ) is increasing on (0, ρ0 ) and decreasing on (ρ0 , ∞) with ρ0 = max{e2/κ − 2, 0}. When |t − r| ≥ ρ0 , by (2.44) we have t+r dρ C min{t, r}f (|t − r|) ≤ . I ≤ f (|t − r|) 1+κ/2 ρ t + rt − rκ/2 |t−r| This means that (2.71) holds. While, if |t − r| ≤ ρ0 , then we have 0 < f (0) ≤ f (|t − r|) ≤ f (ρ0 ) ≤ Cf (0), in particular, f (ρ0 ) ≤ Cf (|t − r|). Therefore we get t+r C min{t, r}f (|t − r|) dρ ≤ . I ≤ f (ρ0 ) 1+κ/2 ρ t + rt − rκ/2 |t−r|

Thus we have shown (2.71). This completes the proof.

End of the proof of Theorem 2.3: As it is mentioned in Remark 2.9, we consider the diﬀerence between the solutions of (2.7) and the homogeneous wave equation. Namely, we handle the following integral equation: w = Lc [|u0 + w|p ] in [0, T ) × Rn ,

(2.74)

where u0 (t, x) = Kc [ϕ, ψ](t, x). Suppose we obtain the solution w(t, x) of (2.74). Then, putting u = w + u0 , we get the solution of (2.7) and its lifespan is the same as that of w. Thus we have reduced the problem to the analysis for (2.74). For ε > 0 and T > 0 we set Z := {w ∈ C([0, T ) × Rn ) : w ≡ wZ(c,p∗ ) ≤ 2p C1 C0p εp },

(2.75)

vZ(c,κ) =

(2.76)

sup (t,x)∈[0,T )×Rn

t + |x| ct − |x| |v(t, x)|. m

κ

Deﬁne a map from Z to C([0, T ) × Rn ) by Ψ[w] := Lc [|u0 + w|p ].

(2.77)

Let us ﬁx ε0 as p2p+1 C1 (C C0 ε0 )p−1 ≤ 1. For ε with 0 < ε ≤ ε0 we take T such that 2

p2p

+1

C1p (C C0 ε0 )p(p−1) log(2 + T ) ≤ 1.

(2.78)

Coupled Systems of Semilinear Wave Equations

137

To apply the contraction mapping principle, we shall show that Ψ[w] ≤

2p C1 (C C0 ε0 )p for w ∈ Z, 1 w1 − w2 for w1 , w2 ∈ Z Ψ[w1 ] − Ψ[w2 ] ≤ 2 for 0 < ε ≤ ε0 , provided T satisﬁes (2.78). Since |s + t|p ≤ 2p−1 (|s|p + |t|p ) for p > 1 and s, t ∈ R, we have

(2.79) (2.80)

Ψ[w] ≤ 2p−1 (Lc [|u0 |p ] + Lc[|w|p ]). We see from (2.41) u0 Z(c,ν) ≤ C0 ε. Since ν > 1/p, (2.49) with a = c, µ = mp and κ = pν yields C0 ε)p . Lc [|u0 |p ]Z(c,p∗ ) ≤ C1 [u0 Z(c,ν) ]p ≤ C1 (C Using (2.65) as a = c, µ = mp and κ = pp∗ = 1, we get Lc[|w|p ]Z(c,p∗ ) ≤ C1 log(2 + T )[wZ(c,p∗ ) ]p ≤ C1 (C C0 ε)p for w ∈ Z, by virtue of (2.78). Therefore (2.79) is valid. Next we consider (2.80). It follows that Ψ[w1 ] − Ψ[w2 ] ≤

p2p−2 (2Lc[|u0 |p−1 |w1 − w2 |] + +Lc[(|w1 |p−1 + |w2 |p−1 )|w1 − w2 |]).

By (2.49) with a = c, µ = mp and κ = (p − 1)ν + p∗ we have p2p−1 Lc [|u0 |p−1 |w1 − w2 |]

≤ p2p−1 C1 [u0 Z(c,ν) ]p−1 w1 − w2 1 ≤ w1 − w2 4 by the choice of ε. (Recall ν > 1/p = p∗ for p = p0 (n).) While, by (2.65) with a = c, µ = mp and κ = 1 we get p2p−2 Lc [(|w1 |p−1 + |w2 |p−1 )|w1 − w2 |] ≤ p2p−2 C1 log(2 + T )[w1 p−1 + w2 p−1 ]w1 − w2 1 ≤ w1 − w2 4 for w1 , w2 ∈ Z, by (2.78). Hence we get (2.80). Moreover, (2.78) implies (2.64). This completes the proof of Theorem 2.3. 2.4. Self-similar solution The aim of this subsection is to study the existence and non-existence of the selfsimilar solution to (2.1) or the associated integral equation (2.7), by following Pecher [89]. We call u(t, x) a self-similar solution, if it satisﬁes (2.1) and Dλ,p u(t, x) ≡ u(t, x)

(2.81)

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for (t, x) ∈ [0, ∞) × Rn and λ > 0, where Dλ,p u is given in (2.4). Taking t = 0 and λ = 1/|x| in (2.81), we have u(0, x) = |x|− p−1 u(0, ω), 2

∂t u(0, x) = |x|− p−1 −1 ∂t u(0, ω) 2

(2.82)

with ω = x/|x|. This means that the initial data of a self-similar solution must be a homogeneous function. Since such an initial data does not belong to Y (p∗ ), we can not apply the existence result stated in Remark 2.7, despite of the fact that 2/(p − 1) = p∗ + m. For this reason, we introduce the following homogeneous version of Y (ν) deﬁned by Y˙ (ν) = {(ϕ, ψ) ∈ C 1 (Rn∗ ) × C(Rn∗ ) : (ϕ, ψ)Y˙ (ν) < ∞}, ν+ n−1 2

(ϕ, ψ)Y˙ (ν) = sup {|x| x∈Rn ∗

ν+ n+1 2

|ϕ(x)| + |x|

(2.83)

(|∇ϕ(x)| + |ψ(x)|)}.

Besides, since the singularity at the origin propagates through the light cone ct = |x|, we modify the norm · X(c,ν) as vX(c,κ) = sup{|t+|x||m |ct−|x||κ |v(t, x)| : (t, x) ∈ [0, ∞)×Rn , ct = |x|}. (2.84) ˙ Now we are in position to state the main results about self-similar solutions. Theorem 2.4. Assume that (ϕ, ψ) ∈ Y˙ (p∗ ) and n = 2, 3. (i) Let p0 (n) 0 and C0 = C0 (p, c, n) > 0 such that if (ϕ, ψ)Y (p∗ ) ≤ ε for 0 < ε ≤ ε0 , then there is a unique solution of (2.7) in X˙ := {v ∈ C(([0, ∞) × Rn ) \ {ct = |x|}) : vX(c,p ≤ 2C C0 ε0 }. ˙ ∗) (ii) Let 1 < p ≤ p0 (n). Then the conclusion (i) does NOT hold. Corollary 2.1. Let n = 2, 3 and let p0 (n) < p < (n + 3)/(n − 1). Let the initial data take the form ϕ(x) = f (ω)|x|− p−1 , 2

ψ(x) = g(ω)|x|− p−1 −1 2

(2.85)

with ω = x/|x|, f ∈ C 1 (S 2 ) and g ∈ C(S 2 ). If (ϕ, ψ) ∈ Y˙ (p∗ ) is suﬃciently small ˙ in Y˙ (p∗ ), then there exists a unique self-similar solution of (2.7) in X. First we show the part (ii ) of Theorem 2.4. Suppose that the conclusion (i) is still valid for 1 0. Then it follows from (2.7) that u(x, t) ≥ Jc [ψ](t, x).

for x ∈ Rn∗ ,

Coupled Systems of Semilinear Wave Equations

139

For all δ > 0 we have ψ(x) ≥ ε|x|−m (|x| + δ)−p∗ −1 (recall (2.63)). Hence Proposition 2.1 implies r+ct ε (λ + δ)−p∗ −1 dλ (2.86) u(t, x) ≥ 2crm |r−ct| ε (|r − ct| + δ)−p∗ − (r + ct + δ)−p∗ , ≥ 2cp∗ rm where r = |x|, since p∗ > 0 for p < (n + 3)/(n − 1). We ﬁx (t, x) ∈ (0, ∞) × Rn such that ct/2 < |x| < ct, and set E = {(λ, s) ∈ [0, ∞)2 : (ct − r)/2c < s < 2(ct − r)/3c,

c(t − s) − r < λ < cs}.

Notice that if (λ, s) ∈ E, then we have 2λ > cs, hence λ+cs ≥ 3(cs−λ). Therefore, for (s, y) ∈ (0, ∞) × Rn such that (|y|, s) ∈ E we get u(s, y) ≥ Cε|y|−m (||y| − cs| + δ)−p∗

(2.87)

from (2.86). Now it follows from Proposition 2.1 that 1 u(x, t) ≥ λm |u(y, s)|p dλ ds 2crm D (r,t) c εp λ−m(p−1) (|λ − cs| + δ)−pp∗ dλ ds ≥ 2crm E 2(ct−r)/3c cs Cεp ≥ ds (cs − λ + δ)−pp∗ dλ. rm |ct − r|m(p−1) (ct−r)/2c c(t−s)−r Since pp∗ ≥ 1 for 1 < p ≤ p0 (n), the λ-integral goes to ∞ as δ → 0. This is a contradiction. Next we show the part (i ) of Theorem 2.4 by modifying the proof of Theorem 2.2. First of all, we introduce the following analogues to Lemmas 2.5 and 2.6. We omit the proof of them, since it is analogous to that of Lemmas 2.5 and 2.6. Lemma 2.11. For κ > 0, there exists a constant C = C(κ) > 0 such that r+t C min{t, r} ρ−1−κ dρ ≤ , (t + r)|r − t|κ |r −t| r+t 1 1 ρ− 2 −κ (ρ + r − t)− 2 dρ ≤ C|r − t|−κ

(2.88) (2.89)

|r −t|

for (r, t) ∈ (0, ∞)2 . Besides, assuming κ < 1/2, we have [t−r]+ 1 1 ρ− 2 −κ (t − r − ρ)− 2 dρ ≤ C|r − t|−κ 0

for (r, t) ∈ (0, ∞)2 .

(2.90)

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Lemma 2.12. Let κ, a, d > 0. If κ < 1, then there exists a positive constant C = C(κ, a) such that d/a 1 1 |σ|−κ (d + σ)− 2 dσ ≤ C|d|−κ+ 2 . (2.91) −d

Next we introduce the following analogue to Proposition 2.3. It can be shown in the same line as Proposition 2.3, if we use Lemmas 2.11 and 2.12 instead of Lemmas 2.5 and 2.6 respectively. Proposition 2.5. Let n = 2, 3 and let a, c, µ and κ > 0. Suppose F ∈ C(([0, T ) × Rn ) \ {ct = |x|}), κ < 1 and µ + κ > m + 2. Then there exists a constant C = C(a, c, µ, κ) > 0 such that sup |t + |x||m |ct − |x||κ0 |Lc [F ](t, x)|

(2.92)

ct= |x|

≤ C sup |t + |x||µ |at − |x||κ |F (t, x)|, ct= |x|

where we put κ0 = µ + κ − m − 2, provided µ + κ < 3 when n = 2. End of the proof of the part (i ) of Theorem 2.4: Proceeding as in the proof of (2.41) and using Lemma 2.11, we can show that for (ϕ, ψ) ∈ Y˙ (ν) with ν > 0 there is a positive constant C0 = C0 (ν, c, n) such that Kc [ϕ, ψ]X(c,ν) ≤ C0 (ϕ, ψ)Y˙ (ν) , ˙

(2.93)

provided 0 < ν < 1/2 when n = 2. In order to show Lc [|u|p ]X(c,p ≤ C[uX(c,p ]p , ˙ ˙ ∗) ∗) we apply Proposition 2.5 by taking a = c, µ = mp and κ = pp∗ . Since µ+κ−m−2 = p∗ , κ < 1 for p > p0 (n), and µ + κ < 3 for p > 3 and n = 2, we get the desired estimate. We omit the further detail, since it is similar to Step 3 in Subsection 2.2. 2.5. Asymptotic behavior We have studied the existence of a classical global solution of (2.1) (see Remark 2.8). The aim of this subsection is to examine asymptotic behaviors as t → +∞ of such a solution u ∈ X(c, p∗ ) of (2.1). To be more precise, we shall show that u(t, x) is asymptotically free. Namely, there is a unique classical solution v(t, x) of the homogeneous wave equation (∂ ∂t2 − c2 ∆)v = 0,

(t, x) ∈ [0, ∞) × Rn ,

(2.94)

such that u(t) − v(t)E(c) → 0 where

u(t)E(c) =

&1/2 (c |∇x u(t, x)| + |∂ ∂t u(t, x)| )dx . 2

Rn

as t → +∞,

2

2

(2.95)

(2.96)

Coupled Systems of Semilinear Wave Equations

141

For the moment, we roughly explain the idea to get this type of result. Let us rewrite the integral equation (2.7) as ∞ Jc [F (u)(s, ·)](t − s, x) ds u = Kc [ϕ, ψ] + 0 ∞ Jc [F (u)(s, ·)](t − s, x) ds in [0, ∞) × Rn , − t

with F (u)(t, x) = |u(t, x)|p . Having this in mind, if we set v(t, x) = u(t, x) − Rc [F (u)](t, x) in [0, ∞) × Rn , ∞ Rc [F ](t, x) = − Jc [F (s, ·)](t − s, x) ds

(2.97) (2.98)

t

then we have ∂t2 − c2 ∆)u − F (u) = 0. (∂ ∂t2 − c2 ∆)v = (∂ Therefore the main step is reduced to show Rc [F (u)](t)E(c) → 0 as

t → +∞.

(2.99)

To realize this procedure, we ﬁrst prepare the following basic estimate for Rc [F ](t, x). Proposition 2.6. Let n = 2, 3 and let a, c, µ > 0. Suppose F ∈ C([0, ∞) × Rn ) and κ > 1. Then there exists a constant C1 = C1 (a, c, µ, κ) > 0 such that sup (t,x)∈[0,∞)×Rn

≤ C1

t + |x|µ−1 |Rc [F ](t, x)|

sup (t,x)∈[0,∞)×Rn

(2.100)

t + |x|µ at − |x|κ |F (t, x)|,

provided µ > m + 1 when a = c, and that µ > 1 when a = c. Jc [g](t, x) by the deﬁnitions (2.10) and (2.11), we Proof. Since Jc [g](−t, x) = −J have ∞ Jc [F (s, ·)](s − t, x) ds.

Rc [F ](t, x) = t

Besides, we may assume c = 1. When n = 3, we have from (2.36) |R1 [F ](t, x)| ≤ I × where we put 1 I= 2r

sup (t,x)∈[0,∞)×Rn

t + |x|µ at − |x|κ |F (t, x)|,

(2.101)

λ + s−µ+1 λ − as−κ dλ ds,

E1 (r,t)

Ec (r, t) = {(λ, s) ∈ [0, ∞)2 : s > t, |c(s − t) − r| ≤ λ ≤ c(s − t) + r}. Changing the variables by (2.51), we have ∞ r−t 1 + a −κ 1 1−a α+ β dα. I= dβ α−µ+1 4r −(t+r) 2 2 t+r

142

H. Kubo and M. Ohta When a = 1, we have I

≤

t + r−µ+1 4r

r−t

∞

dβ −(t+r)

t+r

1 + a −κ 1−a α+ β dα 2 2

≤ Ct + r−µ+1 , since µ > 1 and κ > 1. While, if a = 1, then we have r−t ∞ 1 −κ I ≤ β dβ α−µ+1 dα 4r −(t+r) t+r ≤

Ct + r−µ+1 ,

since κ > 1 and µ > 2. At the last step, we have used Lemma 2.13 below. Hence (2.100) for n = 3 follows from (2.101), assuming the lemma. Lemma 2.13. For κ > 1 there exists a constant C = C(κ) > 0 such that 1 t+r dβ C J := , t ∈ R, r > 0. ≤ κ r t−r β |t| + r

(2.102)

Proof. Without loss of generality, we may assume t ≥ 0. We easily have three bounds for J as follows: J ≤ 2 ; J ≤ 2t − r−κ

for t > r ; J ≤ Cr−1 .

(2.103)

When t + r ≤ 1, (2.102) follows from the ﬁrst estimate in (2.103). When t + r ≥ 1 and t ≥ 2r, (2.102) follows from the second one in (2.103). Finally, if t + r ≥ 1 and t ≤ 2r, then (2.102) follows from the last one in (2.103). This completes the proof. In order to handle the case n = 2, we prepare the following estimates. Lemma 2.14. Let κ > 0, 0 < γ < 1 and d ∈ R. If κ + γ > 1, then there exists a constant C = C(κ, γ) > 0 such that ∞ (1 + σ)−κ (d + σ)−γ dσ ≤ C(1 + |d|)−κ−γ+1 , (2.104) |d |

and

d

(1 + |σ|)−κ (d − σ)−γ dσ ≤ C(1 + |d|)−γ (1 + |d|)[1−κ]+ ,

(2.105)

or equivalently, ∞ (1 + |σ|)−κ (d + σ)−γ dσ ≤ C(1 + |d|)−γ (1 + |d|)[1−κ]+ .

(2.106)

−∞

−d

Proof. First we show (2.104). Since (d + σ)−γ ≤ (σ − |d|)−γ , by integration by parts, we have ∞ ∞ κ −κ −γ (1 + σ) (d + σ) dσ ≤ (1 + σ)−κ−γ dσ, 1 − γ |d | |d |

Coupled Systems of Semilinear Wave Equations

143

which yields (2.104), because κ + γ > 1. Next we show (2.106). When d ≤ 0, (2.106) follows from (2.104). In what follows, we assume d > 0 and set d ∞ −κ −γ P1 = (1 + |σ|) (d + σ) dσ, P2 = (1 + |σ|)−κ (d + σ)−γ dσ. −d

d

Then (2.104) gives P2 ≤ C(1 + |d|)−κ−γ+1 . While, seeing the proof of (2.58) with a = 1, we ﬁnd that P1 is estimated by C(1+|d|)−γ (1+|d|)[1−κ]+ . Thus we have shown (2.106). The proof is complete. Now we turn to the proof of (2.100) for n = 2. It follows from (2.37) that |R1 [F ](t, x)| ≤ (I1 + I2 ) where we put I1

sup (t,x)∈[0,∞)×Rn

= =

(2.107)

√ 1 λλ + s−µ+ 2 λ − as−κ

E1 (r,t)

I2

t + |x|µ at − |x|κ |F (t, x)|,

f1 (r,t) E

dλ ds, (λ − s + t + r)(λ + s + r − t) √ 1 λλ + s−µ+ 2 λ − as−κ dλ ds (s − λ − t − r)(λ + s + r − t)

with ,c (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ λ ≤ c(s − t) − r}. E Changing the variables by (2.51), we have 1 ∞ −κ α−µ+ 2 1−a α + 1+a 1 r−t 2 β √ 2 dβ dα, I1 ≤ 2 −(t+r) β+r+t t+r ∞ 1 −κ α−µ+ 2 1−a α + 1+a 1 −(t+r) 2 β 2 I2 ≤ dβ dα. 2 −∞ −(β + t + r) −β

(2.108) (2.109)

First we evaluate I1 . Suppose a = 1. Then (2.108) implies ∞ β−κ 1 1 r−t √ dβ α−µ+ 2 dα I1 ≤ 2 −(t+r) β + r + t t+r ∞ 3 β−κ √ dβ, ≤ Ct + r−µ+ 2 β+r+t −(t+r) since µ > 3/2. Thus the application of (2.106) with d = r + t gives I1 ≤ Ct + r−µ+1 ,

(2.110)

since κ > 1. On the one hand, when a = 1, we have ∞ 1 1 + a −κ 1 t + r−µ+ 2 r−t 1−a √ α+ β dα, I1 ≤ dβ 2 2 2 β + r + t −(t+r) t+r

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since µ > 1/2. By κ > 1 we also have (2.110) in this case. Next we evaluate I2 . When a = 1, (2.109) yields ∞ 1 β−κ 1 −(t+r) dβ I2 ≤ α−µ+ 2 dα 2 −∞ −(β + r + t) |β | −(t+r) −κ−µ+ 32 β ≤ C dβ −(β + r + t) −∞ ∞ 3 β−κ−µ+ 2 √ = C dβ, β−r−t t+r since µ > 3/2. By (2.104) with d = −r − t we have I2 ≤ Ct + r−κ−µ+2 , since κ + µ − (3/2) > 1/2. By κ > 1, we arrive at I2 ≤ Ct + r−µ+1 .

(2.111)

Finally, we consider the case where a = 1. We have ∞ 1 β−µ+ 2 1 + a −κ 1 −(t+r) 1−a dβ α+ β dα I2 ≤ 2 −∞ 2 2 −(β + r + t) |β | ∞ 1 β−µ+ 2 √ ≤ C dβ, β−r−t t+r since κ > 1. By (2.104) with d = −r − t, we have (2.111), since µ > 1. Thus we have proved (2.100) via (2.107), (2.110) and (2.111). This completes the proof. Remark 2.15. If u ∈ C 2 ([0, ∞) × Rn ) satisﬁes ∂xα uX(c,ν)t + |x|−m ct − |x|−ν |∂ ∂xα u(t, x)| ≤ ∂ for |α| ≤ 2, then Rc [F (u)] ∈ C 2 ([0, ∞) × Rn ), where F (u) = |u|p . In fact, it follows from (2.100) that ∗

sup (t,x)∈[0,∞)×Rn

t + |x|p

+m

|Rc [F (u)](t, x)| ≤ C1 [uX(c,ν)]p .

(2.112)

Here we took a = c, µ = mp and κ = pν with ν the number in Theorem 2.2. (Recall that µ > m + 1 is equivalent to p∗ > 0 by (2.25) and that p > p0 (n) implies pp∗ > 1.) Since ∂xα (F (u))](t, x) ∂xα Rc [F (u)](t, x) = Rc [∂ for |α| ≤ 2, and ∂xα (F (u)(t, x)) is a continuous function in (t, x), we see from ∗ (2.112) that ∂xα Rc [F (u)](t, x) is estimated by Ct + |x|−p −m [|||u|||X(c,ν) ]p , where we put |||u|||X(c,ν) = max ∂ ∂xα uX(c,ν) . |α|≤2

Coupled Systems of Semilinear Wave Equations

145

As for the derivative in t, we have for n = 3 ∞ 1 ∂t Rc [F ](t, x) = − ds F (u)(s, x + c(s − t)ω) dS Sω 4π t |ω |=1 ∞ 3 c − (t − s) ds ωj ∂xj (F (u)(s, x + c(s − t)ω)) dS Sω 4π t |ω |=1 j=1 and ∂t2 Rc [F ](t, x) = F (t, x) ∞ 3 c ds ωj ∂xj (F (u)(s, x + c(s − t)ω)) dS Sω − 2π t |ω |=1 j=1 −

c2 4π

∞

(t − s) ds

t

3

|ω |=1 j,k=1

ωj ωk ∂xj ∂xk (F (u)(s, x + c(s − t)ω)) dS Sω .

Therefore each term under the integral signs is continuous in (t, x) and these integrals converge, by the assumption on u(t, x). For example, ∂t Rc [F ](t, x) is estimated by some constant times t+1 ds |F (u)(s, x + c(s − t)ω)| dS Sω t

|ω |=1

∞

(s − t)ds

+ t+1 ∞

(s − t) ds

+ t

≤

|ω |=1

|F (u)(s, x + c(s − t)ω)| dS Sω

|ω |=1

|∂ ∂x (F (u)(s, x + c(s − t)ω))| dS Sω ∗

C[uX(c,ν)]p + C[|||u|||X(c,ν) ]p t + |x|−p

−m

,

in view of (2.112). Similarly we can treat the case n = 2. Thus we ﬁnd Rc [F (u)] ∈ C 2 ([0, ∞) × Rn ). Moreover, a straight forward computation shows that Rc [F ] satisﬁes (2.1). In order to prove (2.99), we prepare (2.120) below. To this end, we will set up the argument in more general situation. By u(t) we denote a function of t ∈ R with values in D (Rn ), the space of distributions on Rn . Besides, we denote by S(Rn ) the space of rapidly decreasing functions on Rn , and by S (Rn ) the space of tempered distributions on Rn . We consider the initial value problem u (t) − ∆u(t) = F (t) for t ∈ R

(2.113)

u(t) = u (t) = 0 at t = s,

(2.114)

with zero initial data

where s is an arbitrary real number and u (t) stands for the second derivative of u(t), and so on. For a function f ∈ L2 (Rn ) we denote by fˆ and fˆ∗ , respectively

146

H. Kubo and M. Ohta

the Fourier transform of f and the inverse Fourier transform of f such that fˆL2 (Rn ) = fˆ∗ L2 (Rn ) = f L2 (Rn ) . Then the following facts are in essence well known. Proposition 2.7. Assume that F (t) ∈ C(R; L2 (Rn )). (i) Let s ∈ R be ﬁxed. For t ∈ R we deﬁne a functional u(t; s), ϕ on S(Rn ) by s sin(c(t − τ )|ξ|) ˆ F (ξ, τ )ϕˆ∗ (ξ) dξ for ϕ ∈ S(Rn ). dτ u(t; s), ϕ = − c|ξ| t Rn (2.115) Then u(t; s), with s regarded as a parameter, is a solution of the initial value problem (2.113)–(2.114) such that u(t; s) ∈ C 2 (R; H −1 (Rn )). Moreover, the solution is unique in C 2 (R; D (Rn )). (ii) Let n ≥ 2. Assume that there are positive constants θ and C such that F (t)L2 (Rn ) ≤ C(1 + |t|)−1−θ

for t ∈ R.

For t ∈ R we deﬁne a functional u(t), ϕ on S(R ) by ∞ sin(c(t − τ )|ξ|) ˆ F (ξ, τ )ϕˆ∗ (ξ) dξ dτ u(t), ϕ = − c|ξ| n t R

(2.116)

n

for ϕ ∈ S(Rn ). (2.117)

Then u(t) ∈ C 2 (R; S (Rn )),

(2.118)

and we have for each t ∈ R u (t) ∈ L2 (Rn ), and

∇u(t) ∈ L2 (Rn )

u(t)E(c) ≤ (n + 1)

∞

F (τ )L2 (Rn ) dτ.

(2.119)

(2.120)

t

Proof. The ﬁrst part (i) is well known. (For the uniqueness see for instance [61], Lemma 5.1.) First we shall prove (2.118). Let t ∈ R be ﬁxed. Then we claim that u(t) ∈ S (Rn ). To see this we take a positive number δ such that δ < θ and δ < 1. Then sin(c(t − τ )|ξ|) C(τ − t)δ ≤ |ξ| |ξ|1−δ and |ξ|δ−1 ϕˆ∗ (ξ) ∈ L2 (Rn ) for n ≥ 2, hence the integrand in the right-hand side of (2.117) is integrable with respect to (ξ, τ ) over Rn × (t, ∞), according to (2.116). Therefore we have for ϕ ∈ S(Rn ) u(t), ϕ = lim u(t; k), ϕ, k→∞

where u(t; s) is given by (2.115). Since u(t; k) ∈ S (Rn ) for all k = 1, 2, . . ., we ﬁnd by the Banach-Steinhaus’ theorem that u(t) ∈ S (Rn ). Now the desired property (2.118) follows easily from the above procedure.

Coupled Systems of Semilinear Wave Equations

147

1 ∂ Finally we shall prove (2.119) and (2.120). Let Dk = √ (k = 1, . . . , n). −1 ∂xk Then we get by virtue of (2.116) ∞ c|Dk u(t), ϕ| ≤ ϕL2 (Rn ) F (τ )L2 (Rn ) dτ. t

Hence Dk u(t) ∈ L (R ) and 2

n

cDk u(t)L2 (Rn ) ≤

∞

F (τ )L2 (Rn ) dτ.

t

Analogously we obtain (2.119) and (2.120). This completes the proof.

Now we are in position to state the main result of this subsection. Theorem 2.5. Let n = 2, 3 and let p > p0 (n). For |α| ≤ 2 suppose (∂ ∂xα ϕ, ∂xα ψ) ∈ 2 Y (ν) with ν the number chosen in Theorem 2.2. Let u ∈ C ([0, ∞) × Rn ) be a classical solution to (2.1)–(2.2) such that ∂xα u ∈ X(c, ν) for |α| ≤ 2. Then there is a unique classical solution v(t, x) of (2.94) satisfying ∗

u(t) − v(t)E(c) ≤ C[uX(c,ν)]p t−p

for

t ≥ 0.

(2.121)

Remark 2.16. We recall the choice of ν for convenience. When n = 3, we take ν = p∗ . While, when n = 2, we take ν so that ν = p∗ if p < 4, and 1/4 < ν < 1/2 if p ≥ 4. Proof. As we have seen in Remark 2.15, Rc [F (u)](t, x) is a classical solution to (2.1) if u(t, x) is as in Theorem 2.5, where F (u) = |u|p . Therefore, once we establish ∗

Rc [F (u)](t)E(c) ≤ C[uX(c,ν) ]p t−p

for t ≥ 0,

(2.122)

we ﬁnish the proof by setting v = u − Rc [F (u)]. First we show that there is a constant C = C(p, ν, n) such that ∗

F (u)(t)L2 (Rn ) ≤ C[uX(c,ν)]p t−p

−1

for

t ≥ 0.

(2.123)

The left-hand side of (2.123) raised the power 2 is estimated by [uX(c,ν)]2p times I(t) := t + |x|−2pm ct − |x|−2pν dx. Rn

Switching to the polar co-ordinates and recalling (2.25), we get ∞ I(t) = cn t + r−2pm ct − r−2pν r2m dr 0 ∞ ∗ t + r−2(p +1) ct − r−2pν dr. ≤ cn 0

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Since pν > 1 according to the choice of ν, we obtain (2.123). Analogously we see that F (u)(t) ∈ C(R; L2 (Rn )), hence (2.120) in Proposition 2.7 yields ∞ Rc [F (u)](t)E(c) ≤ (n + 1) F (u)(τ )L2 (Rn ) dτ t ∞ ∗ p τ −p −1 dτ, ≤ C[uX(c,ν) ] t

by (2.123). Since p∗ > 0, we arrive at (2.122). This completes the proof.

Thanks to the fact that if F (x, t) ∈ C([0, ∞) × Rn ), then ∞ ∞ dt Rc [F ](t, x)(∂ ∂t2 − ∆)φ(t, x)dx = dt F (t, x)φ(t, x)dx 0

Rn ∞ C0 ((0, ∞)× Rn ),

0

Rn

for all φ ∈ the result obtained in Theorem 2.5 can be formulated in a slightly diﬀerent way as follows: Theorem 2.6. Let n = 2, 3 and let p > p0 (n). Suppose (ϕ, ψ) ∈ Y (ν) with ν the number chosen in Theorem 2.2. Let u ∈ X(c, ν) be a solution to (2.7) obtained in Theorem 2.2. Then there is a unique weak solution v ∈ X(c, ν) to (2.94) in the distributional sense satisfying (2.121). In the rest of this subsection, we discuss the so-called nonlinear scattering. The map W+ from H˙ 1 × L2 to itself deﬁned as follows is called wave operator: Let u and v+ be solutions to (2.1)–(2.2) and (2.94) respectively such that u(t) − v+ (t)E(c) → 0

as t → +∞.

(2.124)

Then we put W+ [(v+ (0), (∂ ∂t v+ )(0))] := (u(0), ∂t u(0)). Similarly, when u and v− are solutions to (2.1)–(2.2) and (2.94) in (−∞, 0] × Rn respectively such that u(t) − v− (t)E(c) → 0 as t → −∞,

(2.125)

we deﬁne W− as ∂t v− )(0))] := (u(0), ∂t u(0)). W− [(v− (0), (∂ Moreover, if W+ is bounded, invertible and the range of W+ contains the range of W− , then one can deﬁne a scattering operator as S := (W W+ )−1 ◦ W− , or (v− (0), ∂t v− (0)) −→ (v+ (0), (∂ ∂t v+ )(0)).

(2.126)

Unfortunately, we do not ﬁnd from Theorem 2.5 or its proof if it is possible to deﬁne W± even on some dense set of a neighborhood of the origin in H˙ 1 × L2 . Nevertheless, we can directly construct a densely deﬁned nonlinear scattering operator S in the following manner. Let us consider u = Kc [ϕ− , ψ− ] + Tc [F (u)] in

R × Rn ,

(2.127)

Coupled Systems of Semilinear Wave Equations where Kc [ϕ, ψ] is deﬁned by (2.8) and we put t Tc [F ](t, x) = Jc [F (s, ·)](t − s, x) ds.

149

(2.128)

−∞

Roughly speaking, it suﬃces to carry out the following two steps: 1st step. For given v− (t, x) := Kc [ϕ− , ψ− ](t, x), ﬁnd a unique solution u(t, x) of (2.127) satisfying (2.125). 2nd step. Find a unique solution v+ (t, x) of the homogeneous wave equation (2.94) satisfying (2.124). Then the scattering operator S related to (2.1) is deﬁned by (2.126). To state the result obtained in this framework, we introduce vV (c,κ) =

sup (t,x)∈R×Rn

|t| + |x|m c|t| − |x|κ |v(t, x)|.

(2.129)

Then we have the following statements. ∂xα ϕ− , ∂xα ψ− ) ∈ Theorem 2.7. Let n = 2, 3 and let p > p0 (n). For |α| ≤ 2 suppose (∂ Y (ν) with ν the number chosen in Theorem 2.2. (i) Then there are constants ε0 = ε0 (p, ν, c, n) > 0 and C0 = C0 (ν, c, n) > 0 such that if ∂xα ϕ− , ∂xα ψ− )Y (ν) ≤ ε for 0 < ε ≤ ε0 , max (∂ |α|≤2

then there is a unique classical solution u of (2.1) in {v ∈ C(R × Rn ) : vV (c,ν) ≤ 2C C0 ε}. Moreover, we have ∂xα u ∈ V (c, ν) (|α| = 1, 2), and ∗

u(t) − v− (t)E(c) ≤ C[uV (c,ν) ]p t−p

for t ≤ 0,

(2.130)

where v− := Kc [ϕ− , ψ− ](t, x). (ii) Let u be as in the part (i) of the theorem. Then there exists a unique classical solution v+ (t, x) to (2.94) such that ∂xα v+ ∈ V (c, ν) (|α| ≤ 2), and ∗

u(t) − v+ (t)E(c) ≤ C[uV (c,ν) ]p t−p

for t ≥ 0.

(2.131)

Proof. First we consider the part (i). Since the procedure is similar to that of Theorem 2.2, we concentrate on establishing the following basic estimate for Tc [F ](t, x). Proposition 2.8. Let n = 2, 3 and let a, c, µ, κ > 0. Suppose F ∈ C(R × Rn ), µ > m + 1 and κ > 1. Then there exists a constant C1 = C1 (a, c, µ, κ) > 0 such that sup (t,x)∈R×Rn

≤ C1

|t| + |x|m ct − |x|κ0 |T Tc [F ](t, x)|

sup (t,x)∈R×Rn

|t| + |x|µ a|t| − |x|κ |F (t, x)|,

where we put κ0 = µ − m − 1, provided µ < 2 when n = 2.

(2.132)

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H. Kubo and M. Ohta

Proof. First suppose t ≤ 0. Since Jc [g](−t, x) = −J Jc [g](t, x) by the deﬁnitions (2.10) and (2.11), it follows from (2.128) and (2.98) that Tc [F ](t, x) = Rc [F (−s, ·)](|t|, x). Therefore (2.100) implies |t| + |x|m ct − |x|κ0 |T Tc [F ](t, x)| m+κ0 |Rc [F (−s, ·)](|t|, x)| ≤ C|t| + |x| ≤C

sup (t,x)∈[0,∞)×Rn

t + |x|µ at − |x|κ |F (−t, x)|,

which is bounded by the right-hand side of (2.132). Next suppose t ≥ 0. Then we have from (2.128) and (2.9) Tc [F ](t, x) = Lc [F ](t, x) + L− c [F ](t, x), where we put L− c [F ](t, x)

0

= −∞

Jc [F (s, ·)](t − s, x) ds.

(2.133)

Since (2.49) implies t + |x|m ct − |x|κ0 |Lc [F ](t, x)| t + |x|µ at − |x|κ |F (t, x)|, ≤C sup (t,x)∈[0,∞)×Rn

it is enough to show t + |x|m ct − |x|κ0 |L− c [F ](t, x)| ≤C sup |t| + |x|µ a|t| − |x|κ |F (t, x)|.

(2.134)

(t,x)∈R×Rn

In what follows, we assume c = 1. When n = 3, we have from (2.133) and (2.36) |L− 1 [F ](t, x)| ≤ I ×

sup (t,x)∈R×Rn

|t| + |x|µ a|t| − |x|κ |F (t, x)|,

where we put I=

1 2r

0

−∞

t−s+r

|t−s−r|

λ − s−µ+1 λ + as−κ dλ ds.

Changing the variables by (2.51), we have ∞ t+r 1 1 − a −κ 1+a I= α+ β dβ, dα β−µ+1 4r t−r 2 2 α∨|r−t| where a ∨ b = max{a, b}.

(2.135)

Coupled Systems of Semilinear Wave Equations

151

When a = 1, we have 1 4r

≤

I

t+r

t−r

α−κ dα

∞

|r −t|

β−µ+1 dβ

≤ Ct + r−1 r − t−µ+2 , since κ > 1 and µ > 2. Here we have used (2.102). Thus we get I ≤ Ct + r−1 r − t−κ0 .

(2.136)

When a = 1, we consider two cases separately. First suppose t ≥ 2r > 0 or 0 < r ≤ 1. It follows that ∞ 1 − a −κ 1+a r − t−µ+1 t+r α+ β dβ. dα I ≤ 4r 2 2 t−r −∞ Cr − t−µ+1 ,

≤

since µ > 1 and κ > 1. This estimate yields (2.136) in this case. Next suppose 2r ≥ t > 0 and r ≥ 1. Since κ > 1, we have C ∞ I≤ β−µ+1 dβ, r |r−t| which implies (2.136) in this case. Thus we see that (2.136) is also valid for the case a = 1. Therefore (2.134) for n = 3 follows from (2.135). Next we prove (2.134) for n = 2 by dividing the argument into two cases. Case 1. a = 1 or t ≥ 2r > 0 or 0 < r ≤ 1. It follows from (2.133) and (2.37) that |L− 1 [F ](t, x)| ≤ (I1 + I2 )

sup (t,x)∈(−∞,0]×Rn

|t| + |x|µ a|t| − |x|κ |F (t, x)|,

(2.137)

where we put √ 1 λ − sλ − s−µ+ 2 λ + as−κ dλ ds (λ − s + t + r)(λ + s + r − t) −∞ |t−s−r| √ 1 ∞ 1−a −κ ββ−µ+ 2 1+a 1 t+r 2 α + 2 β √ √ dα dβ, 2 t−r β+t+r α+r−t α∨|r−t|

I1

= =

0

t−s+r

(2.138)

and I2

(t−r)∧0

t−s−r

= −∞

=

1 2

0

t−r

∞

dα −∞

|α|

√ 1 λ − sλ − s−µ+ 2 λ + as−κ dλ ds (λ − s + t + r)(t − r − λ − s)

√ 1 1−a −κ ββ−µ+ 2 1+a 2 α + 2 β √ √ dβ, β+t+r t−r−α

(2.139)

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H. Kubo and M. Ohta

where we have changed the variables by (2.51). Moreover, we divide I2 into two integrals J1 and J2 deﬁned as follows: t−r β √ 1 1−a −κ ββ−µ+ 2 1+a 2 α + 2 β √ √ 2J J1 = H(t − r) dα (2.140) dβ β+t+r t−r−α 0 −β 1 ∞ t−r √ 1−a −κ ββ−µ+ 2 1+a 2 α + 2 β √ √ 2J J2 = dβ dα, (2.141) β+t+r t−r−α |r −t| −β where H(t) is the Heaviside function, i.e., H(t) = 1 for t > 0, H(t) = 0 for t ≤ 0. We shall show I1 + J1 + J2 ≤ Ct + r− 2 r − t−µ+ 2 , 1

3

provided κ > 1 and 3/2 < µ < 2. Since it holds that for β, r, t > 0 β Cβ β + r + t ≤ t + r , we get for t > r 1 2

t + r J1

t−r

≤ C

−µ+1

β 0

≤ C

t−r

−∞

β

dβ −β

(2.142)

(2.143)

1−a −κ 1+a 2 α + 2 β √ dα t−r−α

(2.144)

β−µ+1 (t − r − β)− 2 dβ 1

≤ Cr − t− 2 r − t[2−µ]+ , 1

by (2.105). This estimate gives (2.142) for J1 . When a = 1, we use (2.143) to get t+r ∞ −κ β−µ+1 1+a α + 1−a 1 2 β √ 2 t + r 2 I1 ≤ C dβ, dα α+r−t t−r |r −t| ∞ t−r −κ β−µ+1 1+a α + 1−a 1 2 β 2 √ 2 dβ t + r J2 ≤ C dα. t−r−α |r −t| −β It follows that ∞ ∞ 1 1 − a −κ 1+a − 12 α+ β (α + r − t) dα ≤ C σ−κ (σ − β ∗ )− 2 dσ, 2 2 ∗ t−r β 1−a 1+a (t − r) + β. Since κ > 1, we see from (2.106) that the last 2 2 ∗ − 12 integral is estimated by Cβ . Hence, ∞ 1 1 1+a 2 t + r I1 ≤ C (r − t)− 2 dβ. β−µ+1 β − 1 − a |r −t| where β ∗ =

1

Analogously, we ﬁnd that t + r 2 J2 is estimated by the same integral as in the above. Since µ > 3/2, we obtain the desired estimate (2.142) by the following lemma.

Coupled Systems of Semilinear Wave Equations

153

Lemma 2.17. Let κ > 1/2 and b, d ∈ R. Then there exists a constant C = C(κ) > 0 such that ∞ 1 1 (1 + σ)−κ (1 + |σ − d|)− 2 dσ ≤ C(1 + |b|)−κ (1 + |b| + |d|) 2 . (2.145) |b|

Proof. Let H(t) be the Heaviside function. If we set ∞ σ 1 P1 = H(|σ − d| − )(1 + σ)−κ (1 + |σ − d|)− 2 dσ, 2 |b| ∞ 1 σ P2 = (1 − H(|σ − d| − ))(1 + σ)−κ (1 + |σ − d|)− 2 dσ, 2 |b| then we see that the left-hand side of (2.145) is estimated by P1 + P2 . It is easy to see that ∞ 1 1 P1 ≤ C (1 + σ)−κ− 2 dσ ≤ C(1 + |b|)−κ+ 2 .

(2.146)

|b|

On the one hand, since |σ − d| ≤ σ/2 is equivalent to 2d/3 ≤ σ ≤ 2d, we see that I2 = 0 if d ≤ 0 and that for d > 0 2d 1 P2 ≤ C(1 + |b|)−κ (1 + |σ − d|)− 2 dσ 2 3d

≤

−κ

C(1 + |b|)

1

(1 + |d|) 2 .

Combining this estimate with (2.146), we get (2.145). The proof is complete.

When a = 1, it follows from (2.138) and (2.141) that t+r ∞ 1 β−µ+ 2 α−κ √ 2I1 ≤ dβ, dα α+r−t t−r |r −t| ∞ t−r 1 β−µ+ 2 α−κ √ dβ 2J J2 ≤ dα. t−r−α |r −t| −β Since µ > 3/2 and κ > 1, we have µ− 32 (I1 + J2 ) ≤ C r − t

∞ t−r

α−κ √ dα + α+r−t − 12

≤ Cr − t

t−r

−∞

α−κ √ dα t−r−α

,

by virtue of (2.105) and (2.106). When t ≥ 2r > 0 or 0 < r ≤ 1, this estimate implies (2.142). Thus we have shown (2.134) for n = 2 in this case. Case 2. a = 1, 0 < t ≤ 2r and r ≥ 1. If we set 1 b dσ √ √ K(a, b, d) = √ π a b−σ σ−a d−σ

(2.147)

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H. Kubo and M. Ohta

for a < b < d, then it follows from (2.133) and (2.23) that I3 + I4 ) |L− 1 [F ](t, x)| ≤ (I where we put

I3

0

t−s+r

= −∞

I4

sup (t,x)∈(−∞,0]×Rn

|t−s−r|

|t| + |x|µ |t| − |x|κ |F (t, x)|,

(2.148)

√ 1 λλ − s−µ+ 2 λ + s−κ

×K((λ − r)2 , (t − s)2 , (λ + r)2 ) dλ ds, √ 1 λλ − s−µ+ 2 λ + s−κ

(t−r)∧0 t−s−r

= −∞

0

×K((λ − r)2 , (λ + r)2 , (t − s)2 ) dλ ds. Using Lemma 2.18 below, we get t+r ∞ β−µ+1+θ α−κ √ √ I3 ≤ C dα dβ, t + r − α β + t − r(α + r − t)θ t−r |r −t| I4

≤

β

dβ −β

0

∞

+C

t−r

dβ |r −t|

≡

t−r

CH(t − r)

−β

√ 1 ββ−µ+ 2 α−κ √ √ dα β+t+r t−r−α

(2.149)

(2.150)

β−µ+ 2 +θ α−κ √ dα r(t − r − α)θ 1

J3 + J4 ,

where we divided I4 into two integrals as we have done it in (2.140) and (2.141). Note that J3 is the same form as in (2.140) with a = 1, and that it was 1 3 evaluated by Ct + r− 2 r − t−µ+ 2 in (2.144) without using the assumption a = 1. Therefore we have only to estimate I3 and J4 . We ﬁx θ as 0 < θ < min{µ − (3/2), 1/2}. Then we have t−r √ −µ+ 32 +θ rJ J4 ≤ Cr − t α−κ (t − r − α)−θ dα. −∞

By (2.105) we get (2.142) for J4 , since κ > 1, and we have assumed 0 < t ≤ 2r and r ≥ 1. Finally, we evaluate I3 . By (2.149) and (2.104) we have t+r 3 1 α−κ (t + r − α)− 2 (α + r − t)−θ dα. (2.151) r − tµ− 2 −θ I3 ≤ C t−r

Since (t + r)/3 ≥ t − r for 0 < t ≤ 2r, we may split the integral at α = (t + r)/3. Then we see that the right-hand side is estimated by ∞ 1 C(t + r)− 2 α−κ (α + r − t)−θ dα +Cr + t−κ

t−r t+r t−r

(t + r − α)− 2 (α + r − t)−θ dα. 1

Coupled Systems of Semilinear Wave Equations

155

Since κ > 1, (2.106) yields ∞ α−κ (α + r − t)−θ dα ≤ Cr − t−θ . t−r

Moreover, since θ < 1/2, we have t+r 1 1 (t + r − α)− 2 (α + r − t)−θ dα ≤ π(2r) 2 −θ . t−r

Therefore we see that the α–integral in (2.151) is estimated by Cr +t− 2 r −t−θ , because r ≥ 1, t > 0 and κ > 1. Hence we get (2.142) for I3 . This completes the proof of Proposition 2.8, assuming Lemma 2.18 below is valid. 1

Lemma 2.18. Let θ > 0, s < 0 and |t − s − r| < λ < t − s + r. Then there is a constant C = C(θ) > 0 such that C βθ √ , K((λ − r)2 , (t − s)2 , (λ + r)2 ) ≤ √ t + r − α β + t − r (α + r − t)θ where α, β are deﬁned by (2.51). Let θ > 0 and 0 < λ < t − s − r. Then it holds that 1 √ K((λ − r)2 , (λ + r)2 , (t − s)2 ) ≤ √ , t−r−α t+r+β and that if β > t − r, then βθ C K((λ − r)2 , (λ + r)2 , (t − s)2 ) ≤ √ . rλ (t − r − α)θ

(2.152)

(2.153)

(2.154)

Proof. First we prove (2.153). It is easy to see from (2.24) that 1 K((λ − r)2 , (λ + r)2 , (t − s)2 ) ≤ , (t − s)2 − (λ + r)2 which implies (2.153). To prove (2.152) and (2.154), we prepare the following inequality: For θ > 0 there is a constant C = C(θ) > 0 such that θ d−a 1 √ K(a, b, d) ≤ C . (2.155) d−b b−a In fact, splitting the integral at σ = (a + b)/2, we have from (2.147) d b √ dσ dσ √ √ √ b − a K(a, b, d) ≤ + . √ σ−a d−σ b−σ d−σ a a We see that the ﬁrst term of the right-hand side is equal to π. While, by using the integration by parts, the second one is estimated by √ θ b d−a d−a dσ 2 b−a √ ≤ 2 + log ≤C + d−b d−b d−a a d−σ for all θ > 0. Thus we obtain (2.155).

156

H. Kubo and M. Ohta Now it follows from (2.155) that K((λ − r)2 , (t − s)2 , (λ + r)2 ) θ 1 (λ + r)2 − (λ − r)2 ≤C 2 (λ + r)2 − (t − s)2 (t − s) − (λ − r)2 θ 4λr 1 √ √ , =C (β + r + t)(α + r − t) t+r−α β+t−r

which implies (2.152), since λ < β and β + r + t > r for s < 0. Moreover, we have K((λ − r)2 , (λ + r)2 , (t − s)2 ) θ (t − s)2 − (λ − r)2 1 ≤C 2 (t − s)2 − (λ + r)2 (λ + r) − (λ − r)2 θ (t − r + β)(t + r − α) 1 √ =C , (t − r − α)(t + r + β) 4rλ which yields (2.154), because t − r < β and −α < β. This completes the proof. Next we consider the part (ii). Setting v+ (t, x) = u(t, x) − Rc [|u|p ](t, x)

for (t, x) ∈ R × Rn ,

one can verify that v+ has the desired properties as in the proof of Theorem 2.7.

3. Semilinear system, I In this section we consider the following system:

2 (t, x) ∈ [0, ∞) × Rn , (∂ ∂t − c21 ∆)u1 = |u2 |p , 2 2 q (∂ ∂t − c2 ∆)u2 = |u1 | , (t, x) ∈ [0, ∞) × Rn ,

(3.1)

where p, q > 1 and n ≥ 2. The initial value problem for (3.1) was studied by Del Santo, Georgiev and Mitidieri in [18]. They found the critical curve F (p, q) = 0 in the p-q plane when c1 = c2 . Here critical curve means that if F (p, q) < 0, then small data global existence holds, and blow-up occurs if F (p, q) > 0. The function F (p, q) is deﬁned as follows:

& q + 2 + p−1 p + 2 + q −1 n−1 , . (3.2) F (p, q) = max − pq − 1 pq − 1 2 The blow-up part was also established by Deng [20] independently (see also [17]). The critical case where F (p, q) = 0 was treated independently by [3], [19] for n = 3 and by [65] for n = 2, 3. In these works the blow-up result was obtained. Next the authors studied the case c1 = c2 in [66]. This work is motivated by the recent results established by Kovalyov [57], Agemi and Yokoyama [5], Hoshiga and Kubo [36] and Yokoyama [114]. In those papers, the small data global existence

Coupled Systems of Semilinear Wave Equations

157

for systems of nonlinear wave equations with diﬀerent propagation speeds has been well developed when the nonlinear terms depend only on the derivatives of the unknown functions but not on the unknown functions themselves (see also [96] and [2] for related results on nonlinear elastic wave equations, and [86] on KleinGordon-Zakharov equations). It was shown in [66] that even if c1 = c2 , the critical curve is the same as in the case c1 = c2 for n = 3. In Theorem 3.1 below we extend the result to the two dimensional case. Therefore we conclude that the unequal propagation speeds do not have a major eﬀect on the system (3.1). Moreover, in the case where F (p, q) ≥ 0, the lifespan of the solution was studied by [67] for n = 3 and by Di Flaviano [16] for n = 2. It is interesting that the discrepancy of p and q has an eﬀect on the order of ε in the estimate of the lifespan only for the critical case (see Theorem 3.3 below). On the one hand, in the case where F (p, q) < 0, the existence of self-similar solution was treated by Kubo and Tsugawa [70], and asymptotic behaviors of the global solution was considered by Kubo and Kubota [62, 63, 64]. This section is organized as follows. In the next subsection, we prove the blow-up result in the same line as in Subsection 2.1. The argument improves the proof in earlier works [65, 66] by employing Lemma 3.3. Subsection 3.2 is devoted to the existence part which can be handled by (2.49), since it is applicable also to the case where the propagation speeds are distinct. In Subsection 3.3 we consider the self-similar solution by making use of (2.92). Subsection 3.4 is concerned with asymptotic behaviors. We shall establish analogous result to those from Subsection 2.5 by using (2.100) and (2.132). 3.1. Blow-up As is well known, a solution (u1 , u2 ) of the system (3.1) with the initial data uj (0, x) = ϕj (x),

∂t uj (0, x) = ψj (x),

x ∈ Rn

(j = 1, 2),

(3.3)

is obtained by solving the following system of integral equations: u1 u2

= =

Kc1 [ϕ1 , ψ1 ] + Lc1 [|u2 |p ] in [0, ∞) × Rn , Kc2 [ϕ2 , ψ2 ] + Lc2 [|u1 |q ] in [0, ∞) × Rn ,

(3.4) (3.5)

where Kc [ϕ, ψ], Lc [F ] are deﬁned by (2.8), (2.9) respectively. Throughout this section, we assume 1 < p ≤ q, and for convenience we set n−1 , (3.6) p∗ = mp − m − 1, q ∗ = mq − m − 1, m = 2 α = pq ∗ − 1, β = qp∗ − 1, Γ ≡ Γ(p, q) = α + pβ. (3.7) We note that β ≤ α and Γ(p, q) = −p(pq − 1)F (p, q) when 1 0 and gj ∈ C(R ) satisﬁes n

gj (x) ≥ 0 for all x ∈ Rn ,

gj (0) > 0

for j = 1, 2.

(3.9)

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Theorem 3.1. Let n = 2, 3, 1 < p ≤ q and Γ(p, q) ≤ 0. Suppose that ε ∈ (0, 1] and g1 , g2 ∈ C(Rn ) satisﬁes (3.9). Then the solution of (3.4)–(3.5) with (3.8) blows up in a ﬁnite time T ∗ (ε). Moreover, there exists a positive constant C ∗ independent of ε such that (3.10) if Γ(p, q) = 0, p < q, T ∗ (ε) ≤ exp Cε−p(pq−1) T ∗ (ε) ≤ exp Cε−p(p−1) (3.11) if Γ(p, q) = 0, p = q, T ∗ (ε) ≤ Cεp(pq−1)/Γ(p,q)

if

Γ(p, q) < 0.

(3.12)

Remark 3.1. We remark that the orders of ε appearing in the estimates (3.10)– (3.12) are optimal, thanks to (3.40)–(3.42) below. In order to prove Theorem 3.1, we ﬁrst prove the following lemma. Lemma 3.2. Let a, c > 0, µ∗ > 0, ν ≥ 0, µ ≤ µ∗ and α > 0. Then, there exists a positive constant C depending only on a, c, µ∗ , ν such that cs − λ λν f dλ ds (3.13) µ c Da (r,t)∩Σ(c,α) (s + λ) ν+1 (at−r)/a Cr aη f (η) dη ≥ 1− (at + r)(at − r)µ−ν−1 α at − r holds for any non-negative function f on [α, ∞) and any (r, t) ∈ Σ(a, α), where Da (r, t) and Σ(c, α) are deﬁned by (2.22) and (2.31). Proof. Let (r, t) ∈ Σ(a, α). We denote the left-hand side of (3.13) by I(r, t), and change the variables by cs − λ . c We divide the argument into two cases; a ≤ c and a > c. First, we consider the case a ≤ c. Then, by Lemma 2.2, we have (at−r)/a at+r (ξ − aη)ν I(r, t) ≥ C dξ f (η) dη ξµ α at−r at+r dξ (at−r)/a ≥ C (at − r − aη)ν f (η) dη µ at−r ξ α (at−r)/a Cr ≥ (at − r − aη)ν f (η) dη (at + r)(at − r)µ−1 α ν (at−r)/a Cr aη = f (η) dη, 1 − (at + r)(at − r)µ−ν−1 α at − r ξ = as + λ,

η=

which implies (3.13). Next, we consider the case a > c. We divide further into two cases, (r, t) ∈ Σ(c, α) and (r, t) ∈ Σ(a, α) \ Σ(c, α). When (r, t) ∈ Σ(c, α), we have

Coupled Systems of Semilinear Wave Equations I(r, t) ≥ C{I1 (r, t) + I2 (r, t)}, where (ct−r)/c I1 (r, t) =

at+r

α at−r (at−r)/a ξ ∗ (η)

I2 (r, t) =

(ct−r)/c

at−r

159

(ξ − aη)ν f (η) dξ dη, ξµ (ξ − aη)ν f (η) dξ dη. ξµ

While, if (r, t) ∈ Σ(a, α) \ Σ(c, α), we have I(r, t) ≥ CII3 (r, t), where (at−r)/a ξ∗ (η) (ξ − aη)ν f (η) I3 (r, t) = dξ dη. ξµ α at−r In the deﬁnitions of I2 (r, t) and I3 (r, t), we put ξ ∗ (η) =

2ac a+c (at − r) − η. a−c a−c

As in the case a ≤ c, we have Cr I1 (r, t) ≥ (at + r)(at − r)µ−ν−1

(ct−r)/c

aη 1− at − r

α

On the other hand, for j = 2, 3, we have (at−r)/a (at − r − aη)ν f (η) Ij (r, t) ≥ C ηj∗

where we put

η2∗

η3∗

= (ct − r)/c and

ν

ξ ∗ (η) at−r

f (η) dη.

(3.14)

dξ dη, ξµ

= α. Since

a+c (at − r), a−c 2c ξ ∗ (η) − (at − r) = (at − r − aη), a−c at − r ≤ ξ ∗ (η) ≤

we have

ξ ∗ (η)

dξ at − r − aη ≥C . ξµ (at − r)µ

at−r

Thus, for j = 2, 3, we have Ij (r, t)

(at−r)/a

≥

C (at − r)µ−ν−1

≥

Cr (at + r)(at − r)µ−ν−1

ηj∗

ν+1 aη f (η) dη (3.15) 1− at − r ν+1 (at−r)/a aη f (η) dη. 1− at − r ηj∗

From (3.14) and (3.15), we see that (3.13) is also valid for the case a > c. This completes the proof. By Lemma 3.2, we have the following proposition.

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Proposition 3.1. Let n = 2, 3, a, c > 0, p ≥ 1 and κ ∈ R. Then there exists a constant C = C(a, c, p) > 0 such that b y η [f c,κ (η)]p dη, y ≥ 1, (3.16) 1− La [|f |p ]a,p∗ (y) ≥ C y η pκ 1 where b = m + (1 − m)p + 1. ˜ y), by (2.21) and (3.13), we have Proof. Let y ≥ 1. For any (t, x) ∈ Σ(a, La [|f |p ](t, x) cs − λ λm+(1−m)p 1 ≥ f c,κ p pκ 2arm c Da (r,t)∩Σ(c,1) (cs + λ) (cs − λ) b (at−r)/a [f c,κ (η)]p Cr1−m aη dη. ≥ 1 − ∗ (at + r)(at − r)p 1 at − r η pκ Since the function

y

1−

y → 1

η y

b

p

dλ ds

[f c,κ (η)]p dη η pκ

˜ y), we have is non-decreasing, for any (t, x) ∈ Σ(a, ∗

|x|−(1−m) (at + |x|)(at − |x|)p |La [|f |p ](t, x)| ≥ C

1

y

b [f c,κ (η)]p η dη, 1− y η pκ

which implies (3.16). Now, we shall give the proof of Theorem 3.1.

Step 1. As in the proof of Theorem 2.1, from (3.8) and (3.9), we see that there exist constants δ ∈ (0, 1) and C0 > 0 such that uj (t, x) ≥ εJ Jcj [gj ](t, x) ≥

C0 ε (t + r)m

holds for |cj t − r| ≤ δ/2, cj t + r ≥ δ and j = 1, 2. Step 2. There exists a constant C1 > 0 such that u1 c1 ,p∗ (y) ≥ C1 εp ,

u2 c2 ,q∗ (y) ≥ C1 εq

for y ≥ 1.

˜ 1 , 1), by (2.21), Step 1 and (3.13), we have Indeed, for any (t, x) ∈ Σ(c u1 (t, x) ≥ ≥ ≥ ≥

Lc1 [|u2 |p ](t, x) c2 s − λ (C C0 ε)p λm χ dλ ds mp [δ/4,δ/2] 2c1 rm c2 Dc1 (r,t) (s + λ) m+1 δ/2 Cεp r1−m c1 η dη 1− (c1 t + r)(c1 t − r)mp−m−1 δ/4 c1 t − r δ/2 Cεp r1−m (1 − η)m+1 dη. (c1 t + r)(c1 t − r)p∗ δ/4

(3.17)

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161

Here χ[δ/4,δ/2] (λ) denotes the characteristic function on [δ/4, δ/2]. Thus we have u1 c1 ,p∗ (y) ≥ Cεp

for y ≥ 1.

In the same way, we have u2 c2 ,q∗ (y) ≥ Cε for y ≥ 1. Step 3. Since q

u1 c1 ,p∗ (y) ≥ Lc1 [|u2 |p ]c1 ,p∗ (y)

and u2 c2 ,q∗ (y) ≥ Lc2 [|u1 |q ]c2 ,q∗ (y),

by Proposition 3.1, there exists a constant C2 > 0 such that b y η 1 [u2 c2 ,q∗ (η)]p 1− dη u1 c1 ,p∗ (y) ≥ C2 y η pq∗ 1 b y η 2 [u1 c1 ,p∗ (η)]q u2 c2 ,q∗ (y) ≥ C2 dη 1− y η qp∗ 1

for y ≥ 1, (3.18) for y ≥ 1, (3.19)

where we put b1 = m + (1 − m)p + 1 and b2 = m + (1 − m)q + 1. Step 4. We put U1 (y) = u1 c1 ,p∗ (y) and U2 (y) = u2 c2 ,q∗ (y) for y ≥ 1. Then, by (3.17), (3.18) and (3.19), we can show that if Γ(p, q) ≤ 0, (U1 (y), U2 (y)) blows up Uj (y) : in a ﬁnite time. Indeed, when Γ(p, q) = 0 and p = q, we put U (y) = min{U j = 1, 2}. Then, we have b y [U (η)]p η dη for y ≥ 1, 1− U (y) ≥ C1 εp , U (y) ≥ C2 y η pp∗ 1 where we put b = m + (1 − m)p + 1. Thus (3.11) follows from Lemma 2.3, since pp∗ = 1 in this case. Next, we consider the case where Γ(p, q) = 0 and 1 < p < q. Note that β < 0 < α and α + pβ = 0 in this case. We put f1 (z) = ε−p U1 (eλz ),

f2 (z) = ε−pq eβλz U2 (eλz ),

λ = ε−p(pq−1) .

Then, for z ≥ 0 we have f1 (z) ≥ C1 , f2 (z) ≥ 0, and z b1 1 − e−(z−ζ) [ff2 (ζ)]p dζ, f1 (z) ≥ C2 0 z b2 f2 (z) ≥ C2 1 − e−λ(z−ζ) eβλ(z−ζ) [f1 (ζ)]q λ dζ, 0

where we used the fact that z → (1 − e−z )b1 is increasing on [0, ∞). Since f1 (z) is non-decreasing and λ ≥ 1, for any z ≥ 1 and 0 < h ≤ 1 we have z b2 f2 (z) ≥ C2 [f1 (z − h)]q 1 − e−λ(z−ζ) eβλ(z−ζ) λ dζ z −h/λ

≥ C2 [f1 (z − h)]q

h

b2 βτ 1 − e−τ e dτ.

0

Since 1 − e

−τ

−1

≥ (1 − e )τ for 0 ≤ τ ≤ 1, we have h b2 βτ (1 − e−1 )b2 eβ b2 +1 h 1 − e−τ e dτ ≥ . b2 + 1 0

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H. Kubo and M. Ohta

Thus, we see that f1 (z) ≥ C1 ,

z

f1 (z) ≥ C3 hp(b2 +1)

1 − e−(z−ζ)

b1

[f1 (ζ − h)]pq dζ

(3.20)

1

holds for any z ≥ 1 and 0 < h ≤ 1. Thus (3.10) follows from Lemma 3.3 below. Finally, we consider the case Γ(p, q) < 0. Note that Γ(p, q) = α + pβ < 0 and β < 0 in this case. We put f1 (z) = ε−p U1 (ε−µ ez ),

f2 (z) = ε−pq (ε−µ ez )β U2 (ε−µ ez ),

µ=−

p(pq − 1) . α + pβ

Then, for z ≥ 0, we have f1 (z) ≥ C1 , f2 (z) ≥ 0, and z b1 1 − e−(z−ζ) f1 (z) ≥ C2 [ff2 (ζ)]p dζ, 0 z b2 1 − e−(z−ζ) f2 (z) ≥ C2 eβ(z−ζ) [f1 (ζ)]q dζ. 0

Thus, as in the previous case, we see that (3.20) holds for any z ≥ 1 and 0 < h ≤ 1. Thus (3.12) follows from Lemma 3.3 below. Lemma 3.3. Let C1 , C2 > 0, a, b ≥ 0 and p > 1. Suppose that f (t) satisﬁes f (t) ≥ C1 ,

f (t) ≥ C2 ha

t

(1 − e−(t−τ ) )b [f (τ − h)]p dτ

(3.21)

1

for any t ≥ 1 and 0 < h ≤ 1. Then, f (t) blows up in a ﬁnite time. Proof. By (3.21) with h = 1, for t ≥ 2, we have f (t) ≥

t

(1 − e

C2 2

=

−(t−τ ) b

) [f (τ − 1)] dτ ≥ p

C1p C2

t

(1 − e−(t−τ ))b dτ

2 t−2

C1p C2

(1 − e−σ )b dσ ≥ C1p C2

0

(1 − e−1 )b (t − 2)b+1 . b+1

Thus, there exists T1 > 2 such that f (t) ≥ A1 for any t ≥ T1 , where we put ⎞ ⎛ ∞ b+1 log γ log j ⎠ , γ = max{ + 2(a + b + 1) A1 = exp ⎝1 + , 1}. j p−1 p C2 (1 − e−1 )b j=1 (3.22) Tk } by Here, we deﬁne sequences {Ak } and {T Ak+1 =

Apk γk 2(a+b+1)

,

Tk+1 = Tk +

2 , k2

k ∈ N.

(3.23)

Coupled Systems of Semilinear Wave Equations

163

Then, for any k ∈ N, we see that f (t) ≥ Ak for t ≥ Tk . Indeed, by (3.21) with h = k −2 , for t ≥ Tk + 2k −2 , we have t −2a f (t) ≥ C2 k (1 − e−(t−τ ) )b [f (τ − k −2 )]p dτ t−k−2

≥ C2 Apk k −2a

k−2

(1 − e−σ )b dσ ≥

0

C2 (1 − e−1 )b p −2(a+b+1) Ak k . b+1

Moreover, by (3.22) and (3.23), we have ⎛ ⎞ k k log γ log j ⎠ log Ak+1 = pk ⎝log A1 − − 2(a + b + 1) j p pj j=1 j=1 ⎛ ⎞ ∞ log j ⎠ log γ − 2(a + b + 1) ≥ pk ⎝log A1 − ≥ pk , j p−1 p j=1 Tk+1

=

T1 +

k 2 j2 j=1

for any k ∈ N. Therefore, f (t) blows up in a ﬁnite time.

3.2. Small data global existence In this subsection we mainly consider the case where Γ ≡ Γ(p, q) > 0,

(3.24)

where Γ is deﬁned by (3.7). One can verify that the condition F (p, q) < 0 with F deﬁned by (3.2) is equivalent to (3.24) when p ≤ q. In addition, we assume the following condition for the sake of simplicity: 0 < p∗ ≤ q ∗ ,

i.e., (n + 1)/(n − 1) 0 as follows. Lemma 3.4. If (3.25) and (3.24) hold, then there are ν and κ satisfying 0 < ν ≤ p∗ ,

(3.26)

∗

0<κ≤q , p∗ − ν + pκ > 1,

q ∗ − κ + qν > 1.

(3.27) (3.28)

Moreover, when n = 2, we can choose them such that ν < 1/2,

κ < 1/2.

(3.29)

Proof. To ﬁnd κ satisfying (3.27) and (3.28) for some ν, we need to assure that (1 − p∗ + ν)/p < q ∗ + qν − 1,

(1 − p∗ + ν)/p < q ∗ ,

0 < q ∗ + qν − 1.

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H. Kubo and M. Ohta

Equivalently, ν > (1 − p∗ − pq ∗ + p)/(pq − 1),

ν < pq ∗ + p∗ − 1,

ν > (1 − q ∗ )/q.

(3.30)

Note that (1 − q ∗ )/q ≤ 0 if q ∗ ≥ 1, and 1 − q∗ 1 − p∗ − pq ∗ + p q ∗ − 1 + q(p∗ − 1) − = ≤0 q pq − 1 q(pq − 1) if q ∗ ≤ 1. Therefore, to take ν satisfying (3.30) together with (3.26), it suﬃces to assure (1 − p∗ − pq ∗ + p)/(pq − 1) < pq ∗ + p∗ − 1, (1 − p∗ − pq ∗ + p)/(pq − 1) < p∗ , 0 < pq ∗ + p∗ − 1. Simplifying the above relations by the aid of (3.7), we get β + qα > 0,

α + pβ > 0,

α + p∗ > 0.

(3.31)

Notice that β≤

Γ ≤α p+1

for 1 < p ≤ q.

(3.32)

So β + qα ≥ α + pβ = Γ for 1 < p ≤ q. Therefore (3.24) implies (3.31). In addition, to choose ν and κ so that (3.29) holds, we need (1 − p∗ + ν)/p < 1/2,

i.e., ν < p∗ + (p − 2)/2,

hence (1 − p∗ − pq ∗ + p)/(pq − 1) < ∗

∗

(1 − p − pq + p)/(pq − 1) <

p∗ + (p − 2)/2,

(3.33)

1/2.

(3.34)

Since (1 − p∗ − pq ∗ + p)/(pq − 1) = −Γ/(pq − 1) + p∗ , we easily have (3.33). While (3.34) is equivalent to p∗ + 3q ∗ + 2p∗q ∗ > 0 when n = 2, which follows from (3.25). Thus we ﬁnish the proof. Proposition 3.2. Assume that (3.24) and (3.25) hold. Let u ∈ X(c1 , ν) and v ∈ X(c2 , κ). If ν and κ satisfy (3.26), (3.27) and the ﬁrst inequality in (3.28), and also (3.29) when n = 2, then there is a constant K0 = K0 (c1 , c2 , n, p, q, ν, κ) such that Lc1 [|v|p ]X(c1 ,ν) ≤ K0 [vX(c2 ,κ) ]p .

(3.35)

Moreover, if ν and κ satisfy (3.26), (3.27) and the second inequality in (3.28), and also (3.29) when n = 2, then we have Lc2 [|u|q ]X(c2 ,κ) ≤ K0 [uX(c1 ,ν) ]q .

(3.36)

Coupled Systems of Semilinear Wave Equations

165

Proof. We shall only prove (3.35), since the other inequality can be shown in an analogous way. We make use of (2.49) by taking c = c1 , a = c2 , µ = m + 1 + ν and κ = p∗ − ν + pκ (> 1) (recall (3.28)). Then we get Lc1 [|v|p ]X(c1 ,ν) ≤ C1

sup (t,x)∈[0,T )×Rn

∗

t + |x|m+1+ν c2 t − |x|p

−ν+pκ

|v(t, x)|p . (3.37)

Since mp = p∗ + m + 1 by (3.6), we have |v(y, s)|p

∗

≤ [vX(c2 ,κ) ]p λ + s−m−1−p λ − c2 s−pκ ∗

≤ C[vX(c2 ,κ) ]p λ + s−m−1−ν λ − c2 s−pκ+ν−p

for (y, s) ∈ Rn × R with λ = |y|, where C is a constant depending only on c2 and ν − p∗ . Therefore we obtain (3.35) from (3.37). Thus the proof is complete. We are now in position to state the main results of this subsection. Theorem 3.2. Let n = 2, 3 and let (3.24), (3.25) hold. Let ν and κ satisfy (3.26), (3.27) and (3.28), and also (3.29) when n = 2. Suppose (ϕ1 , ψ1 ) ∈ Y (ν) and (ϕ2 , ψ2 ) ∈ Y (κ). Then there is a positive constant ε0 = ε0 (c1 , c2 , n, p, q, ν, κ) such that if we assume (ϕ1 , ψ1 )Y (ν) + (ϕ2 , ψ2 )Y (κ) ≤ ε for 0 < ε ≤ ε0 , there exists a unique solution (u1 , u2 ) ∈ X(c1 , ν) × X(c2 , κ) of (3.4)–(3.5). Proof. For ε > 0 we put X := {(u, v) ∈ C([0, ∞) × Rn ) × C([0, ∞) × Rn ) : C0 ε}, (u, v) ≡ uX(c1 ,ν) + vX(c2 ,κ) ≤ 2C where ν, κ are the numbers in the theorem, and C0 is the number in (2.41). We have only to prove that M deﬁned by M [u, v] = (Kc1 [(ϕ1 , ψ1 ], Kc2 [ϕ2 , ψ2 ]) + (Lc1 [|v|p ], Lc2 [|u|q ]) is a contraction in X for suﬃciently small ε. This can be done by making use of (2.41) and Proposition 3.2. We omit the details. At the end of this subsection, we state the estimate for the lifespan from below. This can be shown by combining the argument presented in the Subsection 2.3 and the computation made in [67] and [16]. Theorem 3.3. Let n = 2, 3 and let (3.25) hold. Suppose that (ffj , gj ) ∈ C 1 (Rn ) × C(Rn ) satisﬁes fj (x) = gj (x) = 0 for |x| ≥ R (3.38) for some R > 0 and j = 1, 2. Let us take the initial data as ϕj (x) = εffj (x),

ψj (x) = εgj (x)

for

x ∈ Rn .

(3.39)

Then there is a constant ε0 = ε0 (c1 , c2 , n, p, q, R) > 0 such that there exists a unique solution (u1 , u2 ) ∈ C([0, T ∗ (ε)) × Rn ) × C([0, T ∗ (ε)) × Rn ) of (3.4)–(3.5).

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Moreover, there is a constant C ∗ = C ∗ (c1 , c2 , n, p, q, R) > 0 such that for any ε with 0 < ε ≤ ε0 we have if Γ(p, q) = 0, p = q, (3.40) T ∗ (ε) ≥ exp C ∗ ε−p(pq−1) T ∗ (ε) ≥ exp C ∗ ε−p(p−1) if Γ(p, q) = 0, p = q, (3.41) T ∗ (ε) ≥ C ∗ εp(pq−1)/Γ(p,q)

if Γ(p, q) < 0.

(3.42)

3.3. Self-similar solution We study the existence of the self-similar solution to (3.1). For λ > 0, let us deﬁne uλ1 (t, x) = λ

2(p+1) pq−1

uλ2 (t, x) = λ

u1 (λt, λx),

2(q+1) pq−1

u2 (λt, λx).

If (u1 , u2 ) satisﬁes (3.1), then (uλ1 , uλ2 ) also solves (3.1). We call (u1 , u2 ) a selfsimilar solution, if it satisﬁes (3.1) and uλ1 (t, x) ≡ u1 (t, x),

uλ2 (t, x) ≡ u2 (t, x)

(3.43)

for (t, x) ∈ [0, ∞) × R and λ > 0. Additionally, we put n

p∗ =

2(p + 1) − m, pq − 1

q∗ =

2(q + 1) − m, pq − 1

m=

n−1 . 2

(3.44)

Our main result of this subsection reads as follows. Theorem 3.4. Let n = 2, 3 and let (3.24), (3.25) and p∗ > 0 hold. Suppose (ϕ1 , ψ1 ) ∈ Y˙ (p∗ ) and (ϕ2 , ψ2 ) ∈ Y˙ (q∗ ). Then there is a positive constant ε0 = ε0 (c1 , c2 , p, q, n) such that if we assume (ϕ1 , ψ1 )Y˙ (p∗ ) + (ϕ2 , ψ2 )Y˙ (q∗ ) ≤ ε for ˙ 1 , p∗ ) × X(c ˙ 2 , q∗ ) of 0 < ε ≤ ε0 , there exists a unique solution (u1 , u2 ) ∈ X(c (3.4)–(3.5). Remark 3.5. We can understand the choice of p∗ and q∗ in the connection with the limiting case of (3.28). Namely, for ν = p∗ and κ = q∗ we have p∗ − ν + pκ = 1,

q ∗ − κ + qν = 1.

Corollary 3.1. Let n = 2, 3 and let (3.24), (3.25) and p∗ > 0 hold. Let the initial data take the form ϕ1 (x) = f1 (ω)|x|−

2(p+1) pq−1

− 2(q+1) pq−1

ϕ2 (x) = f2 (ω)|x|

ψ1 (x) = g1 (ω)|x|−

, ,

2(p+1) pq−1 −1

,

(3.45)

− 2(q+1) pq−1 −1

(3.46)

ψ2 (x) = g2 (ω)|x|

with ω = x/|x|, fj ∈ C 1 (S 2 ), gj ∈ C(S 2 ) (j = 1, 2). If (ϕ1 , ψ1 ) ∈ Y˙ (p∗ ) and (ϕ2 , ψ2 ) ∈ Y˙ (q∗ ) is suﬃciently small in Y˙ (p∗ ) and Y˙ (q∗ ) respectively, then there exists a unique self-similar solution of (3.4)–(3.5). To prove Theorem 3.4, the following proposition is crucial. Once we establish it, a standard argument yields the conclusion.

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˙ 1 , p∗ ) and v ∈ Proposition 3.3. Assume that (3.24) and (3.25) hold. Let u ∈ X(c ˙ X(c2 , q∗ ). Then there is a constant K0 = K0 (c1 , c2 , n, p, q) such that p Lc1 [|v|p ]X(c ˙ 1 ,p∗ ) ≤ K0 [vX(c ˙ 2 ,q∗ ) ] ,

(3.47)

Lc2 [|u| ]X(c ˙ 2 ,q∗ ) ≤ K0 [uX(c ˙ 1 ,p∗ ) ] .

(3.48)

q

q

Proof. First we note that (3.24) is equivalent to pq∗ < 1. Moreover, (3.25) implies pq∗ ≥ qp∗ , hence qp∗ < 1 holds. We shall only prove (3.47), since the other inequality can be shown in an analogous way. We make use of Proposition 2.5 by taking c = c1 , a = c2 , µ = mp and κ = pq∗ , so that mp + pq∗ − m − 2 = p∗ . When n = 2, we have mp + pq∗ = (5/2) + p∗ . Therefore, p∗ < 1/2 assures mp + pq∗ < 3. Since 3 < p ≤ q, we get p∗ < 1/2, hence we can apply the proposition. In this way we obtain (3.47). Thus the proof is complete. 3.4. Asymptotic behavior In this subsection we consider asymptotic behaviors of solutions to (3.1) from the same point of view as in the Subsection 2.5. The ﬁrst result claims that the classical solution of (3.1) with (3.3) tends to a solution of the system of homogeneous wave equations

2 (∂ ∂t − c21 ∆)v1 = 0, (t, x) ∈ [0, ∞) × Rn , (3.49) 2 2 (t, x) ∈ [0, ∞) × Rn , (∂ ∂t − c2 ∆)v2 = 0, as t → +∞ in the sense of the energy. Since this result can be shown by following the proof of Theorem 2.5, we omit the proof. Theorem 3.5. Let n = 2, 3 and let (3.24), (3.25) hold. Let ν and κ satisfy (3.26), (3.27) and (3.28), and also (3.29) when n = 2. For |α| ≤ 2 suppose (∂ ∂xα ϕ1 , ∂xα ψ1 ) ∈ α α Y (ν) and (∂ ∂x ϕ2 , ∂x ψ2 ) ∈ Y (κ). Let (u1 , u2 ) be a classical solution to (3.1) with (3.3) such that (∂ ∂xα u1 , ∂xα u2 ) ∈ X(c1 , ν) × X(c2 , κ) for |α| ≤ 2. Then there is a unique classical solution (v1+ , v2+ ) of (3.49) satisfying ∂xα v1+ ∈ X(c1 , ν), ∂xα v2+ ∈ X(c2 , κ) (|α| ≤ 2), and ' ( 12 ∗ u1 (t) − v1+ (t)E(c1 ) ≤ C[u2 X(c2 ,κ) ]p t−p t[1−2pκ]+ for t ≥ 0, (3.50) u2 (t) − v2+ (t)E(c2 ) ≤ C[u1 X(c1 ,ν) ]q t−q

∗

' ( 12 t[1−2qν]+

for t ≥ 0. (3.51)

Remark 3.6. 1) In particular, if we take ν as ν = p∗ , we have κ > 1/p from (3.28). /1 . Hence we can drop the factor (1 + |t|)[1−2pκ]+ 2 in (3.50). 2) If p and q satisfy β = qp∗ − 1 > 0, (3.52) then α = pq ∗ −1 > 0 by (3.32), hence p∗ > 1/q, q ∗ > 1/p. Note that (3.26) through (3.28) holds for all ν and κ satisfying 1/q < ν ≤ p∗ ,

1/p < κ ≤ q ∗ .

(3.53)

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Therefore, choosing ν and κ verifying (3.53), we can drop not only the same factor /1 . as in the item 1) but also the factor (1 + |t|)[1−2qν]+ 2 in (3.51). 3) Even in the case where 2pκ ≤ 1 (resp. 2qν ≤ 1), the right-hand side of (3.50) (resp. (3.51)) tends to zero as t → −∞, since (3.28) implies −p∗ +

1 1 − pκ < − − ν, 2 2

−q ∗ +

1 1 − qν < − − κ. 2 2

(3.54)

The second result is concerned with the nonlinear scattering related to (3.1). We consider the following integral equation − − − (u1 , u2 ) = (Kc1 [(ϕ− Tc1 [|u2 |p ], Tc2 [|u1 |q ]) 1 , ψ1 ], Kc2 [ϕ2 , ψ2 ]) + (T

(3.55)

in R × R , where Tc [F ] was deﬁned by (2.128). Then we can show the result by making use of (2.132). n

Theorem 3.6. Let the assumption of Theorem 3.5 be fulﬁlled. (i) Then there is a constant ε0 = ε0 (p, q, ν, κ, c1 , c2 , n) > 0 such that if α − α − ∂xα ϕ− ∂xα ϕ− max {(∂ 1 , ∂x ψ1 )Y (ν) + (∂ 2 , ∂x ψ2 )Y (κ) } ≤ ε

|α|≤2

for 0 < ε ≤ ε0 ,

then there is a unique classical solution (u1 , u2 ) of (3.1) in {(v1 , v2 ) ∈ C(R × Rn ) × C(R × Rn ) : C0 ε}, v1 V (c1 ,ν) + v2 V (c2 ,κ) ≤ 2C where C0 = C0 (ν, κ, c1 , c2 , n) > 0 is a constant. Moreover, we have ∂xα u1 ∈ V (c1 , ν), ∂xα u2 ∈ V (c2 , κ) (|α| ≤ 2), and ' ( 12 ∗ for t ≤ 0, (3.56) u1 (t) − v1− (t)E(c1 ) ≤ C[u2 V (c2 ,κ) ]p t−p t[1−2pκ]+ u2 (t) − v2− (t)E(c2 ) ≤ C[u1 V (c1 ,ν) ]q t−q

∗

'

t[1−2qν]+

( 12

for t ≤ 0, (3.57)

− where vj− (t, x) := Kcj [ϕ− j , ψj ](t, x) (j = 1, 2). (ii) Let (u1 , u2 ) be as in the part (i) of the theorem. Then there exists a unique classical solution (v1+ , v2+ ) to (3.49) such that ∂xα v1+ ∈ V (c1 , ν), ∂xα v2+ ∈ V (c2 , κ) (|α| ≤ 2), and (3.50), (3.51) holds good.

4. Semilinear system, II In this section we study the following system:

2 (∂ ∂t − c21 ∆)u1 = λ1 |u1 |p1 |u2 |p2 , (∂ ∂t2 − c22 ∆)u2 = λ2 |u1 |q1 |u2 |q2 ,

(t, x) ∈ [0, ∞) × Rn , (t, x) ∈ [0, ∞) × Rn ,

(4.1)

where λ1 , λ2 ∈ R and n ≥ 2. If p1 = 0 and q2 = 0, then the system (4.1) reduces to the system (3.1). For this reason, we assume p1 , p2 , q1 , q2 ≥ 1.

(4.2)

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169

This means that the nonlinearities are involved by a product of u1 and u2 and that they are Lipschitz continuous with respect to u1 and u2 . Based on this structure, we shall derive the eﬀect of the discrepancy of the propagation speeds. For such a purpose, we may assume that the degree of the nonlinearity of the ﬁrst equation is the same as that of the second one. Namely, we consider the case where there is α ≥ 2 such that p1 + p2 = q1 + q2 ≡ α.

(4.3)

When c1 = c2 , it follows from the result about the single wave equation (2.1) that small data global existence holds if α > p0 (n) and that blow-up occurs if 2 ≤ α ≤ p0 (n). Here p0 (n) is the positive root of (2.3). (For the details about (2.1), see Section 2.) Next we turn our attention to the case c1 = c2 . When n = 3, Kubo and Tsugawa [70] ﬁrstly proved small data global existence for all α > 2. Then the authors [68] showed that the same is true for α = 2. Let √us compare these results with those for the case c1 = c2 . Since p0 (3) = 1 + 2,√we ﬁnd that there is a signiﬁcant diﬀerence among them when 2 ≤ α ≤ 1 + 2. Actually, for such α we have a global solution if c1 = c2 , while blow-up occurs if c1 = c2 . This observation shows the eﬀect of the discrepancy between the propagation speeds, which comes from the way of interaction in the nonlinearities (recall that we do not have such an eﬀect for the system (3.1)). In fact, since the right-hand side of the equations in (4.1) are involved by a product of u1 and u2 , one can compensate the deﬁciency of the pointwise decaying order for the powers of u1 and u2 each other, based on the unequal propagation speeds. Recently the extension to the two dimensional case n = 2 was √ done by the authors [69]. The result implies that for 3 < α ≤ p0 (2) = (3 + 17)/2, we also have such an eﬀect of the unequal propagation speeds as in case n = 3. (See also Remark 4.1 below.) Concerning the existence and non-existence of self-similar solution for the system (4.1), the following result was established by [70]. Let n = 3. Then in contrast to the condition on α for the existence of the smooth solution mentioned in the above, it is really necessary to assume α > 2 in order to assure the existence of self-similar solution. Actually, if α = 2, then we can show a counter-example which implies that self-similar solution does not exist in general (see Theorems 4.3 and 4.7 below). On the one hand, the asymptotic behavior of the smooth solution for the system (4.1) was considered by [63]. This section is organized as follows. We shall establish the existence result of the global solution and the self-similar solution in Subsections 4.1 and 4.2 by using (2.49) and (2.92), respectively. As for the peculiar case where α = 2 and n = 3, we show a blow-up result for slowly decaying initial data in Theorem 4.4. In Subsection 4.3 we extend these results to the case where the number of the equations is more than 2.

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4.1. Small data global existence As usual, we treat the system (4.1) in the following integral form: u1 u2

= Kc1 [ϕ1 , ψ1 ] + λ1 Lc1 [|u1 |p1 |u2 |p2 ] in [0, ∞) × Rn , = Kc2 [ϕ2 , ψ2 ] + λ2 Lc2 [|u1 |q1 |u2 |q2 ] in [0, ∞) × Rn ,

(4.4) (4.5)

where Kc [ϕ, ψ], Lc [F ] are deﬁned by (2.8), (2.9) respectively. For convenience, we put n−1 2 − m, m = , (4.6) α∗ = α−1 2 where α is deﬁned by (4.3). The main result of this subsection is as follows. Theorem 4.1. Let n = 2, 3 and let (4.2) and (4.3) hold. Assume that c1 = c2 and that either α > 2,

ν ≥ α∗ ,

ν > 0,

(4.7)

or α = 2,

ν>1

(4.8)

for n = 3, and that 1 (4.9) 2 for n = 2. Suppose (ϕj , ψj ) ∈ Y (ν) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 , n, p1 , p2 , q1 , q2 , ν) such that if we assume (ϕ1 , ψ1 )Y (ν) + (ϕ2 , ψ2 )Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ X(c1 , ν) × X(c2 , ν) of (4.4)–(4.5). α > 3,

ν ≥ α∗ ,

0<ν<

Remark 4.1. By (4.2) we have α ≥ 2. When n = 2, the above theorem does not give any information for the case 2 ≤ α ≤ 3. However the authors [69] recently showed that blow-up occurs in that case. Therefore the assumption α > 3 in the theorem is sharp. We also remark that one can take ν in such a way that α∗ = (5−α)/2(α−1) < ν < 1/2, since α > 3 when n = 2. To prove Theorem 4.1, the following proposition is crucial. Once we establish it, a standard argument yields the conclusion. Proposition 4.1. Let the assumption in Theorem 4.1 be fulﬁlled. Let u ∈ X(c1 , ν) and v ∈ X(c2 , ν). Then there is a constant K0 = K0 (c1 , c2 , n, p1 , p2 , ν) such that Lc1 [|u|p1 |v|p2 ]X(c1 ,ν) ≤ K0 [uX(c1 ,ν) ]p1 [vX(c2 ,ν) ]p2 .

(4.10)

Proof. First we consider the case where α > (n + 1)/(n − 1), ν ≥ α∗ and ν > 0. We use the key assumption c1 = c2 in the following way: If we take (s, y) near the light cone c2 s = |y|, then the weight c1 s − |y| behaves like s + |y|. On the contrary, if (s, y) is close to the other light cone c1 s = |y|, then the weight c2 s − |y| behaves

Coupled Systems of Semilinear Wave Equations

171

like s + |y|. In conclusion, one can extract some additional decay in s + |y|, provided c1 = c2 . This observation leads us to |u(s, y)|p1 |v(s, y)|p2

(4.11) −mα−p1 ν

≤ C[uX(c1 ,ν) ] [vX(c2 ,ν) ] {λ + s p1

p2

−mα−p2 ν

+λ + s

−p1 ν

λ − c1 s

−p2 ν

λ − c2 s

}

for (s, y) ∈ R × R with λ = |y|. For j = 1, 2 we put n

ρj = m(α − 1) − 1 + (pj − 1)ν. Since pj ≥ 1, ν > 0, and m(α − 1) > 1 for α > (n + 1)/(n − 1), we have ρj > 0. Therefore, (4.11) implies |u(s, y)|p1 |v(s, y)|p2 ≤ C[uX(c1 ,ν) ]p1 [vX(c2 ,ν) ]p2 {λ + s−m−1−ν λ − c2 s−ρ1 −p2 ν +λ + s−m−1−ν λ − c1 s−ρ2 −p1 ν . Now we consider two diﬀerent cases. First suppose ν > α∗ . We shall make use of (2.49) by taking µ = m + 1 + ν,

κ = (m + ν)(α − 1) − 1.

Then we have κ > 1 by ν > α∗ and ν > 0. Since ρ1 + p2 ν = ρ2 + p1 ν = κ, (2.49) yields (4.10). Next suppose ν = α∗ . Then we have α∗ > 0 and for n = 2, α∗ < 1/2. Moreover, we have κ = 1. Since ρj > 0, we see that pj ν < 1 for j = 1, 2. Therefore, (4.11) and (2.49) yield (4.10), since mα + p1 ν + p2 ν − m − 2 = α∗ for ν = α∗ . It remains to deal with the case where n = 3, α = 2 and ν > 1. By (4.11) we have |u(s, y)|p1 |v(s, y)|p2 ≤ C[uX(c1 ,ν) ]p1 [vX(c2 ,ν) ]p2 {λ + s−2−ν λ − c2 s−ν +λ + s−2−ν λ − c1 s−ν } for (s, y) ∈ R × R3 with λ = |y|, since p1 = p2 = 1 for α = 2. Therefore (2.49) implies (4.10), because of ν > 1. This completes the proof. At the end of this subsection, we give a uniﬁed statement of Theorems 3.2 and 4.1 except for the limiting case where n = 3 and α = 2. We assume in the system (4.1) that either p1 = q2 = 0,

p2 , q1 > 1,

(4.12)

or p1 , p2 , q1 , q2 ≥ 1.

(4.13)

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Note that in this situation, the assumption (4.3) does not hold in general. On the one hand, the system (4.1) keeps the scaling invariance related to the following exponents: k1 =

2{(q2 − 1) − p2 } − m, (p1 − 1)(q2 − 1) − p2 q1

k2 =

2{(p1 − 1) − q1 } − m. (p1 − 1)(q2 − 1) − p2 q1

Here we assumed (p1 − 1)(q2 − 1) − p2 q1 = 0. Actually, in the case where (4.12) holds, k1 and k2 reduce to k1 =

2(p2 + 1) − m = (p2 )∗ , p2 q1 − 1

k2 =

2(q1 + 1) − m = (q1 )∗ , p2 q1 − 1

(4.14)

in view of (3.44). While, when (4.13) and (4.3) hold, we see from (4.6) that 2 − m = α∗ . α−1 Then we have the following generalization of the theorems. k1 = k2 =

Theorem 4.2. Let n = 2, 3 and let pj , qj (j = 1, 2) satisfy (p1 − 1)(q2 − 1) = p2 q1 and either (4.12) or (4.13). In addition, for j = 1, 2 we assume that kj > 0 and that when n = 2, kj < 1/2. Suppose that for c1 = c2 : k1 p1 + k2 p2 < 1,

k1 q1 + k2 q2 < 1,

(4.15)

k1 q1 < 1,

(4.16)

and for c1 = c2 : k1 p1 < 1,

k2 p2 < 1,

k2 q2 < 1.

Let (ϕj , ψj ) ∈ Y (kj ). Then there is a positive constant ε0 = ε0 (c1 , c2 , pj , qj , n) such that if we assume max (ϕj , ψj )Y (kj ) ≤ ε for 0 < ε ≤ ε0 , then there exists j=1,2

a unique solution (u1 , u2 ) ∈ X(c1 , k1 ) × X(c2 , k2 ) of (4.4)–(4.5). Proof. First suppose that (4.12) holds. Putting p = p2 and q = q1 , we see that both conditions (4.15) and (4.16) are equivalent to k2 p < 1,

k1 q < 1.

We proceed as in the proof of Proposition 3.2 by using (2.49) with µ = mp, κ = pk2 and recalling (4.14). Then we get Lc1 [|u2 |p ]X(c1 ,k1 ) ≤ C[u2 X(c2 ,k2 ) ]p ,

(4.17)

Lc2 [|u1 | ]X(c2 ,k2 ) ≤ C[u1 X(c1 ,k1 ) ] .

(4.18)

q

q

We omit the further details. Next we consider the case where (4.13) and c1 = c2 hold. Similarly to the above, we get Lc1 [|u1 |p1 |u2 |p2 ]X(c1 ,k1 ) ≤ C[u1 X(c1 ,k1 ) ]p1 [u2 X(c2 ,k2 ) ]p2 .

(4.19)

In the application of (2.49), we take µ = m(p1 + p2 ), κ = p1 k1 + p2 k2 . Then we have µ + κ − m − 2 = k1 . We omit the further details.

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173

Finally we treat the case where (4.13) and c1 = c2 hold. Analogously to the proof of Proposition 4.1, we again get (4.19). We omit the further details. This completes the proof. 4.2. Self-similar solution When (4.3) holds, we see that the system (4.1) is invariant with respect to the scale transform uj −→ uλj (j = 1, 2), where 2

uλj (t, x) = λ α−1 uj (λt, λx)

(λ > 0).

(4.20)

We call (u1 , u2 ) a self-similar solution, if it satisﬁes (4.1) and uλj (t, x) ≡ uj (t, x),

(4.21)

for (t, x) ∈ [0, ∞) × R , j = 1, 2, and λ > 0. Our main result of this subsection reads as follows. n

Theorem 4.3. Let n = 2, 3 and let (4.2) and (4.3) hold. Assume that c1 = c2 and that n+3 n+1 <α< . n−1 n−1 Suppose (ϕj , ψj ) ∈ Y˙ (α∗ ) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 , p1 , p2 , q1 , q2 , n) such that if we assume (ϕ1 , ψ1 )Y˙ (α∗ ) +(ϕ2 , ψ2 )Y˙ (α∗ ) ≤ ˙ 1 , α∗ )×X(c ˙ 2 , α∗ ) ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ X(c of (4.4)–(4.5). Corollary 4.1. Let the assumption in Theorem 4.3 be fulﬁlled. Let the initial data take the form ϕj (x) = fj (ω)|x|− α−1 , 2

ψj (x) = gj (ω)|x|− α−1 −1 2

(4.22) ˙ with ω = x/|x|, fj ∈ C (S ), gj ∈ C(S ) (j = 1, 2). If (ϕj , ψj ) ∈ Y (α∗ ) are suﬃciently small in Y˙ (α∗ ), then there exists a unique self-similar solution of (4.4)– (4.5). 1

2

2

To prove Theorem 4.3, the following proposition is crucial. Once we establish it, a standard argument, together with (2.93), yields the conclusion. Proposition 4.2. Let the assumption in Theorem 4.3 be fulﬁlled. Let u ∈ X(c1 , α∗ ) and v ∈ X(c2 , α∗ ). Then there is a constant K0 = K0 (c1 , c2 , n, p1 , p2 , α) such that p1 p2 Lc1 [|u|p1 |v|p2 ]X(c ˙ 1 ,α∗ ) ≤ K0 [uX(c ˙ 1 ,α∗ ) ] [vX(c ˙ 2 ,α∗ ) ] .

(4.23)

Proof. We use the key assumption c1 = c2 in the following way: If we take (s, y) near the light cone c2 s = |y|, then the weight |c1 s − |y|| is equivalent to |s + |y||. On the contrary, if (s, y) is close to the other light cone c1 s = |y|, then the weight |c2 s − |y|| is equivalent to |s + |y||. Therefore we have |u(s, y)|p1 |v(s, y)|p2 p1 p2 −mα−p1 α∗ ≤ C[uX(c |λ − c2 s|−p2 α∗ ˙ 1 ,α∗ ) ] [vX(c ˙ 2 ,α∗ ) ] {|λ + s| +|λ + s|−mα−p2 α∗ |λ − c1 s|−p1 α∗ }

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for (s, y) ∈ R × Rn with λ = |y|. We shall make use of Proposition 2.5 by taking either µ = mα + p1 α∗ ,

κ = p2 α∗

or

µ = mα + p2 α∗ ,

κ = p1 α∗ .

In both cases we have µ+κ−m−2 = α∗ . (Note that α∗ > 0 if α < (n+3)/(n−1).) Besides, when n = 2, we have α∗ < 1/2 since α > 3. Thus µ + κ < 3. Moreover, for j = 1, 2 we have 2 − m < 1. (4.24) pj α∗ = pj α−1 In fact, since α > (n + 1)/(n − 1) is equivalent to (α − 1)m > 1, we have pj (2 − (α − 1)m) < pj . On the one hand, (4.2) and (4.3) imply pj ≤ α − 1. Hence (4.24) holds. Thus Proposition 2.5 yields (4.23). This completes the proof. Remark 4.2. We give a remark about the assumption on α for n = 3 in Theorems 4.1 and 4.3. In the former, α = 2 is admitted in (4.8). While it is excluded in the latter. This can be understood as follows. The global solution obtained in Theorem ˙ 1 , 1) × X(c ˙ 2 , 1), since α∗ = 1 for α = 2. On the contrary, we 4.3 belongs to X(c need to assume ν > 1 for α = 2 in Theorem 4.1 for showing the global solution exists in X(c1 , ν) × X(c2 , ν). Moreover, we can show that blow-up actually occurs for the case where α = 2 and ν = 1 as in Theorem 4.4 below. Theorem 4.4. Let n = 3 and let p1 = p2 = q1 = q2 = 1 (that is α = 2). Assume that c1 = c2 and that (ϕj , ψj ) ∈ Y (1) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 ) such that if we assume (ϕ1 , ψ1 )Y (1) +(ϕ2 , ψ2 )Y (1) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ C([0, T ∗ (ε))×R3 )×C([0, T ∗ (ε))×R3 ) of (4.4)–(4.5). Moreover, there is a constant C > 0, independent of ε, such that T ∗ (ε) ≥ exp(Cε−1 ).

(4.25)

Furthermore, when the coeﬃcients λ1 and λ2 in (4.4) and (4.5) are positive, there exist (ϕj , ψj ) ∈ Y (1) with (ϕ1 , ψ1 )Y (1) + (ϕ2 , ψ2 )Y (1) = ε(> 0) such that there is a constant C ∗ > 0, independent of ε, such that T ∗ (ε) ≤ exp(C ∗ ε−1 ).

(4.26)

Proof. First we prove (4.25). Similarly to (4.11) we have |u(s, y)||v(s, y)|

≤ CuZ(c1 ,1) vZ(c2 ,1) {λ + s−3 λ − c2 s−1 +λ + s−3 λ − c1 s−1 }

for (s, y) ∈ R × R3 with λ = |y|, where · Z(c,κ) is deﬁned by (2.76). Therefore (2.65) with µ = 3 implies Lc1 [|u||v|]Z(c1 ,1) + Lc2 [|u||v|]Z(c2 ,1) ≤ C log(2 + T )uZ(c1,1) vZ(c2 ,1) .

(4.27)

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If we deﬁne a map M [u, v] = (Kc1 [(ϕ1 , ψ1 ], Kc2 [ϕ2 , ψ2 ]) + (Lc1 [|u||v|], Lc2 [|u||v|]), then we see from (2.41) and (4.27) that M is a contraction in {(u, v) ∈ C([0, T ) × R3 ) × C([0, T ) × R3 ) : C0 ε} (u, v) ≡ uZ(c1 ,1) + vX(c2 ,1) ≤ 2C with C0 is the number in (2.41), provided that ε is suﬃciently small and that ε log(2 + T ) is bounded by some suitable constant. This will give us (4.25). Next we prove (4.26). Since (0, x−3 ) ∈ Y (1) when n = 3, we can choose ε ϕj (x) ≡ 0, ψj (x) = x−3 . 2 Then it follows from (2.8), (2.20) and Lemma 2.2 that cj t+r ε uj (t, x) ≥ Kcj [0, ψj ](t, x) = ρρ−3 dρ 4cj r |cj t−r| ≥ C1 ε(cj t + r)−1 (cj t − r)−1 for cj t > r and j = 1, 2. Recalling (2.30), we have uj cj ,1 (y) ≥ C1 ε for y ≥ 1.

(4.28)

Moreover, by Proposition 4.3 below, we have 2 y η u1 c1 ,1 (η)u2 c2 ,1 (η) dη 1− Lcj [|u1 ||u2 |]cj ,1 (y) ≥ C y η 1 for y ≥ 1. Setting U (y) = min{u1 c1 ,1 (y), u2 c2 ,1 (y)}, and recalling (4.28), we arrive at U (y) ≥ C1 ε,

U (y) ≥ C2 1

y

2 U (η)2 η dη 1− y η

for y ≥ 1.

Hence we see from Lemma 2.3 with α = 1, β = 0, κ = 1 and p = 2 that the lifespan of U (y) is bounded by exp(C ∗ ε−1 ), which gives (4.26). Proposition 4.3. Let n = 3, and let a0 > 0, 0 < a1 < a2 , p, q ≥ 1, κ∗ > 0, κ1 ∈ R and κ2 ≤ κ∗ . Then, there exists a constant C = C(a0 , a1 , a2 , p, q, κ∗ ) > 0 such that La0 [|f |p |g|q ]a0 ,p+q+qκ2 −2 (y) 2 y [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q η ≥C dη, 1− y η pκ1 1

(4.29) y ≥ 1.

Proof. From the deﬁnition of f a1 ,κ1 (y), we have |f (t, x)| ≥

f a1 ,κ1 ((a1 t − r)/a1 ) , (a1 t + r)(a1 t − r)κ1

˜ 1 , 1). (t, x) ∈ Σ(a

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˜ 1 , 1), we have (t, x) ∈ Σ(a ˜ 2 , (a1 t − r)/a1 ). Thus, from Since a1 < a2 , if (t, x) ∈ Σ(a the deﬁnition of ga2 ,κ2 (y), we have |g(t, x)| ≥

ga2 ,κ2 ((a1 t − r)/a1 ) , (a2 t + r)(a2 t − r)κ2

˜ 1 , 1). (t, x) ∈ Σ(a

Moreover, since 2a1 (a2 t − r) = (a2 − a1 )(a1 t + r) + (a1 + a2 )(a1 t − r), we have a2 − a1 a2 (a1 t + r) ≤ a2 t − r ≤ (a1 t + r) 2a1 a1 ˜ 1 , 1). Thus, we have for (t, x) ∈ Σ(a |f (t, x)|p |g(t, x)|q ≥ C

[f a1 ,κ1 ((a1 t − r)/a1 )]p [ga2 ,κ2 ((a1 t − r)/a1 )]q (t + r)p+q+qκ2 (a1 t − r)pκ1

˜ 1 , 1). Let y ≥ 1. By Proposition 2.1 and Lemma 3.2, we have for (t, x) ∈ Σ(a C La0 [|f |p |g|q ](t, x) ≥ (a0 t + r)(a0 t − r)p+q+qκ2 −2 2 (a0 t−r)/a0 [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q a0 η × dη 1− a0 t − r η pκ1 1 ˜ 0 , y). Since the function for (t, x) ∈ Σ(a 2 y η [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q dη y −→ 1− y η pκ1 1 ˜ 0 , y), we have is non-decreasing, for any (t, x) ∈ Σ(a (a0 t + |x|)(a0 t − |x|)p+q+qκ2 |La0 [|f |p |g|q ](t, x)| 2 y [f a1 ,κ1 (η)]p [ga2 ,κ2 (η)]q η ≥C dη, 1− y η pκ1 1

which implies (4.29).

Similarly to Theorem 4.2, we can state Theorems 3.4 and 4.3 in a uniﬁed way as follows. Since the proof is done by a straightforward computation, we omit it. Theorem 4.5. Let n = 2, 3 and let pj , qj , kj (j = 1, 2) satisfy the assumptions in Theorem 4.2. Let (ϕj , ψj ) ∈ Y˙ (kj ). Then there is a positive constant ε0 = ε0 (c1 , c2 , pj , qj , n) such that if we assume max (ϕj , ψj )Y˙ (kj ) ≤ ε for 0 < ε ≤ ε0 , j=1,2

˙ 1 , k1 ) × X(c ˙ 2 , k2 ) of (4.4)–(4.5). then there exists a unique solution (u1 , u2 ) ∈ X(c 4.3. Generalization To make clear the essential structure of the system (4.1) we consider the following generalized version of it: N

(∂ ∂t2 − c2j ∆)uj =

pjkl Akl |ul |qjkl , j |uk |

(t, x) ∈ [0, ∞) × Rn , (4.30)

k,l=1

uj (0, x) = ϕj (x),

∂t uj (0, x) = ψj (x),

x ∈ Rn ,

(4.31)

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where 1 ≤ j ≤ N , pjkl , qjkl ≥ 1, and Akl j are real constants. We restrict ourselves to the case where there is a constant α ≥ 2 such that pjkl + qjkl ≡ α

for all j, k, l = 1, · · · , N,

(4.32)

since the general case can be formulated as in Theorems 4.2 and 4.5. Moreover, we assume that the propagation speeds are distinct, that is ck = cl if k = l. The crucial point of the argument in the preceding subsections is to make use of the interaction between uk and ul with k = l. Thus we need to exclude self-interaction terms from the nonlinearities. To do so, we pose the non-resonant assumption on them, i.e., Akk j = 0 for all j, k = 1, · · · , N.

(4.33)

We treat the problem (4.30)–(4.31) in the following integral form: uj = Kcj [ϕj , ψj ] +

N

pjkl Akl |ul |qjkl ] in j Lcj [|uk |

[0, ∞) × Rn ,

(4.34)

k,l=1

where 1 ≤ j ≤ N . We look for solutions of (4.34) in the following spaces: X = {u = (u1 , u2 , · · · , uN ) ∈ C([0, ∞) × Rn )N :

(4.35)

uX ≡ max uj X(cj ,ν) < ∞}, 1≤j≤N

and X˙ = {u = (u1 , u2 , · · · , uN ) ∈ C([0, ∞) × Rn )N : uX˙ ≡ max uj X(c ˙ j ,ν) < ∞}.

(4.36)

1≤j≤N

Similarly to the proof of Theorem 4.1, one can show the following. Theorem 4.6. Let n = 2, 3 and let (4.32) and (4.33) hold. Assume that c1 , . . . , cN are diﬀerent from each other and that either (4.7) or (4.8) holds for n = 3, and that (4.9) holds for n = 2. Suppose (ϕj , ψj ) ∈ Y (ν) (1 ≤ j ≤ N ). Then there is a positive constant ε0 = ε0 (c1 , c2 , n, pjkl , qjkl , ν) such that if we assume max (ϕj , ψj )Y (ν) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution 1≤j≤N

(u1 , . . . , uN ) ∈ X of (4.34). Next we consider the self-similar solution (u1 , . . . , uN ) ∈ X˙ which is a solution of (4.30) satisfying (4.21) for all (t, x) ∈ [0, ∞) × Rn , j = 1, 2, . . . , N , and λ > 0. The following analogue to Theorem 2.4 is concerned with the existence and nonexistence of the self-similar solution. Theorem 4.7. Let n = 2, 3 and let (4.32) and (4.33) hold. Assume that c1 , . . . , cN are diﬀerent from each other and that (ϕj , ψj ) ∈ Y˙ (α∗ ) (j = 1, . . . , N ). (i) Suppose n+3 n+1 <α< . n−1 n−1

178

H. Kubo and M. Ohta Then there is a positive constant ε0 = ε0 (c1 , c2 , n, pjkl , qjkl , ν) such that if we assume max (ϕj , ψj )Y˙ (α∗ ) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique 1≤j≤N

solution (u1 , . . . , uN ) ∈ X˙ of (4.34). (ii) Let α = (n + 1)/(n − 1). Then the statement (i) does NOT hold. Proof. To show the part (ii ), it suﬃces to take the data as ϕj (x) = 0,

ψj (x) = ε|x|− α−1 −1 2

for x ∈ Rn∗

for ε > 0 (see also the proof of Theorem 2.4). While, the part (i ) can be proved in a similar way to the proof of Theorem 4.3.

5. Semilinear system, III As we have seen in the Subsection 4.3, the assumption (4.33) plays an essential role in the analysis of the system (4.30). The aim of this section is to study the system for the case where (4.33) is not satisﬁed. To be more precise, we consider the following one as a simplest example:

2 (t, x) ∈ [0, ∞) × R3 , (∂ ∂t − c21 ∆)u1 = |u1 |p1 |u2 |p2 , (5.1) 2 2 q (∂ ∂t − c2 ∆)u2 = |u1 | , (t, x) ∈ [0, ∞) × R3 , where p1 , p2 ≥ 1, q > 1. This system was studied by the authors in [68]. We can regard (5.1) as an intermediate case between (3.1) and (4.1). The point is that the right-hand side of the ﬁrst equation in (5.1) is involved by a product of u1 and u2 , while that of the second one does not. For simplicity, we focus on the case where p1 = p2 = 1.

(5.2)

The exposition for the general case where p1 ≥ 1, p2 ≥ 1 is complicated, although the real proof for large values of p1 and p2 is easier because of the “smallness” of solutions under our consideration. For this reason, we prefer to take p1 = p2 = 1. The main result of this section is roughly stated as follows. Theorem 5.1. Suppose that c1 = c2 and that (5.2) holds. Then for the initial value problem of (5.1) we have: (i) If 1 < q < 3, then blow-up occurs. (ii) If q > 3, then small data global existence holds. (iii) Let q = 3. If c1 > c2 , then blow-up occurs. While, when c1 < c2 , small data global existence holds. It is remarkable that when q = 3, the result depends on the order of the speeds of propagation. We mainly consider this peculiar case and prove the blowup part in Subsection 5.1 and the existence part in Subsection 5.2. To carry out the proof, the argument prepared in Section 2 is not suﬃcient. In other words, we need additionally a something new approach for the case q = 3.

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5.1. Blow-up First of all, we precisely state the blow-up part of Theorem 5.1. Let us consider the system

2 (∂ ∂t − c21 ∆)u1 = |u1 ||u2 |, (t, x) ∈ [0, ∞) × R3 , (5.3) 2 2 q (t, x) ∈ [0, ∞) × R3 (∂ ∂t − c2 ∆)u2 = |u1 | , with the initial data uj (0, x) = 0,

∂t uj (0, x) = εgj (x),

x ∈ R3 (j = 1, 2).

(5.4)

Here q > 1, cj > 0, ε > 0, and gj ∈ C(R3 ) (j = 1, 2) satisﬁes gj (x) ≥ 0 for all x ∈ R3 ,

g1 (0) > 0.

(5.5)

Then we have the following. Theorem 5.2. Let c1 = c2 and 1 < q ≤ 3. Suppose that g1 , g2 ∈ C(R3 ) satisfy (5.5). Then for suﬃciently small ε > 0 the solution (u1 , u2 ) of (5.3)–(5.4) blows up in a ﬁnite time T ∗ (ε), if either q = 3 and c1 > c2 or 1 < q < 3. Moreover, there exists a constant A > 0, independent of ε, such that ⎧ −3 if q = 3 and c1 > c2 ⎨ exp(Aε ) ∗ −q(2+q)/(3−q)2 Aε if 1 < q < 3 and c1 > c2 . T (ε) ≤ (5.6) ⎩ −2q/(3−q)2 Aε if 1 < q < 3 and c1 < c2 Remark 5.1. As for the case where q = 3 and c1 > c2 , Katayama and Matsumura [45] recently proved that there is a constant B > 0, independent of ε, such that T ∗ (ε) ≥ exp(Bε−3 ).

(5.7)

Proof. We treat the problem (5.3)–(5.4) in the integral form: u1 = εJ Jc1 [g1 ] + Lc1 [|u1 ||u2 |] in Jc2 [g2 ] + Lc2 [|u1 | ] u2 = εJ q

[0, ∞) × R3 ,

(5.8)

in [0, ∞) × R .

(5.9)

3

Basically we follow the proof of Theorem 4.4. In particular, the proof for the case where 1 < q < 3 can be done analogously and less hard. For this reason, we concentrate on the case where q = 3 and c1 > c2 . It is the most delicate one in the sense that the result depends not only on the exponent q but also on the propagation speeds c1 and c2 . By (5.8), (5.9) and (5.5), we have u1 (t, x) ≥ εJ Jc1 [g1 ](t, x),

(t, x) ∈ [0, ∞) × R3 ,

u1 (t, x) ≥ Lc1 [|u1 ||u2 |](t, x), u2 (t, x) ≥ Lc2 [|u1 |3 ](t, x),

(5.10)

(t, x) ∈ [0, ∞) × R , 3

(t, x) ∈ [0, ∞) × R3 .

(5.11) (5.12)

As in the proof of (2.26), from (5.5), we see that there exists a constant C > 0 such that u1 (t, x) ≥ Cεr−1 for (t, x) ∈ E. (5.13)

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Here we put E := {(t, x) ∈ [0, ∞) × R3 : |c1 t − |x|| ≤ δ/2, c1 t + |x| ≥ δ}. Based on this estimate, we shall show u1 c1 ,2 (y) ≥ C1 ε4 ,

u2 c2 ,1 (y) ≥ C2 ε3

for y ≥ 1,

(5.14)

provided 0 < δ ≤ min{c2 , 2c1 (c1 − c2 )/(5c1 + c2 )}. Since δ ≤ c2 , by (5.12), (2.21) and (5.13), we have c2 t+r dξ Cε3 δ/2 Cε3 u2 (t, x) ≥ dη ≥ 2 r −δ/2 (c2 t + r)(c2 t − r) c2 t−r ξ ˜ 2 , 1). Thus the second inequality in (5.14) holds true. for (t, x) ∈ Σ(c To prove the ﬁrst one, we prepare the following estimate: u2 (t, x) ≥

Cε3 (c1 t − r) , (t + r)3

(t, x) ∈ Ω,

(5.15)

where we set Ω = {(t, x) ∈ [0, ∞) × R3 : c1 t − |x| ≥ 0, |x| − c2 t ≥ δ}. By (5.12), (2.21) and (5.13), we have λ2 (r,t) Cε3 0 dλ dη , (t, x) ∈ Ω, u2 (t, x) ≥ 2 r −δ/2 λ1 (r,t) λ where we put λ1 (r, t) =

c1 (r − c2 t), c1 − c2

λ2 (r, t) =

c1 (r + c2 t). c1 + c2

Since λ2 (r, t) − λ1 (r, t) = 2c1 c2 (c1 t − r)/(c21 − c22 ), we get (5.15). By (5.13) and (5.15), we have |u1 (t, x)||u2 (t, x)| ≥

Cε4 (c1 t − r) , r(c1 t + r)3

(t, x) ∈ E ∩ Ω.

Since δ ≤ 2c1 (c1 − c2 )/(5c1 + c2 ), by (5.11) and (2.21), we have c1 t+r dξ Cε4 δ/2 Cε4 η dη ≥ u1 (t, x) ≥ 3 r 0 (c1 t + r)(c1 t − r)2 c1 t−r ξ ˜ 1 , 1), which implies the ﬁrst inequality in (5.14). for (t, x) ∈ Σ(c Unfortunately, the ﬁrst estimate in (5.14) is not enough to show the blow-up result because of the fast decay with respect to (c1 t − r). Thus our next step is to improve it. To this end, for 0 ≤ κ ≤ 2 we set U1,κ (y) = u1 c1 ,κ (y),

U2 (y) = u2 c2 ,1 (y).

Then (5.14) implies U1,2 (y) ≥ C1 ε4 ,

U2 (y) ≥ C2 ε3 ,

y ≥ 1.

(5.16)

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181

Moreover, since c2 < c1 , by Propositions 3.1 and 4.3, we see that there exist positive constants C3 = C3 (c1 , c2 ) and C4 = C4 (c1 , c2 ) such that for any κ ∈ [0, 2] 2 y U1,κ (η)U U2 (η) η dη, y ≥ 1, (5.17) U1,κ (y) ≥ C3 1− y η 1 2 y U1,κ (η)3 η U2 (y) ≥ C4 dη, y ≥ 1. (5.18) 1− y η 3κ 1 Note that C3 and C4 do not depend on κ ∈ [0, 2]. Now (5.16) and (5.17) yield 2 y U1,κ (η) η dη, y ≥ 1, (5.19) U1,κ (y) ≥ 16b 1− y η 1 where b = C2 C3 ε3 /16. Especially (5.16) and (5.19) with κ = 2 give 2 y η U1,2 (η) dη, y ≥ 1 U1,2 (y) ≥ a, U1,2 (y) ≥ 16b 1− y η 1

(5.20)

with a = C1 ε4 . One can show that U1,2 (y) grows in y, by using the following lemma. Lemma 5.2. Let a > 0 and 0 < b ≤ 1. Assume that f (y) satisﬁes 2 y f (η) η dη, y ≥ 1. 1− f (y) ≥ a, f (y) ≥ 16b y η 1 Then we have f (y) ≥

a b y , 4

y ≥ 1.

Proof. Put g(y) = (a/4)y b . Then we have g(y) < f (y) for any y ∈ [1, 41/b ). Moreover, since 0 < b ≤ 1 and & 2

y 1 y/2 b−1 1 y b η b−1 η dη ≥ η dη = −1 , 1− y 4 1 4b 2 1 we have

2 g(η) η dη, y ≥ 41/b . 1− y η 1 By the comparison argument, we see that f (y) ≥ g(y) holds for any y ≥ 1. This completes the proof.

y

g(y) ≤ 16b

Applying Lemma 5.2 to (5.20), we get a U1,2 (y) ≥ y b , 4

y ≥ 1.

(5.21)

˜ 1 , y) with y ≥ 1, we have By (5.21) and the deﬁnition of U1,κ (y), for (t, x) ∈ Σ(c b c1 t − |x| a c1 t − |x| 2 . (c1 t + |x|)(c1 t − |x|) |u1 (t, x)| ≥ U1,2 ≥ c1 4 c1

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Again by the deﬁnition of U1,κ (y), we have U1,2−b (y) ≥

a , 4c1 b

y ≥ 1.

Repeating this procedure 2n times, we obtain U1,2−2nb (y) ≥

a , (4cb1 )2n

y ≥ 1,

which implies (c1 t + |x|)(c1 t − |x|)2−2nb |u1 (t, x)| ≥ U1,2−2nb (y) ≥

a (4cb1 )2n

˜ 1 , y). Combining this with c1 t − |x| ≥ c1 y, we obtain for y ≥ 1 and (t, x) ∈ Σ(c U1,2−nb (y) ≥

a y nb , 42n c1 nb

y ≥ 1.

(5.22)

Let be the smallest natural number satisfying 3(2 − b) ≤ 1. Since a = C1 ε4 and b = C2 C3 ε3 /16, we see that = O(ε−3 ) and a ≥ exp(−C C5 ε−3 ) 42 c1 b

(5.23)

with a positive constant C5 . By (5.22), (5.23) and b ≥ 1, we have C5 ε−3 )y b ≥ exp(−C C5 ε−3 )y, U1,2−b (y) ≥ exp(−C

y ≥ 1.

(5.24)

Thus, we see that U1,2−b (y) ≥ 1 for y ≥ α1 := exp(C C5 ε−3 ). Moreover, by (5.18) with κ = 2 − b, we have 2 y U1,2−b (η)3 η U2 (y) ≥ C4 dη 1− y η 3(2−b) α1 2 y/2 C4 y 1 η ≥ C4 dη ≥ log , y ≥ 2α1 . 1− y η 4 2α1 α1 C4 ). Thus, U2 (y) ≥ 1 for y ≥ α∗ := 2α1 exp(4/C Finally, rescaling as U (z) = min{U1,2−b (α∗ z), U2 (α∗ z)} and using 3(2−b) ≤ 1, we ﬁnd from (5.17) and (5.18) that 2 z U (ζ)2 ζ dζ, z ≥ 1, U (z) ≥ 1, U (z) ≥ C6 1− z ζ 1 where C6 = min{C C3 , C4 }. By Lemma 2.3, we see that U (z) blows up in a ﬁnite time. Hence, the classical solution of (5.3)–(5.4) blows up in a ﬁnite time. Moreover, the lifespan T ∗ (ε) is estimated from above by exp(C ∗ ε−3 ) with a suitable positive constant C ∗ , by the choice of α∗ and α1 . This completes the proof.

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183

5.2. Small data global existence Let us consider the following integral version of the initial value problem (5.3): u1 = Kc1 [ϕ1 , ψ1 ] + Lc1 [|u1 ||u2 |] in [0, ∞) × R3 ,

(5.25)

u2 = Kc2 [ϕ2 , ψ2 ] + Lc2 [|u1 |q ]

(5.26)

in [0, ∞) × R3 .

To deal with the case q = 3, we need the following modiﬁed L∞ -space: W (c, κ) = {u ∈ C([0, ∞) × R3 ) : uW (c,κ) < ∞}, uW (c,κ) =

sup (t,x)∈[0,∞)×R3

t + |x|ct − |x|κ {w(|x|, t)}−1 |u(t, x)|,

where κ > 0, and where we deﬁned ⎧ ⎪ 1, ⎨ c2 t − r w(r, t) = log(2 + c2 t − r), 1+ ⎪ r ⎩ {log(2 + |c1 t − r|)}2 ,

(r, t) ∈ Ω1 , (r, t) ∈ Ω2 ,

(5.27)

(r, t) ∈ Ω3 ,

with Ω1 = {(r, t) ∈ [0, ∞)2 : c2 t < r}, Ω2 = {(r, t) ∈ [0, ∞)2 : (c1 + c2 )t/2 < r < c2 t}, Ω3 = {(r, t) ∈ [0, ∞)2 : r < (c1 + c2 )t/2}. Now we precisely state the existence part of Theorem 5.1. Theorem 5.3. Let c1 = c2 . (i) Let q > 3 and (ϕj , ψj ) ∈ Y (q − 2) (j = 1, 2). Then there is a positive constant ε0 = ε0 (c1 , c2 , q) such that if we assume max (ϕj , ψj )Y (q−2) ≤ ε for 0 < j=1,2

ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ X(c1 , q−2)×X(c2 , q−2) of (5.25)–(5.26). (ii) Let q = 3, 0 < c1 < c2 and 1/3 < κ < 1. Suppose (ϕ1 , ψ1 ) ∈ Y (κ), (ϕ2 , ψ2 ) ∈ Y (1). Then there is a positive constant ε0 = ε0 (c1 , c2 ) such that if we assume (ϕ1 , ψ1 )Y (κ) + (ϕ2 , ψ2 )Y (1) ≤ ε for 0 < ε ≤ ε0 , then there exists a unique solution (u1 , u2 ) ∈ W (c1 , κ) × X(c2 , 1) of (5.25)–(5.26). Proof. First we consider the part (i). Since c1 = c2 , analogously to the proof of Proposition 4.1, we get Lc1 [|u1 ||u2 |]X(c1 ,q−2) ≤ Cu1 X(c1 ,q−2) u2 X(c2 ,q−2) .

(5.28)

In the application of (2.49), it suﬃces to take µ = q, κ = q − 2 (> 1). While, using (2.49) with µ = q, κ = q(q − 2) (> 3), we get Lc2 [|u1 |q ]X(c2 ,q−2) ≤ C[u1 X(c1 ,q−2) ]q .

(5.29)

We omit the further details. Next we consider the part (ii). Since X(c1 , κ) → W (c1 , κ), (2.41) implies Kc1 [ϕ1 , ψ1 ]W (c1 ,κ) ≤ C0 (ϕ, ψ)Y (κ) .

(5.30)

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Therefore, once we establish Propositions 5.1 and 5.2 below, the standard contraction argument gives the conclusion. Proposition 5.1. Let 0 < c1 < c2 and 0 < κ < 1. Then there is a constant C = C(c1 , c2 , κ) > 0 such that Lc1 [f g]W (c1 ,κ) ≤ Cf W (c1 ,κ) gX(c2 ,1) .

(5.31)

Proof. We see from (2.38) that |Lc1 [f g](t, x)| ≤ f W (c1 ,κ) gX(c2,1) I(r, t) for any (t, x) ∈ [0, ∞) × R3 with r = |x|, where λw(λ, s) 1 dλ ds, I(r, t) = 2 c s − λκ c s − λ 2c1 r s + λ 1 2 Dc1 (r,t)

(5.32)

Dc (r, t) = {(λ, s) ∈ [0, ∞)2 : 0 ≤ s ≤ t, |c(t − s) − r| ≤ λ ≤ c(t − s) + r}. Thus it suﬃces to show that I(r, t) ≤ Ct + r−1 c1 t − r−κ w(r, t)

(5.33)

for any (r, t) ∈ [0, ∞) . In the following, we use the notation: 2

ξ = c1 s + λ,

ηj = cj s − λ,

(5.34)

where j = 1, 2. First we suppose (r, t) ∈ Ω1 . In this case we have Dc1 (r, t) ⊂ Ω1 . Moreover, r − c1 t is equivalent to r + t. Hence, c1 s + λ ≥ λ − c1 s ≥ C(t + r)

for

(λ, s) ∈ Dc1 (r, t).

Therefore from (5.32) and (5.27) we have dλ ds C 1+κ . t + r I(r, t) = r λ − c2 s Dc1 (r,t)

(5.35)

(5.36)

Changing the variables by (5.34) with j = 2 and putting c1 + c2 2c1 ξ ∗ (η2 ) = (r − c1 t) + η2 , c2 − c1 c2 − c1 we see that λ = r − c1 (t − s) is equivalent to ξ = ξ ∗ (η2 ). Thus from (5.36) we have C c2 t−r dη2 r+c1 t 1+κ I(r, t) ≤ dξ t + r r −(r+c1 t) η2 ξ∗ (η2 ) 2c1 C c2 t−r c2 t − r − η2 = dη2 c2 − c1 r −(r+c1 t) η2 |η2 | C 0 dη2 , ≤ r −(r+c1 t) η2 since c2 t < r. Therefore t + r1+κ I(r, t) is bounded. This means that (5.33) is valid for (r, t) ∈ Ω1 .

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Next we consider the case (r, t) ∈ Ω2 . Again, r − c1 t is equivalent to r + t, hence (5.35) is still valid. Therefore from (5.32) and (5.27) we have λ dλ ds C (5.37) t + r1+κ I(r, t) ≤ r s + λc2 s − λ Dc1 (r,t)∩Ω2 (c2 s − λ) log(2 + c2 s − λ) C 1 dλ ds + r r + t c2 s − λ Dc1 (r,t) C c2 t−r c2 t − r − η2 ≤ dη2 r −(r+c1 t) η2 c2 t−r C 1 + (c2 t − r − η2 ) log(2 + η2 ) dη2 , r r + t 0 where we have changed the variables by (5.34) with j = 2. It is easy to see that the second term of the right-hand side is estimated by some constant times (c2 t − r)2 log(c2 t − r) ≤ w(r, t) rr + t for (r, t) ∈ Ω2 . While, since 0 < c2 t − r − η2 < 2(c2 t − r) for |η2 | < c2 t − r, and 0 < c2 t − r − η2 < 2|η2 | for η2 < −(c2 t − r) < 0, we see that the ﬁrst term of the right-hand side of (5.37) is estimated by some constant times c2 t − r r

c2 t−r

−(c2 t−r)

1 dη2 + η2 r

−(c2 t−r)

−(c1 t+r)

dη2 ≤ 2w(r, t)

for (r, t) ∈ Ω2 (recall c2 > c1 ). In conclusion, we obtained (5.33) for (r, t) ∈ Ω2 . Finally, we treat the case (r, t) ∈ Ω3 . Note that w(r, t) ≤ C log(2 + |c2 t − r|)

for

(r, t) ∈ Ω1 ∪ Ω2 ,

(5.38)

since 0 < c2 t − r < (c2 − c1 )r/(c2 + c1 ). Moreover, c1 s − λ is equivalent to s + λ if (λ, s) ∈ Ω1 ∪ Ω2 , and c2 s − λ is so if (λ, s) ∈ Ω3 . Having these observations in mind, we see from (5.32) and (5.27) that log(2 + |c2 s − λ|) C dλ ds I(r, t) ≤ r s + λ1+κ c2 s − λ Dc1 (r,t) C {log(2 + |c1 s − λ|)}2 + dλ ds 2 κ r Dc1 (r,t) s + λ c1 s − λ c2 ξ/c1 C c1 t+r 1 log(2 + |η2 |) ≤ dη2 dξ 1+κ r |c1 t−r| ξ η2 −ξ ξ {log(2 + |η1 |)}2 C c1 t+r 1 + dξ dη1 , r |c1 t−r| ξ2 η1 κ −ξ

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where we have changed the variables by (5.34) with j = 2 in the ﬁrst term, and with j = 1 in the second term. Since κ < 1, we have C c1 t+r {log(2 + ξ)}2 dξ I(r, t) ≤ r |c1 t−r| ξ1+κ ≤

C{log(2 + |c1 t − r|)}2 . t + rc1 t − rκ

At the last step, we have used (2.71). Thus we get (5.33) for (r, t) ∈ Ω3 . This completes the proof. Proposition 5.2. Let 0 < c1 < c2 and κ > 1/3. Then there exists C = C(c1 , c2 , κ) > 0 such that Lc2 [f 3 ]X(c2 ,1) ≤ C[f W (c1 ,κ) ]3 .

(5.39)

Proof. We see from (2.38) that |Lc2 [f 3 ](t, x)| ≤ [f W (c1 ,κ) ]3 I(r, t) for any (t, x) ∈ [0, ∞) × R3 with r = |x|, where 1 λ{w(λ, s)}3 I(r, t) = dλ ds. 3 3κ 2c2 r Dc2 (r,t) s + λ c1 s − λ

(5.40)

Thus it suﬃces to show that I(r, t) ≤ Ct + r−1 c1 t − r−1

(5.41)

for any (r, t) ∈ [0, ∞)2 . By (5.38) and the fact that c1 s−λ is equivalent to s+λ if (λ, s) ∈ Ω1 ∪Ω2 , we get λ{log(2 + |c2 s − λ|)}3 C dλ ds I(r, t) ≤ r s + λ3+3κ Dc2 (r,t) C λ{log(2 + |c1 s − λ|)}6 + dλ ds 3 3κ r Dc2 (r,t) s + λ c1 s − λ ξ 1 C c2 t+r dξ {log(2 + |η2 |)}3 dη2 ≤ r |c2 t−r| ξ2+3κ −ξ c1 ξ/c2 C c2 t+r 1 {log(2 + |η1 |)}6 + dξ dη1 , r |c2 t−r| ξ2 η1 3κ −ξ where we have changed the variables by (5.34) with j = 2 in the ﬁrst term, and with j = 1 in the second term. Since 3κ > 1, we have C c2 t+r dξ I(r, t) ≤ , r |c2 t−r| ξ2 which implies (5.41). This completes the proof.

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By (5.30), Propositions 5.1 and 5.2 we can prove the part (ii). This completes the proof of Theorem 5.3.

6. Nonlinear system In the preceding sections we have studied the hyperbolic system for the case where the nonlinearity depends only on the unknowns themselves. From now on, we shall consider the system whose nonlinearity is involved by the derivatives of the unknowns. In this case the following Fujita type exponent comes into play: 2 (n ≥ 2). (6.1) pc (n) = 1 + n−1 Notice that pc (n) < p0 (n), since Φ(n, pc (n)) = −1 with Φ(n, p) the function in (2.3). The number can be introduced by the following heuristic argument. Let us consider the single wave equation ∂t u|p , (∂ ∂t2 − c2 ∆)u = |∂

(t, x) ∈ [0, ∞) × Rn ,

(6.2)

where c > 0 and p > 1. Suppose that we a priori know that the solution to (6.2) is asymptotically free. Namely, it is close to a solution v(t, x) of the homogeneous wave equation (2.94) in the sense of the energy for large values of t. Then, in virtue of Proposition 2.7, we can expect ∞ u(t) − v(t)E(c) ≤ (n + 1) F (∂ ∂t u)(τ )L2 (Rn ) dτ −→ 0, (6.3) t

as t → +∞, where F (u) = |u|p . As it is well known, v(t)E(c) is conserved and n−1 |∂ ∂t v(t, x)| ≤ Ct− 2 for (t, x) ∈ [0, ∞) × Rn . Therefore we have ∞ ∞ n−1 F (∂ ∂t v)(τ )L2 (Rn ) dτ ≤ Cv(0)E(c) τ − 2 (p−1) dτ. t

t

It is clear that when p > pc (n), the last integral tends to zero as t → +∞. Hence the condition p > pc (n) seems to ensure (6.3). We come back to the systems of nonlinear wave equations of the form (∂ ∂t2 − c2i ∆)ui = Fi (u, ∂u, ∂∇u), (t, x) ∈ [0, ∞) × Rn (1 ≤ i ≤ N ),(6.4) (1 ≤ i ≤ N ),(6.5) ui (x, 0) = εffi (x), ∂t ui (x, 0) = εgi (x), x ∈ Rn ∂t , ∇), ∇ = (∂ ∂x1 , . . . , ∂xn ), ε > 0 is a small parawhere ∂ = (∂0 , ∂1 , . . . , ∂n ) = (∂ meter, ci is a positive constant, fi , gi are real-valued functions which belong to C0∞ (Rn ), and u = (u1 , . . . , uN ) is a RN -valued unknown function of (t, x). In what follows, the nonlinear term F = (F F1 , . . . , FN ) is supposed to be a smooth function of (u, ∂u, ∂∇u). Moreover, we assume that F vanishes to the p-th order near the origin with some integer p ≥ 2. Namely, there exist positive constants C and δ such that N |F Fi (u, v, w)| ≤ C(|u|p + |v|p + |w|p ) for |u| + |v| + |w| ≤ δ, (6.6) i=1

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where the variables v = (vaj ; 0 ≤ a ≤ n, 1 ≤ j ≤ N ) ∈ R(1+n)×N , w = (warj ; 0 ≤ a ≤ n, 1 ≤ r ≤ n, 1 ≤ j ≤ N ) ∈ R(1+n)×n×N correspond to (∂a uj ), (∂a ∂r uj ), respectively. In the sequel, we use the abbreviation F (u, ∂u, ∂∇u) = O(|u|p + |∂u|p + |∂∇u|p ). Without loss of generality, we may assume that F is aﬃne with respect to ∂∇u (see, e.g., Courant and Hilbert [13], Chapter I, Section 7). This means F = (F F1 , · · · , FN ) has the following form: Fi (u, v, w) =

N n

αabj i (u, v)wabj + βi (u, v)

(6.7)

a,b=0 j=1

with some p−1 αabj + |v|p−1 ) and βi (u, v) = O(|u|p + |v|p ), i (u, v) = O(|u|

where a, b = 0, . . . , n, i, j = 1, . . . , N . In order to guarantee the existence of the local solution for the system, we assume baj abi αabj i (u, v) = αi (u, v) = αj (u, v).

(6.8)

Besides, we denote by F (p) the p-th order term of Taylor’s expansion for F around the origin (u, v, w) = (0, 0, 0). First we recall known results for the case where the propagation speeds coincide each other, that is, c1 = c2 = · · · = cN . It has been proved by John [39] that the blow-up occurs for the single equations (6.2) with p = pc (3) = 2, n = 3, and (∂ ∂t2 − c2 ∆)u = u ∂t u,

(t, x) ∈ [0, ∞) × R3 .

(6.9)

Agemi [1] extended these results to the case n = 2 (see also [93, 90, 58]). Therefore, when n = 2, 3, we see that the problem (6.4)–(6.5) with p = pc (n) in (6.6) does not admit the global classical solution in general, however small the initial data is. Nevertheless, Christodoulou [12] and Klainerman [54] independently established small data global existence for the case where n = 3 and p = 2 in (6.6) by diﬀerent approaches, provided that F (2) has some algebraic structure called null condition which will be deﬁned in the Deﬁnition 6.1 below. A corresponding result for the case where n = 2 and p = pc (2) = 3 in (6.6) was obtained by Katayama [42]. Moreover, when F is independent of the unknowns themselves, Alinhac [8] showed that small data global existence holds for the case where n = 2 and p = 2 in (6.6), provided that both F (2) and F (3) satisfy the null condition (also see [27, 42] for the scalar case N = 1). For the super critical case p > pc (n), we refer to [94, 53, 106, 74, 75, 31, 113] and references cited therein.

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Deﬁnition 6.1. For µ, ν ∈ RN and X = (X0 , X1 , . . . , Xn ) ∈ Rn+1 we put V (µ, X) = (Xa µj ; 0 ≤ a ≤ n, 1 ≤ j ≤ N ), W (ν, X) = (Xa Xr νj ; 0 ≤ a ≤ n, 1 ≤ r ≤ n, 1 ≤ j ≤ N ). We say F (u, v, w) = (F F1 , . . . , FN )(u, v, w) satisﬁes the null condition if Fi (λ, V (µ, X), W (ν, X)) = 0

(6.10)

holds for all i ∈ {1, . . . , N }, λ, µ, ν ∈ RN and X ∈ Rn+1 satisfying X02 = c2i (X12 + · · · + Xn2 ). On the contrary, when p = 2, the null condition is necessary to ensure small data global existence if F (u, v, w) ≡ F (v, w). In fact, a blow-up result was obtained by Alinhac [6] for n = 2 and by [7] for the cases where either n = 3 or n = 2 and F (2) satisﬁes the null condition, together with precise estimates for the lifespan. (See also [40, 32, 27, 33].) However, when F depends on u itself, the situation is diﬀerent. Indeed, small data global existence holds for the single wave equation (∂ ∂t2 − c2 ∆)u = u ∆u,

(t, x) ∈ [0, ∞) × R3 .

(6.11)

Such an interesting result was shown by Lindblad [77] for the radially symmetric case and by Alinhac [9] for the general case. Next we turn our attention to the case where the propagation speeds are distinct. To our knowledge, the eﬀect of the discrepancy of the propagation speeds was ﬁrstly pointed out by Kovalyov [57] for the case where n = 2, 3, p = pc (n) and F (u, v, w) = F (v). Namely there is a possibility to show small data global existence for such a nonlinearity that does not satisfy the null condition. Firstly let us consider the case n = 3. For the sake of simplicity of the exposition, we shall assume that αabj i (u, v) ≡ 0 in (6.7). In other words, we consider the case where the nonlinear function F is independent of the second derivatives of the unknowns. Then F verifying (6.6) with p = 2 can be written as N

Fi (u, ∂u) =

Aj,k i uj uk +

Bia,j,k uj ∂a uk

(6.12)

a=0 j,k=1

j,k=1

+

n N

N n

Dia,b,j,k ∂a uj ∂b uk + Hi (u, ∂u),

a,b=0 j,k=1

where

Aj,k i ,

Bia,j,k ,

Dia,b,j,k

are constants and

Hi (u, v) = O(|u|3 + |v|3 ).

(6.13)

In order to regard Hi (u, ∂u) as a higher-order perturbation, we need the assumption = 0 for all i, j, k = 1, . . . , N, Aj,k i

(6.14)

because of Theorem 5.2 with q = 3 and c1 > c2 . Moreover we consider the following two cases separately; the second term of the right-hand side in (6.12) vanishes and it does not.

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H. Kubo and M. Ohta First we consider the former case, namely Bia,j,k = 0

for all i, j, k = 1, . . . , N and a = 0, . . . , n.

(6.15)

For the case where Hi (u, ∂u) = Hi (∂u), that is, F is independent of the unknowns themselves, Yokoyama [114] proved that the small data global existence holds for (6.4)-(6.5) if (6.14), (6.15) and the following assumption on the quadratic part of F (u, v, w) are valid: For each i ∈ {1, . . . , N }, ˜ (ν, X)) = 0 Fi (λ, V˜ (µ, X), W

(6.16)

holds for all λ, µ, ν ∈ RN and X = (X0 , X1 , . . . , Xn ) ∈ Rn+1 satisfying X02 = c2i (X12 + · · · + Xn2 ). Here we put V˜ (µ, X) = (Xa δij µj ; 0 ≤ a ≤ n, 1 ≤ j ≤ N ), ˜ (ν, X) = (Xa Xr δij νj ; 0 ≤ a ≤ n, 1 ≤ r ≤ n, 1 ≤ j ≤ N ), W and δij is the Kronecker delta. We remark that Fi satisfying the condition (6.16) does not always verify (6.10). For instance, not only the nonlinearity of the system ∂t u1 )(∂ ∂t u2 ), (∂ ∂t2 − c21 ∆)u1 = a(∂ (∂ ∂t2

−

c22 ∆)u2

(t, x) ∈ [0, ∞) × R3 ,

(6.17)

(t, x) ∈ [0, ∞) × R ,

(6.18)

3

= b(∂ ∂t u1 )(∂ ∂t u2 ),

but also that of the system ∂t u2 )2 , (∂ ∂t2 − c21 ∆)u1 = a(∂ (∂ ∂t2

−

c22 ∆)u2

2

= b(∂ ∂t u1 ) ,

(t, x) ∈ [0, ∞) × R3 ,

(6.19)

(t, x) ∈ [0, ∞) × R3 ,

(6.20)

does not satisfy (6.10) unless a = b = 0, while they enjoy (6.16) for all a, b ∈ R. Thus the above result implies the existence theorem for these systems if c1 = c2 . Another proof based on the work of Klainerman and Sideris [56] was given by Sideris and Tu [98], Sogge [99] and Hidano [30]. Without such a restriction on Hi (u, v), Kubota and Yokoyama [73] ﬁrstly treated the problem, and then Katayama [43] obtained the existence result under the same assumption as in [114]. In conclusion, comparing this with the results for semilinear systems obtained in Sections 4 and 5, we see that the eﬀect of the discrepancy of the propagation speeds strongly appears in the case where the nonlinearity depends only on the derivatives of the unknowns. Next we consider the case where (6.15) does not hold. Katayama [44] tried to relax the condition (6.15), and its extension was obtained by Katayama and Yokoyama [46]. Their result reads as follows: let (6.14) and Bia,i,i = 0 for all i = 1, . . . , N and a = 0, . . . , n, be valid. If Nij (∂uj ) =

n

(6.21)

Dia,b,j,j ∂a uj ∂b uj , which is a part of Fi expressed in

a,b=1

(6.12), satisﬁes Nij (V (µ, X)) = 0

(6.22)

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for all i, j ∈ {1, · · · , N }, µ ∈ R and X ∈ Rn+1 satisfying X02 = c2j (X12 + · · · + Xn2 ), then the small data global existence holds. One may suppose that the condition on Nij for j = i is not necessary, in view of the result in [43]. But it is not true, because of a counter-example obtained by Ohta [85]. In fact, when c1 < c2 , we have a blow-up result for the system (∂ ∂t2 − c21 ∆)u1 = u2 ∂t u1 , (∂ ∂t2

−

c22 ∆)u2

2

= (∂ ∂t u1 ) ,

(t, x) ∈ [0, ∞) × R3 ,

(6.23)

(t, x) ∈ [0, ∞) × R3 .

(6.24)

We discuss the detail in Subsection 6.1 below. Finally, we consider the case n = 2. When F is independent of the unknowns themselves, Agemi and Yokoyama [5] reﬁned the result in [57], and Hoshiga and Kubo [36] extended their result in such a way that the small data global existence holds if F satisﬁes (6.6) with p = pc (2) = 3 and its cubic part veriﬁes (6.16). The novelty of the latter work is to provide an idea to deal with the so-called null form Q0 (φ, ψ) = ∂t φ ∂t ψ − c2 ∇φ · ∇ψ.

(6.25)

(For the detail, see Remark 6.7 below.) As for the lifespan, Hoshiga [34, 35] handled the case where F satisﬁes (6.6) with p = 2 and either its quadratic or cubic part does not verify (6.16). The case where F depends on the unknowns themselves is complicated especially when n = 2, because we do not have suitable L2 -bound for the unknowns themselves. Despite of the diﬃculty, the following partial result was obtained by Hoshiga and Kubo [37]. Let Fi admit the following decomposition: Fi (u, ∂u, ∂∇u) =

N

Nij (u, ∂uj , ∂∇uj )+Ri (u, ∂u, ∂∇u)+H Hi (u, ∂u, ∂∇u), (6.26)

j=1

where Hi is a higher-order term satisfying Hi (u, v, w) = O(|u|4 + |v|4 + |w|4 ),

(6.27)

Nij is a homogeneous polynomial only in (u, ∂uj , ∂∇uj ) of degree 3, and Ri is a homogeneous polynomial in (u, ∂u, ∂∇u) of degree 3 being explicitly written as ijkl Ri (u, ∂u, ∂∇u) = qαβγ (∂ α uj )(∂ β uk )(∇γ ∂ul ). (6.28) j,k,l=1,··· ,N |α|≤1,|β|=1,|γ|≤1 k= l ijkl is a constant. If Nij satisﬁes Here qαβγ

Nij (λ, V (µ, X), W (ν, X)) = 0

(6.29)

for all i, j ∈ {1, · · · , N }, λ ∈ RN , µ, ν ∈ R and X ∈ Rn+1 satisfying X02 = c2j (X12 + · · · + Xn2 ), then the small data global existence holds. The point is to make use of a kind of Hardy’s inequality obtaind in [14]: For s ∈ [0, 1/2), t ≥ 0 and v ∈ C0∞ (Rn ), there is a positive constant C = C(s, n) such that v s (6.30) || · | − t|s 2 n ≤ C|ξ| vˆL2 (Rn ) L (R )

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Then we have the loss of decay with respect to |x| − ci t. But we can absorb such a loss by means of the point-wise decay estimates. 6.1. Blow-up The aim of this subsection is to establish the counter-example mentioned in the above. More precisely, we consider the system

2 (∂ ∂t − c21 ∆)u1 = u2 ∂t u1 , (t, x) ∈ [0, ∞) × R3 , (6.31) 2 2 2 ∂t u1 ) , (t, x) ∈ [0, ∞) × R3 , (∂ ∂t − c2 ∆)u2 = (∂ with the following initial condition:

u1 (x, 0) = 0, ∂t u1 (x, 0) = εψ1 (|x|), u2 (x, 0) = 0, ∂t u2 (x, 0) = 0,

x ∈ R3 , x ∈ R3 ,

(6.32)

where ε > 0, ψ1 (|x|) ∈ C0∞ (R3 ). In addition, we assume that there exists a constant δ > 0 such that ψ1 (r) > 0 for r ∈ [0, δ),

ψ1 (r) = 0 for r ∈ [δ, ∞).

(6.33)

The main result in the present subsection reads as follows. Theorem 6.1. Let c1 < c2 and ε ∈ (0, 1]. Suppose that (6.33) holds. Then the classical solution (u1 , u2 ) of (6.31)–(6.32) blows up in a ﬁnite time T ∗ (ε). Moreover, there exists a positive constant C ∗ , which is independent of ε, such that T ∗ (ε) ≤ exp(C ∗ ε−2 ). Since both equations and initial data in (6.31) and (6.32) are radially symmetric, by the uniqueness of classical solutions, the classical solution (u1 , u2 ) of (6.31)–(6.32) is also radially symmetric. For this reason, we shall write ui (r, t) for ui (t, x) (i = 1, 2) in what follows. Moreover, for v = v(r, t) and c > 0, we denote ∂t2 (rv) − c2 ∂r2 (rv)}. c v = r−1 {∂ Then (6.31)–(6.32) is rewritten in the following form:

(r, t) ∈ [0, ∞)2 , c1 u1 = u2 ∂t u1 , 2 c2 u2 = (∂ ∂t u1 ) , (r, t) ∈ [0, ∞)2 , with

u1 (r, 0) = 0, ∂t u1 (r, 0) = εψ1 (r), u2 (r, 0) = 0, ∂t u2 (r, 0) = 0, The following lemma is well known.

r ∈ [0, ∞), r ∈ [0, ∞).

Lemma 6.2. Let v(r, t) be the classical solution of

c v = f (r, t), (r, t) ∈ [0, ∞) × [0, T ), r ∈ [0, ∞). v(r, 0) = 0, ∂t v(r, 0) = g(r), Then, for (r, t) ∈ [0, ∞) × [0, T ), we have r+c(t−τ ) 1 r+ct 1 t rv(r, t) = ρg(ρ) dρ + ρf (ρ, τ ) dρ dτ. 2c |r−ct| 2c 0 |r −c(t−τ )|

(6.34)

(6.35)

(6.36)

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193

Moreover, if r ≥ ct ≥ 0, then we have r∂ ∂t v(r, t) =

1 {(r + ct)g(r + ct) + (r − ct)g(r − ct)} 2 1 t + {(r + c(t − τ ))f (r + c(t − τ ), τ ) 2 0 +(r − c(t − τ ))f (r − c(t − τ ), τ )} dτ.

Using the above lemma, we have the following result. Lemma 6.3. Assume (6.33). Let T ∗ (ε) be the lifespan of the classical solution (u1 , u2 ) of (6.34)–(6.35). Then we have u2 (r, t) ≥ 0 for all (r, t) ∈ [0, ∞) × [0, T ∗ (ε)). Moreover, we have ∂t u1 (r, t) > 0 if 0 < r − c1 t < δ, and ∂t u1 (r, t) = 0 if r − c1 t ≥ δ. Proof. By Lemma 6.2, for (r, t) ∈ [0, ∞) × [0, T ∗ (ε)), we have t r+c2 (t−τ ) 1 2 ru2 (r, t) = ρ∂ ∂t u1 (ρ, τ ) dρ dτ ≥ 0. 2c2 0 |r −c2 (t−τ )| Moreover, since u1 (r, 0) = ∂t u1 (r, 0) = 0 for r ≥ δ, by the uniqueness of classical solutions of (6.34)–(6.35), we have ∂t u1 (r, t) = 0 for r − c1 t ≥ δ. Finally, we show that ∂t u1 (r, t) > 0 if 0 < r − c1 t < δ. We put Ω = {(ρ, τ ) ∈ [0, ∞) × (0, T ∗ (ε)) : 0 < ρ − c1 τ < δ}. Moreover, for (r, t) ∈ Ω, we put D(r, t) = {(ρ, τ ) ∈ [0, ∞)2 : 0 ≤ τ ≤ t, |ρ − r| ≤ c1 (t − τ )}. By (6.33) and the continuity of ∂t u1 , there exists (r1 , t1 ) ∈ Ω such that ∂t u1 (r, t) > 0 for (r, t) ∈ D(r1 , t1 ) ∩ Ω. Suppose that there exists (r0 , t0 ) ∈ Ω such that ∂t u1 (r0 , t0 ) ≤ 0. Then, by the continuity of ∂t u1 , there exists (r2 , t2 ) ∈ Ω such that ∂t u1 (r2 , t2 ) = 0 and ∂t u1 (r, t) ≥ 0 for (r, t) ∈ D(r2 , t2 ). Since ∂t u1 (r, t) ≥ 0 and u2 (r, t) ≥ 0 for (r, t) ∈ D(r2 , t2 ), by Lemma 6.2, we have ε 0 = r2 ∂t u1 (r2 , t2 ) ≥ (r2 − c1 t2 )ψ1 (r2 − c1 t2 ). 2 On the other hand, since (r2 , t2 ) ∈ Ω, by (6.33), we have (r2 −c1 t2 )ψ1 (r2 −c1 t2 ) > 0. This is a contradiction. Hence, we obtain that ∂t u1 (r, t) > 0 if 0 < r − c1 t < δ. Let 0 < δ1 < δ2 < δ, and we put Σ = {(r, t) ∈ [0, ∞)2 : δ1 ≤ r − c1 t ≤ δ2 },

Σ(t) = {r ∈ [0, ∞) : (r, t) ∈ Σ}.

Moreover, for the classical solution (u1 , u2 ) of (6.34)-(6.35), we deﬁne ∂t u1 (r, t) : r ∈ Σ(t)}, U1 (t) = inf{∂

U2 (t) = inf{ru2 (r, t) : r ∈ Σ(t)}.

Then we see from Lemma 6.3 that U1 (t) ≥ 0 and U2 (t) ≥ 0 for t ∈ [0, T ∗ (ε)). Moreover, we have the following lemma.

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Lemma 6.4. Under the assumptions in Theorem 6.1, there exist positive constants C1 , C2 , C3 such that t U1 (τ )U U2 (τ ) U1 (t) ≥ C1 ε + C2 dτ, t ≥ 1, (6.37) τ 1 t τ U1 (τ )2 δ2 1− dτ, t ≥ U2 (t) ≥ C3 . (6.38) t τ c − c1 2 (c2 −c1 )t/c2 Proof. First we show (6.37). Let (r, t) ∈ Σ. By Lemmas 6.2 and 6.3, we have r∂ ∂t u1 (r, t) ≥

ε (r − c1 t)ψ1 (r − c1 t) 2 1 t + λ(τ )∂ ∂t u1 (λ(τ ), τ ) u2 (λ(τ ), τ ) dτ, 2 0

where we put λ(τ ) = (r − c1 t + c1 τ ). Setting C1 = inf{ρψ1 (ρ)/2 : δ1 ≤ ρ ≤ δ2 }, we get U2 (τ ) 1 t U1 (τ )U dτ, r∂ ∂t u1 (r, t) ≥ C1 ε + 2 0 c1 τ + δ 2 since λ(τ ) ≤ c1 τ + δ2 for (r, t) ∈ Σ. (Note that C1 > 0 by (6.33).) Therefore (6.37) holds. Next we show (6.38). Let (r, t) ∈ Σ with t ≥ δ2 /(c2 − c1 ). Then we have 0 ≤ (c2 t − r)/c2 ≤ (c2 − c1 )t/c2 and c2 t + r ≥ δ2 . By Lemma 6.2, we have t r+c2 (t−τ ) (ρ∂ ∂t u1 (ρ, τ ))2 χΣ(τ ) (ρ) dρ dτ (6.39) ru2 (r, t) ≥ ρ (c2 t−r)/c2 |r −c2 (t−τ )| t 2 ¯ τ ) U1 (τ ) dτ, ≥ (t, c1 τ + δ 2 (c2 −c1 )t/c2 where χΣ(τ ) denotes the characteristic function of Σ(τ ) and we put ¯ τ ) = inf{(r, t, τ ) : r ∈ Σ(t)}, (t,

r+c2 (t−τ )

(r, t, τ ) = |r −c2 (t−τ )|

χΣ(τ ) (ρ) dρ.

¯ τ ) ≥ (δ2 − δ1 )(1 − τ /t) for (c2 − c1 )t/c2 ≤ τ ≤ t. By (6.39), for Then, we have (t, any (r, t) ∈ Σ with t ≥ δ2 /(c2 − c1 ), we have t τ U1 (τ )2 1− ru2 (r, t) ≥ (δ2 − δ1 ) dτ, t c1 τ + δ 2 (c2 −c1 )t/c2 which implies (6.38). This completes the proof.

Once we establish the following lemma, we can conclude that Theorem 6.1 is valid.

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Lemma 6.5. Let C1 , C2 , C3 > 0, α ≥ β > 1, ε ∈ (0, 1]. Assume that (f (t), g(t)) satisﬁes t f (τ )g(τ ) dτ, t ≥ 1, (6.40) f (t) ≥ C1 ε, f (t) ≥ C2 τ 1 t τ f (τ )2 1− dτ, t ≥ α. (6.41) g(t) ≥ C3 t τ t/β Then, (f (t), g(t)) blows up in a ﬁnite time T∗ (ε). Moreover, there exists a positive constant C∗ , which is independent of ε, such that T∗ (ε) ≤ exp(C∗ ε−2 ). Proof. We deﬁne F (s) = ε−1 f (exp(ε−2 s)),

G(s) = ε−2 g(exp(ε−2 s)).

Then we have

s F (s) ≥ C1 , F (s) ≥ C2 F (σ)G(σ) dσ, s ≥ 0, (6.42) 0 s . / 1 − exp(−ε−2 (s − σ)) F (σ)2 dσ, s ≥ log α. G(s) ≥ C3 ε−2 s−ε2 log β

Suppose we have found that F (s) ≥ A > 0 for s ≥ S ≥ 0. Let h ∈ (0, 1]. Then we have for s ≥ max{S + h log β, log α} s . / G(s) ≥ C3 ε−2 A2 1 − exp(−ε−2 (s − σ)) dσ (6.43)

s−ε2 h log β h log β −σ

(1 − e

= C3 A2

) dσ ≥

0

C3 (β − 1) log β 2 2 h A , 2β

where we used the fact that β−1 1 − exp(− log β) 1 − e−σ ≥ σ= σ, log β β log β

0 ≤ σ ≤ log β.

By (6.42) and (6.43), we have for s ≥ max{S + h(1 + log β), log α + h} s C1 C2 C3 (β − 1) log β 3 2 F (s) ≥ C1 C2 h A . G(σ) dσ ≥ 2β s−h Moreover, since F (s) ≥ C1 for s ≥ 0, we have s F (s) ≥ C1 C2 G(σ) dσ

(6.44)

log α

≥

C13 C2 C3 (β − 1) log β (s − log α), 2β

Now we deﬁne constants γ and A1 by 2η }, γ = max{1, C1 C2 C3 (β − 1) log β

s ≥ log α.

A1 = γ exp 1 + 6

∞ k=1

2

−k

log k .

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Then, by (6.44), there exists a constant S1 ≥ log α such that F (s) ≥ A1 for s ≥ S1 . Furthermore we deﬁne sequences {An } and {Sn } by An+1 =

A2n , γn6

Sn+1 = Sn +

1 + log β , n2

n ∈ N.

Then, for any n ∈ N, we have F (s) ≥ An for s ≥ Sn , and Sn = S1 + (1 + log β)

k=1

log An+1 = 2

n

n−1

1 , k2

log A1 − (1 − 2

−n

) log γ − 6

n

2

−k

log k

≥ 2n .

k=1

Therefore, (F (s), G(s)) blows up at some s = S∗ satisfying S∗ ≤ lim Sn < ∞. n→∞ This completes the proof. 6.2. Null condition In this subsection we ﬁrst derive an estimate (6.54) below for a function that verify the null condition after preliminary steps. Then we consider asymptotic behavior of the solution to the problem (6.4)–(6.5) in Theorem 6.2 below as an application of (6.54). It has been observed by Kubota and Yokoyama [73] that the null condition is closely related to the radiation operators which are deﬁned by Ta = ∂a − ωa ∂r where ∂r =

x · ∇, r

ω0 = −ci ,

(0 ≤ a ≤ n), ωl =

(6.45)

xl (1 ≤ l ≤ n). r

Explicitly we have T0 = ∂t + ci ∂r ,

Tl = ∂l −

xl ∂r (0 ≤ l ≤ n). r

Besides, we denote T = (T T0 , T1 , · · · , Tn ). Then we have the following lemma based on the special structure coming from the null condition. Lemma 6.6. Let F (u, v, w) be a homogeneous polynomial of degree 2 in its arguments and u(t, x) be a RN -valued smooth function. If Fi satisﬁes (6.10) for all λ, µ, ν ∈ RN and X ∈ Rn+1 satisfying X02 = c2i (X12 + · · · + Xn2 ), then there is a positive constant C, independent of t, x and u(t, x), such that |F Fi (u, ∂u, ∂∇u)(t, x)| ≤ C( |∇α ∂u(t, x)||T ∇β u(t, x)| + |∂u(t, x)||T ∂r u(t, x)|) |α|+|β|≤1

(6.46) for (t, x) ∈ [0, ∞) × Rn with |x| ≥ 1.

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Proof. We see from (6.10) that F is independent of u. In fact, we easily have Fi (u, 0, 0) ≡ 0 for all u ∈ RN . Namely, Fi does not include terms which are quadratic in u. Moreover, we ﬁnd that it is impossible to include a term which is linear in u (for the detail, see, e.g., the Appendix in [73]). Therefore Fi (1 ≤ i ≤ N ) is expressed as Fi (u, ∂u, ∂∇u)(t, x) =

n N

Ai,j,k a,b ∂a uj (t, x)∂b uk (t, x)

(6.47)

j,k=1 a,b=0

+

n n N

i,j,k Ba,b,c ∂a uj (t, x)∂b ∂c uk (t, x),

j,k=1 a,b=0 c=1 i,j,k where Ai,j,k a,b , Ba,b,c are constants. Taking λ = u, µ = ∂r u, ν = ∂r2 u and X = (ω0 , ω1 , · · · , ωn ), we have from (6.10) Fi (λ, V (µ, X), W (ν, X)) = 0.

Therefore we get from (6.47) Fi (u, ∂u, ∂∇u) = =

Fi (u, ∂u, ∂∇u) − Fi (λ, V (µ, X), W (ν, X)) n N

Ai,j,k Ta uj ∂b uk + ωa ∂r uj Tb uk ] a,b [T

(6.48)

j,k=1 a,b=0

+

n n N

i,j,k Ba,b,c [T Ta uj ∂b ∂c uk + ωa ∂r uj Tb ∂c uk

j,k=1 a,b=0 c=1

+ωa ωb ∂r uj ∂r Tc uk ]. As for the last term, we note that for 1 ≤ c ≤ n 1 [∂ ∂r , Tc ] = [∂ ∂r , ∂c ] = − Tc . r Here [ , ] denotes the usual commutator of linear operators, namely [A, B] = AB − BA. Hence, we obtain (6.46) from (6.48). This completes the proof. In order to extract additional decay by using the above lemma, we introduce the vector ﬁelds: S = t∂ ∂t + x · ∇,

Ωjk = xj ∂k − xk ∂j (1 ≤ j < k ≤ n).

(6.49)

∂t , ∂1 , · · · , ∂n ), We denote by Γ = (Γ1 , · · · , Γn0 ) these ﬁelds together with ∂ = (∂ + n + 2. It is a part of the vector ﬁelds Λ which includes not where n0 = n(n−1) 2 only Γ but also xi Li = ct∂ ∂i + ∂t (1 ≤ i ≤ n) (6.50) c with c > 0. In [54] the ﬁelds Λ play an essential role to handle nonlinearities satisfying the null condition. One of the advantages to use Λ is contained in the

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following Sobolev type inequality which is well known as Klainerman’s inequality: n−1 1 Λα u(t)L2 (Rn ) (6.51) t + |x| 2 ct − |x| 2 |u(t, x)| ≤ C |α|≤[ n 2 ]+1

(for the proof, see [55] or [41]). This inequality will give us a decay estimate for derivatives of the solution to (6.4)–(6.5), once the right-hand side of (6.51) is under the control. This can be realized via the energy estimate if the system (6.4) has a common propagation speed, because ∂, Ω are commute with c for all c > 0, S is so modulo lower order term, i.e., [S, c ] = −2c , and 2 2 (c − c2 )∂ ∂t ∂j = 0 (1 ≤ j ≤ n; 1 ≤ i ≤ N ) (6.52) c i if we choose c as c = c1 = · · · = cN in (6.50). Clearly we cannot expect (6.52) when the propagation speeds are distinct. In the consequence, Lj is not applicable to the system (6.4) with multiple speeds, hence (6.51) is not available. To overcome the diﬃculty, we can make use of the point-wise estimates for the derivatives of the solution. Such estimates is also derived by the evaluation of the fundamental solution as we have done in Section 2, and can be found, for example, in [73] for n = 3 and [37] for n = 2 (see also [57, 5, 114, 43]). We do not go further into this direction. We come back to the estimation of functions verifying the null condition. To this end, we introduce notations. For a RN -valued smooth function v(t, x) we set [Lj , ci ] =

|v(t, x)|k =

N

|Γα vi (t, x)|,

|α|≤k i=1

where k is a non-negative integer, α = (α1 , · · · , αn0 ) is a multi-index, Γα = αn0 1 Γα 1 · · · Γn0 and |α| = α1 + · · · + αn0 . The order of the application of Γj is not essential because of the following commutator relations: [S, ∂a ] = −∂a , [S, Ωjk ] = 0, [Ωjk , ∂a ] = ηka ∂j − ηja ∂k ,

(6.53)

[Ωjk , Ωlh ] = ηkl Ωjh + ηjh Ωkl − ηkh Ωjl − ηjl Ωkh , for a, b = 0, · · · , n and j, k, l, h = 1, · · · , n, where ∂0 = ∂t and η = (ηab ) = diag(−1, 1, · · · , 1). Besides we deﬁne |v(t, x)|2k dx. v(t)2k = R

Then we have the following estimate. Proposition 6.1. Let the assumption of Lemma 6.6 be fulﬁlled. Then there is a positive constant C, independent of t, x and u(t, x), such that t + |x||F Fi (u, ∂u, ∂∇u)(t, x)| ≤ C|∂u(t, x)|1 (|x| − ci t|∂u(t, x)|1 + |u(t, x)|1 ) (6.54)

Coupled Systems of Semilinear Wave Equations

199

for (t, x) ∈ [0, ∞) × Rn with |x|/2 ≤ ci t ≤ 2|x| and |x| ≥ 1. Proof. In view of (6.46), we see that it suﬃces to prove t + r|T u(t, x)| ≤ C(|ci t − r||∂u(t, x)| + |Γu(t, x)|),

(6.55)

provided r/2 ≤ ci t ≤ 2r and r ≥ 1. We easily have from (6.45) Tj u(t, x) = −

n 1 xk Ωjk u(t, x) r2

(6.56)

k=1

for 1 ≤ j ≤ n and r > 0. This implies |T Tj u(t, x)| ≤

n |Ωu(t, x)|. r

While we have ci t − r 1 ∂r u(t, x) + Su(t, x) (6.57) t t for t > 0. Thus, when r/2 ≤ ci t ≤ 2r and r ≥ 1, we get (6.55). This completes the proof. T0 u(t, x) =

Remark 6.7. An analogous estimate to (6.54) for functions which are cubic and which satisfy the null condition has been obtained by [37]. The point of the proof is the relation (6.57) which is ﬁrstly adopted by [36]. On the one hand, if we use all of the ﬁelds Λ, then we can avoid |x| − ci t in the right-hand side of (6.54). In fact, by choosing c = ci in (6.50), we have n ci T0 u(t, x) = (S + ωi Li )u(t, x), ci t + r j=1

(6.58)

instead of (6.57). This observation is useful for the case where the system (6.4) has common propagation speeds. The local existence of the classical solution to (6.4)–(6.5) is known (see for instance Kato [51], Majda [80]). Consequently what we need to do for proving the global existence theorem is nothing like to derive suitable a-priori estimates for the solution. This can be done by making use of the estimate (6.54) together with (6.59) below. If one would like to evaluate the derivatives of functions verifying the null condition, then the following lemma can be useful. Lemma 6.8. For a real-valued smooth function u(t, x) and a non-negative integer k, we have t + r|T u(t, x)|k ≤ Ck (ci t − r|∂u(t, x)|k + |u(t, x)|k+1 ), for (t, x) ∈ [0, ∞) × Rn with |x|/2 ≤ ci t ≤ 2|x| and |x| ≥ 1.

(6.59)

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Proof. When k = 0, (6.59) is nothing like (6.55). To consider the case k = 1, we notice that ci T0 , [Ωjk , T0 ] = 0, [∂l , T0 ] = Tl , [∂ ∂t , T0 ] = 0, [S, T0 ] = −T r 1 xl xk xk [S, Tk ] = −T Tk , [∂l , Tk ] = − δlk ∂r + 3 ∂r − 2 Tl , [∂ ∂t , Tk ] = 0, r r r and [Ωjk , Tl ] = ηkl Tj − ηjl Tk for j, k, l = 1, · · · , n. These relations can be checked by using (6.53) and [∂l , ∂r ] =

1 Tl , r

[Ωjk , ∂r ] = 0,

[S, ∂r ] = −∂ ∂r .

(6.60)

Thus we have |ΓT Ta u(t, x)| ≤ C(|T Ta Γu(t, x)| +

n b=0

1 ∂r u(t, x)|). |T Tb u(t, x)| + |∂ r

In virtue of (6.55), this estimate implies (6.59) for k = 1. Moreover, noting that for all integer m 1 m mxl 1 1 ] = − m , [Ωjk , m ] = 0, [∂l , m ] = − m+2 , rm r r r r we obtain (6.59) for k ≥ 2, inductively. This completes the proof. [S,

As an application of Proposition 6.1, we consider the asymptotic behavior of the classical solution to (6.4)–(6.5). To this end, we restrict ourselves to the case where n = 3, and the nonlinearity Fi (u, v, w) is independent of u and admits the following decomposition: Fi (∂u, ∂∇u) =

N

Nij (∂uj , ∂∇uj ) + Ri (∂u, ∂∇u),

(6.61)

j=1

where Nij (∂uj , ∂∇uj )

=

n

Ai,j,j a,b ∂a uj (t, x)∂b uj (t, x)

(6.62)

a,b=0

+

n n

i,j,j Ba,b,c ∂a uj (t, x)∂b ∂c uj (t, x),

a,b=0 c=1

Ri (∂u, ∂∇u)(t, x) =

n

Ai,j,k a,b ∂a uj (t, x)∂b uk (t, x)

j= k a,b=0

+

n n j= k a,b=0 c=1

i,j,k Ba,b,c ∂a uj (t, x)∂b ∂c uk (t, x)

(6.63)

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201

i,j,k with some constants Ai,j,k a,b , Ba,b,c . We shall show that the solution to the problem verifying (6.65), (6.66) below tends to a solution of the system of homogeneous wave equations:

(∂ ∂t2 − c2i ∆)vi = 0

for (t, x) ∈ [0, ∞) × Rn

(1 ≤ i ≤ N ),

(6.64)

provided all Nij satisfy the null condition. We ﬁnd from the part 2) of Theorem 1.2 in [73] that there exists a global solution to the problem under the above assumption on Fi . Theorem 6.2. Let n = 3 and let the propagation speeds be distinct. Let Fi (u, v, w) be independent of u, and be decomposed as in (6.61). Suppose that u(t, x) is a classical solution to (6.4)–(6.5) satisfying the following properties: There are constants M > 0 and 0 ≤ ν < 1 such that |ui (t, x)|1 t + |x|ci t − |x|ν + |∂ui (t, x)|1 xci t − |x|ν+1 ≤ M, ∂u(t)1 ≤ M,

(6.65) (6.66)

for (t, x) ∈ [0, ∞) × Rn and i = 1, · · · , N . If Nij satisﬁes (6.29) for all i, j ∈ {1, · · · , N }, µ, ν ∈ R and X ∈ Rn+1 satisfying X02 = c2j (X12 + · · · + Xn2 ), then there is a unique classical solution v(t, x) of (6.64) verifying N

ui (t) − vi (t)E(ci ) ≤ CM 2 t−ν

for

t ≥ 0,

(6.67)

i=1

where the constant C is independent of t and M . Proof. If we set vi = ui − Rci [F Fi (u, ∂u, ∂∇u)] (recall that Rc [F ] is deﬁned by (2.98)), then we see from Proposition 2.7 that vi satisﬁes (6.64) and ∞ F Fi (u, ∂u, ∂∇u)(τ )L2 (R3 ) dτ ui (t) − vi (t)E(ci ) ≤ C t

holds. Therefore, in view of (6.61), it suﬃces to show N Nij (∂uj , ∂∇uj )(t)L2 (R3 ) ≤ CM 2 t−ν−1 ,

(6.68)

−ν−1

(6.69)

Ri (∂u, ∂∇u)(t)L2 (R3 ) ≤ CM t 2

.

First we show (6.68). We divide the argument into two cases. When |x|/2 ≤ cj t ≤ 2|x| and |x| ≥ 1, it follows from (6.54) and (6.65) that t + |x||N Nij (∂uj , ∂∇uj )(t, x)| ≤ CM |∂u(t, x)|1 (x−1 |x| − cj t−ν + t + |x|−1 |x| − cj t−ν ) ≤ CM |∂u(t, x)|1 t + |x|−1 . While, in the other case, |x| − cj t is equivalent to t + |x|. Therefore, by (6.65) we have |N Nij (∂uj , ∂∇uj )(t, x)| ≤ CM |∂u(t, x)|1 t + |x|−ν−1 . Since ν < 1, the above estimate is valid for all (t, x) ∈ [0, ∞) × Rn . Thus (6.66) yields (6.68).

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Next we show (6.68). Since j = k, one of the terms |x| − cj t and |x| − ck t is equivalent to t + |x|, at worst. Therefore we have |Ri (∂u, ∂∇u)(t, x)| ≤ CM |∂u(t, x)|1 t + |x|−ν−1 , which implies (6.69) by (6.66). This completes the proof.

Remark 6.9. It is an interesting open problem to determine the condition on the nonlinearity under which the global solution to the problem (6.4)–(6.5) tends to a free solution, because of an example found by Alinhac [9] which admits a global solution whose energy grows as t → ∞. In other words, the example is not a nonlinear perturbation from the system of homogeneous wave equations at all. Many other examples of this type are known for diﬀerent kind of equations. For instance, we refer to J.-M. Delort, D. Fang and R. Xue [15] and Sunagawa [103, 104, 105] for systems of nonlinear Klein-Gordon equations with multiple masses.

Appendix The aim of this appendix is to give an elementary proof of the domain of dependence property (see Theorem 4 in [39] or Appendix 1 in [41]). We consider (∂ ∂t2 − c2i ∆)ui = Fi (t, x, u, ∂u, ∂∇u) for (t, x) ∈ Γ(t0 , x0 ),

(A.1)

where 1 ≤ i ≤ N and Γ(t0 , x0 ) is the backward light cone Γ(t0 , x0 ) = {(t, x) ∈ (0, T ) × Rn : |x − x0 | < t0 − t} F1 , · · · , FN )(t, x, r, q, p) with (t0 , x0 ) ∈ (0, T ) × Rn and T > 0. We assume that (F is locally Lipschitz in (r, q), namely, for any M > 0 there is a number L > 0 such that if 2 (|rj | + |qqj | + |pj |) ≤ M, |t| + |x| + j=1

then for all i = 1, · · · , n, we have |F Fi (t, x, r1 , q1 , p1 ) − Fi (t, x, r2 , q2 , p2 )| ≤ L(|r1 − r2 | + |q1 − q2 |). Example A.1. We give examples of F (t, x, r, q, p) which are locally Lipschitz in (r, q). • Let bijk , cik and fi ∈ C((0, T ) × Rn ). If we set Fi (t, x, r, q, p) =

n N j=0 k=0

bijk (t, x)qqjk +

N

cik (t, x)rk + fi (t, x),

k=0

then F = (F F1 , · · · , FN ) is locally Lipschitz in (r, q). F1 , · · · , FN ) is locally Lipschitz in • If we set Fi (t, x, r, q, p) = |r|αi , then F = (F (r, q) when αi ≥ 1.

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The ﬁrst result in this appendix is concerned with the domain of dependence property. Theorem A.1. Let (t0 , x0 ) ∈ (0, T )× Rn and Fi (t, x, r, q, p) ∈ C((0, T )× Rn × RN × R(n+1)N × Rn(n+1)N ). Suppose F = (F F1 , · · · , FN ) is locally Lipschitz in (r, q) and Fi (t, x, 0, 0, p) = 0

(t, x) ∈ Γ(t0 , x0 ),

for

p ∈ Rn(n+1)N

(1 ≤ i ≤ N ).

If u is the classical solution to (A.1) satisfying u(0, x) = ∂t u(0, x) = 0

for

|x − x0 | < t0 ,

(A.2)

then u ≡ 0 in Γ(t0 , x0 ). Proof. Since u ∈ C 2 (Γ(t0 , x0 )), by the assumption on F , we have |F Fi (t, x, u(t, x), ∂u(t, x), ∂∇u(t, x))| ≤ L(|u(t, x)| + |∂u(t, x)|)

(A.3)

for all (t, x) ∈ Γ(t0 , x0 ). Note that ui satisﬁes (∂ ∂t2 − c2i ∆)ui + ui = ui + Fi (t, x, u, ∂u, ∂∇u)

(A.4)

for (t, x) ∈ Γ(t0 , x0 ). Setting 1 (|ui (t, x)|2 + |∂ ∂t ui (t, x)|2 + c2i |∇u(t, x)|2 ) dx, Ei (t) = 2 |x−x0 |

|x−x0 |
≤C 0

≤C

|x−x0 |
N i=1

(ui (s, x) + Fi (s, x, u, ∂u, ∂∇u))∂ ∂t ui (s, x) dx ds (|u(s, x)||∂u(s, x)| + |∂u(s, x)|2 ) dx ds

t

Ei (s)ds,

0

by (A.3). Since Ei (0) = 0 by (A.2), Gronwall’s inequality gives This yields the conclusion.

*N i=1

Ei (t) ≤ 0.

As a corollary of Theorem A.1, we have the ﬁnite speeds of propagation property. Theorem A.2. Let x0 ∈ Rn and R > 0. Suppose that F = (F F1 , · · · , FN ) satisﬁes the condition in the preceding theorem. If u is the classical solution of (A.1) in [0, T ) × Rn satisfying u(0, x) = ∂t u(0, x) = 0

for

|x − x0 | ≥ R,

(A.5)

then we have u(t, x) = 0

for

|x − x0 | ≥ t + R, 0 ≤ t < T.

(A.6)

204

H. Kubo and M. Ohta

Next we show the uniqueness of the classical solution of (A.1) for the case where F (t, x, r, q, p) is independent of p. Theorem A.3. Let (t0 , x0 ) ∈ (0, T ) × Rn and Fi (t, x, r, q) ∈ C((0, T ) × Rn × RN × R(n+1)N ). Suppose that F = (F F1 , · · · , FN ) is locally Lipschitz in (r, q). If u, v are classical solutions of (A.1) satisfying u(0, x) = v(0, x),

∂t u(0, x) = ∂t v(0, x)

|x − x0 | < t0 ,

(A.7)

(t, x) ∈ Γ(t0 , x0 ),

(A.8)

for

then u ≡ v in Γ(t0 , x0 ). Proof. If we set w := u − v, then we have (∂ ∂t2 − c2 ∆)wi + wi = Gi (t, x, u, v, ∂u, ∂v), where we put Gi (t, x, u, v, ∂u, ∂v) = wi + Fi (t, x, u, ∂u) − Fi (t, x, v, ∂v). Since u, v ∈ C 2 (Γ(t0 , x0 )), we see that sup

{|t| + |x| + |u(t, x)| + |∂u(t, x)| + |v(t, x)| + |∂v(t, x)|}

(t,x)∈Γ(t0 ,x0 )

is bounded. By the assumption on F , we have |Gi (t, x, u, v, ∂u, ∂v)| ≤ C(|w(t, x)| + |∂w(t, x)|) for all (t, x) ∈ Γ(t0 , x0 ). Therefore we obtain w ≡ 0 in Γ(t0 , x0 ), similarly to the proof of Theorem A.1. This completes the proof. Finally, we consider the system of quasilinear wave equations of the form (∂ ∂t2 − c2i ∆)ui =

N n

αabj i (u, ∂u)∂a ∂b uj + βi (u, v),

(A.9)

a,b=0 j=1 1 N (n+1)N ) satisfy in [0, T ) × Rn , where αabj i , βi ∈ C (R × R

αabj i (0, 0) = βi (0, 0) = 0, and baj abi αabj i (u, v) = αi (u, v) = αj (u, v)

(A.10)

for a, b = 0, · · · , n and i, j = 1, · · · , N . Moreover, in order to ensure the hyperbolicity of the equation, we require the following condition: There is a number θ < 1 such that −

N n i,j=1 a,b=1

αabj i (u, v)Xai Xbj ≤ θ

n n i=1 a=1

c2i |Xai |2

(A.11)

Coupled Systems of Semilinear Wave Equations

205

for all (Xaj ) ∈ RnN and (u, v) ∈ RN × R(n+1)N . Then, making use of the modiﬁed ˜i (t) below and proceeding as before, we ﬁnd that the classical solution of energy E (A.9) with the initial data of compact support is determined uniquely:

˜ Ei (t) := ∂t ui (t, x)|2 + c2i |∇ui (t, x)|2 (A.12) |ui (t, x)|2 + |∂ Rn

+

N n

& αabj i (u, ∂u)∂a ui (t, x)∂b uj (t, x)

dx.

j=1 a,b=1

We remark that (A.11) is fulﬁlled for small amplitude solutions, hence we can apply the result to the nonlinear systems of wave equations studied in Section 6. Acknowledgements The authors would like to express their gratitude to Professor Michael Reissig for giving them the occasion to write the article and valuable remarks. They would also like to thank collaborators of the ﬁrst author Professors Koji ˆ Kubota, Akira Hoshiga, and Doctor Kotaro Tsugawa. It is also a pleasure for them to acknowledge Professors Soichiro Katayama, Kazuyuki Yokoyama and Hideaki Sunagawa, and Doctor Jun Kato for valuable discussions during the preparation of the work.

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Operator Theory: Advances and Applications, Vol. 159, 213–299 c 2005 Birkhauser ¨ Verlag Basel/Switzerland

Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains Mitsuhiro Nakao Abstract. In this article we consider the initial-boundary value problem for linear and nonlinear wave equations in an exterior domain Ω in RN with the homogeneous Dirichlet boundary condition. Under the eﬀect of localized dissipation like a(x)ut we derive both of local and total energy decay estimates for the linear wave equation and apply these to the existence problem of global solutions of semilinear and quasilinear wave equations. We make no geometric condition on the shape of the boundary ∂Ω. The dissipation a(x)ut is intended to be as weak as possible, and if the obstacle V = RN \ Ω is star-shaped our results based on local energy decay hold even if a(x) ≡ 0, while for the results concerning the total energy decay we need a(x) ≥ 0 > 0 near ∞. In the ﬁnal section we consider the wave equation with a ‘ half-linear’ dissipation ρ(x, ut ) which is like a(x)|ut |r ut in a bounded area and which is linear like a(x)ut near ∞. Mathematics Subject Classiﬁcation (2000). 35B35, 35B40, 35L05,35L70. Keywords. Nonlinear wave equations, exterior domains, decay, global solutions.

1. Introduction In this contribution we consider the initial-boundary value problem for linear and nonlinear wave equations in an exterior domain Ω in RN with the homogeneous Dirichlet boundary condition. Roughly speaking we derive local and total energy decays for the linear wave equation with some localized dissipation a(x)ut and apply these estimates to the existence problem of global decaying solutions of nonlinear wave equations. The dissipation a(x)ut is intended to be as weak as possible, and if the obstacle V = RN \ Ω is star-shaped some of our results hold

214

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even if a(x) ≡ 0. It may be emphasized that we make no geometrical assumptions on the shape of the boundary ∂Ω. Throughout the paper we employ only standard multiplier methods originated by Protter [55] and Morawetz [35], and in this sense our arguments are elementary (cf. Chen [6], Komornik [22], Lions [25], Zuazua [70] etc.). To specify our assumption on a(x) we introduce a part of the boundary ∂Ω following Russell [59] and Lions [25]: Γ(x0 ) = {x ∈ ∂Ω | ν(x) · (x − x0 ) > 0},

x0 ∈ RN ,

(1.1)

where ν(x) is the outward normal at x ∈ ∂Ω. We note that V is star-shaped with respect to x0 if and only if Γ(x0 ) = ∅. We shall assume Hyp. A. a(x) > ε0 > 0 on a neighborhood ω of Γ(x0 ) for some x0 . a(x) ≡ 0 (no dissipation) is allowed if V is star-shaped. After the introductory and preliminary sections, in Section 3 we are concerned with the decay of local energy to the linear problem: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(1.2)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. (1.3) 0 We shall prove an algebraic decay of the local energy ER (t) = ΩR |ut (t)|2 + |∇u(t)|2 )dx, ΩR = Ω ∩ BR for the ﬁnite energy solutions u(t) of (1.2)–(1.3), where BR denotes the ball centered at the origin with the radius R. When N is odd we can further apply the method due to Morawetz [35] to conclude the exponential decay of ER (t). These results will be applied in the Section 5 to the derivation of Lp estimates of solutions. When a(x) ≡ 0 and V is not star-shaped we can not expect any uniform decay rate like ER (t) ≤ C(E(0))g(t) with limt→∞ g(t) = 0 (Ralston [58]). But, in our case, due to the dissipation a(x)ut we need not assume any geometrical condition on V . Our result is a natural extension of the classical one due to Morawetz [35] to a general domain. The ingredients of Section 3 are taken from Nakao [40]. The same result has been proved by Aloui and Khenissi [1] by a diﬀerent method based on Lax-Phillips Theory. For a dog-bone type obstacle see Bloom and Kazarinoﬀ [4]. For some special cases admitting exponential local energy decay under a certain derivative-loss see [14], [64]. Iwasaki [15] considered the local energy decay for a general hyperbolic system with a general boundary condition which yields a contraction semigroup. But, no decay is given there. In Section 4 we shall derive (total) energy decay like E(t) ≤ CII02 (1 + t)−1 , where I0 = u0 H 1 + u1 , for the same problem (1.2)–(1.3). For this, however, we must assume further Hyp. A . a(x) ≥ ε0 > 0 for |x|

1.

The result is well-known when Ω = RN and a(x) ≥ ε0 > 0 on RN and our result extends it to a more delicate situation. This result is easily generalized also to the

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case of variable coeﬃcients. Indeed, in the next section we treat the wave equation with variable coeﬃcients and prove a unique continuation property as well as a basic inequality concerning energy E(t). General theorems on unique continuation properties based on micro-local analysis are well studied (cf. Tataru [66]). But, here we prove a simple case by use of multiplier method, which makes our argument in this paper more self-contained. The argument here follows Nakao [42], [43]. In Section 6, following Nakao [44], we apply the local energy decay proved in the ﬁrst section to derive Lp estimates for the linear wave equation in exterior domains. For this, we use the so-called ‘cut-oﬀ’ method as in Shibata and Tsutsumi [63]. In Section 7 we apply the estimates established in Sections 2 and 4 to the existence problem of small amplitude global solutions for the semilinear equations: utt − ∆u + a(x)ut = f (u) in Ω × [0, ∞),

(1.4)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(1.5)

where f (u) is a nonlinear resource term like f (u) = |u| u, α > 0. We consider this problem under two types of assumption on a(x) (1) Hyp. A and Hyp. A , (2) Hyp. A. For the ﬁrst case we require only the regularity on the initial data as (u0 , u1 ) ∈ H01 × L2 or (u0 , u1 ) ∈ H 2 ∩ H01 × H01 , while for the second case we need much more regularity as (u0 , u1 ) ∈ H 2M × H 2M−1 , M = [N/2] + 1. Note that the latter is applied to N ≥ 3. Our restrictions on the exponent α do not seem to be optimal compared with the non-dissipative case or the dissipative case as ut in the whole space (Pecher [54], Georgiev [9], Todorova-Yordanov [68], Nishihara [51], Narazaki [50] etc.) and it is desirable to reﬁne our results by making appropriate additional assumptions on the initial data (cf. Ikehata [14], Ono [52]). The arguments of this section are taken from [43], [44]. In Section 8 we consider the quasilinear wave equation: utt − div{σ |∇u|2 ∇u} + a(x)ut = 0 in Ω × [0, ∞), (1.6) α

u(x, 0) = u0 (x),

ut (x, 0) = u1 (x) and u|∂Ω = 0, (1.7) √ 2 where σ(v ) is a function like σ(v ) = 1/ 1 + v 2 . Under two types of assumptions considered in Section 3 and Section 4, respectively, we prove the existence of smooth global solutions for small initial data. The essential idea of the proof is the same as in Section 4. But, more careful analysis will be required. When a(x) ≡ 1 Matsumura [28] proved the global existence of smooth solutions for (1.6)–(1.7) with Ω = RN , Cauchy problem in the whole space, and this result was generalized by Shibata [62] to the exterior problems with N ≥ 3. Our ﬁrst result establishes a global existence result under a weaker assumption on a(x) which admits a(x) to vanish in a large area. When a(x) ≡ 0 and N = 1, 2 we can not generally expect the global existence of smooth solutions of (1.6)–(1.7) even if the initial-data are small and smooth. 2

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Indeed, when Ω = RN nonexistence was proved by Lax [23] and John [16] for the case N = 1 and by Hoshiga [11] for the case N = 2. For the case N ≥ 3, Klainerman and Ponce [21], Shatah [61] proved global existence of small amplitude solutions when Ω = RN and Shibata and Tsutsumi [63] proved similar results for exterior problems under the assumption that the obstacle V := RN \ Ω is non-trapping, especially convex. Recently, Keel, Smith and Sogge [18], [19] have developed the theory in this direction. However, if Ω is a general domain no result on global existence has been known. The reason is that when V is trapping the local energy never decays uniformly and hence it is diﬃcult to expect global solutions for such an exterior domain. In the latter part of this section, by introducing a localized dissipative term a(x)ut as in Hyp. A we treat general exterior domains in odd dimensions and prove global existence theorem for the problem (1.6)–(1.7). Section 8 follows [45]. In the ﬁnal section we consider the initial-boundary value problem (1.4)– (1.5) with the linear dissipation a(x)ut replaced by a nonlinear one ρ(x, ut ). It is an interesting problem to discuss the energy decay property for the nonlinear dissipation like ρ(x, ut ) = a(x)|ut |r ut . Indeed, such a problem has been fully studied for the case of bounded domains (see Nakao [36, 39], Tcheugou´ ´e T´ebou [67], Martinez [27] and the references cited there). Further, corresponding problem for the Klein-Gordon equation was studied by Nakao [37, 41], Mochizuki and Motai [33]. But, for the wave equation under consideration there seem to be almost no results. Mochizuki and Motai [33] treated the case ρ(x, ut ) = |ut |r ut and derived a logarithmic decay rate. Ono [52] treated the case ρ = ut + |ut |r ut and derived an algebraic decay of energy E(t). In [52] the existence of the linear term ut plays an essential role. Some related topics are discussed also in Matsuyama [29]. The diﬃculty for the whole or exterior domains comes from the facts that • Poincare’s ´ inequality fails and • we have very few means to control L2 (Ω) norm well. Here, in Section 9, we consider the case like ρ(x, ut ) = a(x)|ut |r ut on some bounded domain ΩR and ρ(x, ut ) = a(x)ut , linear, for large |x| with a(x) satisfying Hyp. A and Hyp. A . We also present a result on the global existence for the semilinear equations (1.4)–(1.5) with a source term f (u). Section 9 is taken from [47]. In this article we will not discuss the Kirchhoﬀ type quasilinear wave equations in exterior domains. For this topic see Racke [57], Mochizuki [31], Yamazaki [69], Bae and Nakao [2] and the references cited in these papers. The material is organized as follows.

2. Preliminaries Let Ω be a domain in RN . We denote by C0∞ (Ω) and C0∞ (Ω) the sets of inﬁnitely many times diﬀerentiable functions with compact supports in Ω and Ω, respectively. Lp (Ω), 1 ≤ p ≤ ∞, is the Banach space deﬁned in a standard way. L2 (Ω)

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217

norm is often denoted by · . W m,p (Ω) and W0m,p (Ω) are the completions of C0∞ (Ω) and C0∞ (Ω), respectively, with respect to the norm Dα up , |α|≤m

where for a multi-index α = (α1 , . . . , αN ) we set |α| = α1 + · · · + αN and Dα = Dxα11 · · · DxαNN , Dxi = ∂/∂xi . The completion of C0∞ (Ω) with respect to the norm * α ˙ m,p (Ω). We often use the notation W m,2 (Ω) = |α|=m D up is denoted by W ˙ m,2 (Ω) = H˙ m (Ω). The set of L2 (Ω)-valued continuous functions H m (Ω) and W on [0, ∞) is denoted by C([0, ∞); L2 (Ω)). Similar notation will be freely used throughout the article. We begin with the following well-known lemma. Lemma 2.1 (Gagliardo-Nirenberg). Let 1 ≤ r 0, where Dm uq = |α|=m Dα uq , −1 k 1 1 m 1 1 + − + − θ= N r p N r q and we assume 0 < θ ≤ 1 (0 < θ < 1 if p = ∞ and mq = N ). Let f ∈ L2loc ([0, ∞); L2 (Ω)) and let u ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) be a solution of the problem utt − ∆u + a(x)ut = f in Ω × [0, ∞),

(2.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0.

(2.2)

Let η(x) ∈ W 1,∞ (Ω) and h(x) = (h1 (x), . . . , hN (x)) ∈ W 1,∞ (Ω). Then multiplying the equation by ut , η(x)u and h(x) · ∇u and integrating by parts we obtain the following identities: d 2 E(t) + a(x)|ut | dx = f ut dx, (A) dt Ω Ω d η(x)ut udx − η(x)|ut |2 dx + ∇u · ∇(ηu)dx dt Ω Ω Ω + η(x)a(x)ut udx = f η(x)udx (B) Ω

Ω

and N ∂hi ∂u ∂u 1 d ut h(x) · ∇udx + ∇·h(x) |ut |2 − |∇u|2 dx+ dx dt 2 ∂x j ∂xi ∂xj Ω Ω Ω i,j=1 1 − 2

2 ∂u ν · h(x)dS + a(x)ut h(x) · ∇udx = f h(x) · ∇udx. ∂ Ω ∂ν Ω Ω

(C)

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To derive the identity (C) we have used the following calculations: N ∂u ∂ ∂u ∂u hi dx − ∆uh · ∇udx = h · ∇udS − ∂x ∂x ∂x j j i Ω ∂ Ω ∂ν i,j=1 Ω 2 N N ∂u ∂hi ∂u ∂u ∂u ∂2u dx + hi dx − h · ν dS. = ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj ∂ν ∂Ω i,j=1 Ω i,j=1 Ω Here, N ∂u ∂u ∂u ∂2u ∂ hi dx = − hi dx ∂x ∂x ∂x ∂x ∂x ∂x j i j i j j i,j=1 Ω i,j=1 Ω 2 N N ∂u ∂u ∂u ∂hi ∂u ∂u + νj hi dS − dx + h · ν dS ∂xj ∂xi ∂xj ∂xi ∂xj ∂ν ∂Ω i,j=1 ∂ Ω i,j=1 Ω N

and

N i,j=1

Ω

∂u ∂2u 1 hi dx = ∂xj ∂xi ∂xj 2

∂Ω

2 ∂u h · ν dS. ∂ν

Hence, −

∆uh · ∇udx = Ω

2 N ∂u ∂hi ∂u ∂u 1 dx − h · ν dS. ∂xj ∂xi ∂xj 2 ∂Ω ∂ν i,j=1 Ω

We also use some variations of the above identities. These are the main tools throughout the paper.

3. Local energy decay 3.1. Problem and result In this section, following [40] we investigate the decay property of the local energy of the solutions to the initial-boundary value problem for the wave equation with a localized dissipation: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(3.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. Concerning the initial data we assume that (u0 , u1 ) belongs to has a compact support, that is,

(3.1.2)

H01 (Ω) × L2 (Ω)

and

supp u0 ∪ supp u1 ⊂ BL = {x ∈ RN | |x| ≤ L} for some L > 0. As it was already stated we make the following hypothesis on a(x).

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219

Hyp. A. a(·) is a nonnegative function on Ω belonging to L∞ (Ω), and there exist a relatively open set ω in Ω and x0 ∈ RN such that Γ(x0 ) ⊂ ω and a(x) ≥ ε0 > 0 on ω for some ε0 . ˜ > 0. Moreover, throughout this section we assume that supp a(·) ⊂ BL˜ for some L ˜ We may assume L ≤ L. Our main result reads as follows: Theorem 3.1. Under Hyp. A the solutions u(t) ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) of the problem (3.1.1)–(3.1.2) satisfy the estimate ε Eloc (t) ≤ Cε,δ E(0)(1 + t)−1+δ

(3.1.3)

with any 0 < ε, δ < 1, where we set 2 1 ε Eloc (t) := |ut | + |∇u|2 dx. 2 Ω∩BL+εt The constant Cε,δ depends on ε, δ and L. Since a(x) has compact support we can apply the argument in Morawetz [35] to get the exponential decay for the case of odd dimensions. Corollary 3.1. Let N ≥ 3 be odd. Then, under the conditions of Theorem 3.1, we have further ε Eloc (t) ≤ Cε,δ E(0)e−λt for some λ = λ(ε, δ) > 0. Remark 3.1. When a(x) ≡ 1 Dan and Shibata [7] proved by a spectral method that Eloc (t) ≤ CE(0)(1 + t)−N , 0 where Eloc (t) := 12 ΩR (|ut |2 + |∇u|2 )dx for all R 1. Remark 3.2. When a(x) ≡ 0 and V consists of several convex bodies in some location Ikawa [12], [13] proved for the case N = 3 Eloc (t) ≤ C(||u0 ||H 2 + ||u1 ||H 1 )e−λt with some λ > 0. 3.2. Proof of Theorem 3.1. The proof is given by combining the ideas in Morawetz [35], Bloom and Kazarinoﬀ [4] and Zuazua [70]. Let 0 ≤ T0 < T . First, we note that from (A) T a(x)|ut |2 dx dt = E(T T0 ). (3.2.1) E(T ) + T0

Similarly,

T

Ω

T

ta(x)|ut | dx dt = T0 E(T T0 ) + 2

T E(T ) + T0

Ω

E(t)dt. T0

(3.2.2)

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Mitsuhiro Nakao

From (B), T

η |∇u|2 − |ut |2 dx dt T0 Ω T = −(ηu, ut )|TT0 − ηa(x)ut u dx dt − T0

Ω

T

T0

(3.2.3) ∇η · ∇uu dx dt

Ω

for η ∈ W 1,∞ (Ω). Also from (C), T 2 N T |ut | − |∇u|2 dx dt + |∇u|2 dx dt + (ut , (x − x0 ) · ∇u)|TT0 2 T0 Ω T0 Ω 2 0T 0 0T 0 dx dS = − T0 Ω a(x)(x − x0 ) · ∇uut dx dt + 12 T0 ∂ Ω (x − x0 ) · ν(x) ∂u ∂ν (3.2.4) and ⎛ ⎞ T ∂hj ∂u ∂u ⎠ ⎝ 1 ∇ · h |ut |2 − |∇u|2 + dx dt 2 ∂xi ∂xi ∂xj T0 Ω i,j T (3.2.5) a(x)h · ∇uut dx dt − (ut , h · ∇u)|TT0 =− T0 Ω 2 ∂u 1 T + h(x) · ν(x) dx dS 2 T0 ∂ Ω ∂ν for any W 1,∞ vector ﬁeld h = (h1 , . . . , hN ). It follows from (3.2.2), (3.2.3) with η = 1 and (3.2.4) that 1 T E(T ) + ((x − x0 ) · ∇u(T ), ut (T )) + (N − 1)(u(T ), ut (T )) 2 T T 2 + a(x)t|ut | dx dt + a(x)(x − x0 ) · ∇uut dx dt T0 Ω T0 Ω 1 + (N − 1) a(x)|u(T )|2 dx 2 Ω 1 = T0 E(T T0 ), ut (T T0 ) + ((x − x0 ) · ∇u(T T0 ), uT (T T0 )) + (N − 1)(u(T T0 )) 2 2 T ∂u 1 1 + (N − 1) a(x)|u(T T0 )|2 dx + (x − x0 ) · ν dx dS. 2 2 ∂ν Ω

T0

(3.2.6)

∂Ω

Here, T a(x)(x − x0 ) · ∇uut dx dt T0 Ω T a(x) ε T 1 2 |(x − x0 ) · ∇u| dx dt + a(x)t|ut |2 dx dt ≤ 2ε T0 Ω t 2 T0 Ω ¯ 2 ||a(·)||∞ T L ε T 2 ≤ |∇u| dx dt + a(x)t|ut |2 dx dt 2εT T0 2 T0 Ω T0 ΩL

(3.2.7)

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for any T0 > 0 and ε > 0, where we set ¯ = supx∈Ω(L) |x − x0 | = supx∈∂Ω∪∂B |x − x0 |. L L Thus, for any δ > 0, we can take T0 = T0 (δ) > 0 such that T a(x)(x − x0 ) · ∇uut dx dt T0 Ω T T ≤δ |∇u|2 dx dt + δ a(x)t|ut |2 dx dt. T0

Ω(L)

T0

(3.2.8)

Ω

If a(x) ≡ 0, we can take δ = 0 in (3.2.8). To estimate the boundary integral in (3.2.6) we take h(x) such that h(x) = ν(x) on Γ, h(x) = 0 on ω ˜ c and h(x) · ν(x) ≥ 0 for all x ∈ RN , ˜ and ω ˜ ∩ Ω ⊂ ω. Then, by the where ω ˜ is an open set in RN such that Γ(x0 ) ⊂ ω deﬁnition of Γ(x0 ) and the identity (3.2.5), we have 2 2 T ∂u ∂u (x − x0 ) · ν dx dS ≤ (x − x0 ) · ν dx dS ∂ν ∂ν T0 ∂ Ω T0 Γ(x0 ) 2 T T ∂u 2 ˜ dx dS ≤ C ≤L |ut | + |∇u|2 dx dt + CE(0), ∂ν

T

T0

Γ(x0 )

T0

ω ˜ ∩Ω

(3.2.9) where C is a constant depending on sup x∈ω (|h| + |∇Dh|). The kinetic local 0T 0 0T 0 energy T0 ω˜ ∩Ω |ut |2 dx dt in (3.2.9) will be absorbed into T0 Ω a(x)t|ut |2 dx dt in 0T 0 the left-hand side of (3.2.6), and we must estimate T0 ω˜ ∩Ω |∇u|2 dx dt. For this we take η ∈ W 1,∞ (Ω) such that √ η(x) = 1 on ω ˜ ∩ Ω, η(x) = 0 on ω c , 0 ≤ η(x) ≤ 1 on Ω and |∇η|/ η is bounded. Then, we have from (3.2.3) that

T

T0

ω ˜ ∩Ω

T

|∇u|2 dx dt ≤ C T0

2 |ut | + |u|2 dx dt + CE(0),

(3.2.10)

ω

where we have used the Poincar´´e’s inequality ||u||L2 (ω) ≤ C||∇u||L2 (Ω(L)) for u ∈ H01 (Ω) and (3.2.1). Therefore, from (3.2.9) and (3.2.10) we have

T

T0

2 T ∂u C (x − x0 ) · ν dx dS ≤ ta(x)|ut |2 dx dt ∂ν ε0 T0 T0 Ω ∂Ω T +C |u|2 dx dt + CE(0).

T0

ω

(3.2.11)

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Mitsuhiro Nakao

The term ((x − x0 ) · ∇u(T ), u(T )) in (3.2.6) is estimated in a standard manner as follows: (|x0 | + |x|)|∇u(T )||ut (T )|dx |((x − x0 ) · ∇u(T ), ut (T ))| ≤ Ω(L+T ) = |x0 | |∇u(T )||ut (T )|dx + |x||∇u(T )||ut (T )|dx Ω(L+T ) Ω(εT ) + |x||∇u(T )||ut (T )|dx BL+T \BεT T εT 2 2 ≤ (|x0 | + L)E(0) + |∇u| + |ut | dx + |∇u|2 + |ut |2 dx 2 Ω(εT ) 2 BL+T \BεT T ε = (|x0 | + L)E(0) + εT Eloc |∇u(T )|2 + |ut (T )|2 dx, (T ) + 2 BL+T \BεT (3.2.12) where we have assumed εT > L. We also note that |(u(T ), ut (T ))| ≤ ||u(T )||2 + E(0). Summarizing (3.2.6), (3.2.8), (3.2.11) and (3.2.12), we obtain T C ε (1 − ε)T Eloc + 1−δ− a(x)t|ut |2 dx dt ε0 T 0 T Ω 0 T

≤ (C + T0 )E(T T0 ) + C

T

||u(T )||2 +

|u(t)|2 dx dt + ε T0

|∇u|2 dx dt T0

ω

Ω(L)

(3.2.13) and, taking T0 large enough so that (1 − δ)ε0 T0 > C and εT T0 > L, we have T T ε 2 2 ε (1−ε)T Eloc ≤ (C +T T0 )E(T T0 )+C ||u(T )|| + |u(t)| dx dt +δ Eloc (t)dt. T0

ω

T0

(3.2.14) Finally, we must show the boundedness of the second term in the right-hand side of (3.2.14). For this we use a device due to Morawetz [35]. Let h(x) be the unique solution of the elliptic exterior problem ∆h(x) = u1 (x) + a(x)u0 (x) in Ω 2−N

with h|∂Ω = 0 and h(x) = O(|x| Then, setting

(3.2.15)

) as x → ∞ (h(x) = O(log|x|) if N = 2).

w(t) =

t

u(s) ds + h(x) 0

we have wtt − ∆w + a(x)wt = 0 in Ω × (0, ∞), w(x, 0) = h(x), wt (x, 0) = u0 (x) and w|∂Ω = 0.

(3.2.16)

Concerning the solution h(x) of (3.2.15) it is known (cf. Meyers and Serrin [30]) 2 (Ω) and that h ∈ Hloc sup

N −2 |h(x)| |x|≥2L {|x|

+ |x|N −1 |∇h(x)|} ≤ C(L)(||u1 || + ||u0 ||),

(3.2.17)

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223

where we note supp u0 ∩ supp u1 ⊂ BL (|x|N −2 |h(x)| in the above should be replaced by (log(L + |x|))−1 |h(x)| if N = 2). Thus, multiplying (3.2.16) by wt we can prove T 1 2 2 sup a(x)|wt |2 dx dt |wt (t)| + |∇w(t)| dx + 0≤t≤T 2 Ω(L+T ) 0 Ω ≤ CE(0) < ∞ (CE(0)log(2 + T ) if N = 2),

(3.2.18)

which implies the boundedness or logarithmic growth of the terms in the second bracket in the right-hand side of (3.2.14). Thus, changing the notation for δ, i.e., replacing δ/(1 − ε) by δ, we arrive at the inequality T ε ε (T ) ≤ Cε,δ E(0) + δ Eloc (t)dt if N ≥ 3 (3.2.19) T Eloc T0

or

T

ε T Eloc (t) ≤ Cε,δ E(0)log(2 + T ) + δ

ε Eloc (t)dt if N = 2

(3.2.19)

T0

for any 0 < ε, δ < 1. We can easily solve the inequality (3.2.19) or (3.2.19)’ as in Bloom and Kazarinoﬀ [4]. Indeed, setting T −δ ε Eloc (t)dt φ(t) = T T0

we see that (3.2.19) and (3.2.19)’ are equivalent to φ (T ) ≤ Cε,δ E(0)T −δ−1 and φ (t) ≤ Cε,δ E(0)T −δ−1 log(2 + T ), respectively. Thus, we have φ(T ) ≤ Cε,δ E(0), T0 ≤ T < ∞. Returning to (3.2.19) or (3.2.19)’ we obtain ε (T ) ≤ Cε,δ E(0)(1 + δT δ ) T Eloc

for any 0 < ε, δ < 1, which implies the desired result in Theorem 3.1.

3.3. Proof of Corollary 3.1. The proof of Corollary 3.1 is given along the same line of the argument by Morawetz [35]. For convenience of the readers we sketch it brieﬂy. We use the following notation for R > L: 1 |fft (t)|2 + |∇f (t)|2 dx. E(f, R, t) = 2 ΩR The same notation will be employed when f (t) is deﬁned on RN and ΩR is replaced by BR . We split the solution u(t) as u(t) = F0 (t) + W0 (t), where F0 is the free space solution of the wave equation F0 = 0 in [0, ∞) × RN F0tt − ∆F

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Mitsuhiro Nakao

with F0 (0) = u0 and F0t (0) = u1 . Here, we extend F0 (0) and F0t (0) to inside of the body V = RN \ Ω appropriately so that E(0), µ > 0. E(F F0 , ∞, 0) ≤ (1 + µ) Ω

By Huygens’ principle we see F0 (x, t) = 0 if |x| ≤ t − L or |x| ≥ t + L. We also note that supp W0 (t) = supp(u(t) − F0 (t)) ⊂ B3L for t = 2L. We easily show that W0 + a(x)W W0t = −a(x)F F0t = 0 and W0 |∂Ω = 0 if t ≥ 2L W0tt − ∆W and E(W W0 , ∞, 2L) ≤ (4 + 2µ)E(0). Next, we take T ≥ 2L and split W0 (t) as W0 (t) = F1 (t) + W1 (t) for t ≥ T, where F1 is the free solution of the wave equation for t ≥ T with F1 (0) = W0 (T ) and F1t (T ) = W0t (T ). We see F1 (x, t) = 0 if |x| ≤ t − T − L or |x| ≥ t + T + L and W1 + a(x)W W1t = −a(x)F F1t = 0 if t ≥ T + 2L. W1tt − ∆W Noting that the value of F1 (t) on BL , T ≤ t ≤ T + 2L, depends only on the data of F1 (T ) on B3L , we take F˜1 such that F˜1tt − ∆F˜1 = 0 on [T, ∞) × RN with F˜1 (T ) = F1 (T ) and F˜t (T ) = F1t (T ) if |x| ≤ 3L and F˜1 (T ) = F˜1t (T ) = 0 if |x| ≥ 3L + ε0 , F1 , B3L , T ). Now, we 0 < ε0 < 1. We may assume E(F˜1 , B3L+ε0 , T ) ≤ (1 + µ)E(F set ˜ 0 = W1 + F˜1 = W0 − F1 + F˜1 . W Then, ˜ 0tt − ∆W ˜ 0 + a(x)W ˜ 0t = a(x)(F W F1t − F˜1t ) = 0 if T ≤ t ≤ T + 2L ˜ 0 (T ) = W0 (T ), ˜ 0 |∂Ω = W1 |∂Ω + F˜1 |∂Ω = u|∂Ω = 0. We further see that W and W ˜ 0t (T ) = W0t (T ) on B3L ∩ Ω and W ˜ 0t (T ) = W0t (T ) = 0 if |x| ≥ 3L + ε0 . Setting W

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˜ 0 − F˜1 , we have W1 := W0 − F1 = W ˜ 0 , ∞, T + 2L) + 2E(F˜1 , ∞, T + 2L) E(W W1 , ∞, T + 2L) ≤ 2E(W ˜ 0 , ∞, T ) + 2E(F˜1 , ∞, T ) ≤ 2E(W ˜ ≤ 2E(W0 , B3L+ε0 , T ) + 2E(F˜1 , B3L+ε0 , T ) F1 , B3L , T )) ≤ 2(1 + µ)(E(W W0 , B3L , T ) + E(F ≤ kE(W W0 , B3L , T ), k := 2(1 + µ)(2 + µ). Repeating this argument we obtain the splitting: W0 =

n

Fj + Wn ,

u = F0 + W0 ,

n = 1, 2, 3, . . . ,

(3.3.1)

j=1

where Fj is a free solution of the wave equation for t ≥ jT with suppF Fj (t) ⊂ Bt−jT −L ∩ Bt+jT +L and Wn satisﬁes Wntt − ∆W Wn + a(x)W Wnt = 0 for t > nT + 2L with the boundary condition Wn |∂Ω = 0. Further, for Wn , we have Wn−1 , B3L , nT ). E(W Wn , ∞, nT + 2L) ≤ kE(W

(3.3.2)

Applying the decay property of the local energy of Theorem 3.1 to Wn−1 (t) we know Wn−1 , ∞, (n − 1)T + 2L), E(W Wn−1 , B3L , nT ) ≤ p(T − 2L)E(W

(3.3.3)

where p(t) = (1 + t)−1+δ . From (3.3.2) and (3.3.3) we obtain W0 , ∞, 2L) ≤ k(kp(T −2L))n E(0). (3.3.4) E(W Wn , ∞, nT +2L) ≤ (kp(T −2L))n E(W Hence, we have E(W Wn , ∞, t) ≤ k(kp(T − 2L))n E(0) if nT + 2L ≤ t < (n + 1)T + 2L. For 0 < ε < 1 and for t > 2L/(1 − ε) we take n such that ((n + 1)T + 2L)/(1 − ε) > t ≥ (nT + 2L)/(1 − ε). Then, Fj = 0 on ΩL+εt , j = 1, 2, . . . , n, and ε (t) = E(u, ΩL+εt , t) = E(W Wn , ΩL+εt , t) Eloc

≤ E(W Wn , ∞, t) ≤ k(kp(T − 2L))n/(1−ε) kE(0). Thus, taking T suﬃciently large so that kp(T − 2L) < 1, we conclude the expoε nential decay of Eloc (t).

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4. Total Energy decay for the wave equation with a localized dissipation 4.1. Problem and result Following [43], we derive L2 boundedness and (total) energy decay of solutions for the linear problem: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(4.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. (4.1.2) In addition to Hyp. A we make the assumption Hyp. A . Summarizing these we state: ˜ Hyp. A (1) There exists x0 ∈ RN and a relatively open set ω ⊂ Ω such that Γ(x0 ) ⊂ ω and a(x) ≥ ε0 > 0 for x ∈ ω with some ε0 . (2) There exists L > 0 such that a(x) ≥ ε0 > 0 for |x| ≥ L.

Needless to say we may assume V := Ωc ⊂ BL = {x ∈ RN | |x| ≤ L}. Our result on the total energy decay reads as follows: ˜ Then, for each Theorem 4.1. Let a(·) belong to L∞ (Ω) and satisfy Hyp. A. 1 2 (u0 , u1 ) ∈ H0 (Ω) × L (Ω) there is a unique solution u(t) ∈ C 1 ([0, ∞); L2 (Ω)) ∩ C([0, ∞); H01 (Ω)) such that u(t)2 ≤ C0 I0 (4.1.3) and 1 ut (t)2 + ∇u(t)2 ≤ C0 I02 (1 + t)−1 , E(t) := (4.1.4) 2 where C0 is a positive constant independent of u and we set I0 := u0 H01 + u1 . For more regular solutions we can derive sharper decay estimates. The following is a typical one. ˜ and let (u0 , u1 ) ∈ Theorem 4.2. Let a(·) belong to C 1 (Ω) and satisfy Hyp. A H 2 (Ω) ∩ H01 (Ω) × H01 (Ω). Then, the solution u(t) in Theorem 4.1 satisﬁes further, utt (t)2 + ∇ut (t)2 ≤ C1 I12 (1 + t)−2

(4.1.5)

and ∆u(t)2 ≤ C1 I12 (1 + t)−1 , where C1 is a constant independent of u and I1 = u0 H 2 + u1 H 1 .

(4.1.6)

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4.2. Proof of Theorem 4.1 The existence and uniqueness part is well-known, and for the proof of Theorem 4.1 it suﬃces to derive the estimates (4.1.3)–(4.1.4). We note that for each (u0 , u1 ) ∈ H01 × L2 , the solution u(t) is given as the limit of smooth solutions um (t) with um (0) = um,0 ∈ C0∞ (Ω) and umt (0) = um,1 ∈ C0∞ (Ω) such that um,0 → u0 ∈ H01 and um,1 → u1 ∈ L2 . Therefore, in deriving estimates we may assume that u(t) is smooth (H 2 -valued is suﬃcient) and supp u(t) is compact in Ω for each t, which is due to the ﬁnite propagation property of the wave equation. For a simple proof of this property by multiplier technique see John [16]. From (A), d E(t) + a(x)|ut (t)|2 dx = 0. (4.2.1) dt Ω From (B) with η(x) = 1, d 1 d a(x)|u|2 dx = 0. (ut , u) + |∇u|2 − |ut |2 dx + dt 2 dt Ω Ω

(4.2.2)

We deﬁne a Lipschitz continuous function φ(r) on R+ as follows: ⎧ ⎨ε 0 if 0 ≤ r ≤ L, φ(r) = ε0 L ⎩ if r ≥ L, r ˜ Setting h(x) = φ(r)(x−x0 ) where ε0 , L are positive constants appearing in Hyp. A. with r = |x − x0 | in (C) we see d 1 ut φ(x − x0 ) · ∇u dx + (N φ + φ r) |ut |2 − |∇u|2 dx dt Ω 2 Ω 2 φ 1 2 2 ∂u ν · (x − x0 )φ dS + |(x − x ) · ∇u| + φ|∇u| − dx (4.2.3) 0 2 ∂ Ω ∂ν r Ω a(x)ut φ(x − x0 ) · ∇u dx = 0, + Ω

and hence, combining this with (4.2.2) and (4.2.3), we have for α ≥ 0 and k ≥ 0, d α 2 ut φ(x − x0 ) · ∇u dx + α(ut , u) + a(x)|u| dx + kE(t) dt 2 Ω Ω N φ + φ r N φ + φ r + − α + ka(x) |ut |2 dx + + φ + φ r |∇u|2 dx α− 2 2 Ω Ω 2 1 ∂u φν · (x − x0 )dS − ≤ aut φ(x − x0 ) · ∇u dx. 2 Γ(x0 ) ∂ν Ω (4.2.4) Noting that a(x) ≥ ε0 > 0 if |x| ≥ L, we easily see that there exists α > 0 such that N φ + φ r ka(x) + − α > ε1 > 0 (4.2.5) 2 2

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and

N φ + φ r + φ + φ r > ε1 > 0 (4.2.6) 2 for some ε1 , 0 < ε1 < ε0 /2, and for any k ≥ N + 1. Indeed, we can choose, for example, ε0 (2N − 1)ε0 and ε1 = . α= 4 4 Further, we note that C 2 a(x)ut φx · ∇u dx ≤ k a(x)|u | dx + |∇u|2 dx t 4 k α−

Ω

Ω

Ω

with some C > 0. Therefore, we obtain from (4.2.4) that d α ut φ(x − x0 ) · ∇u dx + α(ut , u) + a(x)|u|2 dx + kE(t) dt 2 Ω Ω 2 ∂u k 1 φν · (x − x0 ) dS + a(x)|ut |2 dx + ε1 E(t) ≤ (4.2.7) 4 Ω 2 Γ(x0 ) ∂ν for any large k. In order to control the right-hand side of (4.2.7) we take a vector ﬁeld h(x) as in (3.2.9). Then, from (C) we have 2 ∂u 1 dS ≤ d u h · ∇u dx + ∇ · h |ut |2 − |∇u|2 dx (4.2.8) t ∂ν dt Ω 2 Ω Γ(x0 ) |a(x)ut h · ∇u|dx + |∇u|2 |Dh|dx ω Ω 2 d ≤ ut h · ∇u dx + C |ut | + |∇u|2 dx dt Ω ω ∩Ω +

with some C > 0. Thus, we have from (4.2.7) and (4.2.8) that d α 2 ut (φ(x − x0 ) − C0 h) · ∇u dx + α(ut , u) + a(x)|u| dx + kE(t) dt 2 Ω Ω 2 k ε1 + a(x)|ut |2 dx + E(t) ≤ C |ut | + |∇u|2 dx (4.2.9) 4 Ω 2 ω ∩Ω for k ≥ N + 1 and some C0 , C > 0. The ﬁrst term of the right-hand side of (4.2.9) is easily absorbed into the left-hand side if we take k large. To control the second term we choose a nonnegative function η ∈ W 1,∞ (Ω) as for getting (3.2.10). Then from (C), 2 |∇u| dx ≤ η|∇u|2 dx ω ∩Ω Ω d 1 d 2 ∇η · ∇uu dx − (ut , ηu) + η|ut | dx − a(x)η|u|2 dx. =− dt 2 dt Ω Ω Ω (4.2.10) Since 1 |∇η · ∇uu|dx ≤ η|∇u|2 dx + C |u|2 dx 2 Ω Ω ω

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229

we have from (4.2.10), 2 d |ut | + |u|2 dx − |∇u|2 dx ≤ C a(x)η|u|2 dx . 2(ut , ηu) + dt ω ∩Ω ω Ω (4.2.11) We obtain from (4.2.9) and (4.2.11) that d 1 2 ut (φ(x − x0 ) + h) · ∇u dx + α(ut , u) + a(x)(α + η)|u| dx + kE(t) dt 2 Ω Ω k ε1 + a(x)|ut |2 dx + E(t) ≤ C |u|2 dx, (4.2.12) 8 Ω 2 ω where we have chosen a large k > 0 (for simplicity, we use notations h and η for Ch and Cη, respectively). Finally, we prepare the following proposition which shows the possibility of the right-hand side of (4.2.12) being controlled by the last two terms of the left-hand side of (4.2.12). Proposition 4.1. There exists T0 > 0 independent of u such that if T > T0 , the inequality t+T t+T ε1 t+T 2 2 |u(x, s)| dx ds ≤ C a(x)|ut (x, s)| dx ds + E(s) ds, 4 t t t ω Ω (4.2.13) t > 0, holds, where C is a constant which may depend on T . Proof. When Ω is bounded the inequality (4.2.13) holds without the second term in the right-hand side as it is shown in Zuazua [70]. For the proof of (4.2.13) we modify the argument in [70]. If (4.2.13) would be not true, there would exist a sequence of numbers {tn } and a sequence of solutions {un } such that tn +T tn +T ε1 tn +T |un |2 dx ds ≥ n a(x)|unt |2 dx ds + En (s) ds, 4 tn tn tn ω Ω (4.2.14) where En (t) is deﬁned by E(t) with u replaced by un . Setting tn +T 2 λn = |un |2 dx ds and vn (t) = u(· + tn )/λn tn

we have

ω

ε1 T n a(x)|vnt (t)| dx dt + En (t)dt ≤ 1, (4.2.15) 4 0 0 Ω where En (t) is deﬁned by En (t) with un replaced by vn . Then, we note that T |vn (t)|2 dx dt = 1, (4.2.16) T

2

0

T

Ω

0

ω

|vnt (t)|2 + |∇vn (t)|2 dx dt ≤ 8ε−1 1 < ∞

(4.2.17)

230 and

Mitsuhiro Nakao

T

a(x)|vnt (t)|2 dx dt → 0 as n → ∞.

0

(4.2.18)

Ω

Therefore, along a subsequence, vn (t) is weakly convergent to a function v ∈ L2 ([0, T ]; H01,loc (Ω)) ∩ W 1,2 ([0, T ]; L2loc(Ω)) with ∇v, vt ∈ L2 ([0, T ]; L2(Ω)). Here we note that H01,loc (Ω) denotes the set of functions u such that φ(·)u(·) ∈ H01 (Ω) for any φ(·) ∈ C0∞ (Ω). We may assume, by Rellich’s Lemma, that vn (t) converges to v strongly in L2 ([0, T ] × ω). It is clear that the limit function v(t) solves the equation vtt − ∆v = 0 in [0, T ] × Ω and, further, it satisﬁes the conditions T |v|2 dx dt = 1 0

(4.2.19)

(4.2.20)

ω

and (4.2.21) vt (x, t) = 0 on [0, T ] × supp a(·). c Since supp a(·) includes ω and ΩL = {x ∈ Ω| |x| > L}, by a general result of unique continuation, we see that there exists T0 > 0 such that if T > T0 , vt (x, t) = 0 on Ω × [0, T ].

(4.2.22)

(For a convenient unique continuation property see Remark 4.1 below. A more general result will be proved in Section 5.) The fact (4.2.22) means that v(x, t) = v(x), independent of t, and by (4.2.19), −∆v(x) = 0 in Ω. Since v ∈ H01,loc (Ω) and ∇v ∈ L2 (Ω) we conclude from the above that v(x) ≡ 0 in Ω. This is a contradiction to (4.2.20). Remark 4.1. Let us assume that Γ(x0 ) ⊂ ω and let v(t) ∈ C([0, T ]; H01 (Ω)) ∩ C 1 ([0, T ]; L2 (Ω)) be a solution of the problem (4.1.1)–(4.1.2). Further, we assume v(x, t) = 0 on ω ∪ ΩcL . Then, we know by the inequality (4.2.12), d vt φ(x − x0 ) · ∇vdx + α(vt , v) + kE(t) + ε1 E(t) ≤ 0, dt Ω which implies easily for large k > 0 that E(T ) + T E(T ) ≤ CE(0) = CE(T ) for some C > 0 independent of T , where we have used the inequality |(vt , v)| ≤ Cvt (t)∇v(t) which follows from the facts v(t) ∈ H01 (Ω) and supp v(t) ⊂ BL . Taking T > C we see E(0) = 0, that is, v(x, t) = 0 on [0, T ] × Ω. When v is a weaker solution

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231

we should apply the above to vε (t) = (ρε v)(t), where ρε (t) is the molliﬁer with respect to the t variable. Now, we return to (4.2.12). By Proposition 4.1 we have for large k > 0 and for T > T0 , ε1 t+T k t+T 2 a(x)|ut | dx ds + E(s) ds ≤ 0, (4.2.23) X(t + T ) − X(t) + 8 t 4 t Ω where we set α ut (φ(x − x0 ) + h) · ∇u dx + α(ut , u) + a(x)(1 + η)|u(t)|2 dx + kE(t). X(t) = 2 Ω Ω (4.2.24) Noting that φ(r)(x − x0 ) is bounded, a(x) ≥ ε0 for |x| ≥ L and |u|2 dx ≤ C∇u2 (Poincar´ ´e’s inequality), Ω∩BL

˜ := E(t) + u(t)2 if k is large. When we easily see that X(t) is equivalent to E(t) N Ω = R , that is, we consider the Cauchy problem, we should use the inequality 2 2 2 |u| dx ≤ C ∇u + |u| dx |x|≥L

BL

instead of the above Poincare’s ´ inequality. Thus, we conclude from (4.2.23) that ∞ k ε1 ∞ a(x)|ut |2 dx ds + E(s) ds ≤ X(0). (4.2.25) 8 0 4 0 Ω Returning to the inequality (4.2.23) and using the estimate (4.2.25) we obtain ∞ ∞ sup0≤t<∞ E(t) + u(t)2 + a(x)|ut |2 dx ds + E(s) ds 0

0

Ω

≤ CX(0) ≤

CII02

< ∞.

(4.2.26)

Once the estimate (4.2.26) is established the decay of E(t) can be easily proved. Indeed, we have from the standard energy identity that (1 + t)

d E(t) ≤ 0 dt

and d ((1 + t)E(t)) ≤ E(t), dt which implies

t

E(s) ds ≤ CII02 < ∞.

(1 + t)E(t) ≤ E(0) +

(4.2.27)

0

The proof of Theorem 4.1 is now complete.

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4.3. Proof of Theorem 4.2 To prove Theorem 4.2 we set v = ut . Then, we see

vtt − ∆v + a(x)vt = 0 in Ω × [0, ∞), v(x, 0) = u1 (x), vt (x, 0) = ∆u0 (x) − a(x)u1 (x) =: v1 (x) and v|∂Ω = 0. Replacing u by v in the deﬁnition of X(t) and E(t) we denote these by X1 (t) and E1 (t), respectively. Then, we know by (4.2.23), k t+T ε1 t+T a(x)|ut |2 dx ds + E1 (s) ds ≤ 0, t ≥ 0, X1 (t + T ) − X1 (t) + 8 t 4 t Ω (4.3.1) and (4.3.2) v(t)2 = ut (t)2 ≤ CII02 (1 + t)−1 . Repeating the same argument proving (4.2.26) we see that ∞ E1 (t)dt ≤ CI12 < ∞.

(4.3.3)

0

Next, we show

∞

(1 + s)E1 (s) ds ≤ CI12 < ∞.

(4.3.4)

0

Indeed, multiplying (4.3.26) by (1 + t + T ) we have ε1 t+T (1 + t + T )X1 (t + T ) − (1 + t)X1 (t) + E1 (s) ds ≤ T X1 (t) 4 t

(4.3.5)

and hence, for any natural number n, n−1 n−1 ε1 nT E1 (jT ) + ut (jT )2 (1 + s)E1 (s) ds ≤ T X1 (jT ) ≤ CT 4 0 j=0 j=0 ∞ n−1 (j+1)T ≤ CT (E1 (s) + E(s)) ds ≤ CT (E1 (s) + E(s)) ds ≤ CT I02 < ∞, 0

jT

j=0

which implies (4.3.3). Now, by the energy identity for v = ut , we see dE1 (t) ≤0 dt

(1 + t)2 and hence,

d (1 + t)2 E1 (t) ≤ 2(1 + t)E1 (t). dt

This implies

(1 + t)2 E1 (t) ≤ E1 (0) + 2

∞

(1 + s)E1 (s) ds ≤ CI12 < ∞.

0

Thus, we conclude that utt (t)2 + ∇ut (t)2 ≤ CI12 (1 + t)−2 .

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The estimate for ∆u(t) follows immediately from the equation itself. The proof of Theorem 4.2 is ﬁnished.

5. Linear equations with variable coeﬃcients; Unique continuation property and a basic inequality 5.1. Problem and result In this section we derive some basic inequality for the energy of solutions to the linear wave equation with variable coeﬃcients. This will be useful when we consider quasilinear wave equations. Similar arguments can be applied to the equation in bounded domains and we can get a simple convenient unique continuation property. The equation in bounded domains is easier to treat and we begin with this case. The equation we consider is N ∂ ∂ u = 0 in Ω × [0, T ] (5.1.1) utt − aij (x, t) ∂xi ∂xj i,j=1 with the boundary condition u(x, t)|∂Ω = 0, 0 ≤ t ≤ T , where Ω is a bounded domain in RN with C 2 class boundary ∂Ω. We assume that aij ∈ C 1 (Ω × [0, T ]) and N aij ξi ξj ≥ k0 |ξ|2 for ξ ∈ RN (5.1.2) i,j=1

with some constant k0 > 0. For later use we ﬁrst consider the equation with a forcing term f ∈ L2 ([0, T ]; L2(Ω)), N ∂ ∂ aij (x, t) utt − u = f (x, t) ∂xi ∂xj i,j=1

in

Ω × [0, T ].

(5.1.3)

We derive the following diﬀerential inequality: Proposition 5.1. Let u ∈ C 1 ([0, T ]; L2 (Ω))∩C([0, T ]; H01 (Ω)) be a solution of (5.1.3) and let ω be an open set in Ω such that ω ⊃ Γ(x0 ) for some x0 ∈ RN , where Γ(x0 ) is the part of the boundary ∂Ω deﬁned in the introduction. Then there exists δ0 > 0 such that if |∇aij | ≤ δ0 , it holds T ∇u(t)2 + ut (t)2 dt 0 T (5.1.4) T 2 2 2 |ut | + |u| dx ds + f (t) dt ≤ C E(0) + E(T ) + 0

ω

0

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Mitsuhiro Nakao

with some C > 0 independent of u, where we set ⎞ ⎛ N 1 ∂u ∂u E(t) := ⎝ut (t)2 + aij (x, t) dx⎠ . 2 ∂xi ∂xj Ω i,j=1 Then, by use of the above inequality we can prove the following theorem on the unique continuation: Theorem 5.1. Let ω be an open set in Ω such that ω ⊃ Γ(x0 ). Then, there exist δ1 > 0 and T0 > 0 such that if T > T0 and &

∂ (5.1.5) sup |∇aij (x, t)| + aij (x, t) ≤ δ1 , ∂t i,j;(x,t)∈QT a solution u ∈ C 1 ([0, T ]; L2(Ω))∩C([0, T ]; H01 (Ω)) of (5.1.1) with u ≡ 0 on ω×[0, T ] vanishes identically on QT := Ω × [0, T ]. Although Theorem 5.1 seems to be interesting, for our later use we need the following theorem: Theorem 5.2. Let u ∈ X2 (T ) := C 2 ([0, T ]; L2 (Ω)) ∩ C 1 ([0, T ]; H01 (Ω)) ∩ C([0, T ]; H 2 ∩ H01 (Ω)) be a solution of the problem (5.1.1) with ut (x, t) ≡ 0 on ω × [0, T ], where ω is an open set in Ω satisfying ω ⊃ Γ(x0 ). Then, there exist constants δ2 > 0 and T1 > 0 such that if T > T1 and &

∂ ∂ sup (5.1.6) |∇aij (x, t)| + ∇ aij (x, t) + aij (x, t) ≤ δ2 , ∂t ∂t i,j,(x,t)∈QT then u(x, t) ≡ 0 on Ω × [0, T ]. Next, we consider the wave equation with variable coeﬃcients in an exterior domain Ω with a dissipation: N ∂ ∂u utt − (5.1.7) aij (x, t) + a(x)ut = f (x, t) in Ω × (0, ∞), ∂xi ∂xj i,j=1 u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0.

(5.1.8)

We assume N

aij (x, t)ξi ξj ≥ k0 |ξ|2 ,

ξ ∈ RN ,

i,j=1

with some k0 > 0. We also assume that aij belong to C 1 (Ω × [0, T ]), that Ω is an 1,2 exterior domain with boundary from C 1 , and f ∈ Wloc ([0, ∞); L2 (Ω)). Proposition 5.2. Under the assumption (5.1.6) on aij from Theorem 5.2 with QT replaced by Q∞ there exists a constant T0 > 0 such that if T > T0 , the solutions

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u(t) ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) of the problem (5.1.7)–(5.1.8) satisfy t+T t+T X(t + T ) − X(t) + ε0 E(s) ds + k a(x)|ut |2 dx ds t t Ω (5.1.9) t+T ≤C (|∇u| + |u| + k|ut |) |f | dx ds, t ≥ 0, Ω

t

with some ε0 > 0 and an arbitrarily large k > 0, where we set 1 a(x)|u|2 dx + kE(t). X(t) = (ut , (φ(x − x0 ) + c0 h) · ∇u)+((α + 2c1 η)ut , u)+ 2 Ω (5.1.10) We see easily that X(t) is equivalent to E(t) + u(t)2 for a large k > 0 since |u|2 dx ≤ C ΩL

Ω2L \ΩL

|u|2 dx +

|∇u|2 dx Ω2L

with some C > 0. Note that the above Proposition 5.2 is a generalization of (4.2.23). It is not diﬃcult to derive some energy decay properties from (5.1.9) if f = 0. Indeed, this will be used to derive various types of energy decays for quasilinear wave equations in Section 8. 5.2. Proof of Proposition 5.1 We use a similar multiplier method employed in the previous sections. We ﬁrst consider the solutions u ∈ X2 . The identities (A) to (C) from Section 2 become d 1 2 ˜ E(t) + a(x)|ut | dx = − aij,t uxi uxj dx + f ut dx, (A) dt 2 Ω i,j Ω Ω N i,j=1

0

T

Ω

T ηaij uxi uxj dx dt = (ut , ηu) − 0

T

N Ω i,j=1

aij uxj ηxi u dx dt

˜ (B)

η|ut |2 + f ηu dx dt,

+ 0

0

T

Ω

T ∂ ∂ u h · ∇u dx dt aij − ∂xj 0 Ω i,j ∂xi 1 T =− ((h · ∇)aij + (∇ · h)aij ) uxi uxj dx dt 2 i,j 0 Ω 2 T ∂u 1 T ∂ − aij νi νj (ν · h) dS dt + aij uxj uxk hk dx dt. 2 i,j 0 ∂ Ω ∂n ∂xi Ω i,j 0 ˜ (C)

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˜ that Now, taking h = x − x0 we have from (C) T T 2 N |ut | − aij uxi uxj dx dt + aij uxi uxj dx dt 2 0 Ω 0 Ω i,j i,j 2 T ∂u 1 T = −(ut , h · ∇u) + aij νi νj (ν · h) dS dt 2 ∂n 0 ∂Ω i,j 0 T 1 T − (h · ∇aij )uxi uxj dx dt + (f, h · ∇u) dx dt. 2 i,j 0 Ω 0

(5.2.1)

˜ with η(x) = 1 and using the assumption on aij we have Combining (5.2.1) and (B) T 2 |ut | + aij uxi uxj dx dt 0

Ω

i,j

≤ C ut (T )(∇u(T ) + u(T )) + ut (0)(∇u(0) + u(0))

T

+ sup |∇aij | x,t,i,j

+

0

⎞ 1/2 1/2 2 T T ∂u 2 2 aij νi νj dS dt⎠ + f (t) dt ∇u , ∂ν 0 0 Γ(x0 )

and hence T 0

Ω

2 aij uxi uxj dx dt |ut | + i,j

≤C

aij uxi uxj dx dt

T

0

i,j

Ω i,j

+

i,j

0

T

T

f (t)2 dt + sup |∇aij | x,t,i,j 0 ⎞ 2 ∂u aij νi νj dS dt⎠ . ∂ν

E(T ) + E(0) + Γ(x0 )

T

E(t)dt

(5.2.2)

0

To estimate the boundary integral in (5.2.2) we make the usual device. That is, by choosing h(x) as in (3.1.9) and further choosing η(x) as in (3.1.10) we can derive the estimates 2 T ∂u aij νi νj dS dt ∂ν Γ(x0 ) i,j 0 T

≤C

E(0) + E(T ) + sup |∇aij | x,t,i,j

T

|ut |2 dx dt +

+ 0

ω

E(t) dt

T

f (t)2 dt + 0

(5.2.3)

0

i,j

0

T

ω

⎞ aij uxi uxj dx dt⎠

Decay and Global Existence

237

and i,j

≤C

T 0

ω

aij uxi uxj dx dt

T

E(0) + E(T ) + 0

2 |u| + |ut |2 dx dt +

T

(5.2.4)

f (t) dt . 2

0

ω

The estimate (5.1.4) follows from (5.2.2), (5.2.3) and (5.2.4). Since every term appearing in (5.2.4) is meaningful for ﬁnite energy solutions this is in fact valid for such a solution. 5.3. Proof of Theorems 5.1 and 5.2 Let u(t) be a solution of (5.2.1) with u(x, t) ≡ 0 on ω × [0, T ]. We take δ1 ≤ δ0 . Then, from Proposition 5.1 we have T E(t)dt ≤ C1 (E(0) + E(T )). (5.3.1) 0

Let E(t∗ ) = min0≤t≤T E(t), 0 ≤ t∗ ≤ T . Then, by the identity 1 t E(t) = E(t∗ ) + aij,s (x, s)uxi uxj dx ds, 2 t∗ Ω i,j

˜ (A)

we see

∗

E(t) ≤ E(t ) + C2 sup aij,t L∞ (QT ) i,j

∗

0

T

Ω i,j

aij uxi uxj dx dt

(5.3.2)

T

≤ E(t ) + C2 δ1

E(s) ds 0

for all t, 0 ≤ t ≤ T . Hence, if 2C1 C2 δ1 < 1/2 we have from (5.3.1) and (5.3.2) T (T E(t∗ ) ≤) E(t)dt ≤ C3 E(t∗ ) (5.3.3) 0

for a certain constant C3 > 0. Taking T > T0 := C3 we obtain E(t∗ ) = 0, i.e., ut (t∗ ) = u(t∗ ) ≡ 0. Thus, we conclude from the uniqueness of solutions to the initial-boundary value problem that u(x, t) ≡ 0 on Ω × [0, T ]. Remark 5.1. Modifying the proof of Theorem 5.1 we easily see the inequality T T 2 2 2 |ut | + |u| dx dt + f (t) dt E(0) ≤ C 0

ω

0

for solutions u(t) of (5.1.3), which may be more convenient for applications.

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Let us proceed to the proof of Theorem 5.2. Diﬀerentiating the equation with respect to t we have ∂ ∂ut ∂ ∂u aij = aij,t . (5.3.4) uttt − ∂xi ∂xj ∂xi ∂xj i,j i,j Applying Proposition 5.1 with u replaced by ut to (5.3.4) we have ⎛ ⎞ 2 T T ∂ ∂u dx dt⎠ , E1 (t)dt ≤ C ⎝E1 (0) + E1 (T ) + aij,t ∂xj 0 0 Ω i,j ∂xi (5.3.5) where E1 (t) is deﬁned by E(t) with u replaced by ut . Here, we see 2 T ∂ ∂u dx dt aij,t ∂xj 0 Ω i,j ∂xi 2 2 2 T ∂ u ∂ 2 2 ≤C |∇aij,t | |∇u| + aij dx dt ∂t ∂xi ∂xj 0 Ω T T T ≤ Cδ2 ∆u2 dt ≤ Cδ2 utt 2 dt ≤ Cδ2 E1 (t) dt, 0

0

(5.3.6)

0

where in the last step we have used the equation (5.1.1) and an elliptic regularity result for bounded domains. It follows from (5.3.5) and (5.3.6) that for small δ2 > 0, T E1 (t)dt ≤ C(E1 (0) + E1 (T )). (5.3.7) 0 ∗

Setting again E1 (t ) = min0≤t≤T E1 (t) we have from the equation (5.3.4) that t 1 ∂ ∂ut ∂ut ∂u ∗ aij,t E1 (t) = E1 (t ) + + aij,t utt dx dt 2 ∂xi ∂xj ∂xi ∂xj ∗ Ω i,j t T ≤ E1 (t∗ ) + C δ2 + δ2 E1 (s) ds, 0 ≤ t ≤ T, 0

(5.3.8) and hence, assuming δ2 < 1, we have from (5.3.7) and (5.3.8), T T √ ∗ ∗ (T E1 (t ) ≤) E1 (t)dt ≤ C E1 (t ) + ε2 E1 (t)dt 0

(5.3.9)

0

for some C > 0. Thus, if we take δ2 small enough and large T > 0 we obtain E1 (t∗ ) = 0, which implies ut (x, t) ≡ 0 on Ω × [0, T ]. Returning to the equation (5.1.1) we have ∂ ∂u − aij (x, t) = 0 on Ω × [0, T ] ∂xi ∂xj i,j and we conclude, by the boundary condition, u(x, t) ≡ 0 on Ω × [0, T ].

Decay and Global Existence

239

5.4. Proof of Proposition 5.2 Here we consider the problem (5.1.7) in the exterior domain Ω. We combine the ˜ (B) ˜ and (C) ˜ which are true even for the case of exterior domains identities (A), (in fact we use these identities in forms of diﬀerential equalities). Taking h = φ(r)(x − x0 ) as in the proof of Theorem 4.1 we can show the following inequality (see (4.2.4)): d 1 a(x)|u|2 dx + kE(t) (ut , φ(x − x0 ) · ∇u) + α(ut , u) + dt 2 Ω 0 0 (N φ+φ r) (x −x )(x −x φ(x−x0 )·∇aij i j i j) +φ +φ+α α aij + uxi uxj dx − 2 r 2 i,j Ω N φ + φ r − α + ka(x) |ut |2 dx + 2 Ω 2 ∂u k 1 ≤ aij,t uxi xj dx + aij νi νj φν · (x − x0 ) dS 2 2 i,j Γ(x0 ) ∂ν Ω i,j |a(x)ut φ(x − x0 ) · ∇u| dx. + (φ(x − x0 ) · ∇u + u + kut ) f dx + Ω

Ω

(5.4.1) Here, we see that (xi −x0i )(xj −x0j ) φ(x−x0 )·∇aij (N φ+φ r) +φ +φ+α aij − u xi u xj − 2 r 2 i,j ˜1 |)ε δ (N φ + φ r) (L + |x 0 0 + φ r + φ + α − ≥ − aij uxi uxj , 2 2 i,j (5.4.2) where δ˜1 = k0−1

i,j

∇aij 2∞

1/2 .

Also, assuming (L + |x0 |)δ˜1 < 1/4, we can choose α > 0 and k − and

1 such that

ε0 N φ + φ r (L + |x0 |)ε0 l0 − + φ r + φ + α ≥ 2 2 8 k ε0 N φ + φ r − α + a(x) ≥ . 2 2 8

Finally, note that 1 1 ¯1 ≤ δ a u u dx aij uxi uxj dx ij,t x x i j 2 Ω i,j i,j 2 Ω * with δ¯1 = k0−1 ( i,j aij,t 2∞ )1/2 . We assume δ¯1 is suﬃciently small so that k δ¯1 ≤ ε0 /4.

240

Mitsuhiro Nakao

Then we obtain from (5.4.1) and (5.4.2) that 1 d a(x)|u|2 dx + kE(t) (ut , φ(x − x0 ) · ∇u) + α(ut , u) + dt 2 Ω ε0 k 2 + E(t) + a(x)|ut | dx 8 2 Ω 2 ∂u dS + C (|∇u| + |u| + k|ut |)|f | dx ≤C Γ(x0 ) ∂ν Ω

(5.4.3)

with a constant C > 0. To control the ﬁrst term of the right-hand side of (5.4.3) we use the usual argument obtaining (5.2.3) and (5.2.4). Then we have: 2 ∂u d 2 2 dS ≤ C |∇u| dx + C + |u | |f ||∇u|dx − c0 (ut , h · ∇u) t ∂ν dt Γ(x0 ) ω ∩Ω ω (5.4.4) and 2 d 2 2 |∇u| dx + c1 (ut , ηu) + c1 a(x)η|u| dx ≤ C |ut | + |u|2 dx (5.4.5) dt ω ∩Ω Ω ω for some c1 > 0, C > 0. It follows from (5.4.3)–(5.4.5) that dX(t) ε0 k + E(t) + a(x)|ut |2 dx dt 8 2 Ω (5.4.6) |u|2 dx + (|∇u| + |u| + k|ut |)|f | dx ≤C Ω

ω

with some C > 0 and a large k > 0, where we set

1 X(t) = (ut , (φ(x−x0 )+c0 h)·∇u) + ((α+2c1 η)ut , u) + a(x)|u|2 dx + kE(t). 2 Ω We summarize the above arguments in the following statement:

Proposition 5.3. For any large k, say k > N + 1, if 1/2 1/2 2 2 ∇aij ∞ + aij,t ∞ sup 0≤t<∞

i,j

i,j

(5.4.7)

≤ δ2 =: min{(L + |x0 |)/4, k0 ε0 /8k}, then the inequality (5.4.6) holds for the ﬁnite energy solutions u(t) of the problem (5.1.7)–(5.1.8). To control the ﬁrst integral in the right-hand side of the inequality (5.4.6) we prepare an inequality corresponding to (4.2.13) for the usual wave equation. Proposition 5.4. Under the assumptions (5.1.6), for any ε > 0 there exists a constant Cε > 0 such that t+T t+T t+T a(x)|ut |2 + |f |2 dx ds+ε |u|2 dx ds ≤ Cε E(t)dt (5.4.8) t

ω

for any t > 0.

t

Ω

t

Decay and Global Existence

241

Proof. The proof is given in a quite similar way as the proof of Proposition 4.1. But, we need a unique continuation property as stated in Theorem 5.2. We note that on the limit function v appearing in the proof we may assume that v ∈ C 2 ([0, T ]; L2loc(Ω)), ∇v ∈ C([0, T ]; L2 (Ω)), D2 u ∈ C([0, T ]; L2 (Ω)). Because, we can use a molliﬁer ρ(t) with respect to t to regularize v. Details are left to the readers as an exercise. Without loss of generality we may assume δ2 ≤ δ1 . Then, by combining Propositions 5.2 and 5.3 we arrive at the desired inequality stated in Proposition 5.1. Finally, we note that by a standard density argument, the inequality is valid for ﬁnite energy solutions u(t).

6. Lp estimates for the wave equation in exterior domains 6.1. Problem and result Let us consider again the linear wave equation with a localized dissipation: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(6.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, (6.1.2) where a = a(x) is a continuous function on Ω satisfying Hyp. A. For each (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω) there exists a unique solution u of problem (6.1.1)–(6.1.2) in the class C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)). On the basis of the local energy decay proved in Section 3 we shall prove Lp estimates of the solutions under additional regularity conditions. These results will be used in the next section for the proof of global existence of solutions to semilinear wave equations. We denote the norm of W m,p (Ω) or of W m,p (RN ) by || · ||m,p . We begin with the Cauchy problem in the whole space, utt − ∆u = 0 in RN × [0, ∞), u(x, 0) = u0 (x), The following estimate is known:

ut (x, 0) = u1 (x).

(6.1.3) (6.1.4)

Theorem 6.1. Let N ≥ 2. If u0 , u1 ∈ C0∞ (RN ), the solution u(t) of the Cauchy problem (6.1.3)–(6.1.4) satisﬁes the estimates u(t)p ≤ Ct−b (u0 1,p∗ + u1 p∗ ) and

(6.1.5)

u(t)∞ ≤ Ct−d (u0 M,1 + u1 M−1,1 ) (6.1.6) for any p, 2 < p ≤ p0 , where C denotes a constant independent of u and we set N 2(N + 1) 2N p M= , p∗ = , + 1, p0 = 2 N −1 (N + 1)p + 2 b = (N − 1)(1/2 − 1/p) and

d = (N − 1)/2.

242

Mitsuhiro Nakao

The proof of Theorem 6.1 is not trivial. For the proof see Brenner [5], Pecher [54] and Mochizuki and Motai [32] etc., where more general results are proved. For general more precise estimates concerning the wave equation in the whole space see, e.g., Georgiev [9]. Remark 6.1. By a density argument we see that the above estimates are valid for more general solutions with initial data for which the right-hand sides are ﬁnite. In particular, the compactness of the supports of u0 , u1 is not necessary. By combining Theorem 3.1 from Section 3 and Theorem 6.1 with a ‘cut-oﬀ’ method as in Shibata-Tsutsumi [63] we can prove: Theorem 6.2. Let N ≥ 2 and m be a nonnegative integer. Let a ∈ C 2M+m (Ω) and in addition to Hyp. A we assume supp a(·) is compact. Concerning initial data we assume that u0 ∈ H01 (Ω) ∩ H 2M+m (Ω) ∩ W 2M+m,1 (Ω), u1 ∈ H 2M+m−1 (Ω) ∩ W 2M+m−1,1 (Ω) and that these data satisfy the compatibility condition of (M + m − 1)-th order. Then there exists a unique solution u(t) of the problem (6.1.1)– (6.1.2) in C([0, ∞); H 2M+m (Ω) ∩ H01 (Ω)) ∩ C 1 ([0, ∞); H 2M+m−1 (Ω) ∩ H01 (Ω)) and it satisﬁes (6.1.7) u(t)m,p ≤ C I˜M +m (1 + t)−b for 2 < p ≤ p0 , and u(t)m,∞ ≤ C I˜2M+m (1 + t)−d , (6.1.8) where we set (u0 k+1,i + u1 k,i ), I˜k = i=1,2

⎧ ⎪ ⎨b b = b−δ ⎪ ⎩ b−1−δ

if N is odd and N ≥ 5, if N = 3, if N is even,

and

⎧ ⎪ ⎪(N − 1)/2 if N is odd and N ≥ 5, ⎪ ⎨1 − δ if N = 3, d = ⎪−δ if N is even and N ≥ 4, ⎪ ⎪ ⎩ −1/2 − δ if N = 2. We note that δ > 0 in the above can be chosen arbitrarily small. By Shibata and Tsutsumi [63] we know that when V is convex and a(x) ≡ 0, then ER (t) ≤ CE(0)(1+t)−2(N −1) for solutions with initial data with compact support. Using this we have the following: Corollary 6.1. Let V be convex and let N ≥ 4 be even. We assume that a(x) ≡ 0. Then, the estimates (6.1.7) and (6.1.8) are valid with b = b and d = d. Remark 6.2. sides of (6.1.7) and (6.1.8) can be re*mIt is clear that the*left-hand m placed by k=0 Dtk u(t)p and k=0 Dtk u(t)∞ , respectively, where Dt denotes the partial diﬀerentiation with respect to t.

Decay and Global Existence

243

Remark 6.3. When N is even the estimates (6.1.7) and (6.1.8) do not imply any decay property but show some growth property. These estimates, however, are never trivial since the standard energy inequality E(t) ≤ E(0) gives only t u(t)2 ≤ u0 + ut (s) ds ≤ u0 + E(0) (1 + t). 0

Remark 6.4. By the energy inequality and Sobolev’s embedding theorem we easily see that for 2 ≤ q ≤ ∞, u(t)m,q ≤ C(u0 M+m,2 + u1 M+m−1,2 ),

0 ≤ t ≤ 1.

This fact will be used to estimate the solutions near t = 0. Remark 6.5. See Racke [56], where Lp –Lq estimates are discussed for thermoelastic systems. 6.2. Proof of Theorem 6.2 We follow [44]. First, we take a smooth function φ(x) such that 1 if |x| ≥ L + 2, φ(x) = 0 if |x| ≤ L + 1, ˜(x, t) of the where we assume supp a(·) ⊂ BL . Let us consider the solution u problem u˜tt − ∆˜ u + a(x)˜ ut = 0 in Ω × [0, ∞), (6.2.1) ˜|∂Ω = 0. (6.2.2) u˜(x, 0) = φ(x)u0 (x), u˜t (x, 0) = φ(x)u1 (x) and u We can expect that u˜(x, t) behaves like the original solution u(x, t) if |x| is large. To estimate u ˜(x, t) we introduce the solution v of the Cauchy problem: vtt − ∆v = 0 in RN × [0, ∞), v(x, 0) = φ(x)u0 (x), vt (x, 0) = φ(x)u1 (x). We further take a smooth function ρ(x) such that 1 if |x| ≥ L + 1, ρ(x) = 0 if |x| ≤ L,

(6.2.3) (6.2.4)

and consider ρv, which satisﬁes (ρv)tt − ∆(ρv) = −∇ρ · ∇v − ∆ρv in RN × [0, ∞)

(6.2.5)

and (ρv)(x, 0) = φ(x)u0 (x), (ρv)t (x, 0) = φu1 (x). Note that ρφ = φ. We set w = u˜ − ρv. Then we see that wtt − ∆w + awt = ∇ρ · ∇v + ∆ρv =: g and w(x, 0) = 0,

wt (x, 0) = 0 and w|∂Ω = 0.

(6.2.6)

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Mitsuhiro Nakao

Since v is a solution of the Cauchy problem in RN we know by Theorem 6.1 that for any nonnegative integer m, ||ρvm,p ≤ Ct−b (u0 m+1,p∗ + u1 m,p∗ )

(6.2.7)

for 2 < p ≤ 2(N + 1)/(N − 1) =: p0 , and ρvm,∞ ≤ C(1 + t)−d (u0 m+M,1 + u1 m+M−1,1 + u0 m+M,2 + u1 m+M−1,2 ), (6.2.8) where we recall b = (N − 1)(1/2 − 1/p) and d = (N − 1)/2. We denote the solutions of the linear equation (6.1.1)–(6.1.2) by S(t; (u0 , u1 )). Then, by variation of constants we see from (6.2.6) t w(t) = S(t − s; (0, g)) ds. 0

Thus, we have by Theorem 6.1 that for any integer m ≥ 1, t S(t − s; (0, g))H m (ΩL+2 ) ds w(t)H m (ΩL+2 ) ≤ 0 t ≤C Γ(t − s)gH m−1 (BL+2 \BL+1 ) ds 0 t ≤C Γ(t − s)vH m (BL+2 \BL+1 ) ds, 0

where we set

Γ(t) =

e−λt (1 + t)(−1+δ)/2

if N is odd , if N is even.

(6.2.9)

Needless to say, Theorem 6.1 (Corollary 6.1) implies that if (u0 , u1 ) ∈ H m+1 (Ω) × H m (Ω) have compact support and satisfy the compatibility condition of the m-th order, then the solution u(t) satisﬁes Dm+1 u(t)L2 (Ω∩BR ) ≤ CR (u0 H m+1 + u1 H m )Γ(t),

R

1,

where D = (∇, Dt ). We have used this fact just above. (When V is convex, N ≥ 4 is an even number and a(x) ≡ 0 we take Γ(t) = (1 + t)−(N −1) . Hereafter, we say V is convex for such a case.) Hence, noting b ≤ (N − 1)/(N + 1) < 1, t Γ(t − s)v(s)m,p(BL+2 \BL+1 ) ds w(t)H m (ΩL+2 ) ≤ C (6.2.10) 0 t ≤C Γ(t − s)s−b (v(0)m+1,p∗ + vt (0)m,p∗ ) ds 0 ˆ

≤ C(1 + t)−b (u0 m+1,p∗ + u1 m,p∗ ), where

ˆb =

b b − 1/2 − δ/2

if N is odd, if N is even.

Decay and Global Existence When V is convex we can take ˆb = b. Similarly, we have t Γ(t − s)v(s)m,∞(BL+2 \BL+1 ) ds w(t)H m (ΩL+2 ) ≤ C 0 t ≤C Γ(t − s)(1 + s)−d (v(0)m+M,i + vt (0)m+M−1,i ) ds 0 ˆ

≤ C(1 + t)−d

245

(6.2.11)

i=1,2

(v(0)m+M,i + vt (0)m+M−1,i ),

i=1,2

where

⎧ ⎪ ⎨d ˆ d = 1/2 − δ/2 ⎪ ⎩ −δ/2

if N is odd, if N is even and N ≥ 4, if N = 2.

When V is convex we can take dˆ = d. In particular, replacing m by M + m in (6.2.11) and using Sobolev’s embedding lemma we have ˆ w(t)m,∞(ΩL+2 ) ≤ C(1 + t)−d (u0 2M+m,i + u1 2M+m−1,i ). (6.2.12) i=1,2

To estimate w(t) outside of the domain ΩL+2 we take a smooth function µ(x) such that 1 if |x| ≥ L + 2, µ(x) = 0 if |x| ≤ L + 1, and set µw = w. ˜ Then, w ˜tt − ∆w ˜ = −∇µ · ∇w − ∆µw =: h in RN × [0, ∞)

(6.2.13)

with w(x, ˜ 0) = 0, w ˜t (x, 0) = 0. Denoting by S0 (t; (u0 , u1 )) the solution of the free wave equation in RN with initial data (u0 , u1 ) we see t w(t) ˜ = S0 (t − s; (0, h)) ds 0

and hence, by Theorem 6.1 and (6.2.11), t w(t) ˜ (t − s)−b h(s)m,p∗ (BL+2 \BL+1 ) ds m,p ≤ C 0 t ≤C (t − s)−b w(s)m+1,2(ΩL+2 ) ds 0 t ˆ ≤C (t − s)−b (1 + s)−d (u0 m+M+1,i + u1 m+M,i ) ds 0 −˜ b

≤ C(1 + t)

i=1,2

(u0 m+M+1,i + u1 m+M,i )

i=1,2

(6.2.14)

246 with

Mitsuhiro Nakao ⎧ ⎪ ⎪b ⎪ ⎨ ˜b = b − δ ⎪ b − (1 + δ)/2 ⎪ ⎪ ⎩ b − 1 − δ/2

if if if if

N N N N

is odd and N > 3, = 3, is even and N ≥ 4, = 2,

˜b = b if V is convex. Similarly, t w(t) ˜ (1 + t − s)−d w(s)m+M,2(BL+2 ) ds m,∞ ≤ C 0 t ˆ (t − s)−d (1 + s)−d (u0 m+2M,i + u1 2M−1,i ) ds ≤C 0 −d˜

≤ C(1 + t)

i=1,2

(u0 m+2M,i + u1 m+2M−1,i ),

i=1,2

(6.2.15)

⎧ d ⎪ ⎪ ⎪ ⎨1 − δ/2 d˜ = ⎪ (1 − δ)/2 ⎪ ⎪ ⎩ −(1 + δ)/2

if if if if

N N N N

is odd and N ≥ 5, = 3, is even and N ≥ 4, = 2.

The estimates (6.2.10) and (6.2.14) yield, in particular, ˜ (u0 m+M+1,i + u1 m+M,i ), w(t)m,p ≤ C(1 + t)−b

(6.2.16)

i=1,2

for 2 < p ≤ p0 . The estimates (6.2.12) and (6.2.15) yield ˜ (u0 m+2M,i + u1 m+2M−1,i ). w(t)m,∞ ≤ C(1 + t)−d

(6.2.17)

i=1,2

It follows from (6.2.7), (6.2.8), (6.2.16), and (6.2.17) that ˜ (u0 m+M+1,i + u1 m+M,i ), ˜ u(t)m,p ≤ C(1 + t)−b

(6.2.18)

i=1,2

and ˜

˜ u(t)m,∞ ≤ C(1 + t)−d

(u0 m+2M,i + u1 m+2M−1,i ).

(6.2.19)

i=1,2

Next, we have to estimate u ˆ = u−u ˜. For this we repeat a similar argument used in the derivation of the estimates for u˜(t). In fact, this time the argument is simpler, but the decay property for even and higher-dimensional domains is broken in this step. From (6.2.1) we see uˆtt − ∆ˆ u + a(x)ˆ ut = 0 in Ω × [0, ∞),

(6.2.20)

and u ˆ(x, 0) = (1 − φ(x))u0 (x),

u ˆt (x, 0) = (1 − φ(x))u1 (x) and u ˆ|∂Ω = 0.

Decay and Global Existence Taking a smooth function ρˆ such that 0 if ρˆ(x) = 1 if

247

|x| ≥ L + 4, |x| ≤ L + 3,

we have from Section 3 ρˆu ˆ(t)m,p ≤ C∇ˆ u(t)m,2(ΩL+4 ) ≤ CΓ(t)(u0 m+1,2 + u1 m,2 )

(6.2.21)

ρˆuˆ(t)m,∞ ≤ Cˆ u(t)m+M,2 ≤ CΓ(t)(u0 m+M,2 + u1 m+M−1,2 ).

(6.2.22)

and

Further, setting w ˆ = uˆ − ρˆuˆ, we have w ˆtt − ∆wˆ = −∇ˆ ρ · ∇ˆ u − ∆ˆ ρu ˆ =: gˆ in Ω × [0, ∞)

(6.2.23)

and w(x, ˆ 0) = 0,

wˆt (x, 0) = 0

and w| ˆ ∂Ω = 0.

This problem is very similar to (6.2.6), but, we note, the properties of g and gˆ are diﬀerent. We see in this case, t w(t) ˆ Γ(t − s)ˆ u(s)H m (BL+4 \BL+3 ) ds (6.2.24) H m (ΩL+4 ) ≤ C 0 t ≤C Γ(t − s)Γ(s)(u0 m,2 + u1 m−1,2 ) ds 0

ˆ ≤ Γ(t)(u 0 m,2 + u1 m−1,2 ) with

ˆ = Γ(t)

e−λt (1 + t)δ

if N is odd, if N is even.

ˆ = Γ(t) = (1+t)−(N −1) . In particular, replacing When V is convex we can take Γ(t) m by m + M , we have ˆ w(t) ˆ m,∞(ΩL+4 ) ≤ C Γ(t)(u0 m+M,2 + u1 m+M−1,2 ).

(6.2.25)

¯ = µw, ˆ where µ Finally, to estimate w(t) ˆ in the complement to ΩL+2 , we set w(t) is the function appearing in (6.2.13). Then, ˆ in RN × [0, ∞) w ¯tt − ∆w ¯ = −∇µ · ∇w ˆ − ∆µwˆ =: h with w(x, ¯ 0) = 0,

w ¯t (x, 0) = 0.

(6.2.26)

248

Mitsuhiro Nakao

As in (6.2.14) and (6.2.15), we see by (6.2.24), t ˆ w(t) ¯ (t − s)−b h(s) m,p ≤ C m,2(BL+4 \BL+3 ) ds 0 t ≤C (t − s)−b w(s) ˆ m+1,2(ΩL+2 ) ds 0 t ˆ ≤C (t − s)−b Γ(s)(u 0 m+1,2 + u1 m,2 ) ds 0

(6.2.27)

≤ C(1 + t)−b (u0 m+1,2 + u1 m,2 ) with

b = and

b b−1−δ

t

w(t) ¯ m,∞ ≤ C

if N is odd, if N is even,

ˆ (1 + t − s)−d h(s) m+M−1,2 ds

0

≤ C(1 + t)−d (u0 m+M,2 + u1 m+M−1,2 ) with

d =

d −δ

(6.2.28)

if N is odd, if N is even.

Thus, we obtain

−b (u0 m+1,2 + u1 m,2 ) w(t) ˆ m,p ≤ C(1 + t)

(6.2.29)

and

−d (u0 m+M,2 + u1 m+M−1,2 ). w(t) ˆ m,∞ ≤ C(1 + t)

(6.2.30)

Further, it follows from (6.2.21) and (6.2.29) that ˆ(t)m,p + w(t) ˆ ˆ u(t)m,p ≤ ρˆu m,p

≤ C(1 + t)−b (u0 m+1,2 + u1 m,2 )

(6.2.31)

and from (6.2.22) and (6.2.30) that

ˆ u(t)m,∞ ≤ C(1 + t)−d (u0 m+M,2 + u1 m+M−1,2 ).

(6.2.32)

Summarizing the above estimates (6.2.18), (6.2.19), (6.2.31), and (6.2.32) we ob˜ d }. tain the desired results (6.1.7) and (6.1.8) with b = min{˜b, b } and d = min{d,

Decay and Global Existence

249

7. Semilinear wave equations 7.1. Problem and result In this section we consider semilinear wave equations in exterior domains: utt − ∆u + a(x)ut = f (u) in Ω × [0, ∞),

(7.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(7.1.2)

where f (u) is a nonlinear term like f (u) = |u|α u. We shall consider this problem ˜ = Hyp. A+ Hyp. A on the under two types of assumptions Hyp. A and Hyp. A dissipation a(x)ut . Recall that under Hyp. A we know the Lp decay of solutions for ˜ we know the total energy the linear equation (Theorem 6.2), while under Hyp. A decay of solutions (Theorems 4.1 and 4.2). We apply these results to the semilinear problem (7.1.1)–(7.1.2). The latter case is easier to treat and we begin with this case. The arguments here are taken from [43] and [44]. We make the following assumptions on f (u). Hyp. B. The function f (u) is continuously diﬀerentiable on R and satisﬁes |f (j) (u)| ≤ kj |u|α+1−j , j = 0, 1, for u ∈ R with some kj > 0 and α > 0. ˜ The function f (u) is 2M − 1 times continuously diﬀerentiable on R and Hyp. B. satisﬁes |f (j) (u)| ≤ kj |u|α+1−j , j = 0, 1, 2, · · · , 2M − 1, for u ∈ R with some kj > 0 and α ≥ 2M − 2. Let us state our existence theorems. Theorem 7.1. Let 1 ≤ N ≤ 4 and Hyp. B be satisﬁed with 4/N < α ≤ 2/(N − 2) ˜ there exists δ > 0 such that if (4/N < α < ∞ if N = 1, 2). Then, under Hyp. A 1 2 (u0 , u1 ) ∈ H0 (Ω) × L (Ω) and I0 := u0 H1 + u1 ≤ δ, the initial-boundary value problem (7.1.1)–(7.1.2) admits a unique solution u(t) in the space C([0, ∞); H01 (Ω))∩C 1 ([0, ∞); L2 (Ω)), satisfying the decay or boundedness estimates (7.1.3) E(t) ≤ C0 I02 (1 + t)−1 and u(t) ≤ CII0 < ∞. Remark 7.1. When N = 1 the problem is reduced to the case Ω = (0, ∞) or (−∞, 0) and we may assume Γ(x0 ) = φ. If we seek for more regular solutions the condition on α in Theorem 7.1 can be relaxed. We illustrate this for the so-called H 2 solutions.

250

Mitsuhiro Nakao

Theorem 7.2. Let 3 ≤ N ≤ 6 and 2/(N − 2) < α ≤ 2/(N − 4)+ . We assume, in addition to the conditions of Hyp. B with this α, that f ∈ C 2 (R) and |f (u)| ≤ k1 |u|α−1 . Then, there exists δ1 > 0 such that if (u0 , u1 ) ∈ H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) and I1 := ||u0 ||H 2 + ||u1 ||H 1 < δ1 , the problem admits a unique solution u(t) in X2 := C 2 ([0, ∞); L2 (Ω)) ∩ C 1 ([0, ∞); H01 (Ω)) ∩ C([0, ∞); H 2 (Ω) ∩ H01 (Ω)), satisfying u(t) ≤ C1 I1 < ∞

(7.1.4)

and u(t)X2 := ut (t) + ∇u(t) + utt (t) + ∇ut (t) + ∆u(t) ≤ C1 I1 (1 + t)−1 with some C1 > 0.

(7.1.5)

Remark 7.2. If f (u) is more smooth and further (u0 , u1 ) belongs to H m+1 (Ω) × H m (Ω), m ≥ 2, and satisﬁes the required compatibility conditions, then we can prove that the solutions in Theorems 7.1, 7.2 are more smooth (cf. [48]). Hyp. C. There exists p, 2 (2M − 1)(1 − (1 − θ)(p − 2)/p), where

θ=

1 1 1 + − N p 2(2M − 1)

2M − 1 1 1 + − N p 2

−1 .

√ Remark √ 7.3. When N = 3 and N = 4 Hyp. C is reduced to α > 2 3 − 1 and α > 15 − 2, respectively. Theorem 7.3. We make the assumptions Hyp. A, Hyp. B and Hyp. C. Let N ≥ 3 be odd. Then, there exists ε1 > 0 such that if I˜2M ≤ ε1 , the problem (7.1.1)–(7.1.2) i 2M−i admits a unique solution u(·) ∈ XM := ∩2M−1 (Ω) ∩ H01 (Ω)) ∩ i=0 C ([0, ∞); H 2M 2 C ([0, ∞); L (Ω)) satisfying the estimates u(t)p ≤ C I˜2M (1 + t)−b

and

u(t)∞ ≤ C I˜2M (1 + t)−d , where d = d := (N − 1)/2 and b = b := (N − 1)(1/2 − 1/p) if N ≥ 5 and b = b − δ and d = d − δ if N = 3. Here we recall I˜2M = ||u0 ||2M,2 + ||u1 ||2M−1,2 .

Corollary 7.1. When N is even and N ≥ 4 we assume that V is convex and a(x) ≡ 0. Then, the same statements as in Theorem 7.3 hold with d = d and b = b.

Decay and Global Existence 7.2. Proof of Theorems 7.1 and 7.2 Setting U (t) = (u(t), ut (t))T , the problem (7.1.1)–(7.1.2) is rewritten as t U (t) = S(t)U U0 + S(t − s)F (s) ds,

251

(7.2.1)

0

where U0 = (u0 , u1 )T ∈ H01 (Ω)×L2 (Ω), F (s) = (0, f (u(s)))T , and S(t) denotes the semi-group in H01 (Ω) × L2 (Ω) associated with the linear problem. It is well-known that under Hyp. B with 0 < α ≤ 2/(N − 2)+ , the problem (7.1.1)–(7.1.2) has a unique solution u ∈ C([0, T ); H01 (Ω)) ∩ C 1 ([0, T ); L2 (Ω)) for some T > 0, and to show the global existence in Theorem 7.1 it suﬃces to derive the a priori estimate E(t) ≤ C(II0 )(1 + t)−1 in the interval of existence. Let K > 0 and assume that the following estimates hold in an interval [0, T ), T > 0, ut (t) + ∇u(t) ≤ KII0 (1 + t)−1/2

(7.2.2)

and u(t) ≤ KII0 . (7.2.3) Note that (7.2.2) and (7.2.3) are valid on some interval if we take a large K > 0. By Theorem 4.1 we know S(t)U U0 E ≤ CII0 (1 + t)−1/2 , where (u(t), ut (t))E = ∇u(t) + ut (t). Thus, by (7.2.1), we have t −1/2 U (t)E ≤ CII0 (1 + t) +C (1 + t − s)−1/2 f (u(s))ds.

(7.2.4)

(7.2.5)

0

Here, by Hyp. B and Gagliardo-Nirenberg inequality, (α+1)(1−θ) f (u(s)) ≤ k0 uα+1 ∇u(α+1)θ 2(α+1) ≤ k0 u

(7.2.6)

with θ = N α/2(α + 1). Thus, as long as (7.2.2) and (7.2.3) hold, we have f (u(s)) ≤ k0 K α+1 I0α+1 (1 + s)−N α/4

(7.2.7)

and hence, U (t)E ≤ C0 (II0 + k0 K α+1 I0α+1 )(1 + t)−1/2 (7.2.8) for some C0 > 0 independent of K, I0 , where we have used the assumption N α/4 > 1. Now, we make an assumption on I0 : Q0 (II0 , K) := C0 (1 + k0 K α+1 I0α ) < K.

(7.2.9)

Then we obtain U (t)E ≤ Q0 (II0 , K)II0 (1 + t)−1/2 < KII0 (1 + t)−1/2 .

(7.2.10)

Next, we check the boundedness of u(t). Since u(t) ≤ CII0 for solutions of the linear equation we have from (7.2.1) that t u(s)α+1 I0 + k0 K α+1 I0α+1 ) u(t) ≤ CII0 + Ck0 2(α+1) ds ≤ C0 (I 0

252

Mitsuhiro Nakao

for some C0 , which can be identiﬁed with the one in (7.2.8). Thus, under the assumption (7.2.9), we have u(t) ≤ Q0 (K, I0 )II0 < KII0 .

(7.2.11)

The estimates (7.2.10) and (7.2.11) show that under the assumption (7.2.9), the (local) solution u(t) exists in fact globally in time and these estimates hold for all t ≥ 0. Needless to say, (7.2.9) is valid if we take K > C0 and I0 < δ for a small δ > 0. The proof of Theorem 7.1 is ﬁnished. Next we shall show the proof of Theorem 7.2. The existence of H 2 local solutions are rather standard. Indeed, all of the estimates required for the existence of H 2 local solutions follow more or less easily from similar estimations below. We note that the assumption |f (u)| ≤ k0 |u|α−1 is used in this step. For the proof of Theorem 7.2 it will be suﬃcient to derive the following a priori estimates: u(t) ≤ KI1

(7.2.12)

and ut (t) + ∇u(t) + utt (t) + ∇ut (t) + ∆u(t) ≤ KI1 (1 + t)−1/2

(7.2.13)

for some K > 0, where I1 = u0 H 2 + u1 H 1 . We may assume that the (local) solutions u belong to C 2 ([0, T ); L2(Ω)) ∩ C 1 ([0, T ); H01 (Ω)) ∩ C([0, T ); H 2 (Ω)) and satisfy the estimates (7.2.12) and (7.2.13) for some interval [0, T ), T > 0. Then, we start again from (7.2.1). But, instead of (7.2.6) and (7.2.7), we use the estimates: ˜ (α+1)(1−θ)

˜

f (u(s) ≤ k0 u2N/(N −2) D2 u(α+1)θ ˜

˜

≤ Ck0 ∇u(α+1)(1−θ) (∇u + ∆u)(α+1)θ

with θ˜ = (α(N − 2) + 2)/2(α + 1), where we have used the assumption &

2 2 ,1 < α ≤ . max N −2 (N − 4)+ Thus, we have u(t)E ≤ CII0 (1 + t)−1/2 + Ck0 K α+1 I1α+1 ≤ C1 (1 + k0 K α+1 I1α )I1 (1 + t)−1/2

t

(1 + t − s)−1/2 (1 + s)−(α+1)/2 ds

0

(7.2.14) with some C1 , where uE = ut + ∇u. Similarly, u(t) ≤ C(I1 + Ck0 K α+1 I1α )I1 .

(7.2.15)

Next, diﬀerentiating the equation we have uttt − ∆ut + a(x)ut = f (u)ut in Ω × [0, ∞),

(7.2.16)

ut (x, 0) = u1 (x), utt (x, 0) = u2 (x) := ∆u0 (x) − a(x)u1 (x) and ut |∂Ω = 0. (7.2.17)

Decay and Global Existence Therefore, −1/2

ut (t)E ≤ CI1 (1 + t)

+C

t

253

(1 + t − s)−1/2 f (u(s))ut (s)ds.

(7.2.18)

0

Here, we easily see f (u(s))ut (s) ≤ k0

(N −2)/2N 1/N |ut |2N/(N −2) dx |u|N α dx

Ω

≤ Ck0 ∇ut ∇u

ˆ α(1−θ)

Ω αθˆ

(∇u + ∆u)

with θˆ = (α(N − 2) − 2)/2α. Thus, we obtain ut (t)E ≤ C1 (1 + k0 K α+1 I1α )I1 (1 + t)−1/2

(7.2.19)

for some C1 > 0. Finally, from the equation itself, we see ∆u(t) ≤ C1 (1 + k0 K α+1 I1α )I1 (1 + t)−1/2 .

(7.2.20)

Now, we take a large K such that K > C1 and make the assumption on I1 : Q1 (I1 , K) := C1 (1 + k0 K α+1 I1α ) < K,

(7.2.21)

which is valid if I1 ≤ δ1 for some small δ1 > 0. Then we obtain u(t) ≤ Q1 (I1 , K)I1 < KI1

(7.2.22)

and ut (t) + ∇u(t) + utt (t) + ∇ut (t) + ∆u(t) ≤ Q1 (I1 , K)I1 (1 + t)−1/2 < KI1 (1 + t)−1/2 .

(7.2.23)

2

We conclude that under the assumption (7.2.21) the local H solutions exist globally in time and the estimates (7.2.22) and (7.2.23) hold for all t ≥ 0. The proof of Theorem 7.2 is now complete. 7.3. Proof of Theorem 7.3 In this section we assume Hyp. A and Hyp. C on a(x) and f (u), respectively, and under these assumptions we prove the existence of global solutions. The following well-known result concerning the local existence is the starting point (cf. Kato [17]). Proposition 7.1. We set M = [N/2]+1. Let (u0 , u1 ) belong to H 2M (Ω)×H 2M−1 (Ω) and satisfy the compatibility condition of the (2M − 1)-order. Then, under Hyp. B, there exist T > 0 and a unique solution u(t) of the problem (7.1.1)–(7.1.2) in the i 2M−i (Ω)∩H H01 (Ω))∩C 2M ([0, T ); L2 (Ω)). Further, class X(T ) := ∩2M−1 i=0 C ([0, T ); H *2M−1 Dtk u(t)H 2M −k + this solution can be continued in t as long as the norm k=0 *2M k k=1 Dt u(t)2 is bounded. In order to prove Theorem 7.3 we set for 2 < p ≤ p0 , V (T ) := {u ∈ X(T )| uV (T ) < ∞},

254

Mitsuhiro Nakao

where uV (T ) :=

' sup 0≤t
+

(1 + t)d u(t)∞ + (1 + t)b (u(t)W 1,p + ut (t)p ) Dtk ∇u(t)2M−k−1

+

k=0

2M

( Dtk u(t)2 .

k=1

Let u(t) be a local solution deﬁned in Proposition 7.1. Then, as long as u(t) exists, the problem (7.1.1)–(7.1.2) is equivalent to t u(t) = S(t; (u0 , u1 )) + S(t − s; (0, f (u(s))) ds. (7.3.1) 0

For the proof of Theorem 7.3 it will be suﬃcient to show the a priori estimate uV (T ) ≤ K < ∞ with some K > 0 independent of T > 0 for assumed solutions u(t) ∈ V (T ) of (7.3.1). Note that the boundedness of u(t)2 for any ﬁnite interval [0, T ] follows 0t from the inequality ||u(t)|| ≤ ||u0 || + 0 ||ut (s)||ds. Proposition 7.2. Let (u0 , u1 ) belong to H 2M(Ω)∩W 2M,1(Ω)×H 2M−1(Ω)∩W 2M,1(Ω) and satisfy the compatibility condition of the 2M − 1-th order. Let u(t) be the local solution of (7.1.2) with u(·)V (T ) ≤ K < ∞ with some K > 0. Then, for p satisfying Hyp. C, we have ' ( sup (1 + t)d u(t)∞ + (1 + t)b (u(t)p + Du(t)p ) ≤ C(I˜2M + K α+1 ), 0≤t
(7.3.2) where D denotes both of Dt and ∇. Proof. From (7.3.1) we see

u(t)∞ ≤ C(1+t)−d I˜2M +

t

(1+t−s)−d (f (u(s))2M−1,1 +f (u(s))2M−1,2 ) ds.

0

(7.3.3) Here, f (u(s))2M−1,1 ≤ C(f (u(s))1 + D2M−1 f (u)1 ) 2M−1 |u|α+1 + ≤C f (j) (u)(Du)ν1 · · · (D2M−1 u)ν2M −1 dx, Ω

where

j=1 ν∈Sj

Sj =

ν = (ν1 , . . . , ν2M−1 )|

2M−1

νk = j and

k=1

2M−1

kνk = 2M − 1 .

k=1

For the ﬁrst term of the right-hand side of (7.3.4) we have easily: α+1−p |u|α+1 dx ≤ u∞ upp Ω

(7.3.4)

≤ CK α+1 (1 + t)−d (α+1−p)−pb ≤ CK α+1 (1 + t)−d −ε ,

ε > 0,

Decay and Global Existence

255

under the assumptions that α ≥ p − 1 and α > p − pb /d . Note that the latter condition is equivalent to α > p − pb/d = 2 because we can take δ > 0 arbitrarily small. The second term is treated as follows:

2M−1

f (j) (u)(Du)ν1 · · · (D2M−1 u)ν2M −1 dx

Ω j=1 ν∈S j

⎛

≤C⎝

⎞ 2M−1 2M−1 1 |u|α D2M−1 u dx + |u|α+1−j |Di u|νi dx⎠

⎛

Ω

j=2 ν∈Sj

≤ C ⎝uα 2α Du2M−2,2 +

2M−1

Ω

i=1

α+1−j ||u||∞

j=2 ν∈Sj

2M−1 1

⎞

||Di u||νpii νi ⎠ ,

i=1

2M−1 satisfying where we should choose {pi }i=1

1 ≤ pi ≤ ∞,

2M−1

1/pi = 1 and pi νi ≤ 2N/(N − 4M + 2i)+ .

i=1

Note that r := max {r, 0} and the last inequality on pi νi assures H 2M−i ⊂ Lpi νi . Such a choice of {pi } is certainly possible because +

2M−1 i=1

2M−1 iνi N − 4M 1 1 νi (N − 4M + 2i)+ ≤ + = − < 1. 2N 2N N 2 N i=1

Now, we easily see that α−p/2

uα 2α Du2M−2,2 ≤ Ku∞

p/2

up

≤ CK α+1 (1 + t)−(α−p/2)d −b p/2 ≤ CK α+1 (1 + t)−d −ε ,

ε > 0,

provided that α > p/2 − b p/2d , which is equivalent to α > p/2 − bp/2d. Further, by Gagliardo-Nirenberg inequality, we have i i Di upi νi ≤ CDu1−θ Duθ2M−1,2 p

with

θi =

i−1 1 1 + − N p pi νi

2M − 1 1 1 + − N p 2

−1 .

Hence, α+1−j ||u||∞

2M−1 1 i=1

||D

i

u||νpii νi

≤ CK

α+1

−(α+1−j)d −

(1 + t)

2M P−1 i=1

(1−θi )νi b

.

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Mitsuhiro Nakao

But, we see that 2M−1

−1 2M−1 2M − 1 1 1 2M − 1 1 1 − + + − νi N 2 N N p 2 i=1 −1 2M−1 iνi 2M − 1 1 1 1 + − + + − N p 2 N pi i=1 −1 2M − 1 1 1 2M − 1 1 1 = − + + − j N 2 N N p 2 −1 2M − 1 1 1 −(2M − 1 − N )N −1 + − N p 2 =: ηj .

(1 − θi )νi =

i=1

Therefore, α+1−j ||u||∞

2M−1 1

||Di u||νpii νi ≤ CK α+1 (1 + t)−(α+1−j)d −ηj b .

i=1

By an easy calculation we can show that min

2≤j≤2M−1

=

min

{(α + 1 − j)d + ηj b }

2≤j≤2M−1

(α + 1)d − (2M − 1 − N )b N −1

+d (p − 4M )N −1 p−1

2M − 1 1 1 + − N p 2

2M − 1 1 1 + − N p 2 -

−1

−1

j

= (α + 1 − 2M + 1)d + η2M−1 b ˜ = (α + 2 − 2M )d + (2M − 1)(1 − θ)b with

−1 1 1 2M − 1 1 1 − + − , p 2M − 1 N p 2 where we have used the fact p ≤ p0 ≤ 4M . Summarizing we obtain θ˜ =

f (u(t))2M−1,1 ≤ CK α+1 (1 + t)−d −ε ,

ε > 0,

provided that α ≥ p − 1,

˜ α > 2 and α > (2M − 1)(1 − (1 − θ)b/d).

Similarly, we can show that

f (u(s))2M−1,2 ≤ C +

uα+1 2(α+1)

2M−1

j=2 ν∈Sj

|u| |D 2α

+ Ω

α+1−j u∞

2M−1 1 i=1

2M−1

1/2 u| dx ⎞ 2

Di uν2pi i νi ⎠ .

(7.3.5)

Decay and Global Existence

257

Thus, replacing p by p/2 and repeating a quite similar argument getting (7.3.5), we have (7.3.6) f (u(t))2M−1,2 ≤ CK α+1 (1 + t)−d −ε , ε > 0, provided that α ≥ p/2 − 1, where

α > p/2 − pb/2d and α > (2M − 1)(1 − (1 − θ)b/d),

θ=

1 1 − p 2(2M − 1)

2M − 1 1 1 + − N p 2

−1 .

The second condition is equivalent to α > 1. We note that 0 < θ˜ < θ < 1. Thus, under Hyp. C, we obtain (7.3.5) and (7.3.6), hence the estimate for ||u(t)||∞ . Next, we consider the estimate for u(t)p and Dup , D = (∇, Dt ). We again use the formula (7.3.1) to get ﬁrst t −b u(t)p ≤ CI1,M (1 + t) +C (1 + t − s)−b f (u)M,i ds. s

Also, −b

Du(t)p ≤ CI1,M+1 (1 + t)

i=1,2

+C

(1 + t − s)−b

Ω

f (u)M+1,i ds.

i=1,2

Since 2M − 1 ≥ M + 1 we can apply the estimates (7.3.5) and (7.3.6) to get the desired estimates for f (u)M+1,i and, consequently,

u(t)W 1,p + ut (t)p ≤ C(I˜2M + K α+1 )(1 + t)−b . The proof of Proposition 7.2 is complete.

(7.3.7)

Finally, we shall show: Proposition 7.3. Assume that u0 2M,2 + u1 2M−1,2 ≤ 1. Then, under Hyp. C, the local solution u(t) with u(t)V (T ) ≤ K satisﬁes 2M−1 k=0

Dtk ∇u(t)2M−k−1 +

2M

Dtk u(t) ≤ C(u0 2M,2 + u1 2M−1,2 + K α+1 )

k=1

with a constant C > 0 independent of K, t and (u0 , u1 ). Proof. Let 1 ≤ l ≤ 2M − 1 be an integer. Diﬀerentiating the equation l times with respect to t we have Utt − ∆U + a(x)U Ut = Dtl f (u), (7.3.8) l where U = Dt u. Since α ≥ 1, it is standard to show that U Ut (0)2 + ∇U (0)2 ≤ C(IIM ) < ∞, where IM := u0 H 2M + u1 H 2M −1 . In particular, if IM ≤ 1, we can prove that U Ut (0)2 + ∇U (0)2 ≤ CIIM

(7.3.9)

258

Mitsuhiro Nakao

with some constant C > 0 independent of IM . Now, multiplying the equation by Ut and taking account the boundary condition U |∂Ω = 0, we have t Ut (0)22 + ∇U (0)22 + 2 |Dtl f (u)||U Ut |dx ds U Ut (t)22 + ∇U (t)22 ≤ U 0 Ω t 2 ≤ CIIM + C sup U Ut (s)2 Dtl f (u)2 ds. 0≤s≤t

0

(7.3.10) Here, by the same argument deriving the estimate (7.3.6) we see

Dtl f (u) ≤ CK α+1 (1 + t)−d −ε , ε > 0.

(7.3.11)

Since d + ε > 1 (note that d + ε = 1 − δ + ε > 1 for suﬃciently small δ > 0 if N = 3) we have from (7.3.10) and (7.3.11) that Dtl+1 u(t)2 + ∇Dtl u(t) ≤ C(IIM + K α+1 ) < ∞.

(7.3.12)

Further, returning to the equation −∆Dtl−1 u(t) = −Dtl+1 u(t) − a(x)Dtl u(t) + Dtl−1 f (u) and using the estimate Dtl−1 f (u)2 ≤ CK α+1 we see by elliptic regularity theory that Dtl−1 ∇uH 1 ≤ C(IIM + K α+1 ).

(7.3.13)

Next, diﬀerentiating the equation l − 1 times with respect to space variables and using the inequality uH m+2 ≤ C(∆uH m + uH m+1 ) we obtain 2M−1

Dtk ∇u(t)H 2M −1−k +

k=1

2M

Dtk u(t)2 ≤ C(IIM + K α+1 )

k=1

for some constant C > 0.

Completion of the proof of Theorem 7.3. Let u(t) be the local solution guaranteed by Proposition 7.1. This solution belongs to V (T ) for a small T > 0. From Propositions 7.2, 7.3 we conclude under Hyp. C that if I˜2M ≤ 1 and u(·)V (T ) ≤ K, then u(·)V (T ) ≤ C(I˜2M + K α+1 ). Therefore, if we choose K > 0 such that C(I˜2M + K α+1 ) < K, (7.3.14) we have u(·)V (T ) < K and hence, u(t) is continued to exist in X(T˜ ), T˜ > T , beyond T . This extended solution still satisﬁes the same estimate on [0, T˜ ], i.e., u(·) ˜ ≤ C(I˜2M + K α+1 ) < K. V (T )

Decay and Global Existence

259

Thus, we can conclude that the solution u(t) exists globally on [0, ∞) and satisﬁes the estimate u(·)V (T ) < K for all T > 0. Such a choice of K as in (7.3.14) is of course possible if I˜2M is suﬃciently small. The proof of Theorem 7.3 is now complete.

8. Quasilinear wave equations 8.1. Problem and result Following [45], we consider in this section the initial-boundary value problem for the quasilinear wave equation: utt − div{σ(|∇u|2 )∇u} + a(x)ut = 0 in Ω × [0, ∞),

(8.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(8.1.2)

N where Ω is an exterior domain in the N dimensional Euclidean √ space R with a smooth boundary ∂Ω and σ(v) is a function like σ(v) = 1/ 1 + v. Concerning the dissipation a(x)ut we make two types of assumptions speciﬁed later, which are intended to make the eﬀect of this term as weak as possible. As in the previous ˜ on the dissipation section, under two types of assumptions Hyp. A and Hyp. A a(x)ut , we will prove theorems of global existence of the problem (8.1.1)–(8.1.2). Concerning σ(·) we make the following assumptions.

Hyp. D. The function σ(·) is a diﬀerentiable function on R+ = [0, ∞) and satisﬁes the conditions: σ(v 2 ) ≥ k0 > 0 and σ(v 2 ) − 2|σ (v 2 )|v 2 ≥ k0 > 0, if |v| ≤ L, where L > 0 is an arbitrarily ﬁxed constant and k0 := k0 (L) is a positive constant (we may assume σ(0) = 1 for simplicity). The following result concerning local in time solutions is standard (cf. Kato [17]). We use the notation Im,q := ||u0 ||m,q + ||u1 ||m−1,q and Im := Im,2 . Proposition 8.1. Let m > M := [N/2] + 1 be an integer and assume that σ(·) ∈ C m+1 ([0, ∞)), a(·) ∈ C m+1 (Ω) and ∂Ω is of class C m+1 . Let the initial data (u0 , u1 ) ∈ H m+1 (Ω) × H m (Ω) satisfy the compatibility condition of order m associated with the problem (8.1.1)–(8.1.2). Then, there exists T = T (IIm ) > 0 such that the problem admits a unique solution u(t) on [0, T ) belonging to T Xm :=

m 2 k=0

C k ([0, T ); H m+1−k (Ω) ∩ H01 (Ω))

2

C m+1 ([0, T ); L2 (Ω)).

260

Mitsuhiro Nakao

∞ We set Xm = Xm . From Proposition 8.1 it suﬃces for the existence of global solutions in Xm to derive the a priori estimate

sup 0≤t
m+1

Dtk u(t)H m+1−k (Ω) < ∞

k=0

for all T > 0, where u(t) is an assumed local solution on [0, T ). In what follows we assume that ∂Ω is suﬃciently smooth, i.e., ∂Ω is of class C m+1 . Our ﬁrst result in this section reads as follows: Theorem 8.1. Let N ≥ 1 be any integer and assume that σ(·) ∈ C m+1 (R+ ) and ˜ and Hyp. D, there a(·) ∈ C m+1 (Ω) with m > [N/2] + 1. Then, under Hyp. A exists δ > 0 such that if (u0 , u1 ) ∈ H m+1 (Ω) × H m (Ω) satisfy the compatibility condition of order m and the smallness condition Im := u0 H m+1 + u1 H m < δ, the problem (8.1.1)–(8.1.2) admits a unique solution u(t) in the class Xm . Further, the following estimates hold: 2 Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k ≤ CIIm (1 + t)−k−1

for 0 ≤ k ≤ m

and 2 ∇u(t)2H m−k ≤ CIIm (1 + t)−1

for 0 ≤ k ≤ m.

Remark 8.1. When Ω is a bounded domain we can consider a more delicate situation, where a(x) is localized and further degenerates on (N − 1)-dimensional submanifolds in Ω (see [42]). The results obtained by a second approach are stated separately in the cases N ≥ 4 and N = 3. Theorem 8.2. Let N ≥ 4. When N is even we assume that V is convex. Assume that σ(·) and a(·) are of class C 3M . We assume that (u0 , u1 ) belong to H 3M+1 (Ω)∩ W 2M+1,1 (Ω) × H 3M (Ω) ∩ W 2M,1 (Ω) and satisfy the compatibility conditions of the order 3M associated with the quasilinear problem (8.1.1)–(8.1.2) and also with the linear problem with σ ≡ 1. Further, we assume that a(·) satisﬁes Hyp. A and supp a(·) is compact. Then, under Hyp. D, there exists δ > 0 such that if I˜3M := u0 H 3M +1 + u0 W 2M +1,1 + u1 H 3M + u1 W 2M,1 ≤ δ, there exists a unique solution u(t) in the class Y3M :=

3M 2 k=0

C k ([0, ∞); H 3M+1−k (Ω) ∩ H01 )(Ω) 2

2

C 3M+1 ([0, ∞); L2 (Ω))

W k,∞ ([0, ∞); W M+1−k,∞ (Ω)),

satisfying 3M k=0

Dtk ∇u(t)H 3M −k ≤ C I˜3M < ∞

Decay and Global Existence and

M

261

Dtk ∇u(t)W M −k,∞ ≤ C I˜3M (1 + t)−d

k=0

with d = (N − 1)/2. More interesting is the case N = 3, where the situation is also more delicate. Theorem 8.3. Let N = 3. Assume that σ(·) and a(·) are of class C 4M+2 . We assume that (u0 , u1 ) belong to H 4M+3 (Ω) ∩ W 4M+2,q (Ω) × H 4M+2 (Ω) ∩ W 4M+1,q (Ω) and satisfy the compatibility conditions of order 4M + 2 associated with the quasilinear problem (6.1.1)–(6.1.2) and also with the linear problem with σ ≡ 1. We assume that a(·) satisﬁes Hyp. A and supp a(·) is compact. Then, under Hyp. D, there exists δ such that if I˜4M+2 := u0 H 4M +3 + u0 W 4M +2,q + u1 H 4M +2 + u1 W 4M +1,q ≤ δ, there exists a unique solution u(t) in the class Y4M+3 :=

4M+2 2

C k ([0, ∞); H 4M+3−k (Ω) ∩ H01 (Ω))

2

C 4M+2 ([0, ∞); L2 (Ω)),

k=0

satisfying 4M+3

Dtk ∇u(t)H 4M +3−k ≤ C I˜4M+2 < ∞

k=0

and

M+1

Dtk ∇u(t)W M +1−k,p ≤ C I˜4M+2 (1 + t)−d(p)

k=0

with d(p) = (p − 2)(1 − ε)/p, 0 < ε " 1, where we should take 6 ≤ p < ∞ and q = p/(p − 1). Remark 8.2. Concerning the regularity of the initial data, in Shibata-Tsutsumi [63] the data (u0 , u1 ) are required to belong to H 20 (Ω) × H 19 (Ω) if N = 3, while here we impose (u0 , u1 ) ∈ H 11 (Ω) × H 10 (Ω). 8.2. Energy decay for the quasilinear equation ˜ and Hyp. D. Let u(t) be a local solution on [0, T˜), 0 < T˜ ≤ ∞ We assume Hyp. A of the problem (8.1.1)–(8.1.2) in Proposition 8.1. In this section, on the basis of the arguments from Section 5 we ﬁrst derive the L2 -boundedness and decay estimates for E(t). Next, we also derive the decay of the energy for U = ut . Proposition 8.2. There exists δ2 > 0 such that if sup (Dx2 Dt u(t)∞ + D2 u(t)∞ + Du(t)∞ ) ≤ δ2 ,

D = (∇, Dt ),

(8.2.1)

0≤t
then sup u(t) ≤ CII0

0≤t
(8.2.2)

262

Mitsuhiro Nakao

and E(t) ≤ CII02 (1 + t)−1 , where 1 E(t) = 2

2 |ut (t)| + Ω

0 ≤ t < T˜,

|∇u(t)|2

(8.2.3)

σ(ξ) dξ dx.

0

˜ with f ≡ 0. Proof. We may assume T˜ > T0 . Otherwise, we get the results by (A) We use the notation ε0 and k for ε0 /16 and k/2, respectively. Then, by Proposition 5.2, we have T˜ T˜ E(s) ds + k a(x)|ut |2 dx ds ≤ X(0) ≤ CII02 X(t) + ε0 0

0

which implies, in particular,

T˜

u(t)2 ≤ CII02 < ∞ and

E(s) ds ≤ CII02 ,

(8.2.4)

0

provided that Dx Dt σ(|∇u|2 )∞ + Dσ(|∇u|2 )∞ ≤ δ1 which holds under (8.2.1). The inequality (8.2.4) and the fact dE(t)/dt ≤ 0 implies easily (see (4.2.27)) T˜ (1 + t)E(t) ≤ E(s) ds + E(0) ≤ CII02 . 0

We proceed to the estimation of the second order derivatives as in Section 4. For this, we assume for a moment, sup (1 + t)k+1 Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k

0≤t
(1 + t) Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k dt ≤ K 2 ,

(8.2.5)

k

+ 0

for 0 ≤ k ≤ m, 0 < t < T˜ and sup (1 + t)∇u(t)2H m−k +

0≤t
0

T˜

∇u(t)2H m−k dt ≤ K 2 ,

(8.2.6)

for 0 ≤ k ≤ m, 0 ≤ t < T˜ with some K > 0. First, we note that if u(t) is a local T in time solution in Xm , m > [N/2] + 1 with a small T˜ , then Dtk+1 u(t) + Dtk ∇u(t) ≤ C(IIm )(u0 H k+1 + u1 H k ),

0 ≤ k ≤ m,

which is a standard fact for quasilinear evolution equations (cf. Kato [17]).

Decay and Global Existence

263

Proposition 8.3. We assume that a local solution u(t) ∈ Xm (T˜ ) satisﬁes (8.2.5) and (8.2.6). Then, under the assumption (8.2.1), u(t) satisﬁes the estimates T˜ (1 + t)E1 (t)dt ≤ CI12 , E1 (t) ≤ C(1 + K 2 )I12 (1 + t)−2 (8.2.7) 0

and

T˜

∆u(t)22 ≤ C(1 + K 2 )I12 (1 + t)−1 ,

∆u(t)2 dt ≤ CI12 ,

(8.2.8)

0

where E1 (t) =

1 2

|utt (t)|2 + |∇ut (t)|2 dx.

Ω

Proof. Setting U = ut we have ∂ ∂U Utt − aij (x, t) + a(x)U Ut = 0, ∂xi ∂xj i,j

(8.2.9)

where aij = σ(|∇u|2 )δij + 2σ uxi uxj . Hence, applying Proposition 5.2 to (8.2.9) we obtain under (8.2.1) t+T t+T E1 (s) ds + k a(x)|ut |2 dx ds ≤ 0 (8.2.10) X1 (t + T ) − X1 (t) + ε0 t

Ω

t

if t + T < T˜ , where X1 is deﬁned by X with u replaced by U and we use the notation ⎛ ⎞ 1 ⎝|utt (t)|2 + aij utxi utxj (t)⎠ dx. E1 (t) = 2 Ω ij By (8.2.3) we know already that U (t)2 = ut (t)2 ≤ CII02 (1 + t)−1 . From (8.2.10) we easily see that T˜ E1 (t) dt ≤ CI12 < ∞.

(8.2.11)

0

Let us show the further inequality T˜ (1 + t)E1 (t)dt ≤ CI12 < ∞.

(8.2.12)

0

Indeed, by (8.2.10), (1 + t + T )X1 (t + T ) − (1 + t)X1 (t) + ε0

t+T

(1 + s)E1 (s) ds, ≤ T X1 (t) t

264

Mitsuhiro Nakao

and hence, taking n such that nT < T˜, nT n−1 ε0 (1 + s)E1 (s) ds ≤ T X1 (jT ) + (1 + T )X1 (0) 0

j=1 n−1

≤ CT

(8.2.13)

E1 (jT ) + U (jT )2 .

j=0

Here, noting that d E1 (t) + a(x)|U Ut |2 dx dt Ω 1 aij,t Uxi Uxj dx ≤ C |∇u||∇ut ||∇U (t)|2 dx ≤ Cδ1 E1 (t), = 2 Ω i,j Ω we have

∗

E1 (jT ) ≤ E1 (s + (j − 1)T ) + Cδ1 for any 0 ≤ s∗ ≤ T , and hence, jT T E1 (jT ) ≤ E1 (s) ds+ Cδ1 T (j −1)T

jT

(j −1)T

E1 (s) ds

jT

(j −1)T

(8.2.14)

E1 (s) ds = (1 + Cδ1 T )

jT

(j −1)T

E1 (s) ds,

where we have taken s∗ such that min(j−1)T ≤s≤jT E1 (s) = E1 (s∗ +(j −1)T ). Thus, we have from (8.2.13) and (8.2.11) that nT T˜ ε0 (1 + s)E1 (s) ds ≤ C(1 + T )(E1 (0) + E(0)) + C(1 + δ1 T ) E1 (s) ds ≤ CI12 0

0

which implies the ﬁrst estimate of (8.2.7). Further, by use of (8.2.14), d d (1 + t)2 E1 (t) ≤ 2(1 + t)E1 (t) + (1 + t)2 E1 (t) dt dt ≤ 2(1 + t)E1 (t) + C(1 + t)2 ∇u(t)∞ ∇ut (t)∞ E1 (t).

(8.2.15)

Here, by Gagliardo-Nirenberg inequality and the assumption (8.2.6) we see that ∇u(t)∞ ≤ C∇u(t)1−θ ∇u(t)θH m ≤ CK(1 + t)−1/2 2 with a certain constant 0 < θ < 1. Similarly, ∇ut (t)∞ ≤ CK(1 + t)−1 . Therefore, we have from (8.2.15) d (1 + t)2 E1 (t) ≤ 2(1 + t)E1 (t) + CK 2 (1 + t)E1 (t) dt and after integrating, T˜ (1 + t)2 E1 (t) ≤ CE1 (0) + C(1 + K 2 ) (1 + s)E1 (s) ds ≤ C(1 + K 2 )I12 0

Decay and Global Existence

265

which implies the second estimate of (8.2.7). To show (8.2.8) we have only to return to the original equation and use the regularity theory for elliptic equations. 8.3. Estimation of higher-order derivatives of solutions On the basis of Propositions 8.1 and 8.2 we derive in this section estimates for T˜ . Throughout of this the higher-order derivatives of the (local) solutions u(t) ∈ Xm section we assume (8.2.1), (8.2.5) and (8.2.6). Proposition 8.4. For 2 ≤ k ≤ m we have T˜ (1 + t)k Ek (t) dt + sup (1 + t)k+1 Ek (t) ≤ Cq(I, K),

(8.3.1)

0≤t
0

where Ek (t) is deﬁned by k+1 1 k k Dt u(t) 2 + aij Dt uxi Dt uxj (t) dx 2 Ω i,j and q(I, K) denotes a quantity continuously depending on I = (II0 , I1 , . . . , Im ) and K such that q(0, K) = 0. We note that Ek (t) is equivalent to Dtk+1 u(t)2 + ∇Dtk u(t)2 . Proof. We know already that (8.3.1) is valid if k = 1. To show this for k, 2 ≤ k ≤ m we use the principle of induction and assume that (8.3.1) is valid for all 1 ≤ j ≤ k − 1. We use the notation I for Im . Diﬀerentiating the equation k times with respect to t and setting U = Dtk u(t), we have ∂ ∂U Utt − (8.3.2) aij (x, t) + a(x)U Ut = F (x, t) ∂xi ∂xj i,j with aij = σδij + 2σ (Du)2 and F =∇

k−1

Cj Dtj σDtk−j ∇u + 2∇

j=1

k−1

Cj Dtj (σ (Du)2 )Dtk−j Du,

j=1

Dtl

where denotes any partial diﬀerentiations with respect to t of order l and sum of them. We often use the notation D for (∇, Dt ) and Dx = ∇. We apply Proposition 5.2 to (8.3.2) to obtain t+T t+T Xk (t + T ) − Xk (t) + ε0 Ek (s) ds ≤ C (|∇U | + |U Ut | + |U |)|F |dx ds, t

and Xk (t + T ) − Xk (t) +

ε0 2

Ω

t

t+T

Ek (s) ds ≤ C t

t

t+T

U F + F 2 ds, (8.3.3)

266

Mitsuhiro Nakao

where t + T < T˜ . To estimate F (t) we rewrite F as follows:

F =

k−1

Cj ΓDuDtj D2 uDtk−j Du +

j=1

+

+

+

k−1

Cj (Γ (Du)2 + Γ)Dut Dtj−1 D2 uDtk−j Du

j=1

j k−1

Cjl Dtl (ΓDu)Dtj−l D2 uDtk−j Du

j=2 l=2 k−1

+

k−1

Cj ΓDuDtj DuDtk−j D2 u

j=1

Cjl (Γ (Du)2 + Γ)Dut Dtj−l DuDtk−j D2 u

j=1 j k−1

Cjl (Γ (Du)2 + Γ)Dut Dtj−l DuDtk−j D2 u

j=2 l=2

=: J1 + J2 + J3 + J4 + J5 + J6 , (8.3.4) where Γ := Γ(Du) = 6σ + 4σ · (Du)2 . Estimation of J1 and J4 . Taking appropriate p1 ≥ 1 and p2 ≥ 2 with 1/p1 +1/p2 = 1/2 we have

J J1 ≤ C ≤C

k−1 j=1 k−1

Du∞ Dtj D2 up1 Dtk−j Dup2 Du1−θ DuθH m Dtj D2 uH m−j Dtk−j DuH m+1−k+j

j=1

≤ CII01−θ K 2+θ (1 + t)−ν with a certain 0 < θ < 1 and ν = 1/2 + (1 + j)/2 + (1 + k − j)/2 = (k + 3)/2. Similarly, we see

T˜

(1 + t)k+2 J J1 (t)2 dt ≤ CK 2 0

≤ Similar estimates hold for J4 .

T˜

(1 + t)Du(t)2(1−θ) Du(t)2θ H m dt

0 2(1−θ) 2(2+θ) CII0 K .

Decay and Global Existence

267

Estimation of J2 and J5 . We see that J J2 ≤ C

k−1

Dut Dtj−1 D2 uDtk−j Du||

j=1

≤ Dut ∞ Dtj−1 D2 up1 Dtk−j Dup2 k−1 ≤C Dut 1−θ Dut θH m−1 Dtj−1 D2 uH m−j Dtk−j DuH m−k+j 2 j=1

≤ C(1 + K 2 )I11−θ (1 + t)−1 K(1 + t)−j/2 K(1 + t)(−k+j−1)/2 ≤ C(1 + K 2 )K 2 I11−θ (1 + t)−ν with a certain 0 < θ < 1 and ν = (k + 3)/2. Similarly, we have T˜ 2(1−θ) (1 + t)k+2 J J2 (t)2 dt ≤ C(1 + K 2 )K 4 I1 . 0

The same estimates as for J2 hold for J5 . ˜ = ΓDu, we have Estimation of J3 and J6 . Setting Γ j l k−1 ˜ (r) j−l 2 k−j α1 αs γ1 γs J J3 ≤ C |(Dt Du) · · · (Dt Du) ||Dt D uDt Du| , Γ j=2 l=2 r=1

Sr

where Sr = (α1 , . . . , αs , γ1 , . . . , γs )|1 ≤ α1 < α2 · · · < αs ,

s

αi γi = l and

i=1

s

γs = r .

i=1

Hence, by choosing appropriate {pi }, pi ≥ 2, (cf. [38]), we have J J3 ≤ C ≤C

j k−1 l

Dtα1 Duγp11 · · · Dtαs Duγpss Dtj−l D2 ups+1 Dtk−j Dups+2

j=2 l=2 r=1 Sr j l k−1

θ1 Dtα1 DuγH1m−α Dtα1 Duγ1 (1−θ1 ) · · · 1

j=2 l=2 r=1 Sr θs αs γs (1−θs ) Dtj−l D2 uH m−j+l−1 Dtk−j DuH m−k+j . ×Dtαs DuγHsm−α s Dt Du

Therefore, by induction assumption, J J3 ≤ Cqk (IIm , K)(1 + t)−

P

i

γi (αi +1)/2−(k−l+2)/2

≤ Cq(I, K)(1 + t)−(r+k+2)/2 ≤ Cq(I, K)(1 + t)−ν with ν = (k + 3)/2 and a certain quantity q(I, K) satisfying q(0, K) = 0. We also have T˜ (1 + t)k+2 J J6 (t)2 dt ≤ Cq(I, K) < ∞. 0

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J6 is also treated quite similarly and satisﬁes the same ﬁnal two estimates. Thus, we obtain T˜ k+3 2 sup (1 + t) F (t) + (1 + t)k+2 F (t)2 dt ≤ q(I, K) < ∞. (8.3.5) 0≤t
0

Therefore, under the smallness assumption on δ2 we obtain from (8.3.3) that ε0 t+T Ek (s)ds ≤ Cq(I, K)(1 + t)−k−3/2 , (8.3.6) Xk (t + T ) − Xk (t) + 2 t where we have used again the induction hypothesis U (t)2 ≤ 2Ek−1 (t) ≤ q(I, K)(1 + t)−k . On the basis of the integral inequality (8.3.6) we shall prove the desired energy estimates (8.3.1). For this we ﬁrst show that T˜ (1 + t)k Ek (t)dt ≤ Cq(I, K) < ∞. (8.3.7) 0

Indeed, by the same argument deriving the estimate for E1 (t), we can show from (8.3.6) that T˜ (1 + t)Ek (t)dt ≤ Cq(I, K) < ∞. 0

So, for induction, let us assume T˜ (1 + t)j Ek (t)dt ≤ Cq(I, K) < ∞ 0

for some j, 0 ≤ j < k. Then, we see from (8.3.6) that ε0 t+T (1 + t + T )j+1 Xk (t + T ) − (1 + t)j+1 Xk (t) + (1 + s)j+1 Ek (s)ds 2 t ≤ Cq(I, K)(1 + t)−3/2 + C(1 + t + T )j Xk (t). Since Xk (t) is equivalent to Ek (t) + Dtk u(t)2 we conclude from the assumption of induction that T˜ (1 + t)j+1 Ek (t)dt ≤ Cq(I, K) < ∞. 0

Thus, we conclude (8.3.7). Finally, we return to the equation (8.3.2) to get the energy inequality d Ek (t) ≤ C F (t) Ek (t) + δ˜0 (t)Ek (t) . (8.3.8) dt Here, δ˜0 (t) := sup aij (t)∞ ≤ C∇u(t)∞ ∇ut (t)∞ ≤ Cq(I, K)(1 + t)−3/2 i,j

Decay and Global Existence and F (t)

269

1 (1 + t)F 2 + (1 + t)−1 Ek (t) . Ek (t) ≤ 2

Thus, we obtain from (8.3.5) and (8.3.7) that d d (1 + t)1+k Ek (t) = (1 + k)(1 + t)k Ek (t) + (1 + t)1+k Ek (t) dt dt ≤ C(1 + t)k Ek (t) + C(1 + t)2+k F (t)2 + Cq(I, K)(1 + t)k−1/2 Ek (t) which together with (8.3.7) and (8.3.5) implies

T˜

(1 + t)1+k Ek (t) ≤ Ek (0) + C

(1 + t)2+k F (t)2 dt 0

T˜

(1 + s)k Ek (s) ds ≤ Cq(I, K) < ∞.

+C(1 + q(I, K)) 0

Thus, (8.3.1) is now proved.

Once the energy decay for the higher derivatives are derived, the following assertion follows by a standard argument: Proposition 8.5. Under the assumptions (8.2.1), (8.2.5) and (8.2.6) we have further sup (1 + t)k+1 Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k 0≤t
+ 0

(1 + t)k Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k dt

(8.3.9)

≤ q(I, K) for 0 ≤ k ≤ m, and sup (1 + t)∇u(t)2H m−k +

0≤t
0

T˜

∇u(t)2H m−k dt ≤ Cq(I, K) for 0 ≤ k ≤ m. (8.3.10)

Proof. The preceding proposition means that (8.3.10) is valid for k = m and further sup (1 + t)j+1 Dtj+1 u(t)2 + Dtj ∇u(t)2 0≤t
+

(1 + t)j Dtj+1 u(t)2 + Dtj ∇u(t)2 dt ≤ q(I, K) for 0 ≤ j ≤ m.

0

(8.3.11)

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Mitsuhiro Nakao

We show (8.3.9) by induction and for this we assume that sup (1 + t)j+1 Dtj+1 u(t)2H m−j + Dtj ∇u(t)2H m−j

0≤t
(1 + t)j Dtj+1 u(t)2H m−j + Dtj ∇u(t)2H m−j dt ≤ q(I, K),

+

k+1 ≤ j ≤ m.

0

(8.3.12) To prove (8.3.9) it suﬃces to show for 0 ≤ k ≤ m − 1, sup (1 + t)k+1 Dtk ∇u(t)2H m−k +

0≤t
T˜

(1 + t)k Dtk ∇u(t)2H m−k dt ≤ q(I, K), 0

(8.3.9) under the conditions that (8.3.9) with k = m, (8.3.11) and (8.3.12) hold. Diﬀerentiating the equation k times with respect to t we have ∆(Dtk u) = −D Dtk ((σ − 1)∇u) + Dtk+2 u(t) + a(x)Dtk+1 u(t) k 2 k 2 = Γ(|∇u| )Dt D u(t) + Cj,k Dtj ΓDtk−j D2 u + Dtk+2 u(t) + a(x)Dtk+1 u(t) j=1

=: F0 (t) + F1 (t) + F2 (t) + F3 (t) =: F (t), (8.3.13) where Γ = σ − 1. Since we may assume σ(0) = 1 we have Γ(0) = 0. Then, by the elliptic regularity theory we know Dtk ∇u(t)H m−k ≤ C(Dtk ∇u(t)H m−1−k + F (t)H m−1−k ).

(8.3.14)

First, we note that by Gagliardo-Nirenberg inequality, Dl Dtj ∇u(t) ≤ CDtj ∇u(t)1−θ Dtj ∇u(t)θH m−j ≤ Cq(I, K)(1 + t)−(1+j)(1−θ)/2 (1 + t)K θ (1 + t)−(1+j)θ/2 ≤ Cq(I, K)(1 + t)−(1+j)/2 for 0 ≤ l ≤ m − 1 − j, and hence Dtj ∇u(t)2H m−1−j ≤ Cq(I, K)(1 + t)−1−j

(8.3.15)

for all j, 0 ≤ j ≤ m − 1. Similarly, we have

T˜

(1 + t)j Dtj ∇u(t)2H m−1−j ≤ Cq(I, K),

0 ≤ j ≤ m − 1.

0

Further, we easily see by (8.3.12) that F F2 (t) + F3 (t)H m−1−k ≤ C Dtk+2 u(t)H m−1−k + Dtk+1 u(t)H m−1−k ≤ Cq(I, K)(1 + t)−(1+k)/2

Decay and Global Existence

271

and

T˜

F F2 (t) + F3 (t)2H m−1−k dt

0

T˜

≤C 0

k+2 Dt u(t)2H m−1−k + Dtk+1 u(t)2H m−1−k dt ≤ Cq(I, K).

By use of (8.3.5), (8.3.6), and (8.3.14) we can carry out a similar argument obtaining (8.3.5) to get sup (1 + t)k+1 F F0 (t)2H m−1−k +

0≤t
T˜

0

(1 + t)k F F0 (t)2H m−1−k dt ≤ Cq(I, K) < ∞,

where we have used Γ(0) = 0. The treatment of the term F1 (t) is also delicate. But, noting that F F1 (t)H m−1−k ≤ C

k

Dtj ΓDtk−j D2 u + C

j=1

k

Dm−1−k (Dtj ΓDtk−j D2 u)

j=1

and repeating again a similar argument estimating J1 through J6 as in (8.3.3) we can prove that F1 (t)2 + sup (1 + t)k+1 F

0≤t
T˜

(1 + t)k F F1 (t)2 dt ≤ Cq(I, K). 0

Thus, we conclude (8.3.9)’.

Completion of the proof of Theorem 8.1. The a priori estimates in the preceding sections are suﬃcient for the proof of Theorem 8.1. Under the assumptions (8.3.1), T˜ satisﬁes (8.3.5) and (8.3.6) we have shown that any local solution u(t) ∈ Xm sup (1 + t)k+1 Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k 0≤t
+ 0

(1 + t)k Dtk+1 u(t)2H m−k + Dtk ∇u(t)2H m−k dt ≤ q(I, K) < ∞

(8.3.16) for 0 ≤ k ≤ m, 0 ≤ t < T˜ , where q(I, K) is some quantity depending on I, K in such a way that q(0, K) = 0. Thus, ﬁxing K > 0 arbitrarily and making the additional assumption q(I, K) < K 2

(8.3.17)

we can conclude that under (8.3.17), the local solutions exist in fact on [0, ∞) and the estimate (8.3.16) holds on [0, ∞). Finally, we note that since m > [N/2] + 2,

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Mitsuhiro Nakao

under (8.3.5) and (8.3.6), Dx2 Dt u(t)∞ + D2 u(t)∞ + Du(t) ≤ C Dt u(t)1−θ1 Dt u(t)θH1m−1 + D2 u(t)1−θ2 D2 u(t)θH2m−1 +Du(t)1−θ3 Du(t)θH3m ≤C

3

(II0 + I1 )1−θi K θi

i=1

with a certain 0 < θi < 1, i = 1, 2, 3. Thus, (8.2.1) is satisﬁed under the further additional assumption 3 (II0 + I1 )1−θi K θi < δ2 (8.3.18) C i=1

for the ﬁxed K > 0. Since both of (8.3.17) and (8.3.18) are valid for small I, the proof of Theorem 8.1 is now complete. Remark 8.3. By a careful observation we see that q(I, K) is replaced by q(II0 , K) + 2 2 CIIm . Hence, for any K > CIIm , there exists δ(K) such that q(II0 , K)+CIIm < K 2 if I0 ≤ δ(K). (8.3.17) also holds under these conditions. Thus, the set of initial data assuring the global existence in Theorem 8.1 is in fact unbounded in H m+1 × H m . 8.4. Proof of Theorems 8.2 and 8.3. In this section we assume only Hyp. A. That is, we consider the case, where a(x)ut is eﬀective only at a neighborhood of Γ(x0 ) and supp a(·) is compact, say, supp a(·) ⊂ BL ,

L > 0.

(8.4.1)

First, let us consider the linear wave equation with a localized dissipation: utt − ∆u + a(x)ut = 0 in Ω × [0, ∞),

(8.4.2)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0. (8.4.3) Under our assumptions on a(x) we have shown in Section 3 a local energy decay for the solutions of the linear problem (8.4.2)–(8.4.3) and using this we have proved Lp estimates in Section 6 for the linear equation (cf. [17]). In particular, for the odd dimensional cases we have the following: Proposition 8.6. Let N ≥ 3 be an odd integer and m be a nonnegative integer. We assume that a ∈ C 2M+m (Ω), Hyp. A is valid and supp a(·) is compact. We set M = [N/2] + 1. Concerning the initial data we assume that u0 ∈ H01 (Ω) ∩ H 2M+m (Ω) ∩ W 2M+m,1 (Ω), u1 ∈ H 2M+m−1 (Ω) ∩ W 2M+m−1,1 and these data satisfy the compatibility condition of the order M + m − 1. Then, there exists a unique solution u(t) of the problem (8.4.2)–(8.4.3) in C([0, ∞); H 2M+m (Ω) ∩ H01 (Ω)) ∩ C 1 ([0, ∞); H 2M+m−1 (Ω) ∩ H01 (Ω)) and it satisﬁes m k=0

Dtk u(t)m−k,p ≤ C I˜M +m (1 + t)−b

(8.4.4)

Decay and Global Existence

273

if 2 < p ≤ p0 := 2(N + 1)/(N − 1) and m Dtk u(t)m−k,∞ ≤ C I˜2M−1+m (1 + t)−d ,

(8.4.5)

k=0

where we set b=

Il,1 = (u0 l+1,1 + u1 l,1 ), Il,2 = (u0 l+1,2 + u1 l,2 ), (N − 1)(1/2 − 1/p) 1 − 2/p − δ

and d=

(N − 1)/2 1−δ

if N is odd and N ≥ 5, if N = 3,

if N is odd and N ≥ 5, if N = 3.

We note that δ > 0 can be chosen arbitrarily small. Remark 8.4. Proposition 8.6 is valid for even integer N ≥ 4 if V is convex, where we can take b = (N − 1)(1/2 − 1/p) and d = (N − 1)/2. When N ≥ 4 we set Y3TM :=

3M 2

C k ([0, T ); H 3M+1−k (Ω) ∩ H01 (Ω))

2

C 3M+1 ([0, T ); L2(Ω))

k=0

and

V3M (K, T ) =

3M+1 u ∈ Y3TM Dtk ∇u(t)H 3M +1−k ≤ K k=0

and

M

-

Dtk ∇u(t)M+1−k,∞ ≤ K(1 + t)−d

k=0

for K > 0. When N = 3 we set 4M+2 2 2 T Y4M+2 := C k ([0, T ); H 4M+3−k (Ω) ∩ H01 (Ω)) C 4M+3 ([0, T ); L2(Ω)) k=0

and

V4M+2 (K, T ) =

4M+2 u ∈ Y4TM+2 Dtk ∇u(t)H 4M +2−k ≤ K k=0

and

M+1

Dtk ∇u(t)M+1−k,p

−d(p)

≤ K(1 + t)

k=0

for K > 0 and 6 ≤ p < ∞, where d(p) = d(1 − 2/p) = (1 − ε)(1 − 2/p), 0 < ε " 1. The local existence of the solution for each (u0 , u1 ) satisfying the compatibility condition is standard (cf. Kato [17]). So, for the proof of Theorems 8.2 and 8.3 it suﬃces to derive the desired estimates. We treat mainly the case N = 3 because

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Mitsuhiro Nakao

the other cases are proved in a similar and even simpler manner. We set m = 3M if N ≥ 4 and m = 4M + 2 if N = 3. We may assume u(·) ∈ Vm (K, T ) for some K > 0, T > 0, which will be shown later. For a moment we assume that ∇u(t)∞ + ∇Dt u(t)∞ ≤ δ1

(8.4.6)

with some small δ1 > 0, which will be satisﬁed if Im := u0 H m+1 + u1 H m is small. We begin with the following observation which will make a proof a little simpler for the case N = 3. Proposition 8.7. Let N = 3. If u(t) is a (local) solution from Vm (T ), then it satisﬁes M+1 Dtk Du(t)M+1−k,∞ ≤ CK(1 + t)−d1 (8.4.7) k=0

with d1 =

11(p − 2)(1 − ε) 11p + 6

and

(8.4.8) DtM+2 Du(t)2M−k,∞ ≤ Cq(K)K(1 + t)−d2 with d2 = 7d1 /9, where q(K) is a quantity (in fact a polynomial) depending on K and we recall that D = (∇, Dt ) and 2 < p ≤ p0 = 2(N + 1)/(N − 1). Proof. We ﬁrst note that M = 2 if N = 3. By Gagliardo-Nirenberg inequality we see k θ Dtk Du(t)M+1−k,∞ ≤ CDtk Du(t)1−θ M+1−k,p Dt Du(t)4M+2−k,2

≤ CK(1 + t)d(1−θ) with

−1 6 1 3M + 1 1 1 + − = p N p 2 11p + 6 which implies (8.4.7). To prove (8.4.8) we return to the equation. Note that 2M = M + 2 = 4. Then we see by a standard argument based on Leibniz’ formula that θ=

DtM+2 Du(t)M−2,∞ = DtM D(∇(σ∇u) + aut )∞ 2 i i ≤ Cq(K) Dt Du(t)M,∞ + Dt Du(t)M−1,∞ . i=1

Here, by (8.4.7), Dt Du(t)M,∞ +

2

Dti Du(t)M−1,∞ ≤ K(1 + t)−d1

i=1

and further, by Gagliardo-Nirenberg inequality, 2 θ Dt2 Du(t)M,∞ ≤ CDt2 Du(t)1−θ M−1,∞ Dt Du(t)4M,2

≤ CK(1 + t)−(1−θ)d1 , Thus we have (8.4.8).

θ = 2/9.

Decay and Global Existence

275

We proceed to the proof of Theorem 8.3. Let N = 3. Setting Dtm u(t) = U (t), m = 4M + 2, we have Ut = 0. Utt − div(Dtm (σ∇u)) + a(x)U Then, as in (8.4.3), we can derive d Em (t) ≤ |Dt σ||Dtm ∇u|2 dx dt Ω m−1 + |Dtk (Γ)||Dtm−k ∇u||∇Dtm+1 u| dx =: J1 + J2 ,

(8.4.9)

(8.4.10)

Ω k=1

where Γ = Γ(∇u) := σ(|∇u|2 ) + 2σ (∇u)2 . Here, taking account of (8.4.7), we see J1 ≤ C∇u(t)∞ Dt ∇u(t)∞ Dtm ∇u(t)2 ≤ CK 3 (1 + t)−2d1 Em (t). The treatment of J2 is delicate. We ﬁrst observe J2 ≤ C ∇u(t)∞ ∇ut (t)∞ Dtm−1 ∇u(t)Dtm ∇u(t) +

m−1 k

|Γ(i) ||Dt (∇u(t))2 |ν1 · · ·

Ω k=2 i=1 S i

×|Dtk (∇u(t))2 |νk |Dtm−k ∇u(t)||Dtm+1 ∇u(t)|dx (1)

(2)

=: J2 + J2 . It is easy to see that (1)

J2

≤ CK 3 (1 + t)−2d1

Em (t).

Further, we see (2)

J2

≤C

k m−1 k=2 i=1 Si

1/2 |Dt (∇u)2 |2ν1 · · · |Dtk (∇u))2 |2νk |Dtm−k ∇u|2 dx Em .

Ω

Here, since Dtk (∇u(t))2 =

k

Ckj Dtj ∇uDtk−j ∇u

j=0

and m = 4M + 2, we observe that each product of the above integrand contains the term |Dtj ∇uDtl ∇u|2 with j ≤ M + 1 and l ≤ M + 2 = 2M. Indeed, one of the most delicate terms appears in the case k = 2M + 2, that is, |Dt2M+2 (∇u)2 |2 |Dt2M ∇u|2 .

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Mitsuhiro Nakao

For this term, however, we see Dt2M+2 (∇u)2 =

Cj Dtj ∇uDt2M+2−j ∇u.

0≤j≤M+1

Hence, by (8.4.7), we can prove that (2)

J2

≤ Cq(K)K 2 (1 + t)−d1 −d2

Em (t).

We note that when N ≥ 4 we can use the estimate Dtj ∇u(t)∞ ≤ K(1 + t)−d to prove (2)

J2

≤ Cq(K)K 2 (1 + t)−d

Em (t),

0 ≤ j ≤ M + 1,

with d = (N − 1)/2 > 1. Thus, we obtain d Em (t) ≤ Cq(K)K 2 (1 + t)−d1 −d2 Em (t). dt Since d1 + d2 > 1, this implies Em (t) ≤ C(q(K)K 4 + Em (0)).

(8.4.11)

The same estimate is valid for the case N ≥ 4, where we take m = 3M . Note that a standard argument shows Em (0) ≤ q(K)IIm , where Im := u0 H m+1 + u1 H m . Now, returning to the equation and combining elliptic regularity theory with (8.4.11) we can prove as in Proposition 8.5 that 4M+2

Dtk Du(t)H 4M +2−k ≤ q(K)(K 2 + Im ).

(8.4.12)

k=0

We summarize the estimates (8.4.11) and (8.4.12) in the following proposition: Proposition 8.8. Let u(·) ∈ Vm (K, T ) be a solution of the problem (8.1.1)–(8.1.2) as in Proposition 8.7. Then, we have m

Dtk Du(t)H m−k ≤ q(K)(K + Im ),

D = (∇, Dt ),

k=0

where m = 3M if N ≥ 4 and m = 4M + 2 if N = 3.

(8.4.13)

Decay and Global Existence

277

Next, we proceed to the estimation of Dt ∇u(t)M+1−k,p , 0 ≤ k ≤ M + 1, 6 ≤ p < ∞. For this we begin with Lp estimates for the linear equation. By the known result (8.4.5) and Sobolev’s embedding theorem W l+1,1 ⊂ W 2M+l,2 , we see for the solutions u(t) of the linear equation that Dtk ∇u(t)l−k,∞ ≤ C I˜M +l (1 + t)−d ≤ (u0 3M+l+1,1 + u1 3M+l,1 )(1 + t)−d with any nonnegative integer l. On the other hand, by use of the energy identity, we easily see that Dtk ∇u(t)l−k,2 ≤ C(u0 l+1,2 + u1 l,2 ) ≤ C(u0 3M+l+1,2 + u1 3M+l,2 ). Thus, by interpolation, we obtain for all 2 ≤ p ≤ ∞, Dtk ∇u(t)l−k,p ≤ C(u0 3M+l+1,q + u1 3M+l,q )(1 + t)−d(p) ,

0 ≤ k ≤ l, (8.4.14) where 1/q + 1/p = 1. We use this estimate with l = M + 1 and N = 3. Let u(·) ∈ Vm (K, T ) be a solution of the quasilinear equation with initial data (u0 , u1 ) and let us denote the solution of the linear equation with the same initial data by U (t; u0 , u1 ). Then, by constant variation formula, t U (t − s; 0, F˜ ) ds, (8.4.15) u(t) = U (t; u0 , u1 ) + 0

where

F˜ = ∇ · σ(|∇u|2 ) − 1)∇u(t) =: Γ(|∇u|2 ) · (Du)2 D2 u.

Thus, by (8.4.14) and (8.4.15) we have Dtk ∇u(t)M+1−k,p ≤ CII4M+2,q (1 + t)−d(p) +

t

(1 + t − s)−d(p) F˜ (s)4M+1,q ds.

0

(8.4.16) Here, D4M+1 F˜ =

4M+1

˜ α,β Dα ∇uDβ ∇uD4M+2−j ∇u Γ

(8.4.17)

j=0 |α|≤M+1,|β|≤M+1

+

3M

˜ α,β Dα ∇uDβ ∇uD4M+2−j ∇u Γ

j=0 |α|≤M+1,|β|≥M+2

+

4M+1

˜ α,β Dα ∇uDβ ∇uD4M+2−j ∇u Γ

j=3M+1 |α|≥M+2,|β|≥M+2

=: J1 + J2 + J3 . ˜ = Γ(D ˜ γ u), γ ≤ 2M , and Γ ˜ ∞ ≤ C(K) < ∞, we estimate Ji , Noting that Γ 1 ≤ i ≤ 3, as follows:

278

Mitsuhiro Nakao First, we see that

J J1 q ≤ C

4M+1

|D ∇u| |D ∇u| |D α

4M+1

|D u|

≤

2q/(2−q)

α

4M+3−j

1/q u| dx q

2−q 2−q 2q 2q β 2q/(2−q) dx |D u| dx

Ω

|α|,|β|≤M+1

≤ CK

q

D4M+3−j u2

j=0

×

β

Ω

j=0 |α|,|β|≤M+1

≤ C(K)

q

Ω (1−θ)

Dα ∇u2

|α|,|β|≤M+1 2(1−θ) C(K)K 3 I0 (1

(1−θ)

Dα ∇uθp Dβ ∇u2

Dβ ∇uθp

+ t)−2d(p)θ (8.4.18)

with θ = (p + 2)/2(p − 2). Here, we note that 2d(p)θ = (p + 2)(1 − ε)/p > 1. For J2 we see that J J2 q ≤ C(K) ≤ C(K)

3M

j=0

|α|≤M +1 |β|≥M +2

j≤3M

|α|≤M +1 |β|≥M +2

|D ∇u| |D ∇u| |D α

q

β

q

4M+2−j

1/q ∇u| dx q

Ω

Dα ∇up Dβ ∇u2pq/(p−q) D4M+2−j ∇u2pq/(p−q) . (8.4.19)

Here, we see by Gagliardo-Nirenberg inequality, θ1 1 Dβ ∇u2pq/(p−q) ≤ CDM+1 ∇u1−θ 2pq/(p−q) ∇uH 4M +2

with

β−M −1 p−q 2pq + − N 2pq p−q p(β − M − 1) , = p(4M + 2 − β) − N

θ1 =

p−q 1 4M + 2 − β + − N 2pq 2

−1

and ˜

˜

DM+1 ∇u2pq/(p−q) ≤ CK∇u1−θ1 DM+1 ∇upθ1 with θ˜1 = 2/(p − 2). Thus, we have Dβ ∇u2pq/(p−q) ≤ C(K)(1 + t)−d(p)θ2

(8.4.20)

with θ2 = (1 − θ1 )θ˜1 . Quite similarly, since 4M + 2 − j ≥ M + 2 we can show that ¯

D4M+2−j ∇u2pq/(p−q) ≤ C(K)K(1 + t)−d(p)θ2

(8.4.21)

Decay and Global Existence

279

with θ¯2 = (1 − θ¯1 )θ˜1 , where θ¯1 is deﬁned by θ1 with β replaced by 4M + 2 − j. Thus, we obtain from (8.4.19), (8.4.20) and (8.4.21), ¯

J J2 q ≤ C(K)K 3 (1 + t)−(1+θ2 +θ2 )d(p) .

(8.4.22)

Here, we observe that θ1 , θ˜1 ≤ 3p/(3pM − N ) = p/(2p − 1) and (1 + θ2 + θ¯2 )d(p) ≥ (1 + 2(p − 3)/3p(2p − 1))(1 − ε) > 1. Finally, we can show as in the treatment of J2 , J J3 q ≤ C(K)

4M+1

D4M+2−j ∇up

(8.4.23)

j=3M+1 |α|≥M+2,|β|≥M+2 α ×D ∇u2pq/(p−q) Dβ ∇u2pq/(p−q) 3 −(1+θ2 +θ˜2 )d(p)

≤ C(K)K (1 + t)

,

where θ˜2 is deﬁned by θ2 with β replaced by α, and we know (1 + θ2 + θ˜2 )d(p) ≥ (1 + 2(p − 3)/3p(2p − 1))(1 − ε) > 1. Also we easily see F˜ q ≤ CDu2p Du4q/(2−q) ≤ CK 3 (1 + t)−2d(p) .

(8.4.24)

Since F˜ 4M+1,q ≤ C(D4M+1 F˜ q + F˜ q ), returning to the integral inequality (8.4.16) we obtain Dtk ∇u(t)M+1−k,p ≤ CII4M+2,q (1 + t)−d(p) + c(K)K 3

≤ (II4M+2,q + C(K)K 3 )(1 + t)−d(p) .

t

˜

(1 + t − s)−d(p) (1 + s)−d ds,

d˜ > 1,

0

(8.4.25) We summarize the results concerning the estimate of Dtk ∇u(t)M+1−k,p as follows: Proposition 8.9. Let N = 3 and let u(·) ∈ V4M+2 (K, T ) be a solution of the problem (8.1.1)–(8.1.2) as in Proposition 8.1. Then we have M+1

Dtk Du(t)M+1−k,p ≤ (q(K)K + CII4M+2,q )(1 + t)−d(p) ,

(8.4.26)

k=0

where q(K) is a quantity depending on K continuously in such a way that q(0) = 0. When N ≥ 4, for the solution u(t) ∈ V3M (K, T ) we have Dtk ∇u(t)M+1−k,∞ t ≤ C I˜3M (1 + t)−d + (1 + t − s)−d (F˜ (s)3M,2 + F˜ (s)3M,1 ) ds. 0

(8.4.27)

280

Mitsuhiro Nakao

By use of the inequality M

Dtk ∇u(t)M+1−k,∞ ≤ K(1 + t)−d

k=0

we can easily prove under the assumption (8.4.6) that F˜ (s)3M,2 + F˜ (s)3M,1 ≤ CK 3 (1 + t)−d , and hence we obtain M+1 Dtk Du(t)M+1−k,∞ ≤ (q(K)K + CII3M,1 )(1 + t)−d .

(8.4.28)

(8.4.29)

k=0

Completion of the proof of Theorems 8.2, 8.3. Let us consider the case N = 3. By the proofs of Propositions 8.6, 8.7 we easily see that if K > 2(II4M+2,2 + I4M+1,q ) the local solution u(t) belongs to V3M (K, T ) for some T > 0 and by Propositions 8.8 and 8.9 we know that this is valid for all T > 0 provided that q(K)K + C(II4M+2,2 + I4M+1,q ) < K.

(8.4.30)

Since q(K) continuously depends on K and q(0) = 0 the above condition (8.4.30) is satisﬁed if we take K = C1 (II4M+2,2 + I4M+1,q ) with C1 C and if I4M+2,2 + I4M+1,q ≤ δ for a small constant δ > 0. Thus we arrived at the desired estimates for all T > 0. The case N ≥ 4 (Theorem 8.3) is also proved quite similarly by use of Proposition 8.8 and the estimate (8.4.29).

9. The wave equation with a half-linear dissipation 9.1. Problem and result Following [47], we consider in this ﬁnal section the initial-boundary value problem for the wave equations with a nonlinear dissipation: utt − ∆u + ρ(x, ut ) = 0

in Ω × (0, ∞),

(9.1.1)

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, (9.1.2) N where Ω is an exterior domain in R and ρ(x, v) is a function like ρ(x, v) = a(x)g(v) with a(x) ≥ 0 and g (v) ≥ 0. Let a(x) be a nonnegative bounded function ˜ = Hyp. A+ Hyp. A from Section 4. By use of this a(x) on Ω, satisfying Hyp. A we make the following assumption on the dissipation ρ(x, ut ): Hyp. E. The function ρ(x, v) is a monotone increasing and diﬀerentiable function in v(= 0) and satisﬁes the following conditions: C ρ(x, v) = a(x)v if (x, v) ∈ BR ×R

(9.1.3)

for some 0 < R ≤ L such that V ⊂ BR . If (x, v) ∈ ΩR × R and |v| ≤ 1, then k0 a(x)|v|p+2 ≤ ρ(x, v)v ≤ k1 a(x) |v|p+2 + |v|2 , (9.1.4)

Decay and Global Existence

281

and if (x, v) ∈ ΩR × R and |v| ≥ 1, then

k0 a(x)|v|q+2 ≤ ρ(x, v)v ≤ k1 a(x) |v|q+2 + |v|2 ,

(9.1.5)

where k0 , k1 > 0, −1 < p < ∞ and −1 < q ≤ 2/(N − 2) (− ( 1 < q < ∞ if N = 1, 2). Without loss of generality we may assume further ω ⊂ BR . As an example we can take ρ(x, v) = a(x)φ(x)v + a(x)(1 − φ(x)) max{R−p |v|p , R−q |v|q }v, where φ(x) is a nonnegative smooth function such that 0 ≤ φ(x) ≤ 1, φ(x) = 0 if |x| ≤ R/2 and φ(x) = 1 if |x| ≥ R with R 1. The conditions (9.1.3)–(9.1.5) mean that the dissipative term ρ(x, ut ) is linear C on BR , while it may be nonlinear on ΩR . By reason of such two characters we temporarily call it a ‘half-linear’ dissipation. We also note that our dissipation may be localized near a part of the boundary Γ(x0 ) and near inﬁnity and may vanish in a large area. The main purpose of this section is to derive a decay estimate of the energy for the problem (9.1.1), (9.1.2) in such a delicate situation. As an application of our decay estimates we can discuss the existence of global solutions to the perturbed problem: utt − ∆u + ρ(x, ut ) = f (x, u)

in Ω × (0, ∞),

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω ,

(9.1.6) (9.1.7)

where f (x, u) is a function like f (x, u) = |u|α u, α > 0. More precisely we make the following assumption on f (x, u): Hyp. F. The function f (x, u) is measurable in x ∈ Ω for any u ∈ R, diﬀerentiable in u ∈ R for a.e. x ∈ Ω and satisﬁes |f (x, u)| ≤ k2 |u|α+1 and |ffu (x, u)| ≤ k2 |u|α with k2 > 0 and α > 0. Let us assume Hyp. E. Then, for each (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω) the problem (9.1.1), (9.1.2) has a unique solution u(t) in C 1 ([0, ∞); L2 (Ω)) ∩ C([0, ∞); H01 (Ω)) with ∞

0

ρ(x, ut )ut dx dt < ∞, Ω

2 := and for each (u0 , u1 ) ∈ H01 (Ω) ∩ H 2 (Ω) × H01 (Ω) the solution belongs to Xloc 2,∞ 1,∞ 2 1 ∞ 2 Wloc ([0, ∞); L (Ω)) ∩ Wloc ([0, ∞); H0 (Ω)) ∩ Lloc ([0, ∞); H (Ω)), satisfying the estimate

utt + ∇ut + ∆u(t) ≤ C(∇u1 + ∆u0 + ρ(x, u1 )) =: K0 < ∞, (9.1.8) for 0 ≤ t < ∞ with some C > 0. These results are standard and well-known (cf. Lions and Strauss [26]). For these solutions we prove the bound of u(t) and decay estimate of the energy E(t) := 12 (ut (t)2 + ∇u(t)2 ).

282

Mitsuhiro Nakao

2 Theorem 9.1. Let u(t) ∈ Xloc be a solution of (9.1.1), (9.1.2) satisfying (9.1.8). Under Hyp. E we have the estimates

||u(t)||2 ≤ C(Q0 + u0 2 )(1 + t)ν and E(t) ≤ C(Q0 + u0 2 )(1 + t)ν−1 , (9.1.9) where ν and Q0 are deﬁned corresponding to the cases as follows: (1) If 0 ≤ p < ∞ and 0 ≤ q ≤ 2/(N − 2)+ (0 ≤ q < ∞ if N = 1, 2), then ν = max{p/(p + 2), q(N − 2)+ /(4 − (N − 2)+ q)} (ν = max{p/(p + 2), ε},

0 < ε " 1, if N = 2)

and Q0 = E(0)2/(p+2) + E(0) + (K02 + E(0))(q+1)θ E(0)2(q+1)(1−θ)/(q+2) , where θ = N q/(q + 1)(4 − (N − 2)+ q) (θ = (q + 2)ε/(q + 1) if N = 2). (2) If −1 < p ≤ 0 and 0 ≤ q ≤ 2/(N − 2)+ , then ν = max{−p/(p + 2), q(N − 2)+ /(4 − (N − 2)+ q)} (ν = max{−p/(p + 2), ε},

0 < ε " 1, if N = 2)

and Q0 = E(0)2(p+1)/(p+2) + (K02 + E(0))(q+1)θ E(0)2(q+1)(1−θ)/(q+2) , where θ = N q/(q + 1)(4 − (N − 2)+ q) (θ = (q + 2)ε/(q + 1) if N = 2). (3) If 0 ≤ p < ∞ and −1 < q ≤ 0, then ν = max{p/(p + 2), −q(N − 2)+ /(4 − (N − 2)+ q)} (ν = max{p/(p + 2), ε} if N = 2) and Q0 = E(0)2/(p+2) + E(0) + (K02 + E(0))(q+1)θ E(0)4/(4−(N −2)

+

q)

with θ = −N q/(q + 1)(4 − (N − 2)+ q) (θ = ε/(q + 1) if N = 2). (4) If −1 < p ≤ 0 and −1 < q ≤ 0, then ν = max{−p/(p + 2), −q(N − 2)+ /(4 − (N − 2)+ q)} (ν = max{−p/(p + 2), ε},

0 < ε " 1, if N = 2)

and Q0 = E(0)2(p+1)/(p+2) + E(0) + (K02 + E(0))(q+1)θ E(0)4/(4−(N −2)

+

q)

with θ = −N q/(q + 1)(4 − (N − 2)+ q) (θ = ε/(q + 1) if N = 2). The estimates are valid also for the Cauchy problem, Ω = RN , if N ≥ 3, where the boundary condition u|∂Ω = 0 should be dropped.

Decay and Global Existence

283

We note that if q = 0, then the above estimates do not depend on K0 and hence, by a standard density argument we see that they hold for the ﬁnite energy solutions u(t) ∈ C 1 ([0, ∞); L2 (Ω)) ∩ C([0, ∞); H01 (Ω)). This is also true for the Cauchy problem if N ≥ 3. For convenience we again set I02 := u0 2H 1 + u1 2 and I12 := u0 2H 2 + u1 2H 1 . Theorem 9.2. Let 1 ≤ N ≤ 3. In addition to Hyp. E with p, q satisfying ν < N/2 we assume that Hyp. F is valid with α such that 4 < α < ∞, N − 2ν and in addition, if N = 3, α > 2(1 + ν)/(1 − ν). Then, for each K > 0 there exists an open neighborhood SK of (0, 0) in H 2 (Ω) ∩ H01 (Ω) × H01 (Ω) such that if (u0 , u1 ) ∈ SK , the problem (9.1.6), (9.1.7) admits a 2 unique solution u(t) in Xloc , satisfying utt (t) + ∇ut (t) + ∆u(t) ≤ K and u(t)2 ≤ C(K, I1 )(1 + t)ν and E(t) ≤ C(K, I1 )(1 + t)ν−1 with some constant C(K, I1 ) > 0. The conclusion is valid also for the Cauchy problem in Ω = RN , if N = 3. For the case q = 0 we can prove an existence theorem for the initial data (u0 , u1 ) ∈ H01 (Ω) × L2 (Ω). Theorem 9.3. Let 1 ≤ N ≤ 3. We assume Hyp. E with q = 0 and Hyp. F is valid with α such that 4/(N − 2ν) < α ≤ 2/(N − 2)+ , 0 ≤ ν < N/2. Then there exists δ > 0 such that if I0 < δ, the problem (9.1.6), (9.1.7) admits a unique solution u(t) ∈ C([0, ∞); H01 (Ω)) ∩ C 1 ([0, ∞); L2 (Ω)) satisfying u(t)2 ≤ C(II0 , δ)(1 + t)ν and E(t) ≤ C(II0 , δ)(1 + t)ν−1 with some C(II0 , δ) > 0. We note that ν = |p|/(p + 2) in this situation. When N = 3, the assertion is valid also for Ω = RN . 9.2. A basic inequality Let us consider the problem with a forcing term utt − ∆u + ρ(x, ut ) = f (x, t)

in Ω × (0, ∞),

u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

(9.2.1) (9.2.2)

where we assume f ∈ 1,2 For a moment we assume that f ∈ Wloc ([0, ∞); L2 (Ω)) and u(t) is a solution 2 in Xloc of the problem (9.2.1), (9.2.2). L2loc ([0, ∞); L2 (Ω)).

284

Mitsuhiro Nakao

For such solutions we have familiar identities corresponding to (A), (B), and (C). Our basic inequality reads as follows: Proposition 9.1. There exists T0 > 0 such that if T > T0 and k > 0 is suﬃciently large, then t+T t+T X(t + T ) − X(t) + k ρ(x, ut )ut dx ds + ε1 E(s) ds t t (9.2.3) Ω t+T |ut |2 dx + |ρ(x, ut )|2 dx + (|f ||u| + |f |2 )dx ds ≤C t

ΩR

ω

Ω

with some C > 0, and ε1 > 0, where X(t) is a certain quantity equivalent to E(t) + u(t)2 . The proof is given by the almost same argument deriving (4.2.23) (see also (5.4.6)) and we give an outline of it. First, we obtain for any k > 0, d α ut φ(r)(x − x0 ) · ∇u dx + α ut u dx + a(x)|u|2 dx + kE(t) dt 2 ΩCR Ω Ω N φ(r) + rφ (r) − α |ut |2 dx +k ρ(x, ut )ut dx + 2 Ω Ω N φ(r) + rφ (r) + φ(r) |∇u|2 dx + α− 2 Ω + φ (r)r−1 |(x − x0 ) · ∇u(t)|2 + φ(r)|∇u(t)|2 dx Ω 2 ∂u ν · φ(r)(x − x0 ) dS + ρ(x, ut )φ(r)(x − x0 ) · ∇u dx − ∂ν Ω ∂ Ω +α ρ(x, ut )u dx = f (kut + αu + φ(r)(x − x0 ) · ∇u) dx, ΩR

Ω

(9.2.4)

where we have used the fact that due to the ‘half-linearity’: α d ρ(x, ut )u dx = a(x)u2 dx. α C 2 dt ΩC Ω R R Now we take the function φ(r) deﬁned in Section 4.2. We choose k ≥ N + 1 and an appropriate α > 0. Then, I1 (x) :=

k C N φ(r) + rφ (r) − α + χ BR a(x) ≥ ε1 > 0 2 2

and I2 (x) := α −

N φ(r) + rφ (r) + φ(r) + φ (r)r > ε1 > 0 2

Decay and Global Existence

285

with some ε1 > 0. Noting ρ(x, ut )ut ≥ ε0 |ut |2 if x ∈ ΩC L and using Young’s inequality, we obtain α d 2 ut φ(r)(x − x0 ) · ∇u dx + α ut u dx + a(x)|u| dx + kE(t) dt 2 ΩCR Ω Ω ε1 k ρ(x, ut )ut dx + E(t) + 4 Ω 2 2 ∂u (L + |x0 |)ε0 dS ≤ 2 Γ(x0 ) ∂ν 2 2 2 2 |f ||u| + k |f | dx (|ρ(x, ut )| + |u| ) dx + +C ΩL

Ω

ΩL

ω

(9.2.5) for a large k. The boundary integral can be treated as usual (see Section 4) and we obtain d ut (φ(r)(x − x0 ) − C0 h(x)) · ∇u dx + (α + η 2 )ut u dx dt Ω Ω 1 k ε1 + αa(x)|u|2 dx + kE(t) + ρ(x, ut )ut dx + E(t) 2 ΩCR 4 Ω 2 2 2 2 2 2 ≤C |u| + ρ(x, ut ) dx + |ut | dx + |f ||u| + k |f | dx . Ω

(9.2.6) The ﬁrst term of the right-hand side is the most delicate one to control. But, we can show the following lemma which is proved by use of unique continuation as in Proposition 4.1. Lemma 9.1. There exists T0 > 0, independent of u, such that if T > T0 , then the inequality t+T |u|2 dx ds t ΩL t+T t+T 2 2 2 |ρ(x, ut )| + |f | dx + |ut | dx ds + ε E(s) ds ≤ Cε (T ) Ω

t

C ∪ω BL

t

holds for any ﬁxed number ε > 0. Now we set X(t) =

ut (φ(r)(x − x0 ) − C0 h(x)) · ∇u dx + Ω 1 αa(x)|u|2 dx + kE(t). + 2 ΩCR

(α + η 2 )ut u dx

(9.2.7)

Ω

Then we arrive at the basic inequality (9.2.3), where we have replaced the notations k/8 and ε1 /4 by k and ε1 , respectively. It remains to show that X(t) is equivalent to E(t) + ||u(t)||2 . But, we have:

286

Mitsuhiro Nakao

Lemma 9.2. For a large k > 0 there exist constants C1 > 0 and C2 > 0 such that for any t ≥ 0, C1 (E(t) + ||u(t)||2 ) ≤ X(t) ≤ C2 (E(t) + ||u(t)||2 ).

(9.2.8)

Proof. Since the second inequality of (9.2.8) holds trivially, it is suﬃcient to show the ﬁrst inequality. We easily see, ut (φ(r)(x − x0 ) − C0 h(x)) · ∇u dx −CE(t) ≤ Ω

with some constant C > 0. For any ε > 0 we have (α + η 2 )ut u dx ≥ −ε |u|2 dx − C(ε) |ut |2 dx, Ω

Ω

Ω

and by the assumption a(x) ≥ ε0 > 0 in 1 αε0 2 αa(x)|u| dx ≥ |u|2 dx. 2 BRC 2 ΩCL ΩC L,

Finally, we note that Poincar´ ´e’s inequality 2 2 2 |u| dx ≤ C |∇u| dx ≤ C |∇u| dx ΩL

ΩL

Ω

holds due to the boundary condition u|∂Ω = 0. When Ω = RN this should be replaced by the inequality 2 2 2 |u| dx ≤ C |∇u| dx + |u| dx . BL

Ω2L

B2L \BL

Combining these we obtain the ﬁrst inequality.

9.3. Proof of Theorem 9.1 For convenience we set Ω1R (t) = {x ∈ ΩR | |ut (x, t)| ≤ 1} and Ω2R (t) = ΩR \ Ω1R . 2 of the problem (9.1.1)–(9.1.2). Then, by the Let u(t) be a solution from Xloc Proposition 9.1 with f ≡ 0 it holds t+T t+T X(t + T ) + k ρ(x, ut )ut dx ds + ε1 E(s) ds t Ω t (9.3.1) 0 t+T 0 0 2 2 ≤ X(t) + C t ω |ut | dx + ΩR |ρ(x, ut )| dx ds

for a large k > 0 and some ε1 > 0. To consider the integrals of the right-hand side of (9.3.1) we ﬁrst note that by the assumption on ρ(x, v) and the energy identity, ∞ p+2 q+2 E(t) + a(x)|ut | dx + a(x)|ut | dx dt ≤ CE(0) < ∞, 0 ≤ t. 0

Ω1R

Ω2R

(9.3.2)

Decay and Global Existence We set

t

t |ut | dx ds =: I1 (t) and

|ρ(x, ut )|2 dx ds =: I2 (t).

2

0

287

0

ω

ΩR

We shall estimate I1 (t) and I2 (t) separately in the following four cases: Case (1): 0 ≤ p < ∞ and 0 ≤ q ≤ 2/(N − 2). In this case, by Holder’s ¨ inequality and (9.3.2), we have 2 p+2 t

I1 (t) ≤ C 0 t

p p+2

t

Ω1R

a(x)|ut |p+2 dx ds

dx ds 0

+C a(x)|ut |q+2 dx ds 2 0 ΩR ≤ C E(0)2/(p+2) (1 + t)p/(p+2) + E(0) .

Ω1R

To treat I2 (t) we assume for a moment N ≥ 3. By use of Sobolev’s inequality, t 2 I2 (t) ≤ k1 a(x)|ut |2 dx ds 0 ΩR t t 2 2 2(p+1) 2 +k1 a(x) |ut | dx ds + k1 a(x)2 |ut |2(q+1) dx ds 0

Ω1R

0

Ω2

Rt ≤ C E(0)2/(p+2) (1 + t)p/(p+2) + E(0) + C

0 2(q+1)(1−θ)/(q+2)

Ω1R

a(x)|ut |p+2 dx ds

t 2(q+1)θ +C a(x)|ut |q+2 dx ||ut ||2N/(N −2)(ΩL ) ds 2 ΩR 0 2/(p+2) (1 + t)p/(p+2) + E(0) ≤ C E(0) 2(q+1)(1−θ)/(q+2) t a(x)|ut |q+2 dx ds +C(K02 + E(0))(q+1)θ 0

Ω2R

t 2θ(q+1)−q q+2 · ds 0 ≤ C E(0)2/(p+2) (1 + t)p/(p+2) + E(0) +C(K02 + E(0))(q+1)θ E(0)

2(q+1)(1−θ) q+2

q(N −2)

(1 + t) 4−(N −2)q

with θ = N q/(q + 1)(4 − (N − 2)q), where we have used 0 ≤ q ≤ 2/(N − 2)+ . Thus we obtain I1 (t) + I2 (t) ≤ CQ0 (t + 1)ν

(9.3.3)

with Q0 , ν deﬁned in Theorem 9.1. We can show similar estimates for N = 1, 2 with some trivial modiﬁcation.

288

Mitsuhiro Nakao

Case (2): −1 < p ≤ 0 and 0 ≤ q ≤ 2/(N − 2). In this case we get t t I1 (t) ≤ a(x)|ut |p+2 dx ds + Ω1R

0

and

0

t

a(x)|ut |q+2 dx ds ≤ CE(0),

Ω2R

t

I2 (t) ≤ C

a(x)|ut | dx ds + C 2

0

0

ΩR

+C(K02 + E(0))(q+1)θ E(0)

Ω1R

2(q+1)(1−θ) q+2

a(x)|ut |2(p+1) dx ds q(N −2)

(1 + t) 4−(N −2)q ,

N ≥ 3.

Here, we see that t Ω1R

0

a(x)|ut |2(p+1) dx ds

t

≤C

2(p+1)/(p+2) a(x)|ut |

p+2

0

Ω1R

dx ds

(1 + t)−p/(p+2) .

Hence we have again I1 (t) + I2 (t) ≤ CQ0 (1 + t)ν . Some modiﬁcations give the results for the cases N = 1, 2.

(9.3.4)

Case (3): 0 ≤ p < ∞ and −1 < q ≤ 0. Let N ≥ 3. Then we have, by use of H¨ ¨older’s inequality, t a(x)|ut |2 dx ds 0

≤

Ω2R

C(K02

+ E(0))

−N q 4−(N −2)q

a(x)|ut |

q+2

0

and

Ω2R

t 0

Ω2R

4−(N4−2)q

t dx ds

−(N −2)+

(1 + t) 4−(N −2)q ,

t a(x)2 |ut |2(q+1) dx ds ≤ C 0

Ω2R

a(x)|ut |q+2 dx ds ≤ CE(0).

So we obtain the estimate as (9.3.4) with Q0 , ν as in Theorem 9.1. Case (4): −1 < p ≤ 0 and −1 < q ≤ 0. Utilizing the arguments in the cases (2), (3) above, we can conclude the same type of an inequality as (9.3.4). Now we are ready to complete the proof of Theorem 9.1. First we note that if 0 ≤ t ≤ T , then by (9.3.1) with f = 0 and the same estimations for I1 (t), I2 (t) X(t) ≤ X(0) + CQ0 ,

0 ≤ t ≤ T,

and also X(t) ≤ X(nT ) + CQ0 ,

nT < t < (n + 1)T.

Decay and Global Existence

289

Taking these into account we can show by (9.3.1) and the estimates for I1 (t)+II2 (t) that (9.3.4) X(t) ≤ X(0) + CQ0 (1 + t)ν . In particular we see that ||u(t)||2 ≤ C(Q0 + u0 2 )(1 + t)ν . The diﬀerential inequality d/dt((1+t)E(t)) = E(t)+(1+t)d/dtE(t) ≤ E(t) implies t (1 + t)E(t) ≤ E(0) + E(s)ds. 0

But, from (9.3.1) and (9.3.4) we see that t E(s)ds ≤ CX(0) + CQ0 (1 + t)ν 0

for any t > 0. Hence, we obtain E(t) ≤ CX(0)(1 + t)−1 + CQ0 (1 + t)ν−1 ≤ C(Q0 + u0 2 )(1 + t)ν−1 .

The proof of Theorem 9.1 is complete. 9.4. Proof of Theorems 9.2 and 9.3

Let (u0 , u1 ) ∈ H2 (Ω) ∩ H01 (Ω) × H01 (Ω) and let u(t) be the local (in time) solution, say, in X2 (T˜), T˜ > 0. For the proof of Theorem 9.2 we assume for a moment u(t)2 ≤ δ02 (1 + t)ν ,

E(t) ≤ δ02 (1 + t)ν−1

(9.4.1)

and ∇ut (t) + ∆u(t) ≤ K

(9.4.2)

for 0 ≤ t < T˜ . Under the assumptions (9.4.1), (9.4.2) we shall derive the a priori estimates ˜ 0 + u0 2 )(1 + t)ν , ˜ 0 + u0 2 (1 + t)ν−1 (9.4.3) E(t) ≤ C Q u(t)2 ≤ C(Q and ˜1 ∇ut (t) + ∆u(t) ≤ Q

(9.4.4)

˜ 0 will be deﬁned by Q0 with K0 replaced by K and for 0 ≤ t < T˜ , where Q ˜ ˜ I0 , I1 , K) = 0. By a ˜ Q1 = Q1 (II0 , I1 , K) will be a quantity such that limI0 ,I1 →0 Q(I ˜ 0 +u0 2 ) < δ 2 standard argument, the estimates (9.4.3), (9.4.4) assure that if C(Q 0 ˜ 1 < K, then the local solution u(t) exists in fact on the inﬁnite interval [0, ∞) and Q and satisﬁes the estimates on [0, ∞). These conditions will be satisﬁed if I0 and I1 are small. Then, we deﬁne SK by the set of initial data satisfying all of these conditions. Without loss of generality we may assume T˜ > T0 (otherwise, the

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estimates can be proved much more easily). We can use the basic inequality in Proposition 9.1 with f = f (x, u) to get t+T t+T ρ(x, ut )ut dx ds + ε1 E(s) ds X(t + T ) − X(t) + k t t Ω t+T |ut |2 dx + |ρ(x, ut )|2 dx ds ≤C (9.4.5) t ω ΩR t+T

(|f (x, u)||u| + |f (x, u)|2 ) dx ds .

+ t

Ω

Here, when 0 < α ≤ 4/(N − 2)+ we have by the Gagliardo-Nirenberg inequality, t+T t+T |f (x, u)||u| dx ds ≤ C |u|α+2 dx ds (9.4.6) t t Ω Ω t+T ≤C u(α+2)(1−θ0 ) ∇u(α+2)θ0 ds t t+T α ≤ Cδ0 (1 + t)2˜µ E(s)ds t

˜ = (α + 2)(1 − θ0 )ν + ((α + 2)θ0 − 2)(ν − 1). with θ0 = αN/2(α + 2) and µ We note that (α + 2)θ0 = αN/2 ≥ 2 by the assumption on α. Further, we conﬁrm that µ ˜ ≤ 0 is equivalent to θ0 ≥ (να + 2)/(α + 2), that is, α ≥ 4/(N − 2ν) which is also fulﬁlled under our assumption on α. Thus, we have t+T t+T α |f (x, u)||u| dx ds ≤ Cδ0 E(s) ds. t

Ω

t

Similarly, if 0 < α ≤ 2/(N − 2)+ , then t+T |f (x, u)|2 dx ds ≤ Cδ02α t

Ω

t+T

E(s) ds.

t

Thus, when 0 < α ≤ 2/(N − 2)+ we make the assumption ε1 C(δ0α + δ02α ) ≤ . 4

(9.4.7)

When N ≥ 3 and 4/(N − 2)+ < α ≤ 2/(N − 4)+ we have, instead of (9.4.6), t+T t+T |f (x, u)||u| dx ds ≤ C |u|α+2 dx ds t t Ω Ω t+T (α+2)(1−θ ) u2N/(N −2) 1 D2 u(α+2)θ1 ds ≤C (9.4.8) t t+T ≤C ∇u(α+2))(1−θ1 ) (∇u + ∆u)(α+2)θ1 ds t t+T (α+2)(1−θ1 )−2 ≤ Cδ0 (K + δ0 )(α+2)θ1 E(s) ds t

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with θ1 = (α(N −2)−4)/2(α+2), where we have used the fact (α+2)(1−θ1 )−2 ≥ 0. Similarly, if 2/(N − 2)+ < α ≤ 2/(N − 4)+ , then t+T t+T 2(α+1)(1−θ˜1 )−2 2 2(α+1)θ˜1 |f (x, u)| dx ds ≤ Cδ0 (K + δ0 ) E(s) ds Ω

t

t

with θ˜1 = (α(N − 2) − 2)/2(α + 1). When 2/(N − 2)+ < α ≤ 2/(N − 4)+ we assume ε ˜ 1 2(α+1)(1−θ˜1 )−2 (9.4.9) (K + δ0 )2(α+1)θ1 < . C δ0α + δ0 4 Under the assumption (9.4.7) or (9.4.9) we obtain from (9.4.5) t+T t+T 1 X(t + T ) − X(t) + k ρ(x, ut )ut dx ds + ε1 E(s) ds 2 t t Ω t+T 2 2 ≤C |ut | dx + |ρ(x, ut )| dx ds. t

ΩL

ω

This inequality is essentially the same as (9.3.1) and the proof of Theorem 9.1 yields the same estimates (9.1.9) for u(t) and E(t) with K0 replaced by K. Further, the desired inequality ˜ 0 + u0 2 < δ0 C Q will be satisﬁed if I0 is small. In what follows we assume 1 ≤ N ≤ 3. We shall derive the estimate for the second derivatives of u. Diﬀerentiating the equation we have (9.4.11) uttt − ∆ut + ρv (x, ut )utt − βu (x, u)ut = 0. From this we have d (utt (t)2 + ∇ut (t)2 ) ≤ C |u|α |ut ||utt | dx dt Ω 1/2 ≤C |u|2α |ut |2 dx utt .

(9.4.12)

Ω

Here,

1/2 α/2 |u| |ut | dx ≤ Cuα ∇uα/2 ut if N = 1, ∞ ut ≤ Cu 2α

Ω

2

1/2 |u|2α |ut |2 dx ≤ Cuα 2aα ut 2b ,

Ω

≤ Cu

α(1−θ2 )

∇u

a−1 + b−1 = 1, αθ2

ut

1−θ˜2

a

∇ut

with θ2 = 1 − 1/aα, θ˜2 = 1 − 1/b = 1/a " 1, and 1/2 |u|2α |ut |2 dx ≤ Cuα ∞ ut Ω

≤ C∇uα/2 (∆u + ∇u)α/2 ut if N = 3.

θ˜2

1, if N = 2

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Thus,

⎧ 2 (α+1)/2 ˜ ⎪ (1 + t)−µ ⎨K(Q0 + u0 ) α 1+ε 2 (α+1−ε)/2 ˜ 0 + u0 ) C |u| |ut ||utt | dx ≤ K (Q (1 + t)−µ ⎪ α+2 Ω ⎩ ˜ 0 + u0 2 ) 4 (1 + t)−µ (K + I0 )1+α/2 (Q

where µ is deﬁned by ⎧ 1 ⎪ ⎨ 2 (−(α + 1)ν + α/2 + 1) µ = 12 (−(α + 1)ν + α + 1 − ε(2 − ν)) ⎪1 ⎩ 2 (−(α/2 + 1)ν + α/2 + 1)

if N = 1, if N = 2, ε " 1, if N = 3,

if N = 1, if N = 2, if N = 3.

We note that the condition µ > 1 is equivalent to α > 2(1 + ν)/(1 − 2ν) if N = 1, α > (1 + ν)/(1 − ν) if N = 2, α > 2(1 + ν)/(1 − ν) if N = 3, and these conditions are satisﬁed under our assumptions on α. Now, we obtain from (9.4.12) that if N = 3, then utt (t)2 + ∇ut (t)2

(α+2)/4 ˜ 0 + u0 2 ≤ utt (0)2 + ∇ut (0)2 + C(µ − 1)−1 (K + I0 )1+α/2 Q . (9.4.13)

Similar results hold also for N = 1, 2. Thus, utt (t)2 + ∇ut (t)2

γ ˜ 0 + u0 2 ≤ utt (0)2 + ∇ut (0)2 + C(K, I0 )(µ − 1)−1 Q

(9.4.14)

˜ 0 (E(0), K) is the constant Q0 with K0 ˜0 = Q with some γ > 0. Recall that Q ˜ 0 (0, K) = 0. We easily replaced by K deﬁned in the Theorem 9.1 and satisﬁes Q see utt (0) ≤ ∆u0 + ρ(x, u1 ) + f (x, u0 ) −

−

≤ C(I1 + I0 + u1 (q+1)(1−θ2 ) ∇u1 (q+1)θ2 + u1 1+p + u1 1+q ), (9.4.15) with the notation r− := min{r, 0} and a certain 0 ≤ θ2 ≤ 1, where we have used Gagliardo-Nirenberg inequality and the assumption (9.4.15). From (9.4.14) and (9.4.15) we have utt (t) + ∇ut (t) (q+1)(1−θ2 ) (q+1)θ2 (1+p− ) (1+q− ) I1 + I0 + I0 ≤ C I1 + I0 + I0 γ ˜ 0 + u0 2 =: Q1 (II0 , I1 , K). +C(K, I0 )(µ − 1)−1/2 Q

(9.4.16)

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293

Similarly as in (9.4.15), − − ρ(x, ut ) ≤ C I0 + ut (q+1)(1−θ2 ) ∇ut (q+1)θ2 + ut 1+p + ut 1+q − − (q+1)(1−θ2 ) (q+1)θ2 ˆ 1 (II0 , I1 , K). =: Q Q1 + I01+p + I01+q ≤ C I0 + I0 (9.4.17) We also note that β(x, u) ≤ C(u0 2 +u1 2 ) under our assumption (see (9.4.22) below). Thus, ∆u(t) ≤ utt (t) + ρ(x, t) + f (x, u) ˆ 1 (II0 , I1 , K)) =: Q ˜ 1 (II0 , I1 , K). ≤ C(Q1 (II0 , I1 , K) + Q

(9.4.18)

We conclude that ˜ 1 (II0 , I1 , K), ∇ut (t) + ∆u(t) ≤ Q (9.4.19) ˜ I0 , I1 , K). Since ˜ I0 , I1 , K) for C Q(I where for simplicity we use the same notation Q(I ˜ limI0 →0,I1 →0 Q1 (II0 , I1 , K) = 0 we see that if I0 , I1 are small, then ˜ 1 (II0 , I1 , K) < K Q which is the desired second condition on the initial data. The proof of Theorem 9.2 is now complete. Finally, we give the proof of Theorem 9.3. This is in fact essentially included in the ﬁrst part of the proof of Theorem 9.2. Let q = 0 and let u(t) be a local in time solution in C 1 ([0, T˜ ); L2 (Ω)) ∩ C([0, T˜); H01 (Ω)), T˜ > 0. Assume for a moment u(t)2 ≤ δ02 (1 + t)ν ,

E(t) ≤ δ02 (1 + t)ν−1 , 0 < t < T˜.

(9.4.20)

Then as it is already shown, under the assumption (9.4.7), the estimates (9.4.3) ˜ 0 and ν are given by hold, where Q0 = Q ˜ 0 = C E(0) + E(0)min{2/(p+2),1} and ν = |p|/(p + 2). (9.4.21) Q Thus, in addition to (9.4.7), we make the assumption ˜ 0 + u0 ) < δ02 . C(Q

(9.4.22)

Then we can conclude that the local solution u(t) is in fact global in time and the estimates (9.4.3) hold on [0, ∞).

10. Some open problems We give here some open problems related to the topics discussed in this article. 1. Consider utt − u + ρ(x, ut ) = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0,

294

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where Ω is an exterior domain. When ρ(x, ut ) = a(x)|ut |r ut or ρ(x, ut ) = a(x)(ut + |ut |r ut ), derive some decay rate of local energy, where a(x) ≥ 0 is a localized function on ω. 2. Consider utt − u + a(x)ut = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, where a(x) is localized near ω and near inﬁnity. Under appropriate assumptions on (u0 , u1 ), say, (u0 , u1 ) ∈ H01 (Ω) ∩ Lp (Ω) × L2 (Ω) ∩ Lp (Ω), 1 ≤ p < 2, derive some sharper decay rate of total energy like E(t) ≤ C(1 + t)−1−γ , γ > 0. For the case a(x) ≡ 1 see Ono [53]. 3. Consider utt − u + a(x)ut = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, where V ≡ R \ Ω consists of some convex bodies as in Ikawa [12], [13] and a(x) is localized near inﬁnity. By utilizing the result in [13], derive a decay rate of the total energy under the assumption on a(x) including the case, where a(x) = 0 near the boundary ∂Ω and a(x) ≥ ε0 > 0 near inﬁnity. N

4. Consider utt − u + ρ(x, ut ) = 0 in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, where ρ(x, ut ) is a half-linear localized dissipation treated in the ﬁnal section. Derive a sharper decay estimate for E(t) under additional assumptions on the data (u0 , u1 ). Further, derive a decay estimate for the second-order derivatives D2 u(t), D = (∇, Dt ), and apply it to semilinear wave equations with source term f (u). 5. Consider the periodicity problem utt − u + ρ(x, ut ) = f (x, t) in Ω × [0, ∞), u(x, t + T ) = u(x, t) and u|∂Ω = 0, where f (t, x) is a force term with period T in time. Discuss the existence and stability of periodic solutions. For related results in the case of bounded domains see [46]. 6. Discuss the global existence of small data solutions to the problem utt − u + a(x)ut = |ut |p−1 ut in Ω × [0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and u|∂Ω = 0, ˜ under the hypothesis Hyp. A or under the hypothesis Hyp. A.

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7. Consider the equation with Neumann boundary condition: utt − u + a(x)ut = 0 in Ω × [0, ∞), ∂ u|∂Ω = 0, ∂n where a(x) is localized near ω. Discuss the local energy decay for this problem. For the problem with Neumann boundary condition the standard multiplier technique does not work well and we must employ another method, e.g. microlocal analysis. For such a technique see Bardos, Lebeau and Rauch [3]. Generally speaking, it is a very challenging problem how to combine multiplier technique (energy method) with microlocal analysis for the study of the wave equation with some Neumann type boundary conditions. A problem concerning a Neumann type boundary dissipation in a bounded domain is treated in Lasiecka and Triggiani [24], where energy method and microlocal analysis are well combined. See also [2] for an exterior problem. See Shibata and Zheng [64] for general fully nonlinear hyperbolic equations with a Neumann type dissipation in a bounded domain, where spectral analysis for the linearized stationary problem plays an essential role. u(x, 0) = u0 (x), ut (x, 0) = u1 (x) and

Acknowledgement. The author would like to thank Professor M. Reissig for his kind invitation to this contribution and for many helpful suggestions during the preparation of the manuscript.

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[66] D. Tataru, The Xθs spaces and unique continuation for solutions to the semilinear wave equation, Comm. Partial Diﬀerential Equations 21 (1996), 841–887. [67] L. R. Tcheugou´ ´e T´ ´ebou, Stabilization of the wave equation with localized nonlinear damping, J. Diﬀerential Equations 145 (1998), 502–524. [68] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Diﬀerential Equations 174 (2000), 464–489. [69] T. Yamazaki, Global solvability for quasilinear hyperbolic equation of Kirchhoﬀ type in exterior domains of dimension larger than three, Math. Methods Appl. Sci., to appear. [70] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Diﬀerential Equations 15 (1990), 205–235. [71] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl. 70 (1991), 513–529. Mitsuhiro Nakao Graduate School of Mathematics Kyushu University Ropponmatsu Fukuoka 810-8560, Japan e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 301–385 c 2005 Birkhauser ¨ Verlag Basel/Switzerland

Global Existence in the Cauchy Problem for Nonlinear Wave Equations with Variable Speed of Propagation Karen Yagdjian Abstract. We consider nonlinear wave equations with variable coeﬃcients. Special attention is devoted to the parametric resonance phenomena. Mathematics Subject Classiﬁcation (2000). Primary 35L70; Secondary 58J45. Keywords. Nonlinear wave equation, global existence, variable coeﬃcients, parametric resonance.

1. Introduction This paper is aimed to give a collection of some results for the existence and nonexistence of global solutions to the Cauchy problem for hyperbolic equations with variable coeﬃcients. Developed in the last quarter of the last century the theory of Fourier Integral Operators allows to construct locally in time a parametrix and fundamental solution to the Cauchy problem for the linear strictly hyperbolic operators with coeﬃcients depending both on time and on the spatial variables. Moreover, some speciﬁc classes of those operators are used for the operators with characteristics of variable multiplicity. In [37], [38], [39], [40] one can ﬁnd the theory developed for two speciﬁc classes of the operators with the basic types of the behavior of the characteristics. The main ingredients of that theory are: the splitting of the cotangent space into so-called “pseudo-diﬀerential” ﬀ and “hyperbolic zones”, the investigation of the complete symbol of the operator in both zones, Levi conditions (conditions on the lower order terms of the operator), developing calculus of special classes of symbols, construction of exponents of pseudo-diﬀerential operators, and the investigation of the corresponding Hamiltonian vector ﬁelds.

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It turns out that the basic concepts and technique of the above-mentioned theory developed for equations with multiple characteristics work out in the investigation of the existence of a global solution to the Cauchy problem for some classes of second-order hyperbolic operators with variable coeﬃcients. In particular, if the coeﬃcients of the equation depend on time alone, then the phase function exists globally that allows to get global in time results, like Lp − Lq decay estimates, for second-order linear equations. The papers [28], [29], give several results based on this approach. Moreover these basic concepts and techniques provide us with a key to study global existence results for nonlinear equations. In this paper we describe only few of them, which can be exposited in transparent way highlighting basic links between operators with multiple characteristics and the problem of the existence of global solutions for the strictly hyperbolic equations. We pay special attention to equations with oscillating coeﬃcients. In the theory of linear weakly hyperbolic equations too fast oscillations break down the local well-posedness of the Cauchy problem. Corresponding phenomena in the unbounded in time problem for semi-linear wave equations lead to the blow-up of the solution. In [43] it is detected that in bottom of these two phenomena lies the well-known “parametric resonance” of Physics. To bear in mind the extraordinary importance of the parametric resonance for many branches of the applied sciences we give here a detailed proof of Borg’s theorem, and track down the mathematical consequences of the parametric resonance for nonlinear wave equations. For the sake of a self-contained representation we ﬁrst describe in Section 2 and Section 3 well-known results for the wave equation and then generalize them to operators with variable coeﬃcients. To ﬁnd out the inﬂuence of variable coeﬃcients on the global existence in the Cauchy problem for the wave equation we choose the model equation ∇x u|2 ) = 0 utt − a2 (t, x)∆u + f (u)(u2t − a2 (t, x)|∇ that in the case of constant function a(t, x) = const > 0 is the wave map type equation studied in [26]. The wave map equations appear in physics and are important also for other applications. A good introduction on wave map equations with comprehensive references can be ffound in [33]. In Section 4 we consider an equation with periodic, non-constant, smooth, and positive coeﬃcient a = a(t). The proof of the nonexistence of the global in time solution for that equation (see Theorem 4.1) is given in Section 5. The condition on the function f = f (u) that is assumed through the paper, is discussed in Theorem 4.2. After studying the periodic case we turn in Section 6 to the equation with a coeﬃcient stabilizing to a periodic one. Namely we consider the model equation with a(t) = exp(tα )b(t), t ∈ [1, ∞), for α < 0. Here b = b(t) is deﬁned on R, a periodic, non-constant, smooth, and positive function. We study the global solvability of the Cauchy problem with data prescribed on t = 1. In Section 7 we prove the main result of Section 6, Theorem 6.1, that shows that oscillations, which approach for large time the pure periodic behavior (α < −1), in general

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break the global existence. The global existence for small data for the equation with fast oscillating coeﬃcients, that is for the case α ∈ [1/2, ∞), is proved in Section 11. In Section 8 we give an example of the inﬂuence of the decay of the coeﬃcient a(t, x) on the global existence of the solution of the model equation with a(t) = t−2 exp(−t−1 ). Namely, for arbitrary small initial data we construct blowing up solution. That proves Theorem 8.1, the main result of that section. It is easy to see that this equation has a permanently restricted domain of inﬂuence. In Section 9 we consider the model equation with increasing coeﬃcient a(t, x) = tl , l > 0. According to Theorem 9.1 for the Cauchy problem for this equation small data solutions exist globally. In Section 10 we show that the approach of the previous sections works out also for the model equation with the exponentially growing coeﬃcient a(t, x) = exp(t). Namely, due to Theorem 10.1 small data solutions exist globally. We try to make the proofs elementary, so that they can be read without special background in the theory of hyperbolic equations. We intentionally make sections independent from each other and hope that it makes them more convenient to use for lectures. This also explains some reiterations in the formulas. In fact this paper basically is a fragment of the lecture notes given at the University of Tsukuba from April 1998 till March 2000. I am very grateful to the editors of this volume published by Birkhauser, ¨ Prof. Michael Reissig and Prof. Bert-Wolfgang Schulze, not only for the invitation to write a contribution, but also for the allowing to keep this way of writing.

2. Counterexamples to the global existence Example 1 (Nirenberg’s Example [21]). The following one is an example of an equation without global classical solution to the Cauchy problem with large initial data in R3 : utt − ∆u = |∇ ∇x u|2 − |ut |2 , u(0, x) = 0,

ut (0, x) = ϕ1 (x) ,

x ∈ R3 ,

where further conditions on ϕ1 are given below. If u = u(t, x) is a solution to this Cauchy problem, then the function v = v(t, x) deﬁned by v(t, x) = eu(t,x) solves the following Cauchy problem for the wave equation: vtt − ∆v = 0 , v(0, x) = 1, vt (0, x) = ϕ1 (x) ,

x ∈ R3 .

The solution v is unique and there is an explicit representation for v: t v(t, x) = 1 + ϕ1 (x + ty)ds . 4π |y|=1

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Karen Yagdjian

If v > 0, then one obtains

t u(t, x) = log 1 + 4π

|y |=1

ϕ1 (x + ty)ds

.

For every given positive t0 and point x0 ∈ R3 one can ﬁnd a function ϕ1 (x) ∈ C0∞ (R3 ) such that the solution u(t, x) develops singularity not later than at time t0 at the point x0 . Indeed, ϕ1 has only to satisfy the following relation: t0 ϕ1 (x0 + t0 y)ds = −1 . 4π |y|=1 On the other hand it is easily seen that if ϕ1 (z) = O(|z|−1 ) as |z| → ∞, ∇ϕ1 1 < 4π, ϕ1 ∞ < 1, then u is deﬁned globally, that is for all x ∈ R3 and all t ∈ R. To prove the last statement we note that for “large time”, t > 1, there is some decay of the varying part v(t, x) − 1 of the solution v = v(t, x) to the wave equation. Namely, the L∞ -norm of that part satisﬁes the following estimates: ∞ t d |v(t, x) − 1| = ϕ1 (x + τ y)dτ ds 4π |y|=1 t dτ ∞ t = y · (∇ϕ1 )(x + τ y)dτ ds 4π |y|=1 t z t ∞ ds = · (∇ϕ1 )(x + z) 2 dτ 4π t τ |z |=τ τ ∞ t 1 ≤ dτ |(∇ϕ1 )(x + z)|ds 2 4π t τ |z |=τ 1 = |∇ϕ1 (x + z)|dz 4πt |z|≥t 1 1 ∇ϕ1 1 ≤ < 1 . ≤ 4πt t Analogously for t < −1. For a small time t, |t| ≤ 1, that diﬀerence vanishes at t = 0, and we have t ϕ1 (x + ty)ds ≤ |t|ϕ1 ∞ < 1 . |v(t, x) − 1| = 4π |y|=1 Therefore, v(t, x) is positive for all (t, x) ∈ R × R3 and u = u(t, x) is deﬁned globally in R × R3 . From now on through this paper “blow-up” means the nonexistence of a global in time classical solution u ∈ C 2 ([0, ∞) × Rn ) to the Cauchy problem under consideration.

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Example 2. Assume that u = u(x, t) is a smooth solution of utt − ∆u = |u|p on [0, T ] × Rn with 1 < p < (n + 1)/(n − 1), and ut (x, 0) = g(x).

u(x, 0) = f (x) ,

Suppose that supp f , supp g ⊂ { x ∈ Rn ; |x| ≤ R }. By the domain of dependence property, supp u(t, ·) ⊂ { x ∈ Rn ; |x| ≤ R + t }. By integrating the equation with respect to spatial variables we obtain (utt (t, x) − ∆u(t, x)) dx = |u(t, x)|p dx . Rn

Rn

On the other hand the divergence theorem gives ∆u(t, x) dx = 0 , Rn

while d2 dt2 Hence, d2 dt2

u(t, x) dx = Rn

Rn

∂ 2 u(t, x) dx . ∂t2

u(t, x) dx = Rn

Rn

Let

|u(t, x)|p dx .

u(t, x) dx ,

F (t) = Rn

then

F¨ (t) = Rn

|u(t, x)|p dx .

Using the compact support of u(t, ·) and Hlder’s inequality we get with 1/p+1/q = 1, p/q = p − 1, τn the volume of the unit ball in Rn , p p u(t, x) dx n u(t, x) dx = R |x|≤R+t p/q ≤

|x|≤R+t

= Rn

=

|u(t, x)|p dx

1 dx |x|≤R+t

|u(t, x)|p dx τn (R + t)np/q

F¨ (t)ττn (R + t)n(p−1) .

Thus we have obtained the following diﬀerential inequality: F¨ (t) ≥ τn−1 (R + t)−n(p−1) |F (t)|p

(2.1)

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Karen Yagdjian

for all t in the interval [0, T ]. In particular, F¨ (t) ≥ 0, and so F (t) ≥ F˙ (0)t + F (0). Now ˙ F (0) = ut (0, x) dx = g(x) dx ≡ Cg . Rn

Rn

If Cg > 0, then F (t) ≥ (pos. const.)t ,

t

large .

(2.2)

The next lemma shows that any function satisfying (2.1) and (2.2) cannot remain ﬁnite if 1 < p < (n + 1)/(n − 1). Hence, T < ∞. This is a special case of Kato’s theorem [20]. Lemma 2.1. Suppose F (t) ∈ C 2 ([a, b)), and for a ≤ t < b, F (t) ≥ C0 (k + t)r , F¨ (t) ≥ C1 (k + t)−q F (t)p , where C0 , C1 , and k are the positive numbers. If p > 1, r ≥ 1, and (p− 1)r > q − 2, then b must be ﬁnite. Proof. By the hypotheses of the lemma we get F¨ (t) ≥ C1 (k + t)−q F (t)p ≥ C1 (k + t)−q C0p (k + t)pr ≥ C(k + t)pr−q on [a, b). Upon integration, one has F˙ (t) − F˙ (a) ≥ C

t

(k + s)pr−q ds . a

But pr − q ≥ −1 and so the last inequality implies that unless b is ﬁnite, F˙ (t) must be positive for t suﬃciently large. Thus, one may assume that there exists an a0 such that a < a0 0 for all t ∈ [a0 , b) . By interpolating with θ ∈ (0, 1) between the assumed inequalities, one has F¨ (t)

≥ C1 (k + t)−q F (t)θp+p(1−θ) ≥ C(k + t)−q F (t)θp (k + t)rp(1−θ) = C(k + t)rp(1−θ)−q F (t)θp .

Let α = θp and β = q − rp(1 − θ). Since (p − 1)r > q − 2 and r ≥ 1, pr − q > r − 2 ≥ −1 , Then one can choose q−2 1 <θ <1− p pr

= =⇒

q−2 1 <1− . p pr

α > 1 and β < 2 .

Without loss of generality one can set β ≥ 0. Thus, F¨ (t) ≥ C2 (k + t)−β F (t)α

and F˙ (t) > 0 for all t ∈ [a0 , b) .

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307

Note that we can choose a positive constant C2 so small that −C C2

(k + a0 )−β 1 F (a0 )1+α + F˙ (a0 )2 ≥ 0 1+α 2

implies −C C2

(k + t)−β 1 F (a0 )1+α + F˙ (a0 )2 ≥ 0 1+α 2

for all t ∈ [a0 , b) .

This leads to F˙ (t)F¨ (t) ≥ C2 (k + t)−β F (t)α F˙ (t) . Integration of the last inequality yields t 1˙ 2 2 ˙ ≥ C2 F (t) − F (a0 ) (k + s)−β F (s)α F˙ (s) ds 2 a0 t ≥ C2 (k + t)−β F (s)α F˙ (s) ds a0

≥ C2

−β

(k + t) (k + t)−β F (t)1+α − C2 F (a0 )1+α . 1+α 1+α

Hence, 1 ˙ 2 F (t) 2

≥ ≥ ≥

1 (k + t)−β F (t)1+α − F (a0 )1+α + F˙ (a0 )2 1+α 2 −β (k + t)−β (k + t) 1 F (t)1+α − C2 F (a0 )1+α + F˙ (a0 )2 C2 1+α 1+α 2 −β (k + t) F (t)1+α for all t ∈ [a0 , b) . C2 1+α C2

It follows F˙ (t) ≥ C(k + t)−β/2 F (t)(1+α)/2

for all t ∈ [a0 , b) ,

and therefore F (t)−(1+α)/2 F˙ (t) ≥ C(k + t)−β/2

for all t ∈ [a0 , b) .

One ﬁnal integration yields (α > 1) F (a0 )(1−α)/2 − F (t)(1−α)/2 C(α − 1) t ≥ (k + s)−β/2 ds 2 a0 1 C(α − 1) (k + t)1−β/2 − (k + a0 )1−β/2 = 2 1 − β/2 for all t ∈ [a0 , b). Since 1 − β/2 > 0, it is clear that t cannot be arbitrary large. Indeed, the left-hand side is bounded by F (a0 )(1−α)/2 while the right-hand side tends to ∞ as t → ∞.

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It is easily seen that this lemma can be proved also by Bihari’s inequality [5]. Thus there is the blow-up of solution u ∈ C 2 ([0, T ) × Rn ) to the Cauchy problem considered in Example 2 if 1 < p < (n + 1)/(n − 1), that is T < ∞. The next example shows that this result can be extended to some operators with variable coeﬃcients. Example 2a. Assume that u = u(x, t) is a smooth solution of utt − a2 (t)∆u = m(t)|u|p on [0, T ] × Rn , u(x, 0) = f (x) , ut (x, 0) = g(x) on Rn , where for the speed of propagation a = a(t) and for the function m = m(t) the following conditions are fulﬁlled with some positive constants C and k: t ∞ max a(τ ) ds , 0 < a0 ≤ a(t) , a(t), m(t) ∈ C (R+ ) , A(t) := 0 −c

m(t) ≥ C(k + t)

n(p−1)

(k + A(t))

for large t

τ ≤s

with

c < p +1.

Suppose that supp f , supp g ⊂ { x ∈ R ; |x| ≤ R }. By the domain of dependence property, supp u(t, ·) ⊂ { x ∈ Rn ; |x| ≤ R + A(t) }. By integrating the equation with respect to spatial variables we obtain 2 utt (t, x) − a (t)∆u(t, x) dx = m(t)|u(t, x)|p dx . n

Rn

Rn

Similarly to the consideration of the previous example we obtain d2 u(t, x) dx = m(t)|u(t, x)|p dx . dt2 Rn n R Let F (t) = u(t, x) dx ,

then F¨ (t) =

Rn

m(t)|u(t, x)|p dx . Rn

Using the compact support of u(t, ·) and Hlder’s inequality we get with 1/p+1/q = 1, p/q = p − 1, τn the volume of the unit ball in Rn , p p u(t, x) dx n u(t, x) dx = R |x|≤R+A(t) p/q ≤ = =

|x|≤R+A(t)

|u(t, x)|p dx

1 dx |x|≤R+A(t)

1 (R + A(t))np/q m(t)|u(t, x)|p dx τn m(t) n R 1 ¨ (R + A(t))np/q . F (t)ττn m(t)

Global Existence for Nonlinear Wave Equations

309

Thus

F¨ (t) ≥ τn−1 m(t)(R + A(t))−n(p−1) |F (t)|p for all t in the interval [0, T ]. In particular, F¨ (t) ≥ 0, and so F (t) ≥ F˙ (0)t + F (0). Now F˙ (0) = ut (0, x) dx = g(x) dx ≡ Cg . Rn

Rn

If Cg > 0, then F (t) ≥ (pos. const.)t , t large . The Lemma 2.1 with r = 1 and c = q satisfying (p − 1)r > q − 2, shows that the function F (t) cannot remain ﬁnite. Hence, T < ∞. If we consider the case with a(t) = tl and m(t) = tm for large t, then the assumptions on these functions imply p < [n(l + 1) + 1 + m]/[n(l + 1) − 1]. Similar conclusions one can obtain for a(t) = exp(t), m(t) = tm1 exp(m2 t). Open Problem: Prove local and global existence results for the case with a(t) = tl , m(t) = tm . Then the expected suﬃcient condition for the global existence is p > [n(l + 1) + 1 + m]/[n(l + 1) − 1]. Conjecture: Assume that a(t) is a non-vanishing, non-constant, positive and periodic function. Then for every n, s, p > 1, and positive number ε there are initial data f (x), g(x) ∈ C0∞ (Rn ) with small Sobolev norms f (s+1) + g(s) ≤ ε such that the solution u(t, x) to the problem utt − a2 (t)∆u = |u|p on [0, ∞] × Rn , u(x, 0) = f (x) , ut (x, 0) = g(x) on Rn , blows up in ﬁnite time. Example 3. For x ∈ R3 , n = 3, in Example 2 we have 1 < p < 2. By improving √ the lower bound (2.2) one can obtain the following sharper result: if 1 < p < 1 + 2, then the solution blows up in ﬁnite time. Assume that u = u(x, t) is a smooth solution of utt − ∆u = |u|p on [0, T ] × R3 with 1 < p, and u(x, 0) = f (x) ,

ut (x, 0) = g(x).

Suppose that supp f , supp g ⊂ { x ∈ R ; |x| ≤ R } =: K. Then, let u = u0 (x, t) be a solution to the Cauchy problem for the linear wave equation 3

u0tt − ∆u0 = 0 ,

u0 (x, 0) = f (x) ,

u0t (x, 0) = g(x).

By Kirchhoﬀ’s formula the function u = u0 (x, t) can be represented as follows: 1 u0 (x1 , x2 , x3 , t) = f (y1 , y2 , y3 ) dσ 4πt St (x) ∂ 1 + g(y1 , y2 , y3 ) dσ , ∂t 4πt St (x)

310

Karen Yagdjian

where St (x) := {y ∈ R3 ; |x−y| = t}. It follows also the strong Huyghens’ principle, which states supp u0 ⊂ { (x, t) ; St (x) ∩ K = ∅ } . Hence, supp u0 ⊂ { (x, t) ; t − R ≤ |x| ≤ t + R } . Since u0 solves the linear wave equation, one obtains upon integration, that ∂2 u0 (x, t) dx = 0 for all t ≥ 0 . ∂t2 R3

It follows

R3

u0 (x, t) dx = Cg t + Cf ,

where Cg =

g(x) dx ,

Cf =

R3

f (x) dx . R3

Now we regard u(x, t) as a solution to the linear wave equation with the right-hand side ϕ(x, t) = |u|p : utt − ∆u = ϕ . If vϕ (t, x) solves this equation and takes vanishing initial data, vϕ (0, x) = ∂t vϕ (0, x) = 0, then 1 ϕ(y1 , y2 , y3 , t − |x − y|) dy1 dy2 dy3 vϕ (t, x) = 4π |x − y| Bt (x) |u|p (y1 , y2 , y3 , t − |x − y|) 1 dy1 dy2 dy3 ≥ 0 , = 4π |x − y| Bt (x) where Bt (x) := {y ∈ Rn ; |x−y| ≤ t}. On the other hand this leads to the following estimate: u(x, t) = u0 (x, t) + vϕ (x, t) ≥ u0 (x, t) . Thus

Cg t + Cf

0

=

t−R≤|x|≤t+R

≤ ≤

u0 (x, t) dx

u (x, t) dx = R3

u(x, t) dx t−R≤|x|≤t+R

(p−1)/p

(vol {t − R ≤ |x| ≤ t + R})

≤

C(t + R)2(p−1)/p 0

R3

1/p |u(x, t)| dx p

R3

1/p |u(x, t)|p dx .

Assuming Cg > 0 we get for F (t) = R3 u(x, t) dx F¨ (t) = |u(x, t)|p dx ≥ C(t + R)p−p2(p−1)/p = C(t + R)2−p R3

for large t .

Global Existence for Nonlinear Wave Equations Integrating, one has (1 < p < 3) 1 (t + R)3−p − (a0 + R)3−p for F˙ (t) − F˙ (a0 ) ≥ C 3−p Then with a new constant C we get F˙ (t) ≥ C(t + R)3−p for large t .

311

t ∈ [a0 , ∞) .

Integrating once more we derive F (t) ≥ C(t + R)4−p

for large t .

On the other hand, similarly to Example 2 we obtain the diﬀerential inequality (2.1), F¨ (t) ≥ τn−1 (R + t)3−3p |F (t)|p . It is very suggestive to use immediately Lemma 2.1. To do so we set r = 4−p ≥ 1,

q = 3p − 3 ,

then condition (p − 1)r > q − 2 is satisﬁed due to the inequality (p − 1)(4 − p) > 3(p − 1) − 2

⇐⇒

−(p − 1)2 + 2 > 0 ,

and this completes the consideration.

√

Remark 2.2. According to [31] for n = 3 at the borderline value p = 1 + 2 small data solutions must blow up in ﬁnite time. On the other hand for n = 3 and √ p > 1 + 2 small data solutions exist globally [19]. 0 In the all previous examples to get blow-up we assumed Rn g(x) dx > 0. Next example provides blow-up without this assumption. We assume that n = 3, small data, and consider the real-valued solution. Theorem 2.3 (F. John [19]). For the semilinear wave equation u = u2 in three space dimensions, x ∈ R3 , all smooth data with compact support, ϕ0 , ϕ1 ∈ C0∞ (R3 ) produce singularities in ﬁnite time. Proof. Let us assume that the data are supported in the ball of radius R, and are not both identically zero. We let v = u2 and denote by I(r, t) and J(r, t) the spherical means of u and v about the origin respectively: 1 I(r, t) = u(y, t) dσy , 4πr2 Sr 1 1 J(r, t) = v(y, t) dσy = u2 (y, t) dσy , 4πr2 Sr 4πr2 Sr where Sr := {y ∈ R3 ; |y| = r}. For the function u = u(x, t) one can also deﬁne the spherical means of u about the point x: 1 I(x, r, t) = u(x + y, t) dσy , 4πr2 Sr and then an operator Ωr

: u(x, t) −→ rI(x, r, t) .

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Karen Yagdjian

One can prove that for all h ∈ C(R3 ) 2 ∂ ∂2 ∂2 ∂2 ∆x Ωr h = + + h = Ωr h . Ω r ∂ x21 ∂ x22 ∂ x23 ∂ r2 Therefore, we get for the function Ωr u 2 2 ∂ ∂ ∂2 − ∆ u = − Ωr u = Ω Ωr u . x r ∂ t2 ∂ t2 ∂ r2 Since the commutator [−∆, Ωr ] = 0 vanishes we have Ωr u = Ωr u + [, Ωr ]u = Ωr u2 + [−∆, Ωr ]u = Ωr v . Consequently, rI(x, r, t) tt − rI(x, r, t) rr = rJ(x, r, t)

for all t ≥ 0 ,

r ≥ 0,

x ∈ R3 .

We need this equation only at x = 0, so simplifying the notations we have rI(r, t) tt − rI(r, t) rr = rJ(r, t) . One can continue the functions I and J for negative r by setting I(r, t) := I(−r, t) ,

J(r, t) := J(−r, t)

for all r ≤ 0 .

One still has the equation for these values of r, too. Our aim is to prove that if we assume that a smooth solution exists up to the time 6R, we can derive arbitrary large lower bounds on I for 0 < r < t − 6R, and thus we reach a contradiction. Step 1. Preliminaries. The Cauchy-Schwarz inequality gives 2 1 I 2 (r, t) = u(y, t) dσ y 4πr2 Sr 1 1 2 ≤ 1 dσ u(y, t) dσ , y y 4πr2 Sr 4πr2 Sr 1 2 u(y, t) dσ for all r, t . = y 4πr2 Sr Hence, I 2 (r, t) ≤ J(r, t) . If we denote

1 = φ0 (x + y, t) dσy , 4πr Sr 1 = φ1 (x + y, t) dσy , 4πr Sr

rI(r, 0) = Φ0 (r) rIIt (r, 0) = Φ1 (r)

Global Existence for Nonlinear Wave Equations

313

then due to d’Alembert’s formula and to Duhamel’s principle r+t 1 1 {Φ0 (r + t) + Φ0 (r − t)} + Φ1 (ρ) dρ I(r, t) = 2r 2r r−t t r+(t−τ ) 1 + ρJ(ρ, τ ) dρ dτ . 2r 0 r −(t−τ ) The function ρJ(ρ, τ ) is odd, −ρJ(−ρ, τ ) = −ρJ(ρ, τ ). The function rI(r, t) takes initial data Φ0 (r) and Φ1 (r) which vanish for |r| > R. Therefore, if we choose a point (r, t) such that |t + r| > R and |t − r| > R, then t r+(t−τ ) 1 ρJ(ρ, τ ) dρ dτ . I(r, t) = 2r 0 r −(t−τ ) Since ρJ(ρ, τ ) is odd, the part of the integral, corresponding to the region symmetric with respect to the τ axis, is zero. One can therefore restrict the integration to the region ' ( Z(r, t) := (ρ, τ ) ; τ > 0 and t − r < τ + ρ < t + r and τ − ρ < t − r , that is 1 I(r, t) = 2r

for |t + r| > R and |t − r| > R . (2.3)

ρJ(ρ, τ ) dρ dτ Z (r,t)

Letting r go to zero, we ﬁnd t ρJ(t − ρ, ρ) dρ I(0, t) =

for t > R .

0

Step 2. I(0, t) is not identically zero for R < t < 3R. Indeed, otherwise we would have J(τ, ρ) ≡ 0 for τ + ρ ∈ (R, 3R), and therefore 1 J(τ, ρ) = u2 (τ, ρy) dS ≡ 0 4π |y |=1 implies u(τ, ρy) = 0

∀τ + ρ ∈ (R, 3R) , y ∈ S 2 .

Letting τ → R, we ﬁnd in particular that u(R, ρy) = 0 ,

ut (R, ρy) = 0

for all ρ ∈ (0, 2R) , y ∈ S 2 ,

that is u(R, x) = 0 ,

ut (R, x) = 0

for all |x| ≤ 2R .

If u ∈ C ((0, ∞) × R ), then one can regard the equation u = u2 as a linear equation u−a(t, x)u = 0 with a smooth coeﬃcient a(t, x) = u(t, x) ∈ C 2 ([0, ∞)× R3 ). Then the ﬁnite propagation speed forces 2

3

u(0, x) = 0 ,

ut (0, x) = 0 for all |x| ≤ R ,

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and the solution would then vanish identically. The obtained contradiction implies that I(0, t) is not identically zero for R < t < 3R, and consequently, there is a point (0, t0 ), R < t0 < 3R, near which I does not vanish, I(0, t0 ) > 0 ,

R < t0 < 3R .

Moreover, I(r, t) does not vanish on the half-line {(r, t) ; t = t0 + r, r > 0 }, i.e., I(r, t) > 0

for all t = t0 + r ,

r > 0.

Indeed, if I(r , t ) = 0 for some t = t0 + r , r > 0, then from the representation t r +(t −τ ) 1 I(r , t ) = ρJ(ρ, τ ) dρ dτ 2r 0 r −(t −τ ) with the integration restricted to the trapezoid ( ' Z(r , t ) := (ρ, τ ) ; τ > 0 and t − r < τ + ρ < t + r and τ − ρ < t − r follows that J(ρ, τ ) ≡ 0 in Z(r , t ). The estimate I 2 ≤ J then implies I(ρ, τ ) ≡ 0 in Z(r , t ). But the points (0, t0 ) and (r , t ) are the vertexes of Z(r , t ), so that I(0, t0 ) must vanish. With this conclusion we turn from the theory of partial diﬀerential equations to some integral inequalities. Step 3. Two-variables integral inequality. Thus we have from (2.3) and I 2 ≤ J the following two-variables integral inequality: 1 ρI 2 (ρ, τ ) dρ dτ for |t + r| > R and |t − r| > R . I(r, t) ≥ 2r Z (r,t) Lemma 2.4. The continuous function I(r, t) which is deﬁned in {r ≥ 0, t ≥ 0} and which satisﬁes I(r, t) > 0

for all

t = t0 + r ,

cannot solve the inequality 1 ρI 2 (ρ, τ ) dρ dτ I(r, t) ≥ 2r Z (r,t)

for all

t0 ∈ (R, 3R) ,

t+r > R, t−r > R,

where the domain of integration Z(r, t) is ' ( Z(r, t) := (ρ, τ ) ; τ > 0 and t − r < τ + ρ < t + r, and τ − ρ < t − r . Namely, there exists a constant γ such that I(0, γ) = ∞. Proof. Consider the set ' ( T = (ρ, τ ) ; 4R < τ + ρ < 6R, R < τ − ρ < 3R .

Global Existence for Nonlinear Wave Equations

315

The half-line {(r, t) ; t = t0 + r, r > 0 } (with t0 ∈ (R, 3R)) intersects T . On that half-line both J and I are positive, therefore the following number is positive: 1 ρI 2 (ρ, τ ) dρ dτ > 0 . c := 2 T Let S denote the half-strip ' ( S = (ρ, τ ) ; 6R < τ + ρ , 3R < τ − ρ < 4R . Then T ⊂ Z(r, t) for (r, t) ∈ S. Consequently, 1 1 c 2 I(r, t) ≥ ρI (ρ, τ ) dρ dτ ≥ ρI 2 (ρ, τ ) dρ dτ = . 2r 2r r Z (r,t) T Thus

c for all (r, t) ∈ S . r In the last part of the proof we assume that 0 < r < t − 6R. Let S(r, t) denote the rectangle ' ( S(r, t) := Z(r, t) ∩ S = (ρ, τ ) ; t − r < τ + ρ < t + r , 3R < τ − ρ < 4R . I(r, t) ≥

Then I(r, t) ≥

1 2r

ρJ(ρ, τ ) dρ dτ ≥ S (r,t)

c2 2r

ρ−1 dρ dτ .

S (r,t)

If (ρ, τ ) ∈ S(r, t), then τ + ρ < t + r and ρ < τ − 3R imply 2ρ ≤ t + r − 3R and ρ−1 ≥ 2(t + r − 3R)−1 . Hence, if we set R ≥ 1, then c2 c2 I(r, t) ≥ ρ−1 dρ dτ ≥ (t + r − 3R)−1 dρ 2r r S (r,t) S (r,t) c2 = 1 dρ dτ r(t + r − 3R) S (r,t) c2 (t + r) − (t − r) 4R − 3R c2 2rR √ √ · · · = r(t + r − 3R) r(t + r − 3R) 2 2 2 c2 c2 ≥ ≥ for all 0 < r < t − 6R . t + r − 3R t+r Now we introduce the rectangle ' ( T (r, t) := (ρ, τ ) ; t − r < τ + ρ < t + r , 6R < τ − ρ < t − r , =

contained in Z(r, t). We have 1 1 ρI 2 (ρ, τ ) dρ dτ ≥ ρI 2 (ρ, τ ) dρ dτ I(r, t) ≥ 2r 2r Z (r,t) T (r,t) for all 0 < r < t − 6R. The proof of Lemma 2.4 will be completed if we prove the following statements:

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Lemma 2.5. The continuous function I(r, t) deﬁned on {r ≥ 0, t ≥ 0} cannot solve the inequalities 1 ρI 2 (ρ, τ ) dρ dτ for all 0 < r < t − 6R , I(r, t) ≥ 2r T (r,t) c2 t+r

≥

I(r, t)

for all

0 < r < t − 6R ,

simultaneously. Namely, there exists a constant γ such that I(0, γ) = ∞. Proof. We ﬁnd I(r, t) ≥

1 2r

c4 ρI (ρ, τ ) dρ dτ ≥ 2r T (r,t)

2

ρ T (r,t)

1 dρ dτ . (τ + ρ)2

Setting α = τ + ρ and β = τ − ρ and noting that in T (r, t) ρ≥

1 (t − r − β) , 2

β ≤ t−r,

while α ∈ [t − r, t + r] and β ∈ [6R, t − r], we write ρ T (r,t)

1 dρ dτ (τ + ρ)2

= =

t−r

dα

≥

t+r

= t−r t+r

t−r

dβ ρ 6R

1 dα 2 4α

1 1 2 (τ + ρ) 2

t−r

(t − r − β) dβ 6R

t−r 1 dα (t − r − β) dβ 2 t−r α 6R (t − r − 6R)2 2r . 4 (t + r)(t − r) 2

1 4 1

t+r

Finally, for t ≥ 6R + r we obtain I(r, t) ≥

2r (t − r − 6R)2 c4 (t − r − 6R)2 c4 1 = . 2r 4 (t + r)(t − r) 2 8(t + r) (t − r)

Let us suppose for (r, t), t ≥ 6R + r, r > 0, that I satisﬁes the inequality I(r, t) ≥

C (t − r − 6R)k , t+r (t − r)q

with numbers C, k, q, C > 0,

k ≥ 0,

q ≥ 0,

Global Existence for Nonlinear Wave Equations

317

as well as the ﬁrst one of the lemma. Then acting similar to the previous calculations we have 1 ρI 2 (ρ, τ ) dρ dτ I(r, t) ≥ 2r T (r,t) (τ − ρ − 6R)2k 1 C2 ρ dρ dτ ≥ 2 2r (τ − ρ)2q T (r,t) (ρ + τ ) t−r C 2 t+r 1 (τ − ρ − 6R)2k 1 ≥ dα dβ ρ , 2r t−r 2 (ρ + τ )2 (τ − ρ)2q 6R and consequently, I(r, t)

t+r

≥

C2 2r

≥

C2 8r(t − r)2q

t−r

dα t−r

1 (β − 6R)2k 1 (t − r − β) 2 4 α (t − r)2q t−r 1 2k dα (t − r − β)(β − 6R) dβ . α2 6R

dβ

6R t+r

t−r

The last integrals can be evaluated explicitly: t+r 1 t−r C2 dα (t − r − β)(β − 6R)2k dβ 2q 2 8r(t − r) t−r α 6R (t − r − 6R)2k+2 2 2r C = 8r(t − r)2q (t + r)(t − r) (2k + 1)(2k + 2) 2 2k+2 C (t − r − 6R) = . 4(2k + 1)(2k + 2)(t + r)(t − r)2q+1 Hence, I(r, t)

≥

(t − r − 6R)2k+2 C2 . 4(2k + 2)2 (t + r) (t − r)2q+1

Thus we have done one step that resulted in a substitution C (t − r − 6R)k t+r (t − r)q

→

(t − r − 6R)2k+2 C2 2 4(2k + 2) (t + r) (t − r)2q+1

in the estimate from below. Let us deﬁne the sequences {C Cj }j≥0 , {kj }j≥0 , {qqj }j≥0 by C0 = c2 , Cj +1

k0 = q0 = 0, Cj2 = , kj+1 = 2kj + 2 , 4(2kj + 2)2

qj +1 = 2qqj + 1 .

Then a recursive argument gives I(r, t) ≥ Cj (t + r)−1 (t − r − 6R)kj (t − r)−qj for 0 < r < t − 6R and all j = 0, 1, . . . .

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Karen Yagdjian

It is easy to solve the recursion formulas and to ﬁnd kj = 2j+1 − 2 , qj = 2j − 1 , j Cj = exp 2j log C0 − 21−i log(2i+2 − 4) . i=1

Indeed, Cj

Cj2−1 Cj2−1 Cj2−1 Cj2−1 = = = 4(2kj−1 + 2)2 4(2(2j − 2) + 2)2 16(2j − 1)2 (2j+2 − 4)2

=

implies log Cj = 2 log Cj −1 − log (2j+2 − 4)2 = 2 log Cj −1 − 2 log (2j+2 − 4) (one step). If we make one more step, then we obtain ' ( log Cj = 2 2 log Cj −2 − 2 log (2j+1 − 4) − 2 log (2j+2 − 4) =

2 · 2 log Cj −2 − 2 · 2 log (2j+1 − 4) − 2 log (2j+2 − 4) (two steps).

In j steps we arrive at log Cj = 2 · 2 · 2 log Cj −3 −2 · 2 · 2 log (2j −4)−2 · 2 log (2j+1 −4)−2 log (2j+2 −4) ··· j 21+j−i log (2i+2 − 4) (j-steps) = 2j log C0 − i=1

= 2j log C0 −

j i=1

21−i log(2i+2 − 4) .

Global Existence for Nonlinear Wave Equations

319

Thus I(r, t)

j ≥ exp 2j log C0 − 21−i log(2i+2 − 4) i=1 −1

×(t + r) (t − r − 6R)2 −2 (t − r)−2 +1 j = exp 2j log C0 − 21−i log(2i+2 − 4) j+1

j

i=1 j+1 j t−r (t − r − 6R)2 (t − r)−2 × (t + r)(t − r − 6R)2 j j ≥ exp 2 log C0 − 21−i log(2i+2 − 4)

i=1

t−r 1 j 2 j exp 2 × exp 2 log(t − r − 6R) log (t + r)(t − r − 6R)2 (t − r) t−r exp 2j Wj (t − r) ≥ (t + r)(t − r − 6R)2 for 0 < r < t − 6R and j = 0, 1, . . . , where (t − r − 6R)2 1−i − log C0 + log 2 log(2i+2 − 4) t−r i=1 j

Wj (t − r) :=

∞

≥

log C0 + log

(t − r − 6R)2 1−i − 2 log(2i+2 − 4) =: W∞ (t − r) . t−r i=1

Obviously, there exists a constant γ such that W∞ (t − r) > 0 for t − r ≥ γ. Hence, for j → ∞ we ﬁnd that I(r, t) = ∞ for t − r ≥ γ ,

r ≥ 0.

Lemma 2.5 is proved.

Completion of the proof of Theorem 2.3. From Lemma 2.5 in particular it follows u(0, γ) = I(0, γ) = ∞ . The solution has a life span T ≤ γ, and the theorem is proved.

According to the next theorem there is a blow-up for a system that is similar to the equation from Example 2. Theorem 2.6 (R. Agemi, Y. Kurokawa, H. Takamura [1], D. Del Santo [9], D. Del Santo, V. Georgiev, and E. Mitidieri [10], D. Del Santo and E. Mitidieri [11]). Let f1 , f2 , g1 , g2 ∈ C0∞ (R3 ) with ∅ = supp f1 ∪ supp f2 ∪ supp g1 ∪ supp g2 ⊂ {|x| ≤ R} .

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Let T ∈ (0, ∞) and let (u, v), where u, v ∈ C 2 (R3 × [0, T )), be the solution of the Cauchy problem for the system

utt − ∆u = |v|p , vtt − ∆v = |u|q , with initial data ut (x, 0) = g1 (x) , u(x, 0) = f1 (x) , v(x, 0) = f2 (x) , vt (x, 0) = g2 (x) . Suppose that p, q > 1 and

& p + 2 + q −1 q + 2 + p−1 , max ≥ 1. pq − 1 pq − 1 Then T < ∞. On the other hand due to the next theorem small data solutions exist globally in time. Theorem 2.7 (R. Agemi, Y. Kurokawa, H. Takamura [1], D. Del Santo [9], D. Del Santo, V. Georgiev, and E. Mitidieri [10]). Let f1 , f2 , g1 , g2 ∈ C0∞ (R3 ) with supp f1 ∪ supp f2 ∪ supp g1 ∪ supp g2 ⊂ {|x| ≤ R} . Suppose that p, q ≥ 2 and

max

p + 2 + q −1 q + 2 + p−1 , pq − 1 pq − 1

& < 1.

Then there exists ε0 > 0 such that for each ε ∈ (0, ε0 ) the Cauchy problem

utt − ∆u = |v|p , vtt − ∆v = |u|q ,

u(x, 0) = εf1 (x) , ut (x, 0) = εg1 (x) , v(x, 0) = εff2 (x) , vt (x, 0) = εg2 (x) , has a unique global classical solution (u, v), where u, v ∈ C 2 (R3 × [0, ∞)).

3. Blow-up for the problem with large potential energy of nonlinearity Consider the wave equation utt − ∆u + f (u) = 0 0u in R × R . We deﬁne F (u) = 0 f (s) ds, and the “complete energy” &

1 2 2 u + |∇x u| + F (u) dx . E(t) = 2 t Rn n

Global Existence for Nonlinear Wave Equations

321

Theorem 3.1 (H. Levine [22]). Assume that f is smooth, f (0) = 0, uf (u) ≤ (2 + ε)F (u) for some positive ε, and that u is a smooth solution with E(0) < 0. Further, assume that the data have compact support, u(0, x) = u0 (x) ∈ C0∞ (Rn ) ,

ut (0, x) = u1 (x) ∈ C0∞ (Rn ) .

Then u must develop a singularity in ﬁnite time. Proof. For the energy of the classical solution we have . / d E(t) = ut utt + ∇x u · ∇x ut + ut f (u) dx dt n R . / = ut utt − ∆u + f (u) dx = 0 Rn

and E(t) is therefore constant, E(t) ≡ E(0). Multiplying the equation by u and integrating we ﬁnd uutt + |∇x u|2 + uf (u) dx = 0 . Rn

Adding (with α = ε/4) &

1 2 ut + |∇x u|2 + F (u) dx = −(2 + ε)E(0) −(2 + ε) 2 Rn to both sides, we ﬁnd, uutt + |∇x u|2 + uf (u) dx − (2 + ε) Rn

Rn

that is, Rn

& 1 2 2 u + |∇x u| + F (u) dx 2 t = −(2 + ε)E(0),

−(1 + 2α)u2t − 2α|∇x u|2 + uutt + uf (u) − (2 + 4α)F (u) dx = −(2 + 4α)E(0) .

Hence, after using the assumption on f , (2 + 4α)E(0) = (1 + 2α)u2t + 2α|∇x u|2 − uutt − uf (u) + (2 + 4α)F (u) dx n R (1 + 2α)u2t − uutt dx . ≥ Rn

If we deﬁne now the positive function I(t) =

1 1 β(t + τ )2 + 2 2

u2 dx > 0 , Rn

322

Karen Yagdjian

where β and τ are positive constants, then d d2 I(t) = β(t + τ ) + uut dx , I(t) = β + (u2t + uutt ) dx . dt dt2 Rn Rn Using E(0) < 0 we choose β such that 2E(0) + β < 0 , and then choose τ such that

I (0) = βτ + Rn

u0 u1 dx > 0 .

Our goal is to prove that I(t) cannot remain bounded. We write I (t) = β + (u2t + uutt ) dx n R ' ( (1 + 2α)u2t − uutt dx u2t dx − ≥ β + 2(1 + α) n Rn R ≥ β + 2(1 + α) u2t dx − (2 + 4α)E(0) Rn = 2(1 + α)β + 2(1 + α) u2t dx − (2 + 4α)E(0) − (1 + 2α)β n R 2 = 2(1 + α) β + ut dx − (1 + 2α)(β + 2E(0)) . Rn

Thus

I (t) > 2(1 + α) β + Rn

u2t dx .

This implies I(t)I (t) − (1 + α)(I (t))2 ' (' ( 2 2 > β(t + τ ) + u dx (1 + α) β + u2t dx Rn Rn (2 ' uut dx −(1 + α) β(t + τ ) + Rn = (1 + α) β(t + τ )2 + u2 dx β + u2t dx − β(t + τ ) +

Rn

Rn

2 uut dx

Rn

.

The right-hand side is√nonnegative. Indeed, √ using the Cauchy-Schwarz inequality on the product of (u, β(t + τ )) and (ut , β) in L2 (Rn ) × R we obtain 2 β β(t + τ ) + uut dx ≤ β(t + τ )2 + u2 dx β + u2t dx . Rn

Rn

Rn

Global Existence for Nonlinear Wave Equations

323

Hence, I(t)I (t) − (1 + α)(I (t))2 after multiplication with −αI

−α−2

(I

> 0

implies −α

) (t) < 0 .

On the other hand the parameters have been chosen so that I −α (0) > 0 , I −α (0) < 0 . It follows that the function J(t) := I −α (t) must vanish in ﬁnite time. Indeed, t t s J(t) = J(0) + J (s)ds = J(0) + J (τ )dτ ds ≤ J(0) + J (0)t J (0) + 0

0

0

implies that it happens no later than at tls = −J(0)/J (0), hence limt→tls I(t) = ∞. The theorem is proved. Thus according to this theorem one can say, if the absolute value of the negative initial potential energy of the nonlinearity is larger than the initial total energy of the linear problem, then the problem cannot have global solutions. Next we take two examples for the function f (u) and discuss the condition on E(0). 1) If we take f (u) = −|u|p−1 , then to verify the main condition we write for positive u u up (−|s|p−1 )ds = −(2 + ε) , uf (u) = −u|u|p−1 = −up ≤ (2 + ε)F (u) = (2 + ε) p 0 which leads to 0 < ε ≤ p −2. For negative u we obtain uf (u) = −u|u|p−1 = |u|p

≤

=

(2 + ε)

u

(−|s|p−1 )ds

(2 + ε)F (u) = (2 + ε) 0 |u|

|s|p−1 ds = (2 + ε)

0

|u|p p

which leads to 0 < ε, ε ≥ p−2. Finally we have to assume ε = p − 2 > 0, that is, p > 2. 2) If we take f (u) = −up−1 , then to verify the main condition we write u up p (−sp−1 )ds = −(2 + ε) . uf (u) = −u ≤ (2 + ε)F (u) = (2 + ε) p 0 This holds for negative and positive u simultaneously only for p = 2 + ε, and we get the condition p > 2.

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Karen Yagdjian

3) Consider now the assumption E(0) < 0: &

1 1 2 2 p u1 (x) + |∇x u0 (x)| − |u0 (x)| dx < 0 , E(0) = 2 p Rn where we have assumed u0 (x) ≥ 0. If we replace u0 (x) and u1 (x) by εu0 (x) and εu1 (x), respectively, then for ε > 0 small it will never be satisﬁed because of p > 2. Thus we have an example with blow-up for large data. According to Theorem 6.5.2 [17] the Cauchy problem with the initial data εu0 (x) ∈ C0∞ (Rn ) and εu1 (x) ∈ C0∞ (Rn ) has a C ∞ solution for all t ≥ 0 if n ≥ 4 and ε is suﬃciently small, that is, the existence of global small data solutions. It can be similarly proved the nonexistence of global solutions for the Cauchy problem0 for the equation utt − a(t)∆u + f (u) = 0, where a (t) ≤ 0 and uf (u) ≤ u (2 + ε) 0 f (s)ds.

4. Parametric resonance and wave map type equations In this section we consider global existence and blow-up of the solutions to the problem for the scalar nonlinear wave equation:

∇x u|2 ) = 0 , utt − a2 (t, x)∆u + f (u)(u2t − a2 (t, x)|∇ (4.1) ut (0, x) = u1 (x) , u(0, x) = u0 (x) , where u = u(t, x) is a real-valued unknown function, f (u) is a real-valued continuous function, a(t, x) ≥ 0, while ∆ is the Laplace operator in Rn . The equation in (4.1) is a model and a special case of an equation for wave maps. Let M be a Riemannian manifold with metric (gij ) = g. A wave map u : Rn+1 −→ M , is by deﬁnition a solution of the Euler-Lagrange equations associated with the functional ∂α u, ∂ α ug(u) dσ . u −→ J(u) := Rn

∂x1 )α1 , . . . , (−∂ ∂x1 )αn , Here ∂α = ∂tα0 , ∂xα11 , . . . , ∂xα1n , ∂ α = ∂tα0 , (−∂ ·, ·g(u) is the product in the metric g(u), and the usual Einstein summation convention is in force, while dσ denotes the volume measure on Rn+1 with respect to the standard metric. In local coordinates, u is seen to satisfy the equation ul + Γlbc (u)∂α ub ∂ α uc = 0 ,

l = 1, 2, . . . , m ,

where Γlbc (u) refer to the Riemann-Christoﬀel symbols associated with the metric g. The scalar problem (4.1) is similar to the wave map problem, having the same form (4.1) in the case that the target M is one dimensional. The constant coeﬃcient case, a(t, x) ≡ 1, is studied in [26]. There are given necessary and suﬃcient conditions on f = f (u) for which the Cauchy problem

Global Existence for Nonlinear Wave Equations

325

(4.1) has a global smooth solution for any smooth initial data. The conditions are written as ∞ s 0 s exp f (r)dr ds = ∞ and exp f (r)dr ds = ∞. (4.2) 0

0

−∞

0

For instance, the function f (u) = cuk with c ∈ R and nonnegative integer k, satisﬁes condition (4.2) if and only if c ≥ 0 and k is odd. When n = 3, f = f (u) and a = a(t, x) are constant functions, say f (u) ≡ 1 and a(t, x) ≡ 1, equation (4.1) becomes Nirenberg’s example, which is globally solvable for small data only. Moreover, when a(t, x) ≡ 1 and f = f (u) is smooth enough, it still has a global solution for small data [26] independently of condition (4.2). First we will prove that the condition a(t, x) ≡ 1 is crucial for the global solvability of problem (4.1), namely even for smooth, positive, periodic, non-constant function a = a(t, x) the problem is not globally solvable even for arbitrary small data. Secondly we will show that condition (4.2) is equivalent to the global solvability of the problem (4.1) for arbitrary (in particularly large) initial data. The proof of the second statement is short and almost identical to the one suggested in [26]. It will be given at the end of this section, while the proof of the ﬁrst one based on the properties of the Hill’s equation and takes a lot of eﬀorts. Parametric resonance breaks down the small data solution We give an example of the inﬂuence of the behavior of a time-dependent coeﬃcient a = a(t) from (4.1), and in particular its oscillating behavior, on the global existence of solutions to nonlinear hyperbolic equations. Namely for arbitrary small initial data we will construct blowing up solutions. To this end we consider in R × Rn the equation utt − a2 (t, x)∆u + f (u)(u2t − a2 (t, x)|∇ ∇x u|2 ) = 0 ,

(4.3)

where u = u(t, x) is a real-valued unknown function, f = f (u) is a real-valued smooth function. We restrict ourselves to the case of a(t, x) = b(t): n (uxj )2 = 0 . utt − b2 (t)∆u + f (u) (ut )2 − b2 (t)

(4.4)

j=1

The next theorem shows that if the function b = b(t) diﬀers from the case to be a constant, for instance, oscillates, the situation with the global existence for small data changes dramatically. For the equation (4.4) with f (u) ≡ 1 it was detected in [41], [43]. In the next theorem ϕ(s) denotes the norm of the function ϕ = ϕ(x) from the Sobolev space H s (Rn ).

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Theorem 4.1. Let b = b(t) be a deﬁned on R, a periodic, non-constant, smooth, and positive function. Suppose that (4.2) does not hold. Then for every n, s, and for every positive δ there are data u0 ∈ C0∞ (Rn ) and u1 ∈ C0∞ (Rn ) such that u0 (s+1) + u1 (s) ≤ δ,

(4.5)

but a solution u ∈ C (R+ × R ) to the problem with data 2

n

u(0, x) = u0 (x),

ut (0, x) = u1 (x) ,

x ∈ Rn ,

(4.6)

does not exist. For a simplest example of the equation (4.4) one can take f (u) = uk , k even, while b(t) = 1 + ε sin(2πt), with a number ε ∈ (−1/2, 1/2), ε = 0. The oscillations are responsible for the blow-up of the solutions. Indeed, if, for instance for the simpliﬁed equation with f (u) ≡ 1, we switch oﬀ them, that is if we set ε = 0, then for n ≥ 3 the problem has small data global in time solution. (See, e.g., [17], [27].) The next theorem explains why in Theorem 4.1 we have to suppose that (4.2) does not hold. Indeed, if it is satisﬁed, then there is a global solution even for large data. Large Data Global Solution Theorem 4.2. Let f ∈ C ∞ (R) and let a(t, x) ∈ C ∞ (R+ × Rn ) be a uniformly positive bounded function. Then (4.3), (4.6) has a global classical solution u ∈ C ∞ (R1+n ) for any u0 and u1 ∈ C ∞ (Rn ) if and only if the function f satisﬁes (4.2). Proof of Theorem 4.2. In fact equation (4.3) is transformed into the linear wave equation by u s v = G(u) := exp f (r)dr ds . (4.7) 0

0

Since G ∈ C 2 (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C 2 (a, b) , where we denote a := lim G(u) , u→−∞

b := lim G(u) . u→∞

First we show that (4.2) is a suﬃcient condition for global existence. Assume (4.2) and let ϕ, ψ ∈ C ∞ (Rn ). Denote by W (ϕ, ψ) the solution v of the Cauchy problem for the linear wave equation:

vtt − a2 (t, x)∆v = 0 , v(0, x) = ϕ(x) , vt (0, x) = ψ(x) . Set v = W (G(ϕ), G (ψ)). We claim that u = H(v) is a classical solution of (4.3). Indeed, G (u) = f (u)G (u) , G (H(v))H (v) = 1 ,

Global Existence for Nonlinear Wave Equations

327

imply H (u) = −f (H(v))H (v)2 , while ∂α u = H (v)∂α v, ∂α2 u = H (v)(∂α v)2 + H (v)∂α2 v, Thus, we obtain utt − a2 (t, x)∆u

for α = 0, 1, . . . , n.

=

H (v)(vtt − a2 (t, x)∆v) + H (v)((∂ ∂t v)2 − a2 (t, x)|∇ ∇x v|2 )

=

∇x u|2 ) , −f (u)((∂ ∂t u)2 − a2 (t, x)|∇

(4.8)

so that u is a global smooth solution of (4.3). We choose u0 (x) s ϕ(x) := exp f (r)dr ds , ψ(x) := u1 (x) exp 0

0

u0 (x)

f (r)dr

.

0

Now since t = W (0, 1) the function u(t, x) = H(t) = H(W (0, 1)) is a classical solution of (4.3) with (ϕ, ψ) = (0, 1) for a < t < b and satisﬁes lim u(t, x) = −∞ and

t→a+0

lim u(t, x) = −∞

t→b−0

for any x ∈ Rn . Hence, (4.2) is also a necessary condition for which (4.3) has a global smooth solution for any smooth initial data.

5. Proof of Theorem 4.1: Parametric resonance We follow [8] to prove Borgs’s theorem. 5.1. Some properties of the Hill’s equation Consider the auxiliary ordinary diﬀerential equation with a periodic coeﬃcient wtt + λb2 (t)w = 0 .

(5.1)

It can be written also as a system of diﬀerential equations for the vector-valued function x(t) = t (wt , w): d 0 −λb2 (t) x(t) = A(t)x(t) , where A(t) := . 1 0 dt Let the matrix-valued function Xλ (t, t0 ), depending on λ, be the solution of the Cauchy problem d 1 0 . (5.2) X = A(t)X , X(t0 , t0 ) = 0 1 dt Thus, Xλ (t, t0 ) gives the fundamental solution to the equation (5.1). This matrix is called matrizant. In what follows we often omit subindex λ of Xλ (t, t0 ). There is an explicit representation formula for the matrizant: t1 tk−1 ∞ t X(t, t0 ) = I + A(t1 ) dt1 A(t2 ) dt2 . . . A(tk ) dtk . k=1

t0

t0

t0

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Karen Yagdjian

The Liouville formula

t

W (t) = W (t0 ) exp

S(τ )dτ

,

t0

where W (t) := det X(t, t0 ) ,

S(t) :=

2

Akk (t)

k=1

with S(t) ≡ 0 guarantees the existence of the inverse matrix Xλ (t, t0 )−1 . This allows to prove that 1. The matrizant of any system satisﬁes the identities X(t, t0 ) = X(t, t1 )X(t1 , t0 ) ,

X(t, t0 )−1 = X(t0 , t)

for all t0 , t1 , t ∈ R .

Indeed, both sides of the ﬁrst equality satisfy the equation and coincide at t = t1 . By the uniqueness of the solution to the Cauchy problem both sides coincide identically. 2. If A(t) is a 1-periodic matrix, then X(t + t0 , t0 ) is independent of t0 ∈ N. In particular X(t + 1, 1) = X(n + 1, n) =

X(t, 0) for all t ∈ R , X(1, 0) for all n ∈ N .

Indeed, the matrix Y (t) := X(t+t0 , t0 ), t0 ∈ N, solves the equation dY /dt = A(t)Y and takes initial data Y (0) = X(t0 , t0 ) = I. So does the matrix X(t, 0). Uniqueness in the Cauchy problem implies X(t + t0 , t0 ) = X(t, 0) is independent of t0 ∈ N. Problem. Let A(t) be a 1-periodic matrix. Is X(t + t0 , t0 ) independent of t0 when t0 runs over R? Consider now the matrix C(t, t0 ) := (X(t, t0 )) 1, t0 ), then d C(t, t0 ) = dt = = =

−1

X(t + 1, t0 ) = X(t0 , t)X(t +

d (X(t, t0 ))−1 X(t + 1, t0 ) dt d −1 −1 d (X(t, t0 )) X(t + 1, t0 ) X(t + 1, t0 ) + (X(t, t0 )) dt dt d (X(t, t0 ))−1 X(t + 1, t0 ) + (X(t, t0 ))−1 A(t + 1)X(t + 1, t0 ) dt d −1 −1 (X(t, t0 )) X(t + 1, t0 ) + (X(t, t0 )) A(t)X(t + 1, t0 ) . dt

d −1 Further, the identity (X(t, t0 )) X(t, t0 ) ≡ I leads to (X(t, t0 )−1 X(t, t0 )) = 0, dt so that d −1 −1 d (X(t, t0 )) X(t, t0 ) = 0 X(t, t0 ) + (X(t, t0 )) dt dt

Global Existence for Nonlinear Wave Equations implies

d −1 (X(t, t0 )) dt

Thus, d C(t, t0 ) = dt = =

=

− (X(t, t0 ))

−1

=

− (X(t, t0 ))

−1

=

− (X(t, t0 ))

−1

329

d −1 X(t, t0 ) (X(t, t0 )) dt

A(t)X(t, t0 ) (X(t, t0 ))

−1

A(t) .

d −1 (X(t, t0 )) X(t + 1, t0 ) + (X(t, t0 ))−1 A(t)X(t + 1, t0 ) dt

− (X(t, t0 )) 0.

−1

A(t)X(t + 1, t0 ) + (X(t, t0 ))

−1

A(t)X(t + 1, t0 )

That is, the matrix C(t, t0 ) is independent of t. We set t = t0 in C(t0 ) = (X(t, t0 ))−1 X(t + 1, t0 ) and get C(t0 ) = X(t0 + 1, t0 ) ,

X(t + 1, t0 ) := X(t, t0 )C(t0 ) for all t, t0 ∈ R.

We will write just C for the matrix C(0). Hence, C = X(1, 0) , X(t + 1, 0) := X(t, 0)C for all t ∈ R , X(t + n, 0) := X(t, 0)C n for all t ∈ R , n ∈ Z . 3. For the matrix X(1, 0) we will use the notation b11 b12 Xλ (1, 0) = . b21 b22 This matrix is called a monodromy matrix and its eigenvalues are called multipliers of system (5.2) (see [23],[44]). Thus, the monodromy matrix is the value at t = 1 (the “end”of the period) of the fundamental matrix X(t, 0) deﬁned by the initial condition X(0, 0) = I (i.e., matrizant), and the multipliers are the roots of the equation det [X(1, 0) − µI] = 0 .

(5.3)

4. The matrizant of any 1-periodic system satisﬁes the identities X(t + 1, t0 + 1) = X(t, t0 )

for all t0 , t ∈ R .

Indeed, X(t + 1, t0 + 1) = =

X(t + 1, t0 )X(t0 , t0 + 1) = X(t, t0 )C(t0 )X(t0 + 1, t0 )−1 X(t, t0 )C(t0 )C −1 (t0 ) = X(t, t0 ) .

5. Let a be an eigenvector of the monodromy matrix relative to some multiplier µ, X(1, 0)a = µa .

330

Karen Yagdjian

The vector a is in general complex, even when A(t) is a real-valued matrix. Consider the solution x = x(t) of the system dx/dt = A(t)x with initial condition x(0) = a , then it is expressed in terms of the matrizant by x(t) = X(t, 0)a . Using the properties X(t, t0 ) = X(t, t1 )X(t1 , t0 ) and X(t + 1, t0 + 1) = X(t, t0 ) we have x(t + 1) = = =

X(t + 1, 0)a = X(t + 1, 1)X(1, 0)a X(t, 0)X(1, 0)a = X(t, 0)µa µX(t, 0)a = µx(t) .

Thus, for each multiplier there is a solution x = x(t) of the system satisfying x(t + 1) = µx(t)

for all t ∈ R .

(5.4)

One can derive the characteristic equation (5.3) by determining a nontrivial solution x(t) of the system with the property (5.4), where µ is some number. Indeed, the vector functions x(t+ 1) and µx(t) satisfy the system. If they coincide at t = 0, x(1) = µx(0) ,

(5.5)

then due to the uniqueness it follows that (5.4) is true. Thus for the solutions property (5.5) is equivalent to (5.4). It is thus suﬃcient to ﬁnd a solution with the property x(1) = µx(0). Since x(1) = X(1, 0)x(0), it follows that this property is equivalent to [X(1) − µI] x(0) = 0 , so that µ is a root of the characteristic equation (i.e., a multiplier) and x(0) is an eigenvector of the monodromy matrix. 6. Theorem 5.1 (Lyapunov-Floquet). In system (5.2) let A(t) be a continuous matrix which is periodic of period P , A(t+P ) = A(t). Then the matrizant X(t, 0) of (5.2) has a representation of the form X(t, 0) = Z(t)etB ,

where

Z(t + P ) = Z(t)

is P -periodic .

The matrix B, generally speaking, is complex-valued. If in system (5.2) the matrix A(t) is real-valued with period P , then the (real-valued) matrizant X(t, 0) of system (5.2), the system regarded as a periodic one with period 2P , has a representation of the form X(t, 0) = Z(t)etB1 ,

where

Z(t + 2P ) = Z(t)

is 2P -periodic ,

while B1 is a real-valued matrix. Proof. We have proved that X(t+P, 0) = X(t, 0)C. For every C with the property det C = 0 there is a (non-unique) matrix B such that C = eP B ,

Global Existence for Nonlinear Wave Equations

331

i.e., C has logarithm P B. Hence, X(t + P, 0) = X(t, 0)eP B . Deﬁne Z(t) by

Z(t) = X(t, 0)e−tB ,

then =

X(t + P, 0)e−(t+P )B = X(t + P, 0)e−P B e−tB X(t, 0)Ce−P B e−tB = X(t, 0)eP B e−P B e−tB

=

X(t, 0)e−tB = Z(t, 0) .

Z(t + P, 0) =

Thus, it holds Z(t + P, 0) = Z(t, 0) as claimed. We shall now assume that A(t) is a real matrix, X(t, t0 ) is a matrizant, and C is a monodromy matrix, so that X(t + P, 0) = X(t, 0)C and C is real. We have X(t + 2P, 0) = X(t, 0)C 2 . One can easily prove that there exists a real matrix B1 such that e2P B1 = C 2 .

The theorem is proved.

7. If µ ∈ C is a multiplier for the Hill’s equation (5.1), then 1/µ is also a multiplier. Proof. Due to Liouville’s formula for the determinant W (t) of the matrizant X(t, t0 ) we have W (t) ≡ 1. Hence, det X(1, 0) = 1. From characteristic equation (5.3) of the deﬁnition of the multipliers and from Vieta’s Theorem for the products of the multipliers µ1 and µ2 , we get µ1 µ2 = 1, which proves the statement. If y1 (t) and y2 (t) are two solutions to (5.1), which take initial data y1 (0) = 0 ,

y1 (0) = 1 ,

y2 (0) = 1 ,

y2 (0) = 0 ,

then we can give a “more explicit” formula for the monodromy matrix and for the multipliers. Indeed, y1 (t) y2 (t) y1 (1) y2 (1) Xλ (t, 0) = = =⇒ Xλ (1, 0) = , y1 (t) y2 (t) y1 (1) y2 (1) and the characteristic equation (5.3) becomes µ2 − [y1 (1) + y2 (1)]µ + 1 = 0 . The function

∆(λ) = y1 (λ, 1) + y2 (λ, 1) = trace Xλ (1, 0) is called the discriminant of Hill’s equation. 8. Consider now the family of equations wtt + λ2 α(t)w = 0, where λ is a complex parameter and α is smooth, π-periodic such that α(t) ≥ 0

for all t ∈ R .

(5.6)

332

Karen Yagdjian

In the following lemma the periodicity assumption can be omitted. Lemma 5.2. The discriminant ∆(λ), ∆(λ) := y1 (λ, π) + y2 (λ, π) = trace Xλ (π, 0), is an entire function such that ∆(0) = 2 and (∂λ2 ∆)(0) = −2π

π

α(s)ds .

0

Proof. Let y1 (λ; t) and y2 (λ; t) be two solutions to equation (5.6), which take initial data y2 (λ; 0) = 1 , y2 (λ; 0) = 0 . y1 (λ; 0) = 0 , y1 (λ; 0) = 1 , First we note that these functions are holomorphic in λ ∈ C. Then y1 (0; t) = t and y2 (0; t) = 1, so that ∆(0) = 2. By deﬁnition we have ∂λ ∆(λ) = ∂λ y1 (λ, π) + ∂λ y2 (λ, π) ,

∂λ2 ∆(0) = (∂λ2 y1 )(0, π) + (∂λ2 y2 )(0, π) .

Further, from equation (5.6) we get ∂λ y1tt + λ2 α(t)∂λ y1 = −2λα(t)y1 , t t ∂λ y1t (λ, t) − ∂λ y1t (λ, 0) + λ2 α(s)∂λ y1 (λ, s)ds = −2λ α(s)y1 (λ, s)ds , 0 0 t t α(s)∂λ y1 (λ, s)ds = −2λ α(s)y1 (λ, s)ds ∂λ y1t (λ, t) + λ2 0

0

since the initial data for y1 are independent of λ. Hence, π 2 α(s)y1 (0, s)ds = −2 (∂λ y1t )(0, π) = −2 0

π

α(s)s ds .

0

Similarly ∂λ y2tt + λ2 α(t)∂λ y2 = −2λα(t)y2 t t 2 ∂λ y2t (λ, t) − ∂λ y2t (λ, 0) + λ α(s)∂λ y2 (λ, s)ds = −2λ α(s)y2 (λ, s)ds 0 0 t t α(s)∂λ y2 (λ, s)ds = −2λ α(s)y2 (λ, s)ds ∂λ y2t (λ, t) + λ2 0 0 t τ t τ 2 dτ α(s)∂λ y2 (λ, s)ds = −2λ dτ α(s)y2 (λ, s)ds ∂λ y2 (λ, t) + λ 0

implies (∂λ2 y2 )(0, π)

0

0

0

π τ α(s)y2 (0, s)ds = −2 dτ α(s)ds 0 0 0 0 π π π = −2 α(s)ds dτ = −2 α(s)(π − s)ds 0 s 0 π π α(s)ds + 2 α(s)sds = −2π π

= −2

τ

dτ

0

0

Global Existence for Nonlinear Wave Equations

333

which completes the proof. 9.

Theorem 5.3 (Floquet). (a) Assume that ∆2 > 4 for some λ. Then there exist a real number γ and two continuous functions p1 (t) and p2 (t), which are π-periodic or π-semi-periodic functions as ∆ > 2 or ∆ < −2, in such a way that any solution to equation (5.6) can be written as follows: w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt , with some C1 , C2 ∈ C. (b) Assume that ∆2 = 4. Then equation (5.6) admits at least one 2π-periodic solution. Namely, it has a π-periodic solution when ∆ = 2 and π-semi-periodic solution when ∆ = −2. Proof. (a) Assume that ∆ > 2. Then the eigenvalues µ1 and µ2 of the monodromy matrix Xλ (π, 0), are positive, and there are nontrivial solutions y1 (t) and y2 (t) such that y1 (t + π) = µ1 y1 (t) = eln µ1 y1 (t) ,

y2 (t + π) = µ2 y2 (t) = eln µ2 y2 (t)

for all t ∈ R. Then the functions p1 (t) := e−t(ln µ1 )/π y1 (t) and p2 (t) := e−t(ln µ2 )/π y2 (t) are π -periodic. Indeed, pk (t + π) = =

e−(t+π)(ln µk )/π yk (t + π) = e−(t+π)(ln µk )/π eln µk yk (t) e−t(ln µk )/π yk (t) = pk (t) ,

k = 1, 2 .

t(ln µ1 )/π

and y2 (t) = p2 (t)et(ln µ2 )/π form a fundamenThe functions y1 (t) = p1 (t)e tal system of solutions. Assume now that ∆ < −2, then µ1 and µ2 are negative, so that ln µk = ln |µk | + iπ, k = 1, 2, and deﬁned by p1 (t) := e−t(ln µ1 )/π y1 (t) and p2 (t) := e−t(ln µ2 )/π y2 (t) functions are not π-periodic. Nevertheless one can write yk (t) = pk (t)et(iπ+ln |µk |)/π = pk (t)eit et(ln |µk |)/π , pk (t) = yk (t)e−it e−t(ln |µk |)/π , pk (t)eit = yk (t)e−t(ln |µk |)/π , where the functions pk (t)eit , k = 1, 2, are π-semi-periodic. Indeed, pk (t + π)ei(t+π)

=

yk (t + π)e−(t+π)(ln |µk |)/π

=

µk yk (t)e−(t+π)(ln |µk |)/π

=

eln µk yk (t)e−(t+π)(ln |µk |)/π

=

eln |µk |+iπ yk (t)e−(t+π)(ln |µk |)/π

=

eiπ yk (t)e−t(ln |µk |)/π

=

−pk (t)eit ,

k = 1, 2 .

It remains to denote γ := ln |µ1 | = − ln |µ2 |, when ln |µ1 | > 0.

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Karen Yagdjian

(b) If ∆ = 2, then the characteristic equation has a double zero at µ = 1. The function X(t, 0)c, where c is an eigenvector of C, is a π-periodic solution. If ∆ = −2, then the characteristic equation has a double zero at µ = −1. The function X(t, 0)c, with c chosen as an eigenvector of C, is a π-semi-periodic solution. The case ∆2 < 4 can be considered similarly, and we leave it to the reader. 5.2. Borg’s Theorem 10. The previous theorem describes a structure of the solutions by means of the discriminant ∆ of Hill’s equation, that is in an implicit way, assuming a knowledge of the two independent solutions. On the other hand one prefers to obtain the same conclusion starting from the coeﬃcients of the equation. This is a more diﬃcult problem and only a partial answer will be given in the next theorem. The following theorem proves the existence of at least one instability zone Λ, that is, the existence of parametric resonance. The main feature of the parametric resonance is the exponentially increasing amplitude of oscillatory systems, whereas in the case of an ordinary resonance they increase with a power law. Theorem 5.4 (G. Borg’46 [6]). Assume that α is non-constant, positive, and πperiodic function. Then there exists an open interval Λ ⊂ (0, ∞) such that for every given λ ∈ Λ any solution of (5.6) can be written in the form w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt ,

C1 , C2 ∈ C ,

where γ is a positive number and both functions p1 (t) and p2 (t) are π-periodic or π-semi-periodic functions. Proof. The proof consists of the following four steps: Step 1. The function ∆2 (λ) − 4 has only real zeros. Indeed, assume that ∆2 (λ0 ) = 4. Then by Floquet’s theorem, there exists a nontrivial solution w0 (t) to the equation wtt + λ20 α(t)w = 0, which is π-periodic or π-semi-periodic. Multiplying by w0 (t) and integrating we derive π π w0tt (s)w0 (s)ds + λ20 α(s)w0 (s)w0 (s)ds = 0 . 0

0

It follows

w0t (π)w0 (π) − w0t (0)w0 (0) −

π

w0t (s)w0t (s)ds + 0

Hence,

0

Thus λ0 is real.

π

π

α(s)|w0 (s)|2 ds = 0 . 0

|w0t (s)|2 ds + λ20

−

λ20

π

α(s)|w0 (s)|2 ds = 0 . 0

Global Existence for Nonlinear Wave Equations

335

Step 0 π 2. The entire function ∆(λ) has exponential growth with type not greater than α(s)ds, that is, for every positive δ there is a constant C(δ) such that 0 & π α(s)ds + δ |λ| for all λ ∈ C . |∆(λ)| ≤ C(δ) exp 0

For a given λ ∈ C we deﬁne the energy E(t; λ) of the solution w of Hill’s equation by E(t; λ) = |λ|2 α(t)|w(t)|2 + |w (t)|2 . Then we get d E(t; λ) = |λ|2 α (t)|w(t)|2 + |λ|2 α(t)2$(w(t)w (t)) + 2$(w (t)w (t)) dt = |λ|2 α (t)|w(t)|2 + |λ|2 α(t)2$(w(t)w (t)) − 2$(w (t)λ2 α(t)w(t)) |α (t)| E(t; λ) + 2|λ| a(t)(|λ| α(t)|w(t)|)|w (t)| ≤ α(t) +2|λ| α(t)(|λ| α(t)|w(t)|)|w (t)| |α (t)| + 2|λ| α(t) E(t; λ) . ≤ a(t) It follows

t

E(t; λ) ≤ E(0; λ) exp 0

& |α (s)| + 2|λ| α(s) ds . α(s)

This inequality holds for any solution w(t). If we set w = y1 and w = y2 , then we get for the energies E1 (π; λ) and E2 (π; λ) of that solutions & π |α (s)| + 2|λ| α(s) ds E1 (π; λ) ≤ E1 (0; λ) exp α(s) 0 & π |α (s)| = exp + 2|λ| α(s) ds , α(s) 0 & π |α (s)| + 2|λ| α(s) ds E2 (π; λ) ≤ E2 (0; λ) exp α(s) 0 & π |α (s)| = |λ|2 α(0) exp + 2|λ| α(s) ds , α(s) 0 respectively, since E1 (0; λ) = 1 and E2 (0; λ) = |λ|2 α(0). Hence for the discriminant we have |∆(λ)|

= ≤ ≤

|y1 (λ, π) + y2 (λ, π)| ≤ |y1 (λ, π)| + |y2 (λ, π)| 3 E2 (π; λ) E1 (π; λ) + |λ|2 α(π) & π

1 |α (s)| + |λ| α(s) ds . 1 + α(0)/α(π) exp 2 α(s) 0

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Then for δ = 0 (we reserve δ = 0 in the next estimate for Remark 5.5) we obtain & π α(s)ds + δ |λ| , |∆(λ)| ≤ C(δ) exp 0

which proves the claimed statement. Step 3. If ∆2 (λ) ≤ 4 for any real λ, then there exists a complex number B such that ∆(λ) = 2 cos(Bλ) . Indeed, the function 4−∆2 (λ) does not vanish (Step 1) for non-real λ. Assume 2 that it is nonnegative on R. Then all zeros of 4 − ∆ (λ) are of even multiplicity. 2 Hence, 4 − ∆ (λ) is an entire function, too. According to Step 2 the function 4 − ∆2 (λ) has exponential growth. Remind ∆(0) = 2. Denote f (λ) := 1 − ∆2 (λ)/4 and g(λ) := ∆(λ)/2 , g(0) = 1 . The proof of this step will be completed if we apply the following theorem from complex analysis: Let f and g be entire functions with exponential growth. If f (z)2 + g(z)2 = 1 and g(0) = 1, then there is a complex number B ∈ C such that f (z) = sin(Bz) and g(z) = cos(Bz). To prove the last theorem we consider g + if and g − if , then (g + if )(g − if ) ≡ 1, so that they do not vanish anywhere on C. Therefore g + if = exp(A + Bz) and g − if = exp(−A − Bz) while g(0) = 1 leads to A = 0. Step 4. Completion of the proof. If the statement of the theorem is not true, then by Floquet’s theorem ∆2 (λ) ≤ 4 on [0, ∞). But ∆(λ) is an even function, so that ∆2 (λ) ≤ 4 on R. By Step 3 this implies ∆(λ) = 2 cos(Bλ) for some B ∈ C. It follows 1 B 2 = − (∂λ2 ∆)(0) . 2 Moreover, by Lemma 5.2 we have π 1 2 2 B = − (∂λ ∆)(0) = π α(s)ds . 2 0 By Step 2 the entire function ∆(λ) has exponential growth with type not greater 0π than 0 α(s)ds. Hence, 2 π π π |B| ≤ α(s)ds =⇒ = π α(s)ds ≤ α(s)ds . 0

0

0

At the same time, by the Cauchy-Schwarz inequality 2 π π π α(s)ds ≤ α(s)ds 1ds = π 0

π

0

π

π

α(s)ds = 0

0

α(s)ds

0

2 ,

0

π

α(s)ds

= =⇒

Global Existence for Nonlinear Wave Equations which can hold only for constant function α(t). The theorem is proved.

337

Remark 5.5. Borg’s theorem can be proved for non-negative functions α as well. In this case δ = 0 helps. (See, also, [8].) According to Borg’s theorem for every given λ ∈ Λ ⊂ (0, ∞) any solution of (5.6) can be written in the form w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt ,

C1 , C2 ∈ C ,

where γ is a positive number and p1 (t) and p2 (t) are two π-periodic or π-semiperiodic functions. Then according to the following corollary, ∆(λ) > 2 or ∆(λ) < −2, respectively. Corollary 5.6. If for a given λ ∈ C any solution of (5.6) can be written in the form w(t) = C1 p1 (t)eγt + C2 p2 (t)e−γt ,

C1 , C2 ∈ C ,

where γ is a positive number and p1 (t) and p2 (t) are two π-periodic or π-semiperiodic functions, then ∆(λ) > 2 or ∆(λ) < −2, respectively. Proof. Indeed we have w(0) = w (0) =

C1 p1 (0) + C2 p2 (0) , C1 p1 (0) + C1 p1 (0)γ + C2 p2 (0) − C2 p2 (0)γ .

In particular for the solution y1 this implies y1 (0) = y1 (0) =

a1 p1 (0) + a2 p2 (0) = 0 , a1 p1 (0) + a1 p1 (0)γ + a2 p2 (0) − a2 p2 (0)γ = 1 .

One can easily obtain from the last system of equations p2 (0) , p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) p1 (0) , p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0)

a1

= −

a2

=

since p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) = 0. Similarly from y2 (0) = b1 p1 (0) + b2 p2 (0) = 1 , y2 (0) = b1 p1 (0) + b1 p1 (0)γ + b2 p2 (0) − b2 p2 (0)γ = 0 , it follows b1 b2

p2 (0) − γp2 (0) , p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) p1 (0) + γp1 (0) . = − p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0) =

338

Karen Yagdjian

We denote D = 1/(p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp1 (0)p2 (0)), then ∆(λ)

= y1 (π) + y2 (π) = a1 p1 (π)eγπ + a2 p2 (π)e−γπ + a1 p1 (π)γeγπ − a2 p2 (π)γe−γπ + b1 p1 (π)eγπ + b2 p2 (π)e−γπ = a1 p1 (0)eγπ + a2 p2 (0)e−γπ + a1 p1 (0)γeγπ − a2 p2 (0)γe−γπ + b1 p1 (0)eγπ + b2 p2 (0)e−γπ −p2 (0)p1 (0)eγπ +p1 (0)p2 (0)e−γπ −p2 (0)p1 (0)γeγπ −p1 (0)p2 (0)γe−γπ = + (p2 (0) − γp2 (0))p1 (0)eγπ − (p1 (0) + γp1 (0))p2 (0)e−γπ D = eγπ p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp2 (0)p1 (0) + e−γπ p1 (0)p2 (0) − p2 (0)p1 (0) − 2γp2 (0)p1 (0) D = eγπ + e−γπ .

Hence we have proved that for the periodic functions p1 (t) and p2 (t) one has ∆(λ) > 2. The case of the anti-periodic functions can be considered in a similar way. The corollary is proved. 5.3. Construction of an exponentially increasing solution to Hill’s equation 11. About the structure of the monodromy matrix. According to the next key lemma one can ﬁnd in the instability zone Λ a number λ such that a non-diagonal element of the monodromy matrix does not vanish. Moreover, this property is stable under small perturbations of λ. Lemma 5.7 (K. Yagdjian [41], [42]). Let b(t) deﬁned on R be a non-constant, positive, smooth function which is 1-periodic. Then there exists an open subset Λ0 ⊂ Λ such that b21 = 0 for all λ ∈ Λ0 . Proof. If we assume that b21 = 0, then the function w(t) = x21 (t, 0), where x21 (t, 0) is the (2, 1)-element of the matrix x11 (t, 0) x12 (t, 0) Xλ (t, 0) = , x21 (t, 0) x22 (t, 0) solves equation (5.1) and takes boundary values w(0) = w(1) = 0 . Hence, if λ does not belong to the spectrum of the Sturm-Liouville problem (5.1), w(0) = w(1) = 0, then w(t) vanishes identically. It follows dx21 /dt(0, 0) = x11 (0, 0) = 0 and we get a contradiction to the initial condition x11 (0, 0) = 1 of the deﬁnition of the fundamental solution. It remains to take into consideration that the above mentioned spectrum is discrete. The lemma is proved. Next we use the periodicity of b = b(t) and the eigenvalues µ0 , µ−1 of 0 the matrix Xλ (1, 0) to construct solutions of (5.1) with prescribed values on a discrete set of time. The eigenvalues of the matrix Xλ (1, 0) are µ0 and µ−1 0 with

Global Existence for Nonlinear Wave Equations

339

−1 ∆(λ) = b11 + b22 = µ0 + µ−1 0 . Hence (b11 − µ0 ) + (b22 − µ0 ) = −µ0 + µ0 implies −1 |b11 − µ0 | + |b22 − µ0 | ≥ |(b11 − µ0 ) + (b22 − µ0 )| = |µ0 − µ0 | > 0. This leads to 1 max{|b11 − µ0 |, |b22 − µ0 |} ≥ |µ0 − µ−1 0 | > 0. 2 Without loss of generality we can suppose 1 1 −1 |b22 − µ−1 |b11 − µ0 | ≥ |µ0 − µ−1 0 | > 0, 0 | ≥ |µ0 − µ0 | > 0 , 2 2 because of b11 − µ0 = −(b22 − µ−1 0 ). Further,

1−

b21 b12 − b22 µ0 − b11

µ−1 0

=

1 − b22 µ0 − b11 µ−1 0 + b11 b22 − b21 b12 (µ−1 − b 22 )(µ0 − b11 ) 0

=

1 − b22 µ0 − b11 µ−1 0 +1 −1 (µ0 − b22 )(µ0 − b11 )

=

−1 −1 µ0 µ−1 0 − b22 µ0 − b11 µ0 + µ0 µ0 −1 (µ0 − b22 )(µ0 − b11 )

=

−1 µ0 (µ−1 0 − b22 ) − (b11 − µ0 )µ0 (µ−1 0 − b22 )(µ0 − b11 )

=

µ0 (b11 − µ0 ) − (b11 − µ0 )µ−1 0 (µ−1 − b )(µ − b ) 22 0 11 0

= =

(µ0 − µ−1 0 )(b11 − µ0 ) −1 (µ0 − b22 )(µ0 − b11 ) 1 (µ0 − µ−1 = 0 . 0 ) b22 − µ−1 0

Thus, 1−

b12 b21 1 = (µ0 − µ−1 = 0 . 0 ) µ − b µ−1 − b b − µ−1 11 22 0 22 0 0

Lemma 5.8. [41] Let W = W (t) be a solution to (5.1) with parameter λ such that b21 = 0. Suppose that W = W (t) takes the initial data W (0) = 0 ,

Wt (0) = 1.

Then for every positive integer number M ∈ N one has b21 −M W (M ) = (µM ). 0 − µ0 µ0 − µ−1 0

(5.7)

Proof. Let w = w(t) and z = z(t) be the solutions of (5.1) with initial data w(0) = 1 ,

wt (0) = b12 /(µ0 − b11 ) ,

z(0) = b21 (µ−1 0 − b22 ) ,

Then for every positive integer number M ∈ N they satisfy

wt (M ) = µM w(M ) = µM 0 , 0 b12 /(µ0 − b11 ) , −M z(M ) = µ0 b21 /(µ−1 − b zt (M ) = µ−M . 22 ) , 0 0

zt (0) = 1.

(5.8)

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Karen Yagdjian

Indeed, for the function w(t) we have d d w(0) dt w(M ) dt = X(M, 0) w(M ) w(0)

X(M, M − 1)X(M − 1, M − 2) · · · X(1, 0) 566 7 4

=

M−multipliers

X(1, 0)X(1, 0) · · · X(1, 0) 4 566 7

=

d dt w(0)

d dt w(0)

w(0)

.

w(0)

M−multipliers

The matrix

B=

b12 /(µ0 − b11 ) 1 1 b21 /(µ−1 0 − b22 )

(5.9)

with det B

=

B −1

=

b21 b12 1 − 1 = (µ−1 = 0 , 0 − µ0 ) µ − b µ−1 − b b − µ−1 0 11 22 22 0 0 1 −1 b21 /(µ−1 0 − b22 ) , −1 b12 /(µ0 − b11 ) det B

is a diagonalizer for X(1, 0), that is, µ0 0 B −1 X(1, 0)B = =: M , 0 µ−1 0 It follows

d dt w(M )

X(1, 0)B = B

µ0 0

= X(1, 0)X(1, 0) · · · X(1, 0) 4 566 7

d dt w(0)

.

w(0)

M−multipliers

= B (B 4

w(M )

−1

0 µ−1 0

X(1, 0)B)(B

−1

X(1, 0)B) · · · (B 566

−1

X(1, 0)B) B 7

−1

d dt w(0)

w(0)

M−multipliers

= B MM ·56 ·6· MM7 B 4

−1

d dt w(0)

w(0)

M−multipliers M

= BM B M µ0 =B 0

b12 /(µ0 − b11 ) 1 0 b12 /(µ0 − b11 ) −1 . B 1 µ−M 0

−1

On the other hand −1 b12 b21 − b22 ) −1 b21 /(µ−1 −1 0 −1 B = , −1 b12 /(µ0 − b11 ) µ−1 0 − b22 µ0 − b11

Global Existence for Nonlinear Wave Equations so that B

−1

b12 /(µ0 − b11 ) 1

−1 b12 b21 −1 = −1 µ0 − b22 µ0 − b11 1 = . 0

Hence,

d dt w(M )

341

µM 0 0

b21 (µ−1 0 −b22 )

−1

0 µ−M 0

1 0

−1

b12 (µ0 −b11 )

b12 (µ0 −b11 )

1

= B M b12 /(µ0 − b11 ) 1 µ0 = 0 1 b21 /(µ−1 0 − b22 ) b12 /(µ0 − b11 ) = µM , 0 1

w(M )

and the ﬁrst line of (5.8) is proved. The solution z(t) can be considered in a similar way which leads to d d dt z(M ) dt z(0) = BMM B −1 z(M ) z(0) ⎛ ⎞ M 1 µ0 0 ⎠. b21 B −1 ⎝ = B 0 µ−M −1 0 (µ0 − b22 ) On the other hand 1 −1 B b21 (µ−1 0 −b22 )

−1 b12 b21 = − 1 µ−1 0 − b22 µ0 − b11 0 = 1 implies

d dt z(M )

z(M )

b21 (µ−1 0 −b22 )

−1

−1 b12 (µ0 −b11 )

1 b21 (µ−1 0 −b22 )

0 µM 0 0 1 0 µ−M 0 b12 /(µ0 − b11 ) 1 0 µ−M . 0 1 1 b21 /(µ−1 − b ) 22 0

= =

B

It follows the second line of (5.8). Further, we have W (t) = cw w(t) + cz z(t), and W (0) = cw w(0) + cz z(0) = 0 ,

Wt (0) = cw wt (0) + cz zt (0) = 1 ,

342

Karen Yagdjian

where the numbers cw , cz are easily calculated: z(0) w(0) , cz = − , wt (0)z(0) − w(0)zt (0) wt (0)z(0) − w(0)zt (0) b21 b21 −1 b21 (µ−1 − b ) (µ 22 0 0 − b22 ) cw = , = =− 1 b12 b21 (µ0 − µ−1 −1 0 ) (µ − µ ) − 1 0 0 (µ0 − b11 ) (µ−1 b22 − µ−1 0 0 − b22 ) cw =

cz = −

1 b12 b21 −1 (µ0 − b11 ) (µ−1 0 − b22 )

=

(b22 − µ−1 0 ) −1 . (µ0 − µ0 )

Hence (5.8) and the representation W (t) = w(t)

b21 (b22 − µ−1 0 ) + z(t) −1 −1 (µ0 − µ0 ) (µ0 − µ0 )

imply W (M ) = = =

w(M )

b21 (b22 − µ−1 0 ) −1 + z(M ) −1 (µ0 − µ0 ) (µ0 − µ0 )

b21 b21 (b22 − µ−1 −M 0 ) + µ 0 −1 −1 −1 (µ0 − µ0 ) (µ0 − b22 ) (µ0 − µ0 ) b21 −M (µM ). 0 − µ0 µ0 − µ−1 0

µM 0

The lemma is proved. 5.4. Construction of blow-up solutions In fact equation (4.4) is transformed into the linear wave equation by u s v = G(u) := exp f (r)dr ds . 0

(5.10)

0

Since G ∈ C 2 (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C 2 (a, b)), where we denote a := lim G(u) , u→−∞

b := lim G(u) . u→∞

Since condition (4.2) does not hold, one has a > −∞ or b < ∞. If u(t, x) is a solution of (4.4) and takes initial values (4.6), then the function (5.10) solves the linear equation vtt − b2 (t)∆v = 0 ,

(5.11)

Global Existence for Nonlinear Wave Equations and takes initial values u0 (x) exp v(0, x) = 0

s

f (r)dr ds ,

vt (0, x) = u1 (x) exp

0

343

u0 (x)

f (r)dr

.

0

(5.12) Now let us choose initial data (the positive number S > 2n is ﬁxed) 1 x u0 (x) = ∈ C0∞ , χ MS M2 u0 (x) A x exp − u1 (x) = χ f (r)dr cos(x · y) ∈ C0∞ , MS M2 0 where y ∈ Rn , |y|2 = λ while χ ∈ C0∞ (Rn ) is a non-negative cut-oﬀ function, χ(x) = 1 when |x| ≤ 1. The number A ∈ R, |A| = 1, independent of the large parameter M ∈ N, will be chosen later. Let u = u(t, x) be the classical solution which takes these initial data. Then the function v(t, x) = G u(t, x) solves the equation (5.11) and takes values 1S χ( x2 ) s M M v(0, x) = exp f (r)dr ds ∈ C0∞ , 0

0

vt (0, x)

=

Consider the function 1S M V (t, x) = exp 0

0

A x cos (x · y) ∈ C0∞ . χ MS M2 s

A f (r)dr ds + W (t) S cos(x · y) ∈ C ∞ ([0, ∞] × Rn ) . M

The function V (t, x) solves equation (5.11) while s 1S M A exp f (r)dr ds, Vt (0, x) = S cos(x · y) for all x ∈ Rn . V (0, x) = M 0 0 On the other hand for v(t, x) we have s 1S M A exp f (r)dr ds, vt (0, x) = S cos(x · y) when |x| ≤ M 2 . v(0, x) = M 0 0 The ﬁnite propagation speed in the Cauchy problem (5.11), (5.12) implies V (t, x) = v(t, x)

ΠM := [0, M ] × {x ∈ Rn ; |x| ≤ M 3/2 }

in

for large M . Hence 1S M exp v(t, x) =

s

f (r)dr ds + W (t)

0

0

In particular,

1 MS

v(M, 0) = 0

exp

s

f (r)dr ds +

0

A cos(x · y) in ΠM . MS

b21 A −M (µM ). 0 − µ0 M S µ0 − µ−1 0

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Karen Yagdjian

Assume now that b < ∞. Then global existence of u means u(t,x) s v(t, x) = exp f (r)dr ds b as well as (4.5) for u0 , u1 . The theorem is proved. Open problems: (i) Find all u0 , u1 (possible with small Sobolev norms) such that with these data a classical solution exists globally. (ii) Prove that if u0 ≡ 0, then for every u1 ∈ C0∞ (Rn ) there is no global solution.

6. Coeﬃcient stabilizing to a periodic one. Parametric resonance dominates After studying the periodic case the next question is that for corresponding results for equations with a coeﬃcient stabilizing to a periodic one. Equations with coeﬃcient being a product of a periodic function and a function stabilizing to a constant, are considered in many papers and books (see, e.g., Ch.4 [12]). Following this let us restrict to the model equation (6.1) utt − exp(2tα )b2 (t)∆u + f (u) (ut )2 − exp(2tα )b2 (t)|∇ ∇x u|2 = 0 , t ∈ [1, ∞), x ∈ Rn , for α ∈ R, α < 0. Here f ∈ C ∞ (R) is a real-valued function while b = b(t) is deﬁned on R, a periodic, non-constant, smooth, and positive function. We study the global solvability of the Cauchy problem with data prescribed on t = 1: u(1, x) = u0 (x), ut (1, x) = u1 (x) . (6.2) First we note that condition (4.2), ∞ s exp f (r)dr ds = ∞ 0

0

0

and

s

exp −∞

f (r)dr ds = ∞,

0

is equivalent to the following: for every given numbers n1 , n2 , n3 , n4 ∈ R, s n3 s ∞ exp f (r)dr ds = ∞ and exp f (r)dr ds = ∞. n1

n2

−∞

n4

(6.3) In Section 11 we prove the global existence for small data for the equation (6.1) with at most fast oscillating coeﬃcients, that is for the case α ∈ [1/2, ∞). The next theorem shows that oscillations, which approach for large time the pure periodic behavior, in general break the global existence. For the equation (6.1) with f (u) ≡ 1 see [41], [43]. Theorem 6.1. Let α ∈ (−∞, −1) while b = b(t) deﬁned on R, be a periodic, non-constant, smooth, and positive function. Suppose that (6.3) does not hold.

Global Existence for Nonlinear Wave Equations

345

Then for every n, s, and for every positive δ there are data u0 ∈ C0∞ (Rn ) and u1 ∈ C0∞ (Rn ) such that the inequality u0 (s+1) + u0 (s) ≤ δ

(6.4)

is fulﬁlled, but a solution u ∈ C 2 ([1, ∞) × Rn ) to the Cauchy problem (6.1), (6.2) does not exist. In [30] this statement is proved for equation (6.1) with α ∈ [−1, 0). Open problem: Consider the case α ∈ (0, 1/2). The parametric resonance can be a mechanism of the blow-up phenomenon in hyperbolic systems, too. To illustrate this consider the system

∇x u|2 + cu (ut )2 + du u = 0 , utt − (1 + v 2 )∆u − (1 + v 2 )|∇ 2 −1 vtt − (1 + (1 + u ) )∆v − (1 + (1 + u2 )−1 )|∇ ∇x v|2 + cv (vt )2 + dv v = 0, (6.5) where cu = dv = 1, du = cv = 0. Then for every n, s, p, and for every positive δ there are data u0 , u1 , v0 , v1 ∈ C0∞ (Rn ) such that u1 s,2 + ∇u0 s,2 + u1 s,p + ∇u0 s,p + v1 s,2 + ∇v0 s,2 + v1 s,p + ∇v0 s,p < δ, but a solution (u, v), u, v ∈ C 2 (R+ × Rn ), to the Cauchy problem u(0, x) = u0 (x), ut (0, x) = u1 (x), v(0, x) = v0 (x), vt (0, x) = v1 (x), does not exist. (Here ϕs,p denotes the Sobolev norm of ϕ, the element of the Sobolev space W s,p (Rn ), 1 ≤ p ≤ ∞.) We will prove this in a forthcoming paper.

7. Proof of Theorem 6.1: Perturbation theory 1. Perturbation theory for the fundamental solutions and conclusions Consider again the auxiliary ordinary diﬀerential equation (5.1) with a periodic coeﬃcient b = b(t). As in the previous section, X(t, t0 ) denotes the fundamental solution to the equation (5.1). Then X(t + 1, t) is independent of t ∈ N. Set for t0 ∈ N b11 b12 X(t0 + 1, t0 ) = X(1, 0) = . (7.1) b21 b22 The continuous dependence of the fundamental solution X(t, t0 ) on the coeﬃcient b2 (t) on the interval [t0 , t0 + 1] leads together with Lemma 5.7 to the following conclusions. Lemma 7.1. Let us consider instead of (5.1) the ordinary diﬀerential equation wtt + λ0 c2 (t)w = 0, where c = c(t) is a real-valued smooth function deﬁned on R. Then to each ε > 0 there exists a δ = δ(ε) such that if the continuous function c(t) is a perturbation of b(t) on the interval [T, ∞) in the sense that supt∈[T,∞) |b(t) − c(t)| < δ , then the fundamental solution X(t, t0 ) has at the point t = t0 + 1 a real-valued eigenvalue µ = µ(t0 ) which satisﬁes |µ − µ0 | < ε uniformly for all t0 ∈ [T, ∞). Moreover, one has |b21 | ≥ const > 0 uniformly for all t0 ∈ [T, ∞).

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Karen Yagdjian

2. Representation of solutions of special ordinary diﬀerential equations We use Lemma 7.1 to estimate the solutions for a class of Cauchy problems for special ordinary diﬀerential equations. We prescribe Cauchy data on t = t0 , where for M ∈ N the number t0 = t0 (M ) will be denoted by tM , while the integer number tM is deﬁned as follows: !

1 " &− |α| 1 1 tM := ln 1 + o(1) + 1 , o(1) → 0 as M → ∞ . (7.2) 2 M The sequence o(1) will be speciﬁed more exactly later in Lemma 7.4 by the condition o(1)M |α|−1 → ∞ as M → ∞. The constant M , M > 1, will be chosen later such that tM ∈ [T, ∞), T = T (α, δ), T is large and max t∈[T (α,δ),∞)

| exp(2tα ) − 1| = | exp(2T (α, δ)α ) − 1| < δ(ε),

where |µ0 | − ε > 1, ε > 0. It is obvious that for α ∈ (−∞, −1) 1 max | exp(2tα ) − 1| ≤ o(1) , o(1) → 0 as M → ∞ . t∈[tM ,∞) M Next we construct solutions with prescribed asymptotics on a discrete set of time. Lemma 7.2. Consider the ordinary diﬀerential equation wtt + λ exp(2tα )b2 (t)w = 0 ,

(7.3)

where b(t) deﬁned on R, is a 1-periodic, non-constant, smooth, and positive , (t) be a solution to equation (7.3) with initial data function. Let W , (tM ) = 0 , W

,t (tM ) = 1 . W

Then there are positive numbers C0 , µ ,0 , 1 < µ ,0 < µ0 , and a real-valued function C(M ) such that one has M , (tM + M ) = C(M ), W µ0 ,

|C(M )| ≥ C0 > 0 ,

for every suﬃciently large positive integer number M ∈ N. Proof. Let the matrix-valued function Xk (t, 0), k = 1, . . . , M , be the solution of the Cauchy problem d 0 −λ0 exp(2(tM + k − 1 + t)α )b2 (tM + t) X= X, X(0, 0) = I, 1 0 dt where I is identity matrix. We set Xk (1, 0) =

b11 (k) b12 (k) b21 (k) b22 (k)

.

It is easily seen for the diﬀerences ∆k (t) := Xk (t, 0) − X(t, 0) that with X(t, 0) from (5.2) the following estimate 1 for all t ∈ [0, 1] , ∆k (t) = Xk (t, 0) − X(t, 0) ≤ o(1) M

Global Existence for Nonlinear Wave Equations

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with o(1) → 0 as M → ∞, is fulﬁlled. Using Corollary 7.1 with a given arbitrary small positive ε and T = T (α, δ(ε)) leads to the following two properties for the matrices Xk (1, 0), k = 1, . . . , M : • Each matrix Xk (1, 0) has a real-valued eigenvalue νk = νk (M ) which satisﬁes |νk (M ) − µ0 | ≤ γ0 , where γ0 = (|µ0 | − 1)/2 > 0 uniformly for all M ∈ N suﬃciently large. • For all k = 1, 2, . . . , M , M ∈ N suﬃciently large, the estimate |bij (k) − bij | < ε, i, j = 1, 2, is fulﬁlled. Further, the matrix (5.9) is a diagonalizer for X(1, 0) of (7.1). On the other hand each matrix Xk (1, 0) also has a diagonalizer b12 (k)/(νk − b11 (k)) 1 Bk = . 1 b21 (k)/(νk−1 − b22 (k)) The properties of Xk (1, 0), k = 1, . . . , M , lead to the following estimate for the diagonalizers Bk , k = 1, . . . , M with some constant C2 : Bk − B ≤ o(1/M ) and Bk + Bk−1 ≤ C2 for k = 1, . . . , M.

(7.4)

For any solution w

= w(t)

of the equation (7.3) one can write d d

M + M) w(t

M) dt w(t dt = XM (1, 0)XM−1 (1, 0) · · · X1 (1, 0) . w(t

M + M) w(t

M) If we denote Gk := Bk−1 Bk−1 − I , k ≥ 2 , then d dt w(tM + M ) w(tM + M ) νM−1 0 νM 0 ) = BM (I + G (I + GM−1 ) · · · M −1 −1 0 νM−1 0 νM d ν2 0 ν1 0 w(tM ) −1 dt . (I + G2 ) B1 ×(I + G3 ) 0 ν2−1 0 ν1−1 w(tM ) Taking into consideration the above properties for the diagonalizers Bk , then suﬃciently small ε , suﬃciently large T = T (α, δ(ε)), respectively, lead to Gk = Bk−1 Bk−1 − I ≤ o(1)

1 M

for all k = 2, . . . , M ,

with o(1) → 0 as M → ∞. From (7.4) and from the special choice of the initial data, we see that in order to prove the statement of this lemma we have to consider the (1, 1)-element x11 and the (2, 1)-element x21 of the matrix νM 0 νM−1 0 ν1 0 (I + GM ) · · · (I + G2 ) . −1 −1 0 νM 0 νM−1 0 ν1−1

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Karen Yagdjian

To ﬁnd out the properties of the matrix νM−1 νM 0 ) (I + G M −1 0 νM 0 = νM νM−1 . . . ν2 1 1 0 × ) (I + G M −2 0 0 νM

0 −1 νM−1

0 −2 νM−1

···

ν2 0

···

0 ν2−1

1 0 0 ν2−2

(I + G2 ) (I + G2 )

it is suﬃcient to consider the matrix 1 0 1 0 1 0 (M) a := (I + GM ) · · · (I + G2 ) (I + G1 ). −2 −2 0 νM−1 0 νM 0 ν1−2 Lemma 7.3. Assume that the matrices Gk , k = 1, . . . , M , satisfy Gk ≤ M −1 o(1)

where

o(1) → 0

as

M → ∞,

(7.5)

while |νk | ≥ const > 1, k = 1, . . . , M . Then there is a positive constant such that for suﬃciently large M one has (M)

(M)

(M)

(M)

a11 = 1 + o(1) , a21 = o(1) , a12 = o(1) , |a22 | ≤ q ,

q <1 (7.6)

where o(1) → 0 as M → ∞. Proof. Let us denote for k = 1, . . . , M , 1 0 1 0 1 (k) a := (I + Gk ) · · · (I + G2 ) −2 0 νk−1 0 νk−2 0

0 ν1−2

Then we estimate the norm of a(k) : (k) a ≤ (I + Gk ) · · · (I + G2 ) (I + G1 ) ≤ (1 + Gk ) · · · (1 + G2 )(1 + G1 ) k M 1 C ≤ 1+ o(1) ≤ 1+ M M ≤ C0 , k = 1, . . . , M, where C0 is independent of M . Further we write 1 0 a(k) = (I + Gk )a(k−1) , −2 0 νk

k = 2, . . . , M ,

so that the representation (k)

(k−1)

a11 − 1 = a11

(k−1)

− 1 + a11

(k−1)

Gk,11 + a21

Gk,12

(1)

together with |a11 − 1| = |G1,11 | ≤ Co(1)/M implies 2C o(1) k = 2, . . . , M , M and, step by step we obtain the ﬁrst statement of (7.6). (k)

(k−1)

|a11 − 1| ≤ |a11

− 1| + C0

(I + G1 ).

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To prove the second statement of (7.6) we write a21 = νk−2 (1 + Gk,22 )a21 (k)

(k−1)

+ νk−2 Gk,21 a11

(k−1)

,

a21 = ν1−2 G1,21 . (1)

On the other hand due to (7.5) and to the conditions of the lemma one has νk−2 (1 + Gk,22 ) ≤ 1 (k)

for suﬃciently large M . Step by step we obtain the statements for a21 of (7.6). Consider now for k = 2, . . . , M , the elements (k)

= (1 + Gk,11 )a12

(k)

= νk−2 (1 + Gk,22 )a22

a12

a22

(k−1)

(k−1)

+ Gk,12 a22

(k−1)

(1)

,

a12 = G1,12 ,

+ νk−2 Gk,21 a12

(k−1)

,

a22 = ν1−2 (1 + G1,22 ) . (1)

It is easily seen that with some constant q , q < 1, one has |νk−2 (1 + Gk,22 )| ≤ q for all k = 2, . . . , M , and that the inequalities (k)

|a22 | ≤ q ,

(k) |a12 | ≤ q M −1 o(1) (1 + M −1 o(1))k−2 (1 + 2M −1 o(1)) +(1 + M −1 o(1))k−3 + (1 + M −1 o(1))k−4 + . . . + 1

hold for all M suﬃciently large. These imply the last statements of the lemma. Lemma 7.3 is proved. The completion of the proof of Lemma 7.2 consists of the following steps: Step 1.

First we choose for the solution of (7.3) the initial data w(tM ) = 1 ,

wt (tM ) = b12 (1)/(ν1 − b11 (1))

and obtain d 1 dt w(tM + M ) = νM νM−1 . . . ν1 BM 0 w(tM + M )

1 0 0 (I + GM ) −2 −2 0 νM−1 νM 1 0 1 ×(I + GM−1 ) · · · (I + G3 ) (I + G2 ) . 0 0 ν2−2

We denote 1 0 1 ) Z(M ) = (I + G M −2 0 νM 0

0 −2 νM−1

1 0 (I + GM−1 )· · · (I + G2 ) 0 ν2−2

for the matrix with the properties described by Lemma 7.3. Then we consider the matrix BM Z(M ): b12 (M) b12 (M) νM −b11 (M) z11 (M ) + z21 (M ) νM −b11 (M) z12 (M ) + z22 (M ) . (M) (M) z11 (M ) + ν −1b21 z (M ) z12 (M ) + ν −1b21 z (M ) −b (M) 21 −b (M) 22 M

Hence one can write

22

w(tM + M ) = νM νM−1 . . . ν1 z11 (M ) +

M

22

b21 (M ) z21 (M ) . − b22 (M )

−1 νM

350

Karen Yagdjian

Due to Lemma 7.3 we have for large M z11 (M ) − 1 = o(1),

z21 (M ) = o(1) ,

where o(1) → 0

as M → ∞.

Hence due to Lemma 5.7, Lemma 7.3, and (7.4) w(tM + M ) = νM νM−1 . . . ν1 1 + o(1) , where o(1) → 0 as M → ∞ . On the other hand b (M ) d 12 w(tM + M ) = νM νM−1 . . . ν1 z11 (M ) + z21 (M ) dt νM − b11 (M ) implies with o(1) → 0 as M → ∞, d w(tM + M ) = νM νM−1 . . . ν1 dt Step 2.

b12 (M ) + o(1) . νM − b11 (M )

Then we choose for the solution of (7.3) the initial data v(tM ) = b21 (1)/(ν1−1 − b22 (1)) ,

vt (tM ) = 1 ,

and get d 1 dt v(tM + M ) = νM νM−1 . . . ν1 BM 0 v(tM + M )

1 0 0 ) (I + G M −2 −2 0 νM−1 νM 1 0 0 ×(I + GM−1 ) · · ·(I + G3 ) . ) (I + G 2 1 0 ν2−2

By means of the last column of the matrix BM Z(M ) we obtain b (M ) d 12 v(tM + M ) = νM νM−1 . . . ν1 z12 (M ) + z22 (M ) , dt νM − b11 (M ) b21 (M ) v(tM + M ) = νM νM−1 . . . ν1 z12 (M ) + −1 z22 (M ) . νM − b22 (M ) Lemma 7.3 describes the behavior of z12 (M ) and of z22 (M ) as M → 0. , (t) = cw w(t) + cv v(t), where Step 3. Further, we write W cw

= −v(tM )/(w(tM )vt (tM ) − wt (tM )v(tM )) ,

cv

= w(tM )/(w(tM )vt (tM ) − wt (tM )v(tM )) ,

, (tM + M ): and obtain the following representation for W , (tM + M ) = W

1

νM νM−1 . . . ν1 b12 (1) b21 (1) ν1 − b11 (1) ν1−1 − b22 (1) b21 (1) b21 (M ) ν1−1 − b22 (1) z22 − 1 + o(1) . × −1 −1 ν1 − b22 (1) b21 (1) νM − b22 (M ) 1−

Global Existence for Nonlinear Wave Equations

351

On the other hand we have with positive numbers C0 , q, q1 , independent of M b21 (1) 1 − b12 (1) ≥ C −1 > 0 , |ν −1 − b22 (1)| ≥ C −1 > 0 , 0 1 0 ν1 − b11 (1) ν1−1 − b22 (1) b21 (M ) ν1−1 − b22 (1) z |z22 | ≤ q < 1 b21 (1) ν −1 − b (M ) 22 ≤ q1 < 1 , 22 M

for M suﬃciently large. Lemma 7.2 is proved. 3. Estimate in [1, tM ] of the energy of the solutions

Lemma 7.4. Consider for a positive number λ the ordinary diﬀerential equation wtt + λ exp(2tα )b2 (t)w = 0 ,

(7.7)

where the parameter α ∈ (−∞, −1) while the function b = b(t) deﬁned on R, is 1-periodic, smooth, and positive. Let w(t) be a solution to equation (7.7). Then with o(1) of (7.2) appropriately chosen, for the energy E(w; t) := |wt (t)|2 + λ exp(2tα )b2 (t)|w(t)|2 of the solution w the following estimate holds: E(w; 1) ≤ E(w; tM ) exp (o1 (1)M ) 1 − |α|

with o1 (1) = o(1)

M

1 −1 |α|

(7.8)

→ 0 as M → ∞.

Proof. If we denote a2 (t) := λ exp(2tα )b2 (t), then for the energy E(w; t) := (|wt (t)|2 + a2 (t)|w(t)|2 )/2 we have d E(w; t) = a (t)a(t)w(t)w(t) ¯ . dt Hence,

d E(w; t) ≤ |a (t)| E(w; t) dt a(t)

leads to

E(w; 1) ≤ E(w; tM ) exp

tM

1

|a (t)| dt a(t)

.

Further, 1

tM

|a (t)| dt a(t)

≤ 2|α|

tM

t

α−1

tM

dt + 2

1

1

|b (t)| dt b(t)

≤ 2tα M + 2tM max |b (t)|/b(t) [0,1]

1 − |α|

≤ o(1) implies (7.8). The lemma is proved.

M

1 |α|

352

Karen Yagdjian

Lemma 7.5. For a given positive number ε the function , (t)/(|W , (1)| + |W ,t (1)|) W (ε) (t) := εW solves the equation (7.3) and there exist positive constants q, q1 such that |W (ε) (tM + M )| ≥ εq1 exp(qM ) for all suﬃciently large M . Proof. We use Lemma 7.2 to write with C0 > 0 , (1)| + |W ,t (1)|) |C(M )|ε, µ0 M /(|W # , ; 1) . C0 ε, µ0 M / E(W

|W (ε) (tM + M )| ≥ ≥

, ; 1) we use Lemma 7.4 applied to W ,: Then to estimate E(W # M , ; tM ) exp o1 (1)M |W (ε) (tM + M )| ≥ C0 ε, µ0 / E(W ,0 − o1 (1)/2) . ≥ C0 ε exp M (ln µ It remains to note that µ ,0 − ε1 > 1 with ε1 > 0. The lemma is proved.

4. Construction of blow-up solutions If u(t, x) is a solution of (6.1) and takes initial values u(1, x) = u0 (x) , then the function

ut (1, x) = u1 (x) ,

u(t,x)

s

exp

v(t, x) = G(u(t, x)) := 0

f (r)dr ds

0

solves the linear equation vtt − exp(2tα )b2 (t)∆v = 0 , and takes initial values u0 (x) v(1, x) = exp 0

s

f (r)dr ds ,

(7.9)

vt (1, x) = u1 (x) exp

0

u0 (x)

f (r)dr

.

0

Since G (u) > 0, the inverse transformation, u(t, x) = H v(t, x) , exists, is smooth, and has the property H(0) = 0. Now let us choose the initial data (the positive number S will be chosen later) 1 x A x (ε) u0 (x) = H + W (1) S χ cos (x · y) ∈ C0∞ (Rn ), χ MS M2 M M2 u0 (x) A x (ε) W χ (1) cos (x · y) exp f (r)dr ∈ C0∞ (Rn ), − u1 (x) = t MS M2 0

Global Existence for Nonlinear Wave Equations

353

where y ∈ Rn , |y|2 = λ while χ ∈ C0∞ (Rn ) is a non-negative cut-oﬀ function, χ(x) = 1 when |x| ≤ 1. The parameter A ∈ R, |A| = 1, independent of M , will be chosen later. Due to Lemma 7.5 we have u0 , u1 ∈ C0∞ (Rn ). Moreover, for every given s, p, and δ the inequality (6.4) is fulﬁlled for all suﬃciently large M and S. Assume that the solution u = u(t, x) to the Cauchy problem for equation (6.1) with above chosen initial data u0 (x) and u1 (x) exists for all t ≥ 1 and all x ∈ Rn . For the function v(t, x) = G u(t, x) this leads to the global existence in the Cauchy problem for (7.9) with the initial data 1 x A x + W (ε) (1) S χ cos (x · y) ∈ C0∞ , χ v0 (x) = S 2 M M M M2 A x (ε) cos (x · y) ∈ C0∞ v1 (x) = Wt (1) S χ M M2 prescribed at t = 1. Consider now the function 1 A + W (ε) (t) S cos (x · y) . S M M The function V (t, x) solves the equation V (t, x) =

Vtt − exp(2tα )b2 (t)∆V = 0

in

[1, ∞) × Rn ,

(7.10)

while A 1 + W (ε) (1) S cos (x · y) , S M M A (ε) Vt (1, x) = Wt (1) S cos (x · y) M for all x ∈ Rn . Compare the last relations with v0 and v1 we conclude V (1, x)

=

V (1, x) = v(1, x) ,

Vt (1, x) = vt (1, x) when

|x| ≤ M 2 .

Hence, due to the ﬁnite propagation speed in the Cauchy problem for (7.10), V (t, x) = v(t, x)

in ΠM := [1, tM + M ] × {x ∈ Rn ; |x| ≤ M 3/2 }

for large M . Thus v(t, x) =

1 A + W (ε) (t) S cos(x · y) in MS M

ΠM .

In particular, 1 A + W (ε) (tM + M ) S . MS M ∞ s b := exp f (r)dr ds < ∞ .

v(tM + M, 0) = Now assume that

0

0

Due to Lemma 7.5 we can choose A ∈ R, |A| = 1, such that for M large enough one has v(tM + M, 0) > b as well as (6.4) for u0 , u1 . The theorem is proved.

354

Karen Yagdjian

8. Nonexistence for equations with permanently restricted domain of inﬂuence In this section we give an example of the inﬂuence of the decay of a time-dependent coeﬃcient on the global existence of the solution to nonlinear hyperbolic equations. Namely, for arbitrary small initial data we will construct blowing up solutions. To this end we consider in R+ × Rn the equation utt − t−4 exp(−2t−1 )∆u + f (u)(u2t − t−4 exp(−2t−1 )|∇ ∇x u|2 ) = 0 ,

(8.1)

where u = u(t, x) is a real-valued unknown function, f is a real-valued smooth function. Thus the equation (8.1) has multiple characteristics at inﬁnity, or, in other words, is weakly hyperbolic at inﬁnity, or degenerates at inﬁnity. The conditions on f for which the Cauchy problem for equation (8.1) has a global smooth solution for any smooth initial data are s 0 s ∞ exp f (r)dr ds = ∞ and exp f (r)dr ds = ∞. (8.2) 0

−∞

0

0

The Cauchy problem for (8.1) does not have a global solution for small initial data when f ≡ 1 [36]. The theorem below implies blow-up of the solution of equations with non-constant functions f . To formulate that theorem we need some auxiliary functions v0 and v1 deﬁned by initial data u0 and u1 . We transform the equation (8.1) into the linear equation by change u(t,x) s v(t, x) = G(u(t, x)) := exp f (r)dr ds . 0 ∞

0

Since G ∈ C (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C ∞ (a, b) , where we denote a := lim G(u) ≥ −∞ , u→−∞

b := lim G(u) ≤ ∞ . u→∞

(8.3)

The above transformation applied to equation (8.1) leads to the linear equation vtt − t−4 exp(−2t−1 )∆v = 0

(8.4)

for the function v = v(t, x). Moreover, v takes the initial values v(0, x) = v0 (x) and vt (0, x) = v1 (x), where u0 (x) s u0 (x) v0 (x) = exp f (r)dr ds, v1 (x) = u1 (x) exp f (r)dr . 0

0

0

(8.5) It is easily seen that the supports of v0 and v1 are compact sets, and that v0 (s+1) + v1 (s) is small if and only if u0 (s+1) + u1 (s) is small, provided that s is large enough.

Global Existence for Nonlinear Wave Equations

355

Further, in the next theorem we use notations of Section 6.11.2 from [4] for the conﬂuent hypergeometric function, (0+) 1 −iaπ e Ψ(a, c; z) = Γ(1 − a) e−zt ta−1 (1 + t)c−a−1 dt (8.6) 2πi ∞eiφ −π/2 < φ + arg z < π/2, arg t = φ at the starting point, and Γ(α) is Euler’s function. In what follows the constant γ is Euler’s constant (a positive number belonging to the interval (0,1)) and ψ is the digamma function (psi function of Gauss): ψ(z) := Γ (z)/Γ(z). Theorem 8.1. Suppose that (8.2) does not hold. Assume also that the real-valued functions u0 , u1 ∈ C0∞ (Rn ) fulﬁll the condition . / π ei(x−y)·ξ Im ln |2ξ| + i + ψ(1/2) − 2γ ei|ξ| Ψ(1/2, 1; −2i|ξ|) v0 (y) dy dξ 2 Rn Rn . / + ei(x−y)·ξ εIm ei|ξ| Ψ(1/2, 1; −2i|ξ|) v1 (y) dy dξ =: vst (x) ≡ 0 . Rn Rn

In both cases, if −∞ < a and there is a x0 ∈ Rn such that vst (x0 ) < 0, or if b < ∞ and there is a x0 ∈ Rn such that vst (x0 ) > 0, a solution u ∈ C 2 (R+ × Rn ) to the Cauchy problem with data u(0, x) = u0 (x),

ut (0, x) = u1 (x) ,

x ∈ Rn ,

(8.7)

does not exist. In particular, if we set u0 ≡ 0, then the second assumption of the theorem reads as . / ei(x−y)·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u1 (y) dy dξ ≡ 0 , Rn Rn

or

Rn

. / eix·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u 1 (ξ) dξ ≡ 0 .

The last condition is due to relation (15) of Section 6.9.1 [4] between the conﬂuent hypergeometric function Ψ(1/2, 1; −2i|ξ|) and the Bessel function J0 equivalent to eix·ξ J0 (|ξ|) u1 (ξ) dξ ≡ 0 . Rn

Hence we get an evident consequence of the theorem. Corollary 8.2. Suppose that −∞ < a and b < ∞. For any real-valued function u1 ∈ C0∞ (Rn ), u1 ≡ 0, a solution u ∈ C 2 (R+ × Rn ) to the Cauchy problem with data x ∈ Rn , u(0, x) = 0, ut (0, x) = u1 (x) , does not exist.

356

Karen Yagdjian

Proof. If for a given function u1 ∈ C0∞ (Rn ), u1 ≡ 0, we assume that . / eix·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u 1 (ξ) dξ ≡ 0 , .

Rn

/ then.Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) u 1 (ξ) ≡ 0 implies u1 (x) ≡ 0, since the functions / iz Im e Ψ(1/2, 1; −2iz) and J0 (z) have isolated zeros in z ∈ R+ . According to the next corollary the choice of a function u0 ≡ 0 does not improve essentially the situation even for small data. Corollary 8.3. Suppose that (8.2) does not hold. Then for every n, s, and for every positive δ there are data u0 ∈ C0∞ (Rn ) and u1 ∈ C0∞ (Rn ) such that u0 (s+1) + u1 (s) ≤ δ

(8.8)

but a solution u ∈ C 2 (R+ × Rn ) to the Cauchy problem for (8.1) with data (8.7) does not exist. The next statement shows that for the Cauchy problem for (8.1) with “large datum” u1 one can manage a blow-up at a prescribed time t = t0 > 0. Corollary 8.4. Suppose that −∞ < a and b < ∞. For every given positive t0 one can ﬁnd a real-valued function u1 ∈ C0∞ (Rn ), u1 ≡ 0, such that the solution u(t, x) to the Cauchy problem for (8.1) with data u(0, x) = 0,

ut (0, x) = u1 (x) ,

x ∈ Rn ,

develops a singularity not later than at time t0 . Proof of Theorem 8.1. Consider the ordinary diﬀerential equation y + λ2 (t)|ξ|2 y = 0 ,

(8.9)

where t ∈ [0, ∞), ξ is a parameter ranging over R, while Λ(t) = exp(−t−1 ), λ(t) = t−2 exp(−t−1 ). In accordance with the discussion in Subsection 2.1.2 [40], we apply Liouville transformation y(t, |ξ|) = tf (t, |ξ|) and τ = |ξ|Λ(t) to the unknown function and to the independent variables, respectively. Then equation (8.9) becomes 1 (8.10) fττ + fτ + f = 0 . τ One can use either the properties of the Bessel equation (8.10) and the Bessel function J0 (z), or reduce it to the Kummer equation and make use of hypergeometric functions. We follow the last approach since it gives also a parametrix as well as the propagation of singularities [2], [40]. Let us put z z exp , where z = 2iτ , w(z) = f 2i 2 then (8.10) becomes zw (z) + (1 − z)w (z) − αw(z) = 0,

(8.11)

Global Existence for Nonlinear Wave Equations

357

that is the Kummer equation with γ = 1,

α=

1 . 2

This is the so-called logarithmic case (see, Section 6.7.1 from [4]). In the logarithmic case linear independent solutions can be chosen as follows: w1 = Ψ(α, 1, z) ,

w2 = ez Ψ(1 − α, 1, −z) ,

(8.12)

that is y5 , y7 , respectively, in notations of Section 6.7 [4]. Lemma 8.5 (G. Alexandrian [2]). The functions y1 (t, |ξ|)

= te−iΛ(t)|ξ| Ψ(1/2, 1; 2iΛ(t)|ξ|) ,

y2 (t, |ξ|)

= teiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|)

form for t ≥ 0 a fundamental system of solutions of (8.9). Hence, any solution y(t, ξ) to equation (8.9) can be written as follows: y(t, |ξ|) = p0 (t, |ξ|)y(0, |ξ|) + p1 (t, |ξ|)y (0, |ξ|) . Taking into account of the consideration of Subsection 2.1.2 from [40] we easily obtain for p0 (t, |ξ|) and p1 (t, |ξ|) : π t ' − ln |2ξ| − i + ψ(1/2) − 2γ e−iΛ(t)|ξ| Ψ(1/2, 1; 2iΛ(t)|ξ|) p0 (t, |ξ|) = √ i π 2 ( π + ln |2ξ| + i + ψ(1/2) − 2γ eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) , 2 ( t ' iΛ(t)|ξ| e Ψ(1/2, 1; −2iΛ(t)|ξ|) − e−iΛ(t)|ξ| Ψ(1/2, 1; 2iΛ(t)|ξ|) . p1 (t, |ξ|) = √ i π Thus according to page 108 of [40] we conclude ( ' 2t π p0 (t, |ξ|) = √ Im ln |2ξ| + i + ψ(1/2) − 2γ eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) , 2 π ( ' 2t p1 (t, |ξ|) = √ Im eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) , π and we arrive at v(t, x)

=

2t √ (2π)−n/2 π

Rn

2t + √ (2π)−n/2 π

eix·ξ Im

Rn

'

ln |2ξ| + i

π + ψ(1/2) − 2γ 2

( ×eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v0 (ξ) dξ ' ( eix·ξ Im eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v1 (ξ) dξ .

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Since Λ(t) → 1 as t → ∞, for every v0 , v1 ∈ C0∞ (Rn ) we conclude ' π −n/2 (2π) eix·ξ Im ln |2ξ| + i + ψ(1/2) − 2γ 2 Rn

−n/2

+(2π)

Rn

( ×eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v0 (ξ) dξ ' ( eix·ξ Im eiΛ(t)|ξ| Ψ(1/2, 1; −2iΛ(t)|ξ|) v1 (ξ) dξ

→ vst (x) ∈ C0∞ (Rn ) , where vst (x)

:=

(2π)−n/2

Rn

+(2π)−n/2

π + ψ(1/2) − 2γ 2 ( i|ξ| ×e Ψ(1/2, 1; −2i|ξ|) v0 (ξ) dξ ' ( eix·ξ Im ei|ξ| Ψ(1/2, 1; −2i|ξ|) v1 (ξ) dξ .

eix·ξ Im

Rn

'

ln |2ξ| + i

Further, since the condition (8.2) is not fulﬁlled, for the numbers a and b of (8.3) we have min{|a|, b} < ∞. Hence, for large t due to the condition vst (x) ≡ 0, we / (a, b) for some x0 ∈ Rn . Then a global solution u ∈ C 2 (R+ × Rn ) to get v(t, x0 ) ∈ the Cauchy problem for (8.1) does not exist. The theorem is proved. Concluding remark. We make a conjecture that the results of this section are valid also for the equation (with a(t) ≥ 0) ∇x u|2 = 0 utt − a2 (t)∆u + f (u) u2t − a2 (t)|∇ 0∞ provided that R := 0 a(t)dt < ∞, that is, if the domain of inﬂuence for any point x0 ∈ Rn is permanently restricted. In particular that implies that possible singularities of the initial data at x = x0 are trapped in the ball of radius R centered at x0 ∈ Rn . In that sense the hyperbolic equation behaves like an ordinary diﬀerential equation utt + f (u)u2t = 0.

9. Global existence for a model equation with a polynomially growing coeﬃcient We consider the equation

∇x u|2 = 0 utt − a2 (t)∆u + f (u) u2t − a2 (t)|∇

(9.1)

with a(t) = tl , where l > 0. Here f ∈ C ∞ (R) is a real-valued function. Theorem 9.1. For the Cauchy problem for the equation (9.1) with initial conditions u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

(9.2)

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359

small data solutions exist globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ

(9.3)

the solution u ∈ C ([0, ∞) × R ) to the Cauchy problem (9.1), (9.2) exists. 2

n

Proof. The proof of this theorem consists of four steps. Step 1: Reduction to a linear equation We transform the equation (9.1) into the linear equation by change s u exp f (r)dr ds . v = G(u) := 0

0

Since G ∈ C ∞ (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C ∞ (a, b) , where we denote a := lim G(u) ,

b := lim G(u) .

u→−∞

u→∞

Then the equation (9.1) leads to the linear equation vtt − a2 (t)∆v = 0

(9.4)

for the function v = v(t, x). Moreover, v takes initial values s u0 (x) exp f (r)dr ds , vt (0, x) = u1 (x) exp v(0, x) = 0

0

u0 (x)

f (r)dr

.

0

(9.5) It is easily seen that v0 (s+1) + v1 (s) is small if and only if u0 (s+1) + u1 (s) is small, provided that s is large enough, s > n/2. For the Cauchy problem for the hyperbolic equation (9.4) with the data prescribed at t = 0, v(0, x) = v0 (x) ,

vt (0, x) = v1 (x) ,

(9.6)

there is a representation of its solution (see [40]) v(t, x) = V1 (t, (−∆)1/2 ; l)v0 (x) + V2 (t, (−∆)1/2 ; l)v1 (x) ,

(9.7)

where ∆ is the Laplace operator in Rn .We derive (9.7) in the next step. Step 2: Ordinary diﬀerential equations with a parameter Consider the ordinary diﬀerential equation y + t2l |ξ|2 y = 0

(9.8)

with a positive integer l, while ξ is a parameter ranging over R . Set τ = ωtl+1 |ξ| or τ = Λ(t)|ξ| with ω = 1/(l + 1), Λ(t) := ωtl+1 . Then the function v(τ ; l) = y(t, ξ; l) is a solution of the equation n

d2 v dv +v = 0, + lωτ −1 dτ 2 dτ

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provided that y solves (9.8) (t = 0, ξ = 0). If we introduce the new unknown function z z exp (z = 2iτ ) , w(z) = v 2i 2 then for w(z) equation (9.8) becomes 1 zw (z) + (lω − z)w (z) − ωlw(z) = 0 . 2 If we use the notations γ = lω = l/(l + 1), α = coincides with the following one

1 2 ωl,

(9.9)

then the last equation

zw (z) + (γ − z)w (z) − αw(z) = 0 ,

(9.10)

which is called conﬂuent hypergeometric equation (see, for example, Section 6.3 of [4]). Any solution of this equation is called conﬂuent hypergeometric function. There is a solution F (α; γ; z) to equation (9.10) which can be represented in the form F (α; γ; z) =

1 Γ(γ) Γ(α)Γ(γ − α) (1 − e2πi(γ−α) )(1 − e2πiα ) (1+,0+,1−,0−) × ezζ ζ α−1 (1 − ζ)γ−α−1 dζ

(9.11)

C

with F (α; γ; 0) = 1. The function F (α; γ; z) is an entire function of z. In (9.9), (9.10) the parameter γ is not an integer, so (see Section 6.3 of [4]) another linear independent solution is z 1−γ F (α − γ + 1; 2 − γ; z). Thus for the symbols V1 (t, |ξ|; l) and V2 (t, |ξ|; l), which are introduced in (9.7), we obtain l+1 V1 (t, |ξ|; l) = e−iωt |ξ| F (α; γ; 2iωtl+1 |ξ|) , l+1 V2 (t, |ξ|; l) = te−iωt |ξ| F (1 + α − γ; 2 − γ; 2iωtl+1 |ξ|), or

V1 (t, |ξ|; l) = e−iΛ(t)|ξ| F (α; γ; 2iΛ(t)|ξ|) , V2 (t, |ξ|; l) = ω −ω Λ(t)ω e−iΛ(t)|ξ| F (1 + α − γ; 2 − γ; 2iΛ(t)|ξ|),

where γ = l/(l + 1), α = l/(2(l + 1)), 1 + α − γ = (l + 2)/(2(l + 1)), while F (a; c; z) =

N (a)n (a − c + 1)n −a−n z + O(|z|−a−N −1 ) n! n=0

3 3 − π < arg z < π , 2 2

as |z| → ∞ ,

(a)n := Γ(a + n)/Γ(a) .

Hence the estimates |F (α; γ; 2iΛ(t)|ξ|)| ≤ C|Λ(t)ξ|−l/(2(l+1))

as

−(l+2)/(2(l+1))

|F (1 + α − γ; 2 − γ; 2iΛ(t)|ξ|)| ≤ C|Λ(t)ξ|

Λ(t)|ξ| → ∞ , as

Λ(t)|ξ| → ∞ ,

Global Existence for Nonlinear Wave Equations

361

imply for the multipliers V1 and V2 |V V1 (t, |ξ|; l)| ≤

C|Λ(t)ξ|−l/(2(l+1))

|V V2 (t, |ξ|; l)| ≤

CΛ(t) |Λ(t)ξ| ω

as Λ(t)|ξ| → ∞ ,

−(l+2)/(2(l+1))

as Λ(t)|ξ| → ∞ .

Step 3: L1 − L∞ estimate for the linear equation Theorem 9.2. Assume v0 , v1 ∈ S. Then the solution v = v(t, x) to (9.4) belongs to C ∞ and satisﬁes the a priori estimate v(t, ·)∞ ≤ Cε Dn+ε v0 1 + Dn−ω+ε v1 1 for all ε > 0. Proof. To obtain the required inequality one has to consider two cases (zones) : small Λ(t)|ξ| and large Λ(t)|ξ|. Thus let t ≤ 1 and Λ(t)|ξ| ≤ 1. Then by Plancherel’s formula v(t, ·)2

= V V1 (t, (−∆)1/2 ; l)v0 (x) + V2 (t, (−∆)1/2 ; l)v1 (x)2 = V V1 (t, |ξ|; l)ˆ v0 (ξ) + V2 (t, |ξ|; l)ˆ v1 (ξ)2 ≤ V V1 (t, |ξ|; l)∞ ˆ v0 2 + V V2 (t, |ξ|; l)∞ ˆ v1 2 ≤ C(v0 2 + v1 2 ) .

If t ≤ 1 and Λ(t)|ξ| ≥ 1, then |ξ| ≥ l + 1, and we derive v(t, ·)2

≤

V V1 (t, |ξ|; l)∞ v0 2 + |ξ|ω V2 (t, |ξ|; l)∞ |ξ|−ω vˆ1 2

≤

C(v0 2 + v1 (−ω) ) .

Now the Sobolev inequalities w2 ≤ Cε (Dε+n/2 w1

and w∞ ≤ Cε (Dε+n/2 w2 ,

for w ∈ S lead to the desired estimate. If t > 1 we estimate directly between L1 and L∞ and obtain v(t, ·)∞

≤

V V1 (t, |ξ|; l)ˆ v0 (ξ) + V2 (t, |ξ|; l)ˆ v1 (ξ)1

≤

V V1 (t, |ξ|; l)ξ−n 1 ξn vˆ0 (ξ)∞ +ξ−n+ω V2 (t, |ξ|; l)1 ξn−ω vˆ1 ∞ C Dn v0 1 + Dn−ω v1 1 .

≤ The theorem is proved.

Step 4: Completion of the proof of Theorem 9.1 If we choose u0 and u1 small enough in the corresponding Sobolev norms, then the norms of v0 and v1 are small. Hence, according to Theorem 9.2 the solution v = v(t, x) to the linear Cauchy problem (9.4), (9.5) is small and takes values in the domain of the operator H, v(t, x) ∈ (a, b) for all t ∈ [0, ∞), x ∈ Rn . Due

362

Karen Yagdjian

to (4.8) the smooth function u = H(v) is a global solution to the Cauchy problem. The theorem is proved.

10. An example with an exponentially growing coeﬃcient In this section we show that the approach of the previous section works out also for a model equation with an exponentially growing coeﬃcient. Namely, we consider the equation (10.1) ∇x u|2 = 0 utt − a2 (t)∆u + f (u) u2t − a2 (t)|∇ with a(t) = exp(t). Here f ∈ C ∞ (R) is a real-valued function. Thus we have the following global existence theorem. Theorem 10.1. For the Cauchy problem for the equation (10.1) with initial conditions (10.2) u(0, x) = u0 (x) , ut (0, x) = u1 (x) , small data solution exists globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ

(10.3)

the solution u ∈ C ([0, ∞) × R ) to the Cauchy problem (10.1), (10.2) exists. 2

n

Proof. The proof of this theorem consists of six steps. Step 1: Reduction to a linear equation We transform the equation (10.1) into the linear equation by change s u exp f (r)dr ds . v = G(u) := 0

(10.4)

0

Since G ∈ C ∞ (R) and G > 0, there exists the inverse of G: H := G−1 ∈ C ∞ (a, b), where we denote a := lim G(u) , u→−∞

b := lim G(u) . u→∞

Then the equation (10.1) leads to the linear equation vtt − exp(2t)∆v = 0

(10.5)

for the function v = v(t, x). Moreover, v takes initial values u0 (x) s v(0, x) = exp f (r)dr ds , vt (0, x) = u1 (x) exp 0

0

u0 (x)

f (r)dr .

0

(10.6) It is easily seen that v0 (s+1) + v1 (s) is small if and only if u0 (s+1) + u0 (s) is small, provided that s is large enough.

Global Existence for Nonlinear Wave Equations

363

Step 2: Ordinary diﬀerential equations with a parameter Galstian in [13], [14] suggested the model equation (10.5) to get precise Lp − Lq decay estimates. She gave a representation of the solution to the Cauchy problem by means of Bessel functions. Wirth [35] extended this approach to a model equation depending on a parameter µ vt = 0, v + µ > 0, 1+t that is, a wave equation with weak dissipation, which is obtained from (10.5) after transformation of the time variable if µ = 1. We also follow the papers [13], [14] and do appeal to 0the Bessel equation. If v solves (10.5), then the partial Fourier transform Fp (v) = Rn eixξ v(t, x)dx solves Fp (v)tt + e2t |ξ|2 Fp (v) = 0.

(10.7)

Fp (v)(ξ, 0) = F (v0 )(ξ), Fp (v)t (ξ, 0) = F (v1 )(ξ).

(10.8)

One has also If V = V (t, |ξ|) is a solution of (10.7), then the function W(τ, |ξ|)) = |ξ|V (t, |ξ|) with τ = et |ξ| satisﬁes the Bessel equation 1 Wτ τ + Wτ + W = 0 . τ If J0 (τ ) and Y0 (τ ) are the ﬁrst and second kind Bessel functions, then for the Wronskian W (J J0 (τ ), Y0 (τ )) according to formula (28) of [4], vol. 2, page 91, one has 2 . W (J J0 (τ ), Y0 (τ )) = πτ Hence, the fundamental system for the equation (10.7) can be written in the form π V1 (t, |ξ|) = − |ξ| J0 (et |ξ|)Y Y1 (|ξ|) − Y0 (et |ξ|)J J1 (|ξ|) , (10.9) 2 π V2 (t, |ξ|) = − J0 (et |ξ|)Y Y0 (|ξ|) − Y0 (et |ξ|)J J0 (|ξ|) , (10.10) 2 where V1 (0, |ξ|) = 1, V1,t (0, |ξ|) = 0,

V2 (0, |ξ|) = 0, V2,t (0, |ξ|) = 1 .

Finally, for the solution to the Cauchy problem for the equation (10.5) with the initial conditions v(0, x) = v0 (x) ∈ C0∞ (Rn ) ,

vt (0, x) = v1 (x) ∈ C0∞ (Rn )

(10.11)

we obtain the representation: ' π − |ξ| J0 (et |ξ|)Y Y1 (|ξ|) − Y0 (et |ξ|)J J1 (|ξ|) F (v0 )(ξ) v(t, x) = Fp−1 ( π2 J0 (et |ξ|)Y Y0 (|ξ|) − Y0 (et |ξ|)J J0 (|ξ|) F (v1 )(ξ) . (10.12) + − 2 The last representation allows to get an L1 − L∞ estimate for the solution v = v(t, x) if we make use of the following proposition.

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Karen Yagdjian

Proposition 10.2. The symbol V1 (t, |ξ|) satisﬁes the estimates ⎧ for et |ξ| ≤ 1 , ⎪ C ⎪ ⎪ ⎨ t |V V1 (t, |ξ|)| ≤ for et |ξ| ≥ 1 and |ξ| ≥ 1 , Ce− 2 ⎪ ⎪ 1/2 ⎪ ⎩ ≤ C for et |ξ| ≥ 1 and |ξ| ≤ 1 , C πe2t |ξ|

(10.13)

while for V2 (t, |ξ|) one has ⎧ C(1 + t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ t t C|ξ|−1 e− 2 ≤ Ce− 2 |V V2 (t, |ξ|)| ≤ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C 1 − ln |ξ|

ffor

et |ξ| ≤ 1 ,

for

et |ξ| ≥ 1

and

|ξ| ≥ 1 ,

for

et |ξ| ≥ 1

and

|ξ| ≤ 1 .

(10.14)

The proof of this proposition is given in the next two steps. Step 3: Consideration in the pseudo-diﬀerential zone: et |ξ| ≤ 1 To ﬁnd out the asymptotic behavior at zero of the symbol V1 (t, |ξ|) of (10.9) we use the following standard formulas in a neighborhood of the origin: ∞

(−1)m = m!Γ(m + 2) m=0

J1 (|ξ|)

∞

(−1)m = m!Γ(m + 1) m=0

J0 (|ξ|)

|ξ| 2 |ξ| 2

2m+1 =

|ξ| + O(|ξ|3 ) , 2

|ξ| ∈ [0, 1] ,

2m = 1 + O(|ξ|2 ) ,

|ξ| ∈ [0, 1] ,

and Y0 (|ξ|)

=

Y1 (|ξ|)

=

=

|ξ| |ξ| 2 2 γ + ln J0 (|ξ|) + O(|ξ|2 ) = γ + ln + O(|ξ|2 ) , π 2 π 2 2 |ξ| J1 (|ξ|) ln π 2 1+2l −1 ∞ |ξ| 1 |ξ| ψ(l + 2) + ψ(l + 1) 1 − − (−1)l π 2 π 2 l!(l + 1)! l=0 −1 2 |ξ| |ξ| 1 |ξ| |ξ| ln + O(|ξ|3 ) ln − π 2 2 2 π 2 1 |ξ| (ψ(2) + ψ(1)) + O(|ξ|3 ) , − |ξ| ∈ [0, 1] . π 2

Global Existence for Nonlinear Wave Equations

365

Here the function ψ is the digamma function : ψ(z) := Γ (z)/Γ(z). Thus, for the ﬁrst solution V1 (t, |ξ|) we obtain in the zone et |ξ| ≤ 1 −1 1 |ξ| π 2 |ξ| |ξ| − (γ + t) − |V V1 (t, |ξ|)| = 2 π 2 π 2 1+2l ∞ |ξ| ψ(l + 2) + ψ(l + 1) |ξ| 1 l t 2 + O((e |ξ|) ) , − (−1) π 2 l!(l + 1)! 2 l=0

which leads with |ξ| ≤ e−t to the ﬁnal estimate −1 1 |ξ| 1 |ξ| π (C + t) + ≤C |V V1 (t, |ξ|)| ≤ |ξ| 2 π 2 π 2

for et |ξ| ≤ 1 ,

(10.15)

in the pseudo-diﬀerential zone. Consider now the second solution V2 (t, |ξ|): |ξ| π 2 t 2 |V V2 (t, |ξ|)| = γ + ln J0 (|ξ|) + O(|ξ|) ) J0 (e |ξ|) 2 π 2 t e |ξ| 2 t t 2 − γ + ln J0 (e |ξ|) + O((e |ξ|) ) J0 (|ξ|) π 2 π π J0 (et |ξ|)J = − tJ J0 (|ξ|) + J0 (et |ξ|)O(|ξ|)2 ) − J0 (|ξ|)O((et |ξ|)2 ) . 2 2 Hence, |V V2 (t, |ξ|)| ≤ C(1 + t)

for

et |ξ| ≤ 1 .

(10.16)

The last estimate completes the consideration in the pseudo-diﬀerential zone. Step 4: Consideration in the hyperbolic zone: et |ξ| ≥ 1 We have to split that consideration into two cases. Case et |ξ| ≥ 1, |ξ| ≥ 1. For ν = 0, 1 and large z we have 1/2 ∞ 2 1 1 (−1)m (ν, 2m) cos(z − νπ − π) · Jν (z) ∼ πz 2 4 (2z)2m m=0

Yν (z) ∼

∞ 1 (−1)m (ν, 2m + 1) 1 , − sin(z − νπ − π) · 2 4 (2z)2m+1 m=0 1/2 ∞ 2 1 1 (−1)m (ν, 2m) sin(z − νπ − π) · πz 2 4 (2z)2m m=0 ∞ 1 1 (−1)m (ν, 2m + 1) . + cos(z − νπ − π) · 2 4 (2z)2m+1 m=0

(10.17)

(10.18)

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Karen Yagdjian

Here, following Hankel, we write

(ν, m) :=

Γ(ν + m + 12 ) . m!Γ(ν − m + 12 )

Using (10.17), (10.18) we obtain for the ﬁrst solution after substitution

V1 (t, |ξ|)

1/2 ∞ 2 π π (−1)m t cos(e ) · ∼ − |ξ| |ξ| − 2 πet |ξ| 4 m=0 (2et |ξ|)2m − sin(et |ξ| − ×

2 π|ξ|

1/2

sin(|ξ| −

∞ 3π (−1)m (1, 2m) )· 4 (2|ξ|)2m m=0

+ cos(|ξ| − −

2 πet |ξ|

1/2

sin(et |ξ| −

∞ 3π (−1)m (1, 2m + 1) )· 4 (2|ξ|)2m+1 m=0

∞ π (−1)m )· 4 m=0 (2et |ξ|)2m

+ cos(et |ξ| − ×

2 π|ξ|

1/2

∞ (−1)m π )· 4 m=0 (2et |ξ|)2m+1

∞ (−1)m π )· 4 m=0 (2et |ξ|)2m+1

∞ 3π (−1)m (1, 2m) )· cos(|ξ| − 4 (2|ξ|)2m m=0

∞ 3π (−1)m (1, 2m + 1) )· − sin(|ξ| − . 4 (2|ξ|)2m+1 m=0

Therefore, |V V1 (t, ξ)| ∞ ≤2e−t/2

∞ ∞ ∞ 1 1 2m 2m + 1 . + + (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0 (2|ξ|)2m m=0 (2|ξ|)2m+1 m=0

It follows the second estimate for the function V1 (t, ξ) stated in Proposition 10.2.

Global Existence for Nonlinear Wave Equations

367

Consider now the second solution V2 (t, |ξ|). Using (10.17), (10.18) we obtain V2 (t, |ξ|) 1/2 1/2 2 2 π ∼− 2 πet |ξ| π|ξ| ∞ ∞ (−1)m π (−1)m π t ) · × cos(et |ξ| − ) · − sin(e |ξ| − t 2m 4 m=0 (2e |ξ|) 4 m=0 (2et |ξ|)2m+1 ∞ ∞ π (−1)m π (−1)m × sin(|ξ| − ) · + cos(|ξ| − ) · 4 m=0 (2|ξ|)2m 4 m=0 (2|ξ|)2m+1

∞ ∞ (−1)m π (−1)m π t ) · + cos(e |ξ| − − sin(et |ξ| − ) · 4 m=0 (2|et ξ|)2m 4 m=0 (2et |ξ|)2m+1 ∞ ∞ π (−1)m π (−1)m × cos(|ξ| − ) · − sin(|ξ| − ) · . 4 m=0 (2et |ξ|)2m 4 m=0 (2|ξ|)2m+1

Therefore, |V V2 (t, |ξ|)| 1/2 1/2 2 2 ≤π πet |ξ| π|ξ| ∞ ∞ ∞ ∞ 1 1 1 1 . × + + (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0 (2|ξ|)2m m=0 (2|ξ|)2m+1 m=0 It follows the second estimate for the function V2 (t, ξ) stated in Proposition 10.2.

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Karen Yagdjian

Case et |ξ| ≥ 1, |ξ| ≤ 1. Using (10.17) , (10.18) and the formulas from the beginning of Step 3 we obtain V1 (t, |ξ|) π ∼ − |ξ| 2

2 πet |ξ|

1/2

∞ π (−1)m cos(e |ξ| − ) · 4 m=0 (2et |ξ|)2m t

− sin(et |ξ| − 2 |ξ|

∞ (−1)m π )· 4 m=0 (2et |ξ|)2m+1

|ξ| |ξ| + O(|ξ|3 ) ln 2 2 −1 1 |ξ| 1 |ξ| − (ψ(2) + ψ(1)) + O(|ξ|3 ) − π 2 π 2 1/2 ∞ π (−1)m 2 t )· sin(e |ξ| − − πet |ξ| 4 m=0 (2et |ξ|)2m ×

π 2

ln

+ cos(et |ξ| − Therefore, |V V1 (t, ξ)| ≤

≤

∞ (−1)m |ξ| π × )· + O(|ξ|3 ) . t 2m+1 4 m=0 (2e |ξ|) 2

1/2 ∞

∞ 1 1 + (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0 −1 2 |ξ| |ξ| 1 |ξ| |ξ| 3 ln + O(|ξ| ) ln + × π 2 2 2 π 2 1 |ξ| (ψ(2) + ψ(1) + π) + O(|ξ|3 ) + π 2 1/2 2 . C πet |ξ|

π |ξ| 2

2 πet |ξ|

It follows the last estimate for the function V1 (t, ξ) stated in Proposition 10.2.

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Consider now the second solution V2 (t, |ξ|). Using (10.17), (10.18) and the formulas from the beginning of Step 3 we obtain V2 (t, |ξ|) 1/2 2 2 |ξ| π 2 2 ∼− γ + ln (1 + O(|ξ| )) + O(|ξ| ) 2 πet |ξ| π 2 ∞ ∞ (−1)m π (−1)m π t × cos(et |ξ| − ) · ) · − sin(e |ξ| − 4 m=0 (2et |ξ|)2m 4 m=0 (2et |ξ|)2m+1 ∞ π (−1)m − sin(et |ξ| − ) · 4 m=0 (2|et ξ|)2m

∞ m (−1) π (1 + O(|ξ|2 )) . + cos(et |ξ| − ) · 4 m=0 (2et |ξ|)2m+1

Therefore, |V V2 (t, |ξ|)| 1/2 ∞ ∞ π 1 1 2 ≤ + 2 πet |ξ| (2et |ξ|)2m m=0 (2et |ξ|)2m+1 m=0 |ξ| 2 2 2 2 × γ + ln (1 + O(|ξ| )) + O(|ξ| ) + (1 + O(|ξ| )) π 2 1/2 |ξ| 2 2 2 2 2 ≤C γ + ln (1 + O(|ξ| )) + O(|ξ| ) + (1 + O(|ξ| )) . πet |ξ| π 2 It follows the last estimate for the function V2 (t, ξ) stated in the proposition. This completes the proof of Proposition 10.2.

Step 5: L1 − L∞ estimate for the linear equation Theorem 10.3. Assume v0 , v1 ∈ S. Then the solution v = v(t, x) to the Cauchy problem for the equation (10.5) with initial conditions (10.11) belongs to C ∞ and satisﬁes the a priori estimate v(t, ·)∞ ≤ Cε Dn+ε v0 1 + Dn+ε v1 1 , for all ε > 0.

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Proof. According to Proposition 10.2 one can write for every given positive ε v(t, ·)∞

≤ ≤

V V1 (t, |ξ|)ˆ v0 (ξ) + V2 (t, |ξ|)ˆ v1 (ξ)1 V V1 (t, |ξ|)ˆ v0 (ξ)1 + V V2 (t, |ξ|)ˆ v1 (ξ)1

≤

V V1 (t, |ξ|)ξ−n−ε 1 ξn+ε vˆ0 (ξ)∞ +V V2 (t, |ξ|)ξ−n−ε 1 ξn+ε vˆ1 (ξ)∞ Cε Dn+ε v0 1 + Dn+ε v1 1 ,

≤

that completes the proof of the theorem.

Step 6: Completion of the proof of Theorem 10.1 If we choose u0 and u1 small enough in the corresponding Sobolev norms, then the norms of v0 and v1 are small. Hence, according to Theorem 10.3 the solution v = v(t, x) to the linear Cauchy problem (10.5), (10.6) is small and takes values in the domain of the operator H, v(t, x) ∈ (a, b) for all t ∈ [0, ∞), x ∈ Rn . Due to (4.8) the smooth function u = H(v) is a global solution to the Cauchy problem (10.1), (10.2). The theorem is proved.

11. Fast oscillating coeﬃcients: no resonance ?! The counterexample from [7] shows that if we want the Cauchy problem for linear weakly hyperbolic equations to be C ∞ well-posed, then we have to restrict the oscillations. On the other hand in the theory of linear hyperbolic equations with multiple characteristics one can allow some oscillations in the coeﬃcients. These oscillations are described by means of the monotonic part of the coeﬃcient (see, for instance, [40]). In this section we show how the approach of the previous sections and of [38], [40], works out also for model quasi-linear equations with coeﬃcients simultaneously containing monotonically increasing and oscillating factors. To track down the condition on the uniformly positive and oscillating function b(t) in the equation (11.1) ∇x u|2 = 0 , utt − λ2 (t)b2 (t)∆u + f (u) u2t − λ2 (t)b2 (t)|∇ where f = f (u) is a smooth and real-valued function, λ = λ(t) is a smooth, positive, monotone function, λ (t) ≥ 0, we appeal to the linear weakly hyperbolic equation vtt − λ2 (t)b2 (t)∆v = 0 . In the case λ(0) = 0 a condition on the oscillating term is according to Section 3.11.1 [40] k λ(t) (k) | ln λ(t)| for all t ∈ (0, T ] , k = 0, 1, 2 . (11.2) |b (t)| ≤ ck Λ(t)

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0t Here Λ(t) := 0 λ(τ )dτ . This estimate is a key to the next conditions which guarantee a global existence result for the Cauchy problem for the quasi-linear equation (11.1). The logarithmic term in the condition (11.2) points also on a critical case in the theory of operators with multiple characteristics (Section 3.11.1 [40]). Analogously there is a corresponding critical case in the global solvability problem for the quasi-linear equation (11.1), which reveals itself in the dependence of the global solvability on the dimension of the x variables. That is an interesting point about that critical case that only a micro-local consideration allows to prove the wellposedness for the linear weakly hyperbolic equation with fast oscillations. That micro-local approach developed for linear weakly hyperbolic equations hints at the tools which will be used in this section for nonlinear wave equations. Since some steps of this section are similar to those which are already introduced in the previous section, we will skip the details of many calculations trying to minimize the unnecessary reiterations. To formulate the result we need some assumptions on the functions λ = λ(t) and b = b(t). Following Section 3.11.1 [40] we assume concerning the monotone factor that the derivatives of order k = 0, 1, 2 , satisfy k λ(t) λ (t) λ(t) λ(t) (k) ≤ ≤ c0 for all t ∈ (t0 , ∞), (11.3) , c |λ (t)| ≤ ck λ(t) Λ(t) Λ(t) λ(t) Λ(t) 0t where Λ(t) := 0 λ(s)ds, t0 > 0. The oscillations are narrowed by the condition k λ(t) | ln λ(t)| for all t ∈ (t0 , ∞) , k = 0, 1, 2 . (11.4) |b(k) (t)| ≤ ck Λ(t) Because of the last condition the oscillations are called fast. If λ(0) = 0 (that is the equation (11.1) is weakly hyperbolic), then we assume additionally that (11.3) and (11.4) are fulﬁlled in some neighborhood of t0 = 0, with possibly other constants. Theorem 11.1. Assume that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3) and (11.4), respectively. Moreover, suppose that the dimension n is large enough, that is 1 16 3 + N C12 + n> inf Cb2 Cb,sec + 2C , (11.5) {N | N >2Cb , N >2} 2 3 N where 0 < C0 := min b(t), and the constants C1 , Cb , and Cb,sec are determined by 1 Dt b(t) Λ(t) C1 := max b(t), Cb := inf sup (11.6) , {T | 1≤T <∞, λ(T )≥2} [T,∞) 4 b(t) λ(t) ln λ(t) (t)) 2 1 Dt b(t) Λ2 (t) inf sup Cb,sec := . (11.7) 2 {T | 1≤T <∞, λ(T )≥2} [T,∞) 16 b(t) λ2 (t) ln λ(t) Then for the Cauchy problem for the equation (11.1) with initial conditions u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

(11.8)

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small data solution exists globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ a unique solution u ∈ C 2 ([0, ∞) × Rn ) to the Cauchy problem (11.1), (11.8) exists. Mild oscillations. If we want to avoid the condition (11.5) on the space dimension n, then we narrow possible oscillations. Namely, for the oscillating function b(t) now we suppose with some positive constants C0 , C1 the inequalities C0 ≤ b(t) ≤ C1

for all t ∈ (0, ∞) , λ(t) k ln λ(t) β |b(k) (t)| ≤ Ck+1 for all k = 0, 1, 2, Λ(t)

(11.9) t ∈ [t0 , ∞) ,

(11.10)

respectively, where Ck are suitable non-negative constants and β ∈ [0, 1). Because of the last condition the oscillations are called mild. Further, if the function b = b(t) satisﬁes (11.10) with β = 0, the oscillations are called slow. If λ(0) = 0 (that is the equation is weakly hyperbolic), then we assume additionally that (11.2) and (11.3) are fulﬁlled in some neighborhood (0, T ] of t0 = 0 with β = 1 and possibly other constants. According to the next theorem if the oscillations are mild, then the Cauchy problem with small data has a global solution for every space dimension n > 1. Theorem 11.2. Assume that n > 1 and that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3), (11.9), and (11.10). Then for the Cauchy problem for the equation (11.1) with initial conditions u(0, x) = u0 (x) ,

ut (0, x) = u1 (x) ,

(11.11)

small data solutions exist globally. That is, there are s ∈ N and δ > 0 such that for every given u0 , u1 ∈ C0∞ (Rn ) satisfying the inequality u0 (s+1) + u1 (s) ≤ δ a unique solution u ∈ C 2 ([0, ∞)×Rn ) to the Cauchy problem (11.1), (11.11) exists. The case without oscillations, that is b = const, follows immediately from Theorem 11.2. The proof of Theorem 11.2 can be derived from Theorem 11.1 if we use the “λ − b scaling”. That means the following. For the equation (11.1) with the property (11.10) the numbers Cb and Cb,sec are vanishing. Then, the equation is invariant with respect to the transformation λ(t) → ε−1 λ(t), b(t) → εb(t), where ε is a positive number. That allows to make C1 arbitrary small and to reach condition (11.5) for every given n > 1. After that scaling one can apply Theorem 11.1.

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Example. Consider the equation

utt − exp(2tα )b2 (t)∆u + f (u) (ut )2 − exp(2tα )b2 (t)|∇ ∇x u|2 = 0 ,

(11.12)

t ∈ [0, ∞), x ∈ Rn , with α ∈ R, α > 0. Here f ∈ C ∞ (R) is a real-valued function while b = b(t) is deﬁned on R, a periodic, non-constant, smooth, and positive function. The condition (11.10) is satisﬁed if and only if α > 1/2. According to Theorem 11.2 for such α the Cauchy problem (11.11) with small data has a global solution. The case α = 1/2 is a critical case. According to Theorem 11.1 for that value of α global existence can be guaranteed only for large dimension n of the spatial variable x ∈ Rn . Open problems: It was mentioned in Section 6 that the Cauchy problem for equation (6.1) with α ≤ 0 has no global solution. There is no any Lp − Lq decay estimate (see [29]) for the corresponding linear equation if α ∈ (−∞, 1/2). According to Theorem 11.2 the problem for equation (6.1) has a global solution for small data if α ∈ (1/2, ∞), n > 1. For α = 1/2 it is known for large n only, Theorem 11.1. Thus for the equation (11.12) the case α ∈ (0, 1/2] is still open. The one-dimensional case deserves special attention. The proof of Theorem 11.1 is given in the next section.

12. Linear wave equations with oscillating coeﬃcients The equation (11.1) leads to the linear equation vtt − λ2 (t)b2 (t)∆v = 0

(12.1)

for the function v(t, x) = G(u(t, x)) deﬁned by means of the operator G of (10.4). For the Cauchy problem with the data prescribed at t = t0 v(t0 , x) = v0 (x) ,

vt (t0 , x) = v1 (x) ,

(12.2)

we apply Fourier transform and obtain a representation vˆ(t, ξ) = V1 (t, ξ)ˆ v0 (ξ) + V2 (t, ξ)ˆ v1 (ξ) . The next proposition proved in Section 3.11.1 [40] allows to overcome the diﬃculties generated by the multiplicity of the characteristics at the point t = 0. Proposition 12.1. Suppose that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3) and (11.4), respectively, with t0 = 0. Then for every given T > 0 there exist constants k and CT such that the multipliers V1 (t, ξ) and V2 (t, ξ) of the Cauchy problem with t0 = 0 satisfy the estimate |V V1 (t, ξ)| + |V V2 (t, ξ)| ≤ CT (1 + |ξ|)k

for all t ∈ [0, T ] , ξ ∈ Rn .

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Keeping in mind the last proposition we focus on the case with data given at large t0 . Step 1: Zones. Next we split the dual space into the pseudo-diﬀerential zone Zpd and the hyperbolic zone Zhyp , deﬁned for a given positive number N ≥ 2 by (compare with (2.1.71)[40]) Zpd (N )

= {(t, ξ) ∈ (t0 , ∞) × Rn | Λ(t)|ξ| ≤ N ln λ(t)} ,

(12.3)

Zhyp (N )

= {(t, ξ) ∈ (t0 , ∞) × Rn | Λ(t)|ξ| ≥ N ln λ(t)} .

(12.4)

Then for every given point ξ ∈ Rn we deﬁne the function t = tξ as a solution to the equation Λ(tξ )|ξ| = N ln λ(tξ ) .

(12.5)

Because of the monotonicity of the function Λ(t)/ ln λ(t) the point t = tξ is well deﬁned for all ξ ∈ Rn and depends on N . Indeed, for t0 large enough one has λ2 (t) ln λ(t) > Λ(t)λ (t)

for all t ≥ t0 /2 .

Evidently one has ln Λ(tξ ) ≤ 2 ln N − ln |ξ| for all tξ ≥ t0 /2 , for all tξ ≥ t0 /2 . ln |ξ| ≤ ln N , − ln |ξ| ≤ ln Λ(tξ ) In what follows we also need the ball BR(t0 ) (0) with the radius R(t0 ) < 1 determined by the equation Λ(t0 )R(t0 ) = N ln λ(t0 ) .

(12.6)

Proposition 12.2. Suppose that the functions λ = λ(t) and b = b(t) satisfy the conditions (11.3) and (11.4), respectively. Then the multipliers V1 (t, ξ) and V2 (t, ξ) satisfy for every given small ε > 0 with suﬃciently large positive t0 and N , N > 2C Cb , the following estimates: in the pseudo-diﬀerential zone, 2 |V V1 (t, ξ)| + |ξ||V V2 (t, ξ)| ≤ C(N, t0 , ε)|ξ|−ε − N C1

for all (t, ξ) ∈ Zpd (N ),

while in the hyperbolic zone, for all (t, ξ) ∈ Zhyp (N ), |V V1 (t, ξ)| + |ξ||V V2 (t, ξ)| ⎧ 1 1 16 ⎪ ⎪ − − ε − N C12 − Cb,sec + 2C Cb2 + ε ⎪ ⎪ 3 N ⎨ C(N, t0 , ε)|ξ| 2 for all |ξ| ≤ R(t0 ) , ≤ ⎪ ⎪ 1 ⎪ ⎪ − ⎩ C(N, t0 )λ(t) 2 for all |ξ| ≥ R(t0 ) . The last proposition provides us with the key tool to prove the next theorem that brings an L1 − L∞ estimate for solutions to the linear equation (12.1).

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Theorem 12.3. Assume that the dimension n is suﬃciently large, that is the condition (11.5) is satisﬁed. Then for suﬃciently large t0 the solution v = v(t, x) to the Cauchy problem for equation (12.1) with the conditions v(t0 , x) = v0 (x) ∈ C0∞ (Rn ) ,

vt (t0 , x) = v1 (x) ∈ C0∞ (Rn ) ,

satisﬁes the following a priori estimate: v(t, ·)∞ ≤ C(t0 , ε) Dn+ε v0 1 + Dn−1+ε v1 1 for every ε > 0 with a suitable constant C(t0 , ε). Proof of Theorem 12.3. The Fourier image vˆ = vˆ(t, ξ) of the function v = v(t, x) satisﬁes the inequality ˆ v (t, ·)∞ ≤ v(t, ·)1 . Further, one has V1 (t, ξ)ˆ v0 1 + V V2 (t, ξ)ˆ v1 1 . v(t, ·)∞ ≤ v(t, ·)1 ≤ V On the other hand, v0 1 V V1 (t, ξ)ˆ =

|V V1 (t, ξ)ˆ v0 | dξ +

(t,ξ )∈Zpd (N )

|V V1 (t, ξ)ˆ v0 | dξ (t,ξ )∈Zhyp (N )∩{|ξ|

|V V1 (t, ξ)ˆ v0 | dξ .

+ (t,ξ )∈Zhyp (N )∩{|ξ|≥R(t0 )}

The estimates of Proposition 12.2 and condition (11.5) imply 2 V V1 (t, ξ)ˆ v0 1 ≤ C(N, t0 , ε) |ξ|−ε − N C1 |ˆ v0 (ξ)| dξ (t,ξ )∈Zpd (N )

|ˆ v0 (ξ)|

+ C(N, t0 , ε) (t,ξ )∈Zhyp (N )∩{|ξ|
×|ξ|− 2 −ε−N C1 − 3 (Cb,sec +2Cb +ε) N dξ 1

+ C(N, t0 )λ(t)− 2 1

≤ ≤

2

16

2

1

|ˆ v0 (ξ)| dξ (t,ξ )∈Zhyp (N )∩{|ξ|≥R(t0 )} C(N, t0 , ε)ˆ v0 ∞ + C(N, t0 , ε)ξ−n−ε 1 ξn+ε vˆ0 ∞ C(N, t0 , ε)ξn+ε vˆ0 ∞ .

Thus v0 1 ≤ C(N, t0 , ε)Dn+ε v0 1 . (12.7) V V1 (t, ξ)ˆ Similarly for V V2 (t, ξ)ˆ v1 1 the estimates of Proposition 12.2 and condition (11.5) imply v1 1 V V2 (t, ξ)ˆ

≤ C(N, t0 , ε)ξn−1+ε vˆ1 ∞ .

Finally v1 1 ≤ C(N, t0 , ε)Dn−1+ε v1 1 . (12.8) V V2 (t, ξ)ˆ The last estimate and (12.7) imply the estimate of the statement of theorem. The theorem is proved.

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Proof of Proposition 12.2. Step 2: Matrizant. First we write the equation in form of a system (Here Dt := −id/dt.) 0 1 v v Dt = , λ2 (t)b2 (t)|ξ|2 0 Dt v Dt v and then we make the transformation v W := H(t, ξ) , Dt v

H(t, ξ) :=

λ(t)|ξ| 0

0 1

.

Let us consider the fundamental solution V = V(t, ξ) of the related ﬁrst-order system for the vector-valued function W = t (λ(t)|ξ|v, Dt v): Dt V = A(t, |ξ|)V, where

A(t, |ξ|) :=

V(t0 , ξ) = I ,

0 λ(t)|ξ| λ(t)b2 (t)|ξ| 0

Dt λ(t) + λ(t)

(12.9)

1 0

0 0

.

There is an explicit representation formula for the matrizant: tk−1 t1 ∞ t V(t, |ξ|) = I + A(t1 , |ξ|) dt1 A(t2 , |ξ|) dt2 . . . A(tk , |ξ|) dtk . k=1

t0

t0

t0

(12.10) Finally, for the multipliers V1 (t, ξ) and V2 (t, ξ) we obtain V1 (t, ξ) =

λ(t0 ) V11 (t, |ξ|) , λ(t)

V2 (t, ξ) =

1 V12 (t, |ξ|) , λ(t)|ξ|

where V11 (t, |ξ|) and V12 (t, |ξ|) are the entries of the matrix V(t, |ξ|). Step 3: Estimate in the pseudo-diﬀerential zone. The representation (12.10) leads to the estimate t V(t, |ξ|) ≤ exp A(τ, |ξ|) dτ . t0

This estimate implies in the pseudo-diﬀerential zone the estimate t t 2 λ (τ ) 2 V(t, |ξ|) ≤ exp C1 |ξ| dτ ≤ C(t0 )λ(t)1+N C1 λ(τ ) dτ + t0 t0 λ(τ ) with the constants C(t0 ) and C1 from (11.6). Step 4: Diagonalization in the hyperbolic zone. In this step we once again follow Section 3.11.1 [40]. Let us deﬁne the matrices 1 1 1 b(t) −1 −1 , M (t) := M (t) := . −b(t) b(t) b(t) 1 2b(t)

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Substituting U = M V some calculations transform the system (12.9) into the ﬁrst-order system Dt (λ(t)b(t)) −1 0 1 0 U− U Dt U − λ(t)b(t)|ξ| 0 1 0 1 2λ(t)b(t) 1 Dt λ(t) Dt b(t) 0 1 − − U = 0. 1 0 2 λ(t) b(t) We denote Dt (λ(t)b(t)) Dt (λ(t)b(t)) , τ2 (t, ξ) := λ(t)b(t)|ξ| + . τ1 (t, ξ) := −λ(t)b(t)|ξ| + 2λ(t)b(t) 2λ(t)b(t) With some positive number c1 we have |ττ2 (t, ξ) − τ1 (t, ξ)| ≥ c1 λ(t)|ξ| ,

τk (t, ξ) ∈ S{1, 1, 0}N,t0 ,

1 Dt λ(t) Dt b(t) b12 (t) := − − ∈ S{0, 0, 1}N,t0 . 2 λ(t) b(t) We have used the following deﬁnition (compare with Deﬁnition 2.1.24 [40]).

while

Deﬁnition 12.4. By S{m1 , m2 , m3 }N,t0 we denote the set of all continuous functions a(t, ξ) ∈ C(Zhyp (N )) satisfying m3 λ(t) m1 m2 | ln λ(t)| for all (t, ξ) ∈ Zhyp (N ) , t ≥ t0 . |a(t, ξ)| ≤ Ca |ξ| λ (t) Λ(t) Thus we got the diagonalization mod S{0, 0, 1}N,t0 : Dt V − D(t, ξ)V + B(t)V = 0 , where D(t, ξ) :=

τ1 (t, ξ) 0

0 τ2 (t, ξ)

,

B(t) := −

1 2

Dt λ(t) Dt b(t) − λ(t) b(t)

0 1 1 0

.

We will restrict ourselves to the diagonalization modulo S{−1, −1, 2}N,t0 . Namely, we choose 1 b12 (t)/(ττ1 (t, ξ) − τ2 (t, ξ)) N1 (t, ξ) := , b21 (t)/(ττ2 (t, ξ) − τ1 (t, ξ)) 1 where b12 (t) and b21 (t) are the entries of the symmetric matrix B(t): 1 Dt λ(t) Dt b(t) b12 (t) = b21 (t) = − − . 2 λ(t) b(t) Thus, b12 (t) 1 −n(t, ξ) N1 (t, ξ) := , n(t, ξ) := . n(t, ξ) 1 τ2 (t, ξ) − τ1 (t, ξ) Further, 3 2 for all (t, ξ) ∈ Zhyp (N ) det N1 (t, ξ) = 1 − |n(t, ξ)| ≥ 4

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since for a suﬃciently large t0 and for every given ε > 0 one has b12 (t) τ1 (t, ξ) − τ2 (t, ξ) 1 Dt λ(t) Dt b(t) Λ(t) ln λ(t) ≤ + 2 λ(t) b(t) 2|ξ|Λ(t)λ(t) ln λ(t) 1 Dt λ(t) Dt b(t) Λ(t) 1 N ln λ(t) ≤ sup + N [T,∞) 4 λ(t) b(t) λ(t) ln λ(t) |ξ|Λ(t) ≤

1 N ln λ(t) (C Cb + ε) , N |ξ|Λ(t)

where Cb is deﬁned in (11.6). If 2C Cb < N we obtain 1 b12 (t) < for all (t, ξ) ∈ Zhyp (N ) . |n(t, ξ)| = τ1 (t, ξ) − τ2 (t, ξ) 2 Hence for the matrix N1 (t, ξ) and for the reciprocal matrix 1 1 n(t, ξ) N1−1 (t, ξ) = −n(t, ξ) 1 1 − |n(t, ξ)|2 we obtain N N1−1 (t, ξ) + N N1 (t, ξ) ≤ C

for all (t, ξ) ∈ Zhyp (N )

with some constant C independent of t0 and N . Then we are going to ﬁnd the matrix R1 such that N1 (t, ξ) = N1 (t, ξ)(Dt − D(t, ξ) − R1 (t, ξ)) . (Dt − D(t, ξ) + B(t))N It follows: N1 (t, ξ)R1 (t, ξ) = iN N1t (t, ξ) − [N [N1 (t, ξ), D(t, ξ)] − B(t)N N1 (t, ξ) . On the other hand,

[N1 (t, ξ), D(t, ξ)] = −n(t, ξ)(ττ2 (t, ξ) − τ1 (t, ξ)) [N while

B(t)N N1 (t, ξ) = b12 (t)

so that

N1 (t, ξ)R1 (t, ξ) = int (t, ξ)

n(t, ξ) 1 1 −n(t, ξ)

0 −1 1 0

0 1 1 0

+ n(t, ξ)b12 (t)

1 0 0 −1

since n(t, ξ)(ττ2 (t, ξ) − τ1 (t, ξ)) = b12 (t). Thus R1 (t, ξ) nt (t, ξ) n(t, ξ)b12 (t) n(t, ξ) −1 −1 =i − 2 2 1 n(t, ξ) n(t, ξ) 1 + n (t, ξ) 1 + n (t, ξ)

n(t, ξ) 1

.

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Completion of the proof of Proposition 12.2. Let E2 = E2 (t, r, ξ) be the matrixvalued function E2 (t, r, ξ) ⎛ t

& 1 (λ(s)b(t)) −λ(s)b(s)|ξ| + exp i ds 0 ⎜ λ(s)b(s) r

2i & =⎜ t ⎝ 1 (λ(s)b(t)) λ(s)b(s)|ξ| + 0 exp i ds 2i λ(s)b(s) r where t ≥ r ≥ t0 . Hence, it can be written as follows: ⎛ t 3 λ(s)b(s) d s |ξ| 0 exp −i λ(t)b(t) ⎜ r ⎜ t E2 (t, r, ξ) = λ(r)b(r) ⎝ 0 exp i λ(s)b(s) d s |ξ|

⎞ ⎟ ⎟, ⎠

⎞ ⎟ ⎟. ⎠

r

Then E2 (t, r, ξ), t, r ≥ t0 , solves the Cauchy problem (Dt − D(t, ξ))E(t, r, ξ) = 0,

E(r, r, ξ) = I (identity matrix) .

Thus for t0 such that λ(t0 ) ≥ 1 we conclude 3 + 3 λ(t)b(t) C1 λ(t) ≤ E2 (t, r, ξ) = λ(r)b(r) C0 λ(t0 )

for all

t, r ≥ t0 .

Here C0 := min b(t) ,

C1 := max b(t) .

Further, consider the system (Dt − D(t, ξ) − R1 (t, ξ))W = 0. Let us denote R1 (t, r, ξ) := E2 (r, t, ξ)R1 (t, ξ)E2 (t, r, ξ) when t, r ≥ max{t0 , tξ }. By the aid of R1 we deﬁne the matrix-valued function

Q1 (t, r, ξ) :=

∞ j=1

i

j

t

R1 (t1 , r, ξ)dt1 r

t1

tj−1

R1 (t2 , r, ξ)dt2 . . . r

R1 (tj , r, ξ)dtj r

for t, r ≥ max{t0 , tξ }. The function Q1 (t, r, ξ) solves the following Cauchy problem: Dt Q(t, r, ξ) − R1 (t, r, ξ)Q(t, r, ξ) − R1 (t, r, ξ) = 0 , Q(r, r, ξ) = 0,

t, r ≥ max{t0 , tξ }.

The construction procedure yields for R1 (t, r, ξ) the estimate R1 (t, r, ξ)) ≤ R1 (t, ξ)) .

380

Karen Yagdjian

Then R1 (t, r, ξ) nt (t, ξ) n(t, ξ)b12 (t) n(t, ξ) −1 −1 n(t, ξ) i = − 1 + n2 (t, ξ) 1 n(t, ξ) n(t, ξ) 1 1 + n2 (t, ξ) 4 4 n(t, ξ) −1 −1 n(t, ξ) + n n(t, ξ)b ≤ (t, ξ) (t) t 12 1 n(t, ξ) 3 n(t, ξ) 1 3 4 ≤ (|nt (t, ξ)| + |n(t, ξ)b12 (t)|) . 3 Further 2 1 Dt λ(t) Dt b(t) 1 |b12 (t)|2 |n(t, ξ)b12 (t)| ≤ − = 2|ξ|λ(t) 4 λ(t) b(t) 2|ξ|λ(t) λ(t) ln2 λ(t) , ≤ 4 Cb2 + ε |ξ|Λ2 (t) while |nt (t, ξ)| = ≤

Dt λ(t) Dt b(t) 1 1 ∂ − 4|ξ| ∂ t λ(t) b(t) λ(t) λ(t) ln2 λ(t) 4 Cb,sec + Cb2 + ε . |ξ|Λ2 (t)

Thus R1 (t, r, ξ) ≤

λ(t) ln2 λ(t) 16 . Cb2 + 2ε Cb,sec + 2C 3 |ξ|Λ2 (t)

It follows that for all (t, ξ) ∈ Zhyp (N ) , |ξ| ≤ R(t0 ), one has ∞ ∞ λ(t) ln2 λ(t) 16 Cb,sec + 2C dt . R1 (t, ξ)dt ≤ Cb2 + 2ε 3 |ξ|Λ2 (t) tξ tξ On the other hand, ∞ λ(t) ln2 λ(t) dt |ξ|Λ2 (t) tξ

=

ln2 λ(tξ ) + |ξ|Λ(tξ )

∞

tξ

2

≤

ln λ(tξ ) + Cλ |ξ|Λ(tξ )

2λ (t) ln λ(t) dt |ξ|λ(t)Λ(t) ∞

tξ

2λ(t) ln2 λ(t) dt , |ξ|Λ2 (t)

where for a large T Cλ := sup [T,∞)

λ (t)Λ(t) λ2 (t) ln λ(t)

<

ε . 2

Hence for all (t, ξ) ∈ Zhyp (N ) , |ξ| ≤ R(t0 ), we obtain ∞ 1 1 N ln2 λ(tξ ) 1 1 λ(t) ln2 λ(t) dt ≤ ≤ ln λ(tξ ) . 2 |ξ|Λ (t) 1 − ε N |ξ|Λ(t ) 1 − εN ξ tξ

Global Existence for Nonlinear Wave Equations

381

Similarly, for all (t, ξ) ∈ Zhyp (N ) , |ξ| ≥ R(t0 ), we derive

∞

t0

λ(t) ln2 λ(t) dt ≤ C(t0 ) . |ξ|Λ2 (t)

For the matrix I +Q1 (t, r, ξ) for all (t, ξ) ∈ Zhyp (N ) , |ξ| ≤ R(t0 ), the last estimate leads to t

I + Q1 (t, r, ξ)

≤ exp

R1 (s, r, ξ)ds

≤ ≤ ≤ ≤

tξ

1 1 16 2 Cb,sec + 2C ln λ(tξ ) Cb + 2ε exp 3 1−εN 1 1 16 Cb,sec + 2C Cb2 + 2ε 3 1−εN λ(tξ ) 1 1 16 Cb,sec + 2C Cb2 + 2ε 1−εN Λ(tξ ) 3 1 1 16 Cb,sec + 2C Cb2 + 2ε N ln λ(tξ ) 3 1−εN . |ξ|

Thus for every given ε and for all (t, ξ) ∈ Zhyp (N ), |ξ| ≤ R(t0 ), the inequality 1 16 − Cb,sec + 2C Cb2 + ε N I + Q1 (t, r, ξ) ≤ C(N, t0 , ε)|ξ| 3 is fulﬁlled, while I + Q1 (t, r, ξ) ≤ C(N, t0 ) for all (t, ξ) ∈ Zhyp (N ) , |ξ| ≥ R(t0 ) . Step 5: Estimate in the hyperbolic zone. To complete the estimates in Zhyp (N ) for all |ξ| ≥ R(t0 ) we write: V(t, ξ) = M −1 (t)N N1 (t, ξ)E2 (t, t0 , ξ)(I + Q(t, t0 , ξ))N N1−1 (t0 , ξ)M (t0 )V(t0 , ξ), and conclude V(t, ξ) ≤ CC(N, t0 ) λ(t)

for all

(t, ξ) ∈ Zhyp (N ),

|ξ| ≥ R(t0 ) .

For all |ξ| ≤ R(t0 ) in Zhyp (N ) we write: V(t, ξ) = M −1 (t)N N1 (t, ξ)E2 (t, tξ , ξ)(I + Q(t, tξ , ξ))N N1−1 (tξ , ξ)M (tξ )V(tξ , ξ).

382

Karen Yagdjian

Then

V(t, ξ) ≤ C(N, t0 , ε) λ(t)(I + Q(t, tξ , ξ))V(tξ , ξ) 1 16 Cb,sec + 2C − Cb2 + ε N V(tξ , ξ) ≤ C(N, t0 , ε) λ(t)|ξ| 3 16 1 − Cb2 + ε Cb,sec + 2C 2 N Λ1+N C1 (tξ ) ≤ C(N, t0 , ε) λ(t)|ξ| 3 1 16 Cb,sec + 2C −1 − ε − N C12 − Cb2 + ε 3 N. ≤ C(N, t0 , ε) λ(t)|ξ|

The proposition is proved.

Completion of the proof of Theorem 11.1. Proposition 12.1 and the well-known a priori estimate for strictly hyperbolic equations for every given t0 > 0 imply the existence of a solution to the linear equation (12.1) and an estimate in the Sobolev norms like v(t, ·)2(s+1) + Dt v(t, ·)2(s) ≤ C v(0, ·)2(s+1+k) + Dt v(0, ·)2(s+k) for all t ∈ [0, t0 ]. Then we apply Theorem 12.3 to choose v0 = v(0, x) and v1 = vt (0, x) such that v(t, x) belongs to the domain of the operator H for all t and x. Then u(t, x) = H(v(t, x)) is a global solution to the Cauchy problem (11.1), (11.8). The theorem is proved.

References [1] R. Agemi, Y. Kurokawa, and H. Takamura, Critical curve for p-q systems of nonlinear wave equations in three space dimensions. J. Diﬀerential Equations 167 (2000), no. 1, 87–133 [2] G. R. Aleksandrian, The parametrix and propagation of the wave front of the solution of the Cauchy problem for a model equation. Izv. Akad. Nauk Armyan. SSR Ser. Mat. 19 (1984), no. 3, 219-232, [3] S. Alinhac, Blow-up for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications, 17. Birkh¨ ¨ auser Boston, Inc., Boston, MA, 1995. [4] H. Bateman and A. Erdelyi, Higher Transcendental Functions. vol. 1, 2, McGrawHill, New York 1953. [5] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of diﬀerential equations. Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94. [6] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Diﬀerentialgleichung durch die Eigenwerte. Acta Math. 78 (1946). 1–96. [7] F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C ∞ . Acta Math. 148 (1982), 243–253 [8] F. Colombini and S. Spagnolo, Hyperbolic equations with coeﬃcients rapidly oscillating in time: a result on nonstability. J. Diﬀerential Equations 52 (1984), 24–38.

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[9] D. Del Santo, Global existence and blow-up for a hyperbolic system in three space dimensions. Rend. Istit. Mat. Univ. Trieste 29 (1997), no. 1-2, 115-140 (1998). [10] D. Del Santo, V. Georgiev, and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems. in “Geometrical optics and related topics (Cortona, 1996)”, 117–140, Progr. Nonlinear Diﬀerential Equations Appl., 32, Birkh¨ ¨ auser Boston, Boston, MA, 1997. [11] D. Del Santo and E. Mitidieri, Blow-up of solutions of a hyperbolic system: the critical case. Diﬀerential Equations 34 (1998), no. 9, 1157–1163 (1999). [12] M.S.P. Eastham, The asymptotic solution of linear diﬀerential systems. Clarendon Press, Oxford, 1989. [13] A. Galstian, Lp -Lq decay estimates for the equation with exponentially growing coefﬁcient. Preprint 2001/24, ISSN 1437-739X, Arbeitsgruppe “Partielle Diﬀerentialgleichungen und Komplex Analysis”, Institut f¨ fur Mathematik, Univ. Potsdam, 2001. [14] A. Galstian, Lp -Lq decay estimates for the wave equations with exponentially growing speed of propagation. Appl. Anal. 82 (2003), no. 3, 197-214. [15] H. Hochstadt, Instability intervals of Hill’s equation. Comm. Pure Appl. Math., vol. 17 (1964), 251–255. [16] H. Hochstadt, On the determination of a Hill’s equation from its spectrum. Arch. Ration. Mech. Anal. 19 (1965), 353–362. [17] L. H¨ ¨ ormander, Lectures on nonlinear hyperbolic equations. Springer-Verlag, Berlin et al. 1997. [18] F. John, Nonlinear wave equations, formation of singularities. Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. University Lecture Series, 2. American Mathematical Society, Providence, RI, 1990. [19] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28 (1979), no. 1-3, 235–268. [20] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations. Comm. Pure Appl. Math. 33 (1980), no. 4, 501–505. [21] S. Klainerman, Global existence for nonlinear wave equations. Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. [22] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = −Au + F (u). Trans. Amer. Math. Soc. 192 (1974), 1–21. [23] W. Magnus and S. Winkler, Hill’s equation. Interscience Publishers, New York/London/Sydney, 1966. [24] E. Mitidieri and S. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial diﬀerential equations and inequalities. Tr. Mat. Inst. Steklova 234 (2001), 1–384. [25] S. Mizohata, Theory of Partial Diﬀerential Equations. Cambridge University Press, Cambridge 1973. [26] K. Nakanishi and M. Ohta, On global existence of solutions to nonlinear wave equations of wave map type. Nonlinear Anal. 42 (2000), no. 7, Ser. A: Theory Methods, 1231–1252.

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[27] R. Racke, Lectures on nonlinear evolution equations. Initial value problems. Aspects of Mathematics, E19. Friedr. Vieweg & Sohn, Braunschweig, 1992. [28] M. Reissig and K. Yagdjian, Lp -Lq decay estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coeﬃcients. Math. Nachr. 214 (2000), 71-104. [29] M. Reissig and K. Yagdjian, Lp -Lq estimates for the solutions of strictly hyperbolic equations of second order with time dependent coeﬃcients — Oscillations via growth —. Technische Universitat ¨ Bergakademie Freiberg, Preprint 98-5, 1998, TU Bergakademie Freiberg, Germany. [30] S. Samarchian and K. Yagdjian, Counterexample to the global existence for the wave equation with the speed stabilizing to periodic. Submitted for publication. [31] J. Schaeﬀer, The equation utt − ∆u = |u|p for the critical value of p. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), no. 1-2, 31–44. [32] T.C. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Diﬀerential Equations 52 (1984), no. 3, 378–406. [33] J. Shatah and M. Struwe, Geometric wave equations. Courant Lecture Notes in Mathematics, 2. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. [34] W. Strauss, Nonlinear wave equations. CBMS Regional Conference Series in Mathematics, 73. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1989. [35] J. Wirth, Solution representations for a wave equation with weak dissipation. Math. Methods Appl. Sci. 27 (2004), no. 1, 101–124. [36] J. Wirth, About the solvability behaviour for special classes of nonlinear hyperbolic equations. Nonlinear Anal. 52 (2003), no. 2, 421–431. [37] K. Yagdjian, Necessary and suﬃcient conditions for the Cauchy problem to be wellposed for operators with multiple characteristics. Izv. Akad. Nauk Armyan. SSR Ser. Mat. 20 (1985), no. 1, 3–25. [38] K. Yagdjian, Pseudodiﬀerential operators with a parameter and a fundamental solution of the Cauchy problem for operators with multiple characteristics. Izv. Akad. Nauk Armyan. SSR Ser. Mat. 21 (1986), no. 4, 317–344. [39] K. Yagdjian, Necessary conditions for the well-posedness of the Cauchy problem for operators with multiple characteristics. Soviet J. Contemporary Math. Anal. 23 (1988), no. 5, 36–61. [40] K. Yagdjian, The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach. Akademie Verlag, Berlin, 1997. [41] K. Yagdjian, Parametric resonance and nonexistence of global solution to nonlinear wave equations. Mathematical Research Note, Institute of Mathematics, University of Tsukuba, 99-001 March, 1999. [42] K. Yagdjian, Parametric resonance and nonexistence of global solution to nonlinear hyperbolic equations. Proceedings Workshop “Partial Diﬀerential Equations”, Villa Gualino,Torino, 8-10 Maggio 2000, Edited by L. Rodino, Dipartimento di Matematica Dell’ Universita di Torino, (2000)157–170.

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[43] K. Yagdjian, Parametric resonance and nonexistence of global solution to nonlinear wave equations. Journal of Mathematical Analysis and Applications 260 (2001), no.1, 251–268. [44] V.A. Yakubovich and V.M. Starzhinskii, Linear Diﬀerential Equations with Periodic Coeﬃcients, v.I, II. J.Wiley&Sons, New York, 1975. Karen Yagdjian Department of Mathematics University of Texas – Pan American 1201 W. University Drive Edinburg, TX 78541-2999 USA and Institute of Mathematics National Academy of Sciences of Armenia Marshal Baghramian Avenue 24B Yerevan, 375019 Armenia e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 387–448 c 2005 Birkhauser ¨ Verlag Basel/Switzerland

On the Nonlinear Cauchy Problem Massimo Cicognani and Luisa Zanghirati Abstract. Our aim is to describe how to obtain, with the same procedure, several results of local existence, uniqueness and propagation of regularity for the solution of a quasilinear hyperbolic Cauchy Problem. Mathematics Subject Classiﬁcation (2000). 35L75; 35B65; 35A20. Keywords. Nonlinear hyperbolic Cauchy problem, regularity of solutions.

1. Introduction Let us consider the quasilinear Cauchy problem ⎧ ⎨ P (t, x, Dm u, Dt , Dx )u = f (t, x, Dm u), (QCP ) ⎩ j Dt u|t=0 = gj , 0 ≤ j < m, (t, x) ∈ [0, T ] × Rn , for the operator

P (t, x, Dm u, Dt , Dx ) =

(P )

α aα (t, x, Dm u)Dt,x ,

|α|≤m

α α α u; |α| ≤ m ), m < m, Dt,x = −i∂ ∂t,x . Dm u := (Dt,x We are concerned with the well-posedness of Problem (QCP) and with the propagation of regularity of the solution. For a space X of functions v(x) in Rn , we say that Problem (QCP) is well posed in X if for every gj ∈ X there is T ∗ > 0 such that (QCP) has a unique solution u ∈ C m−1 ([0, T ∗ ]; X). The coeﬃcients aα (t, x, w) and the function f (t, x, w) are deﬁned for t ∈ [0, T ], x ∈ Rn , w ∈ W , W a neighborhood in RN of Cauchy data. Their reg ularity has to be such that for every u ∈ C m ([0, T ]; X) the composed func tion f (t, x, Dm u) belongs to C([0, T ]; X) and such that the linear operator P = P (t, x, Dm u, Dt , Dx ) maps C m ([0, T ]; X) into C([0, T ]; X) continuously.

388

Massimo Cicognani and Luisa Zanghirati

Here X will be C ∞ (Rn ), the Sobolev space H ∞ (Rn ), a Gevrey class γ s , s > 1, ) s s γs = γA , γA = {v(x); vγAs := sup |Dα v(x)|A−|α| |α!|−s < ∞}, x∈Rn , α∈Zn +

A>0

or a Sobolev-Gevrey space (the deﬁnition of such spaces is given in Section 3). In view of the Lax-Mizohata Theorem, recently proved also for nonlinear equations in [51] and [50], we assume that P = P (t, x, Dm u, Dt , Dx ) is hyperbolic, that is the principal symbol Pm (t, x, w, τ, ξ) = aα (t, x, w)(τ, ξ)α |α|=m

does not vanish for (τ, ξ) = (1, 0), so we may assume aα = 1 for α = (m, 0), and has only real roots in the variable τ . For any parametrization of the roots, we have m 1 (τ − λk (t, x, w, ξ)), λk ∈ R (H) Pm (t, x, w, τ, ξ) = k=1

for every t ∈ [0, T ], x ∈ Rn , w ∈ W , ξ ∈ Rn . When all the roots are simple for every (t, x, w, ξ), ξ = 0, we have a strictly hyperbolic operator and the Cauchy problem (QCP) is well posed in all the spaces X that we consider under suitable regularity assumptions with respect to t. In proving this well-known result, e.g., [48], by pseudodiﬀerential operators calculus, one uses the fact that in the parametrization λ1 (t, x, w, ξ) < λ2 (t, x, w, ξ) < . . . < λm (t, x, w, ξ),

ξ = 0

the roots inherit the same regularity of the coeﬃcients aα , in particular they are symbols after a modiﬁcation in a neighborhood of ξ = 0. This fact holds true if Pm (t, x, w, τ, ξ) has d < m distinct roots λ1 (t, x, w, ξ) < λ2 (t, x, w, ξ) < · · · < λd (t, x, w, ξ),

ξ = 0

each one of constant multiplicity mk , k = 1, . . . , d, so that (H) becomes ⎧ d 1 ⎪ ⎪ ⎪ (τ − λk (t, x, w, ξ))mk , λk ∈ R, ⎨ Pm (t, x, w, τ, ξ) = (CM ) k=1 ⎪ ⎪ ⎪ ⎩ λk (t, x, w, ξ) = λk (t, x, w, ξ), k = k , ξ = 0. Linear equations, that is with aα and f independent of w in (QCP) and (P), have been widely studied. The Cauchy problem for a non-strictly hyperbolic (weakly hyperbolic) linear operator P may not be well posed and the lower order terms play an essential role, remaining arbitrary only when dealing with Gevrey classes γ s of index s ≤ r/(r − 1), r being the largest multiplicity, [7], [31]. Every condition for well-posedness is called a Levi condition after the pioneering paper [39]. General Levi conditions for any weakly hyperbolic linear operator have not yet been obtained, but the case of constant multiplicity is quite well understood

On the Nonlinear Cauchy Problem

389

both in C ∞ and in Gevrey classes. Among many others, we quote [42], [27], [49], [8], [9], [34], [30]. The necessary and suﬃcient Levi condition for well-posedness in C ∞ stated in [27] says that for every real solution ϕ(t, x) of ∂t ϕ = λj (t, x, ∇x ϕ)

(E) we have (L)

e−iϕ P (eiϕ ) = O(m−mj ), → +∞, j = 1, . . . , d.

The suﬃcient Gevrey-Levi condition for well-posedness in γ s stated in [30] says that for every real solution ϕ(t, x) of (E), for every real C ∞ function ψ(t, x) and every C ∞ function h(t, x) with compact support we have (Ls)

P (heiϕ+i

1/s

ψ

) = O(m−mj (1−1/s) ), → +∞, j = 1, . . . , d.

We also use the equivalent forms of these conditions given in [41], [34], [30]. A proof of the equivalence is given in [43]. Still dealing with linear equations, another kind of degenerating problem is the case of less than Lipschitz continuous coeﬃcients with respect to the time variable t whose study begins with the paper [20]. Even for strictly hyperbolic operators the Cauchy problem may be not well posed in C ∞ . For instance, taking Holder ¨ continuous coeﬃcients of exponent χ ∈]0, 1[ one has well-posedness only in Gevrey classes γ s with s < 1/(1 − χ) and the threshold for the modulus of continuity in order to have a C ∞ well-posed problem is the Log-Lipschitz regularity |a(t + τ ) − a(t)| ≤ C|τ || log |τ ||, t, t + τ ∈ [0, T ], τ = 0, [20], [24], [11], [1]. Counterexamples show that these results cannot be improved, [20], [24]. As it concerns linear weakly hyperbolic equations with coeﬃcients in C 0,χ , χ ∈]0, 1], the Cauchy problem is well posed in Gevrey classes γ s for s ≤ 1 + (χ/r), see [23], also for counterexamples, and [44],[45]. When the characteristic roots have the same Holder ¨ continuity of coeﬃcients, in particular when they are of constant multiplicity, the bound for s can be enlarged to s < r/(r − χ), [45], [10], [13]. Recently, after the paper [21], another way to weaken the Lipschitz regularity has been considered. Namely, to compare the growth of the ∂t derivative with q powers 1/ |t − t¯| , q ≥ 1, as t tends to a point t¯ ∈ [0, T ], say t¯ = T . For strictly hyperbolic linear problems, we have C ∞ well-posedness assuming that each coeﬃcient in the principal part satisﬁes a condition of the type a ∈ C 1 (0, T [), |a (t)| ≤ C/(T − t), t ∈ [0, T [ whereas we have Gevrey well-posedness in classes γ s , s ≤ q/(q − 1), assuming a ∈ C 1 ([0, T [), |a (t)| ≤ C/(T − t)q , q > 1, t ∈ [0, T [, [21],[36], [22], [29], [12], [13].

390

Massimo Cicognani and Luisa Zanghirati

Now, if on the one hand we ﬁnd a wide range of literature on the degenerating hyperbolic linear Cauchy problem, weakly hyperbolic or with non-Lipschitz coeﬃcients in time, on the other hand this is not true for nonlinear equations. Problem (QCP) was considered in [38] assuming that the full operator P is a product of strictly hyperbolic factors. For such an operator the roots are smooth even if they are not necessarily of constant multiplicity and, more importantly, the linear Cauchy problem in the variable v ⎧ ⎨ P (t, x, Dm u, Dt , Dx )v = f (t, x), (LCP ) ⎩ j Dt v|t=0 = gj , 0 ≤ j < m, is well posed in X = C ∞ and in every Gevrey class X = γ s , s > 1, given any function u such that Dm u ∈ C k ([0, T ]; X), k suﬃciently large to allow all the compositions in the factorization of P . As far as we know, the only paper, before [14], dealing with a class of quasilinear operators whose linearized problem satisﬁes the Levi condition (L) is [28]. As far as Gevrey classes are concerned, we quote [32] where γ s well-posedness was proved for s ≤ r/(r − 1) without any condition on lower order terms, which is the same result as for linear equations. We considered Gevrey-Levi conditions for quasilinear problems in [15]. Nonlinear problems with less than Lipschitz regularity in the time variable have been considered in [33], [35] and, with the method we are going to describe, in [16], [13], [3]. Here we show how to obtain with the same procedure several results of local existence, uniqueness and propagation of regularity of the solution of the quasilinear problem (QCP) under diﬀerent assumptions on the lower order terms of P (t, x, Dm u, Dt , Dx ) and on the regularity of the coeﬃcients aα in (P) with respect to the time variable t. We have a uniﬁed approach to linear equations which allows us to solve the quasilinear problems with a standard argument of ﬁxed point based on energy estimates of strictly hyperbolic type, [46], after having controlled the degeneracy of the problem by means of a reduction to a ﬁrst-order system. Given any suﬃciently smooth u(t, x), following [26], [41] and [37], from (CM) we get a factorization (F )

P = P1 ◦ P2 ◦ . . . ◦ Pd + R,

where each Pk has principal symbol (τ −λk )mk and the remainder R can be taken of any ﬁxed negative order −M performing a ﬁnite number of operations depending on M , m, r and the dimension n. From (F), the scalar problem (QCP) is equivalent to a ﬁrst order system ⎧ ⎨ Dt U − ∆(t, x, U, Dx )U + R(t, x, U, Dx )U = F (t, x, QU ), (S) ⎩ U|t=0 = G,

On the Nonlinear Cauchy Problem

391

with ∆(t, x, w, ξ) a real diagonal matrix of symbols of order 1, R(t, x, w, ξ) of order ∈ [0, 1[, Q = Q(t, x, U, Dx ) of order m − m + r(1 − ) + . The index is determined in the factorization procedure (F) by the Levi conditions on the lower-order terms in P or by the regularity of the coeﬃcients in the principal part and it identiﬁes the largest space of well-posedness for the linear Cauchy problem (LCP) in the scale A ⊂ γ s ⊂ C ∞ , A being the space of all analytic functions. Precisely, we have that (LCP) is well posed in C ∞ for = 0, in γ 1/ for > 0. Then we ﬁnd a unique local solution of (S) in • • C ∞ if the orders of R and Q are not positive that is if = 0 and m ≤ m−r; • • γ s if the orders of R and Q are not larger than 1/s. We obtain the largest s = 1/ assuming m ≤ m − r(1 − 1/s). So, the nonlinear Levi conditions (N LL)

m ≤ m − r

and (N LLs)

m ≤ m − r(1 − 1/s)

appear in a natural way. We describe here ﬁve realizations of our general scheme: A) The operator P satisﬁes conditions (CM), (NLL) and, in addition, (L) for any u ∈ C ∞ ([0, T ] × Rn ), Dm u ∈ W . In particular the linear problem (LCP) is well posed in C ∞ . Then the nonlinear Problem (QCP) is also well posed in H ∞ and C ∞ , see Theorem 2.5 and Theorem 2.19. B) The operator P satisﬁes conditions (CM), (NLLs) and (Ls), this latest for any u ∈ C ∞ ([0, T ]; γ s ), Dm u ∈ W . In particular the linear problem (LCP) is well posed in γ s . Then the nonlinear Problem (QCP) is also well posed in Sobolev-Gevrey spaces of index s and in γ s , see Theorem 3.5 and Theorem 3.15. C) The operator P is strictly hyperbolic and the coeﬃcients aα , |α| = m, of the principal part are smooth in the variables x, w but only Log-Lipschitz ∂t aα | ≤ C/t. In this case the in the variable t or only C 1 in ]0, T ] with |∂ T matrix R in K = ∆ + R in (S) is such that |R(t, x, w, ξ)|dt ≤ δ logξ, 0

ξ = (1 + |ξ|2 )1/2 , and we get the well-posedness of (QCP) in H ∞ and C ∞ (with a δ-loss of derivatives), see Theorem 4.1. D) The operator P satisﬁes condition (CM) and the coeﬃcients aα , |α| = m, of the principal part are regular in the variables x, w, ξ but only H¨ o¨lder continuous of exponent χ < 1 in the variable t. In this case we ﬁnd wellposedness of (2.3) in Sobolev-Gevrey spaces of index s and in γ s , provided that s ≤ r/(r − χ), see Theorem 5.1.

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Massimo Cicognani and Luisa Zanghirati

E) In all these situations we can prove the propagation of analytic regularity of the solution u in domains of inﬂuence whenever the Cauchy data and the equation are locally analytic, Theorems 2.20, 2.22, 3.16. In A), one needs only a ﬁnite diﬀerentiability of the coeﬃcients to solve (QCP) in a ﬁxed Sobolev space H µ . In both cases A) and B), the conditions (L) or (Ls) are satisﬁed for any given function u if and only if the coeﬃcients aα (t, x, w) fulﬁll some universal relations, so that it is possible to check them looking only at the symbol P (t, x, w, τ, ξ) without taking care of the values after the substitution w = Dm u(t, x), see Examples 2.9, 2.10, 3.9 and Remarks 2.12, 3.10. We : can apply our method to a product of operators of constant multiplicity P = j Aj simply performing (F) for each Aj , so we recover the operators of variable multiplicity of [38] as a particular case, taking also into account that the conditions of nonlinear type in [38] are exactly (NLL) and (NLLs). In both cases C) and D) we have no Levi conditions either of linear or of nonlinear type, in particular we can take m = m − 1. In D) the hypothesis (CM) is not essential. What we really need is to be able to choose a regular parametrization of the roots in (H), see [10] and [13] for the linear problem under this more general assumption. As far as the propagation of analytic regularity is concerned, the reduction to the system (S) allows us to apply here the technique used in [2] for strictly hyperbolic equations. We refer to [17] for the case of Gevrey solutions in γ s , s ≤ r/(r − 1), where Levi conditions are not needed, and to [18], [19] for solutions in γ s , s > r/(r − 1), of some semilinear equations. Classes of semilinear weakly hyperbolic equations with non-regular roots are considered in [47], [18], [25].

2. Well-posedness in C ∞ 2.1. Function and symbol spaces

; We use the standard notation H µ (Rn ), µ ∈ R, H ∞ (Rn ) = µ H µ (Rn ), for the usual Sobolev spaces in Rn , often in the shortened form H µ and H ∞ . For k ∈ Z+ , we denote by B k (Rn ) the space of multipliers for H k (Rn ) consisting of all C k functions f in Rn which are bounded together with all their derivatives ∂ α f , |α| ≤ k. We use bounded pseudodiﬀerential operators p(x, D), D = (1/i)∂, in Sobolev spaces with symbol p(x, ξ) ∈ Sm , m ∈ R, ∈ Z+ , deﬁned as the class of all functions in Rn × Rn such that pSm :=

sup

sup |∂ ∂xβ ∂ξα a(x, ξ)|ξ|α|−m < ∞, ξ = (1 + |ξ|2 )1/2

|α|+|β|≤ R2n

On the Nonlinear Cauchy Problem

393

and, as usual, we write Sm . S m = lim ← →+∞

We need to consider the composition

p(x, Dm u(x), ξ), Dm u = (∂ α u; |α| ≤ m ) for a family of symbols p(x, w, ξ) ∈ B ∞ (W ; S m ) depending on a parameter w ∈ W , W is a neighborhood of the origin in RN , and the derivatives Dm u of a function u ∈ H µ+m . Provided µ > n/2, we can write p(x, u(x), ξ) = p0 (x, ξ) + p1 (x, ξ), p0 (x, ξ) = p(x, 0, ξ) ∈ S m , where p1 has a Sobolev regularity H µ with respect to the variable x. This leads us to consider the following spaces as in [48], Chapter IV: Deﬁnition 2.1. We say p(x, ξ) ∈ H M S m provided pH M S m := sup sup Dξα p(·, ξ)H M ξ|α|−m < ∞ |α|≤M

ξ

One needs only ﬁnitely diﬀerentiability of a symbol for continuity in Sobolev spaces H s , when s is bounded, and to perform any ﬁnite number of operations like compositions, adjoint operators, commutators. Besides the well-known results for p(x, ξ) ∈ Sm , in a similar way one can prove: Proposition 2.2. If p(x, ξ) ∈ H M S 0 and M > (n/2) + µ, then p(·, D)uH s ≤ Cµ pH M S m uH s , u ∈ H s , |s| ≤ µ. Proposition 2.3. Given any M , m1 , m2 there is a µ such that, if pj (x, D) ∈ OP H µ S mj , j = 1, 2, we have p1 (x, D)p2 (x, D) ∈ OP H M S m1 +m2 , pj (x, D)∗ ∈ [ 1 (x, D), p2 (x, D)] ∈ OP H M S m1 +m2 −1 . OP H M S mj , and [p So, for a symmetric hyperbolic system ([48]), ⎧ ⎨ ∂t U (t, x) = K(t, x, Dx )U (t, x) + F (t, x), ⎩

(2.1) U (0, x) = G(x),

by the usual energy method one gets: Proposition 2.4. Given any µ there is an M such that, if K(t, x, ξ) ∈ B 0 [0, T ]; H M S 1 and K(t) + K ∗ (t) has a symbol

a(t, x, ξ) ∈ B 0 [0, T ]; H M S 0 ,

then the system (2.1) has a unique solution U ∈ C ([0, T ]; H s) to given G ∈ H s , F ∈ C ([0, T ]; H s ), and supposing |s| ≤ µ. Such a solution satisﬁes for |s| ≤ µ the estimate t 2 2 F (τ )2H s dτ , t ∈ [0, T ], (2.2) U (t)H s ≤ (1 + Ct) GH s + 0

394

Massimo Cicognani and Luisa Zanghirati

where C depends on ﬁnitely many seminorms of K(t) in H M S 1 and of K(t)+K ∗(t) in H M S 0 , on µ and on n. M depends on µ and n. 2.2. Levi conditions Let us consider the quasilinear Cauchy problem ⎧ α ⎪ aα (t, x, Dm u)Dt,x u = f (t, x, Dm u), ⎪ ⎨ ⎪ ⎪ ⎩

|α|≤m

(2.3)

Dtj u(0, x) = gj (x), 0 ≤ j < m,

where t ∈ [0, T ], x ∈ Rn , m < m, gj ∈ H ∞ (Rn ), j = 1, . . . , m − 1, f, aα ∈ B ∞ ([0, T ] × Rn × W ), |α| ≤ m, W a neighborhood in RN of the set {∂ ∂xβ gj (x); x ∈ n R , j + |β| ≤ m }. With a change of the variable u, one can reduce to null Cauchy data, so we may assume that the origin of RN belongs to W without loss of generality. Moreover, since we look for local solutions, we may assume that f (t, x, 0) has compact support. For the principal symbol Pm of the linear operator P (t, x, w, Dt , Dx ) =

m

Pj (t, x, w, Dt , Dx ),

j=0

Pj (t, x, w, Dt , Dx ) =

α aα (t, x, w)Dt,x

|α|=j

we assume

⎧ d 1 ⎪ ⎪ ⎪ P (t, x, w, τ, ξ) = (τ − λj (t, x, w, ξ))mj , ⎪ m ⎪ ⎨ j=1

λ ∈ R, j = 1, . . . , d, ⎪ ⎪ j ⎪ ⎪ ⎪ ⎩ λh = λk for every (t, x, w, ξ), ξ = 0, h = k,

(2.4)

so, in particular, P is hyperbolic. The largest multiplicity r = max{mj ; j = 1, . . . , d} plays an important role. Notice that r = 1 means that P is strictly hyperbolic and the well-posedness in H ∞ of the Cauchy problem (2.3) is well known in this case. When r ≥ 2, assumption (2.4) is not suﬃcient in general, even for linear problems that is with aα and f independent of w. So we impose further conditions on the lower-order terms, starting from the nonlinear one m ≤ m − r.

(2.5)

On the Nonlinear Cauchy Problem

395

Then, for any smooth u such that Dm u takes values in W , we require that the linear problem in the unknown v ⎧ ⎨ P (t, x, Dm u, Dt , Dx )v(t, x) = f0 (t, x), ⎩

Dtj v(0, x) = gj (x), 0 ≤ j < m,

satisﬁes the necessary and suﬃcient Levi condition for well-posedness in C ∞ stated in [27], that is for every real solution ϕ(t, x) of

∂t ϕ = λj (t, x, Dm u, ∇x ϕ) we have e−iϕ P (eiϕ ) = O(m−mj ), → +∞, j = 1, . . . , d. (2.6) There are several equivalent forms of the linear Levi condition, e.g., [42], [41], [34]. We use some of them in (2.11) and in the Examples 2.9, 2.10 after Theorem 2.5. There we show that to require (2.6) for any given u leads to analytic relations for the coeﬃcients aα (t, x, w), |α| > m − r. It is clear that the terms Pj of P with j ≤ m − r are not involved in (2.6). Now we can state the main result of this section: Theorem 2.5. Assume that conditions (2.4), (2.5) and (2.6) are fulﬁlled. Then, for every gj ∈ H ∞ , j = 1, . . . , m − 1, there is T ∗ ≤ T such that the Cauchy problem (2.3) has a unique solution u ∈ C ∞ ([0, T ∗ ]; H ∞ ) . Remark 2.6. In fact we prove that there is s0 such that, for any gj ∈ H s+m−1−j , j = 1, . . . , m − 1, with s > s0 , there is a unique solution u∈

m−1 2

C j [0, T ∗ ]; H s+m−r−j .

j=0

The solution satisﬁes the estimate ⎧ m−1 ⎪ ⎪ ∂ ⎪ ∂tj u(t, ·)2H s+m−r−j ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎡ ⎤ t m−1 ⎪ ≤C⎣ ⎪ gj 2H s+m−1−j + f (τ, ·, Dm−r u(τ, ·))2H s dτ ⎦ , ⎪ ⎪ ⎪ 0 ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t ∈ [0, T ∗].

(2.7)

Remark 2.7. Using the condition (2.4) and the estimate (2.7), one can prove that the equation has cones of dependence, e.g., [32]. In fact, there is a positive constant c such that for every (t0 , x0 ), 0 < t0 < T , if u1 and u2 are solutions in the backward cone Γ = {(t, x); 0 ≤ t ≤ t0 , |x − x0 | ≤ c(t0 − t)} (2.8)

396

Massimo Cicognani and Luisa Zanghirati

with base ω0 = {x; |x − x0 | ≤ ct0 }, then Dtj u1 = Dtj u2 in ω0 , j = 0, . . . , m − 1 =⇒ = u1 = u2 in Γ.

(2.9)

Remark 2.8. Theorem 2.5 holds with ﬁnite diﬀerentiability in the time variable of the coeﬃcients, f, aα ∈ B k ([0, T ]; B ∞(Rn × W )), |α| ≤ m, with k suﬃciently large.

Example 2.9. Let Qm−1 = Qm−1 (t, x, Dm u, τ, ξ) denote the subprincipal symbol of P : n Qm−1 = Pm−1 + (i/2)∂ ∂τ ∂t Pm + (i/2) ∂xj ∂ξj Pm . j=1

For r = 2, the linear Levi condition (2.6) is equivalent to Qm−1 ≡ 0 for τ = λj , mj = 2. From this analytic relation for the coeﬃcients, one can construct examples. For instance, P = (τ 2 − a2 (u)ξ 2 )2 + b(u, Du)τ 3 + c(u, Du)ξ 3 (n = 1, a(u) = 0) is the symbol of an operator which satisﬁes all the assumptions of Theorem 2.5 if and only if b(u, Du) = −(4/a(u))a (u)Dt u,

c(u, Du) = 4a3 (u)a (u)Dx u.

Example 2.10. From [40], the operator P satisﬁes condition (2.4) if and only if it can be factorized as follows: r

P = Ar11 · · · Add + Rm−1 , where the Aj ’s are strictly hyperbolic diﬀerential operators and the remainder Rm−1 is of order m−1. We have {m1 , . . . , md } = {r1 , . . . , rd }, so r = max1≤j≤d rj . For 1 ≤ k ≤ r, let us deﬁne (r −k)+

Qk = A1 1

(r −k)+

· · · Ad d

, (z)+ = max{z, 0}, A0 = I.

P satisﬁes also the Levi condition (2.6) if (and only if in case of analytic coeﬃcients) P = Q0 +

m

Bj Qj ,

j=1

where Bj denotes an arbitrary diﬀerential operator of order m − j − ord Qj , [26], [34], [30], [43]. For instance, an operator with triple and double roots which satisﬁes all the assumptions of Theorem 2.5 is given by P = A31 A22 + B1 A21 A2 + B2 A1 + B3 , where A1 = A1 (t, x, Dµ1 −1 u, Dt , Dx ) and A2 = A2 (t, x, Dµ2 −3 u, Dt , Dx ) are strictly hyperbolic operators of respective orders µ1 and µ2 , m = 3µ1 + 2µ2 , and

On the Nonlinear Cauchy Problem

397

Bj = Bj (t, x, Dm−3 u, Dt , Dx ), j = 1, 2, 3, are arbitrary operators of respective orders µ1 + µ2 − 1, 2µ1 + 2µ2 − 2, m − 3. The orders of the nonlinear terms µ1 in A1 , µ2 − 3 in A2 , m − 3 in Bj , j = 1, 2, 3, are determined by the nonlinear Levi condition (2.5). There is a small intersection between Theorem 2.5 and the results of [28]. The proof that we are going to give through Subsections 2.3, 2.4, 2.5 and 2.6, is a slight modiﬁcation of that one given in [14]. 2.3. Factorization Let u∈

k0 +m−r 2

C j H µ0 +m−r−j , µ0 > n/2,

j=0

be a given function with Dm−r u(t, x) ∈ W , W is a neighborhood of the origin in RN . For a family of symbols p(t, x, w, ξ) ∈ B ∞ (W ; S m ) depending on parameters w ∈ W , t ∈ [0, T ], we have p(t, x, Dm−r u(x), ξ) = p0 (t, x, ξ) + p1 (t, x, Dm−r u, ξ), p0 (t, x, ξ) = p(t, x, 0, ξ) with p0 (t) ∈ S m , p1 (t) ∈ H µ0 S m so p(t) ∈ S m + H µ0 S m . Let the linear operator P , P (t, x, Dm−r u, Dt , Dx ) =

α aα (t, x, Dm−r u)Dt,x ,

|α|≤m

aα ∈ B k0 ([0, T ]; B ∞(Rn × W )) , satisfy the hyperbolicity condition (2.4) and the linear Levi condition (2.6). After a modiﬁcation in a neighborhood of ξ = 0, the d distinct characteristic roots are symbols λj (t) ∈ S 1 + H µ0 S 1 , so one can carry out the factorization (2.4) to the operators level. From Proposition 2.3, for any ﬁxed µ > n/2 one can take k0 and µ0 = µ + M0 + m − r large enough in order to have P = (Dt − λd )md ◦ · · · ◦ (Dt − λ2 )m2 ◦ (Dt − λ1 )m1 + R(0) , R(0) =

m

(0)

(0)

rj Dtm−j , rj

(0)

= rj (t, x, DM0 +m−r u, Dx ),

j=1 (0)

rj

∈ C [0, T ]; S j−1 + C [0, T ]; H µS j−1 ,

398

Massimo Cicognani and Luisa Zanghirati

where k0 depends only on m, r and n, M0 depends only on µ, m, r and n. This is the ﬁrst step of a well-known factorization performed in [41] and [37] for P with smooth coeﬃcients till to obtain a regularizing remainder of order −∞. In order to allow ﬁnitely many operations, we are going to assume each time k0 ≥ kj ≥ kj−1 , µ0 ≥ µj ≥ µj−1 , µj = µ + Mj , j = 1, 2, . . ., where kj depends only on m, r and n, Mj depends only on µ, m, r and n. In particular, performing ﬁnitely many steps of the factorization of [41] and [37], we obtain: Proposition 2.11. Let P satisfy conditions (2.4) and (2.6). For every µ there are k1 and µ1 = µ + M1 such that, if k0 ≥ k1 and µ0 ≥ µ1 , then ⎧ m ⎪ ⎪ P = P ◦ · · · ◦ P ◦ P + R, R = r Dtm− , ⎪ d 2 1 ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (j) (j) Pj = (Dt − λj )mj + a1 (Dt − λj )mj −1 + · · · + amj , ⎪ ⎪ ⎪ ⎪ (j) (j) ⎪ M +m−r ⎪ u, Dx ) ∈ C [0, T ]; S 0 + C [0, T ]; H µ S 0 , ⎪ ak = ak (t, x, D 1 ⎪ ⎪ ⎪ ⎪ ⎩ r = r (t, x, DM1 +m−r u, Dx ) ∈ C [0, T ]; S −m + C [0, T ]; H µ S −m . (2.10) Remark 2.12. One needs only condition (2.4) to get the factorization of P with (j)

= max{(ord ak )/(mj − k); 1 ≤ k ≤ mj , 1 ≤ j ≤ d} strictly smaller than 1. The index is invariant under permutations of the indices 1, 2, . . . , d and the Levi condition (2.6) is equivalent to ≤ 0.

(2.11)

2.4. The linear problem From the factorization (2.10), we can reduce the linear scalar equation in the unknown v P (t, x, Dm−r u, Dt , Dx )v = f (2.12) to an equivalent system. Without loss of generality, but only to have a simpler notation, let us consider the case d = 2 of an operator with two multiple characteristic roots. Applying Proposition 2.11, we have both P = P2 ◦ P1 + R and, by permutation, ˜ P = P˜1 ◦ P˜2 + R. (j) (j) The factors Pj and P˜j have respective coeﬃcients ak and a ˜k of order 0 whereas ˜ have respective coeﬃcients r and ˜ of order − m. Then, the remainders R and R

On the Nonlinear Cauchy Problem

399

given the scalar function v, let us deﬁne the vector V = (v0 , . . . , vm−1 , vm , . . . , v2m−1 ) by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

v0 = v v1 = (Dt − λ1 )v .. .

vm1 −1 = (Dt − λ1 )m1 −1 v vm1 = P1 v P1 v vm1 +1 = (Dt − λ2 )P .. . vm−1 = (Dt − λ2 )m2 −1 P1 v

vm = v vm+1 = (Dt − λ2 )v .. . vm+m2 −1 = (Dt − λ2 )m2 −1 v vm+m2 = P˜2 v vm+m2 +1 = (Dt − λ1 )P˜2 v .. . v2m−1 = (Dt − λ1 )m1 −1 P˜2 v.

(2.13)

The scalar equation (2.12) is equivalent to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(Dt − λ1 )vj = vj+1 (0 ≤ j ≤ m1 − 2) (Dt − λ1 )vm1 −1 = vm1 −

*m1

(1)

k=1

ak vm1 −k

(Dt − λ2 )vj = vj+1 (m1 ≤ j ≤ m − 2) (Dt − λ2 )vm−1 = f − Rv −

*m2 k=1

(2)

ak vm−k

⎪ ⎪ (Dt − λ2 )vm+j = vm+j+1 (0 ≤ j ≤ m2 − 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ *m2 (2) ⎪ ⎪ (Dt − λ2 )vm+m2 −1 = vm+m2 − k=1 a ˜k vm+m2 −k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (D − λ )v ⎪ t 1 m+j = vm+j+1 (m2 ≤ j ≤ m − 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ * ⎩ (1) ˜ − m1 a (Dt − λ1 )v2m−1 = f − Rv k=1 ˜k v2m−k ,

(2.14)

˜ We have v = v0 and the remainders R, R ˜ so we need to consider the terms Rv, Rv. are of order 0 in Dt,x but they contain derivatives Dtj v0 up to j = m − 1. Anyway, from (2.13), by induction on j = 0, . . . , m − 1, it is easy to obtain: Lemma 2.13. Let u and P (t, x, Dm−r u, Dt , Dx ) be as in Proposition 2.11 and, given the scalar function v, let the vector V = (v0 , . . . , vm−1 , vm , . . . , v2m−1 )

400

Massimo Cicognani and Luisa Zanghirati

be deﬁned by (2.13). For every µ there are k2 and µ2 = µ+M M2 such that, if k0 ≥ k2 and µ0 ≥ µ2 , then ⎧ j ⎪ (j) ⎪ j ⎪ D v = c (t, x, DM2 +m−r u, Dx )vj− + vj , ⎪ t ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎨ j (2.15) (j) j ⎪ ⎪ Dt v = c˜ (t, x, DM2 +m−r u, Dx )vm+j− + vm+j , ⎪ ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎩ j = 0, . . . , m − 1, with

(j) (j) c , c˜ ∈ C [0, T ]; S + C [0, T ]; H µS . This gives immediately ⎧ m−1 ⎪ ⎪ Rv = ⎪ bj (t, x, DM3 +m−r u, Dx )vj , ⎪ ⎪ ⎪ ⎪ j=0 ⎨ ⎪ Rv ˜ = *m−1 ˜bj (t, x, DM3 +m−r u, Dx )vm+j , ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎩ bj , ˜bj ∈ C [0, T ]; S 0 + C [0, T ]; H µ S 0

(2.16)

for any given µ, provided that k0 ≥ k3 , and µ0 ≥ µ3 = µ + M3 . So, from (2.16) and (2.14), we have that the scalar problem ⎧ ⎨ P (t, x, Dm−r u, Dt , Dx )v(t, x) = f0 (t, x), (2.17) ⎩ j Dt v(0, x) = gj (x), 0 ≤ j < m, is equivalent to

⎧ ⎨ ∂t V − K(t, x, DM3 +m−r u, Dx )V = F0 , ⎩

(2.18) V (0) = G,

*m−1 F0 = (0, . . . , 0, f0 , 0, . . . , 0, f0 ), GH µ ≤ C j=0 gj H µ+m−1−j , for a symmetric hyperbolic 2m × 2m system ∂t − K to which we can apply Proposition 2.4 since K = i∆ + R0 with ∆ is a real diagonal matrix pseudodiﬀerential operator of order 1, R0 is a matrix pseudodiﬀerential operator of order 0. In order to get an energy estimate for the scalar function v from (2.2), we use the following proposition. During its proof, it becomes clear why we need to consider ˜ so obtaining a both the factorizations P = P2 ◦ P1 + R and P = P˜1 ◦ P˜2 + R 2m × 2m system (in general a dm × dm system, d the number of the characteristic roots, using the cyclic permutations of (1, 2, . . . , d)).

On the Nonlinear Cauchy Problem

401

Proposition 2.14. Let the vector V = (v0 , . . . , vm−1 , vm , . . . , v2m−1 ) be deﬁned by (2.13). For every µ there are k4 and µ4 = µ + M4 such that, if k0 ≥ k4 and µ0 ≥ µ4 , then there is a matrix Q = Q(t, x, DM4 +m−r u, Dx ) such that Dm−r v = QV, Q ∈ C [0, T ]; S 0 + C [0, T ]; H µ S 0 . (2.19) Proof. During this proof, all the symbols denoted by letters a, b, c, q depend on (t, x, DM4 +m−r u, ξ) and belong to spaces C [0, T ]; S δ + C [0, T ]; H µS δ . The order δ will be speciﬁed for each of them. The ﬁnitely many operations we need to perform, will be allowed taking k0 ≥ k4 and µ0 ≥ µ4 = µ + M4 with suﬃciently large k4 and M4 . From condition (2.4) we may assume |λ2 − λ1 | > cξ, c > 0 after a modiﬁcation in a neighborhood of ξ = 0. So, for 0 ≤ k ≤ m − 1, we can write τ −λ τ − λ2 m−1−k τ − λ1 τ − λ2 k 1 τk = − λ2 − λ1 , λ2 − λ1 λ2 − λ1 λ2 − λ1 λ2 − λ1 and, by Newton’s formula, we have τk =

m−1−k

k

ak1 ,k2 (τ − λ1 )k1 +k2 (τ − λ2 )m−1−k1 −k2 , (k)

k1 =0 k2 =0 (k)

where ord ak1 ,k2 ≤ k + 1 − m. In this sum, for k1 + k2 < m1 , let us write (τ − λ2 )m−1−k1 −k2 = (τ − λ2 )m2 [(τ − λ1 ) + (λ1 − λ2 )]m1 −1−k1 −k2 and, for k1 + k2 ≥ m1 , k1 +k2 −m1 (τ − λ1 )k1 +k2 = (τ − λ1 )m . 1 (τ − λ2 ) + (λ2 − λ1 )]

Applying again Newton’s formula, we obtain τk =

m 1 −1

(k)

b1,j (τ − λ1 )j (τ − λ2 )m2 +

j=0

m 2 −1

(k)

b2,j (τ − λ2 )j (τ − λ1 )m1 ,

j=0

(k)

ord bi,j ≤ k + mi − m − j ≤ k − m + r. From (2.13), this gives ⎧ m m m−1 1 −1 2 −1 (k) ⎪ (k) (k) k ⎪ ⎪ D v = b v + b v + c Dt v, ⎨ t 1,j m+m2 +j 2,j m−1+j ⎪ ⎪ ⎪ ⎩

j=0 (k)

ord c

≤ k − − 1.

j=0

=0

(2.20)

402

Massimo Cicognani and Luisa Zanghirati

In the third sum, we can substitute Dt v with the expression given by (2.20) itself. Repeating this process k + 2m − r − 1 times, we obtain ⎧ m m m−1 1 −1 2 −1 (k) ⎪ (k) (k) k ⎪ ˜ ˜ ⎪ v = v + v + c˜ Dt v, b b D m+m +j m−1+j ⎨ t 2 1,j 2,j j=0 j=0 =0 (2.21) ⎪ ⎪ ⎪ ⎩ (k) (k) ord ˜bi,j ≤ k − m + r, ord ˜ ≤ −2m + r + 1. Now, we use (2.15) for Dt v in the third sum of (2.21) in order to get for all k = 0, . . . , m − 1 Dtk v =

2m−1

(k)

(k)

qj vj , ord qj

≤ k − m + r.

(2.22)

j=0

We have (2.19) applying Dxα , |α| ≤ m − r − k, to both sides of the above equality (2.22). Up to now, from Proposition 2.4 applied to the system (2.18), and taking Proposition 2.14 into account, we have proved the following result of well-posedness in Sobolev spaces for the linear equivalent problems (2.17) and (2.18): Proposition 2.15. For every µ there are k5 and µ5 = µ + M5 , such that, if k0 ≥ k5 and µ0 ≥ µ5 , then the Cauchy problem (2.18) has a unique solution V ∈ C ([0, T ]; H µ ) to any given G ∈ H µ and F0 ∈ C ([0, T ]; H µ ). The solution V satisﬁes the estimate t Cu ) G2H µ + F F0 (τ ))2H µ dτ , t ∈ [0, T ]. V (t)2H µ ≤ (1 + tC 0

From this, the scalar Cauchy problem (2.17) has a unique solution v∈

m−1 2

C j [0, T ]; H µ+m−r−j

j=0

to any given gj ∈ H µ+m−1−j , j = 0, . . . , m − 1, and f0 ∈ C ([0, T ]; H µ ). The solution v satisﬁes the estimate ⎡ ⎤ t m−1 m−1 j ∂ ∂t v(t)2H µ+m−r−j ≤ Cu ⎣ gj 2H µ+m−1−j + ff0 (τ ))2H µ dτ ⎦ , t ∈ [0, T ]. j=0

j=0

0

In the above both estimates, Cu denotes a locally bounded function of the norms ∂ ∂tj u(t, ·)H µ0 +m−r−j , 0 ≤ j ≤ k0 .

On the Nonlinear Cauchy Problem

403

2.5. Commutators Proposition 2.15 shows that the reduction of the scalar problem (2.17) to the system (2.18) is not yet suﬃcient to look for a ﬁxed point. In order to solve (2.18) in H µ , we need u(t) ∈ H µ+M5 +m−r even if the symbol of K depends only on Dµ+M3 +m−r with M3 < M5 . As it concerns the equivalent scalar problem, we loose M5 derivatives in the map u → v. So we introduce a further reduction. For µ0 = µ + M + m − r, β = (β0 , β ), we apply derivatives ∂ β = ∂tβ0 ∂xβ , β0 ≤ k0 , |β| ≤ M , in (2.17) so obtaining problems ⎧ ⎪ P v (β) + [∂ β , P ]v = f (β) , ⎨ (2.23) ⎪ j (β) (β) ⎩ Dt v|t=0 = gj , 0 ≤ j < m, |β| ≤ M, for v (β) = ∂ β v, [∂ β , P ] denotes the commutator ∂ β P − P ∂ β . Then, we introduce (β) V˜ = (vj ; 0 ≤ j ≤ 2m − 1, β0 ≤ k0 , |β| ≤ M )

the vector of components have the following:

(β) vj

(2.24)

deﬁned by (2.13) taking there v (β) in place of v. We

Proposition 2.16. For every µ there are k6 and M6 , such that for all M ≥ M6 , if k0 ≥ k6 and µ0 = µ+M , then there is a matrix of functions F = F (t, x, DM+m−r u) ˜ = Q(t, ˜ x, DM6 +m−r u, Dx ) of operators such that and a matrix Q ⎧ β ˜ V˜ , ⎪ [∂ , P ]v; β0 ≤ k0 , |β| ≤ M = F Q ⎪ ⎪ ⎪ ⎨ F ∈ C ([0, T ]; B ∞) + C ([0, T ]; H µ) , (2.25) ⎪ ⎪ ⎪ ⎪ ⎩ ˜ Q ∈ C [0, T ]; S 0 + C [0, T ]; H µ+M−M6 S 0 . Proof. During this proof, all the symbols denoted by letters a, b, q depend on (t, x, DM6 +m−r u, ξ) and belong to spaces C [0, T ]; S h + C [0, T ]; H µ S h . The order h will be speciﬁed for each of them. The ﬁnitely many operations we need to perform, will be allowed taking k0 ≥ k6 and µ0 = µ + M + m − r with M ≥ M6 . For |β| ≤ M , β0 ≤ k0 , let us deﬁne (β)

V (β) = (vj ; 0 ≤ j ≤ 2m − 1), so that (2.24) becomes V˜ = (V (β) ; β0 ≤ k0 , |β| ≤ M ). Applying (2.19) to v (β) we have Dm−r ∂ β v = QV (β) , so in [∂ β , P ]v =

|α|≤m 0<γ≤β

fα,γ (t, x, Dm−r+|γ| u)∂ α+β−γ v

(2.26) (2.27)

404

Massimo Cicognani and Luisa Zanghirati

(α = (α0 , α ), α0 ≤ m−1), we obtain already the desired representation of all terms such that |α| + |β| − |γ| ≤ M + m − r since for them ∂ α+β−γ v is a component of a vector ∂ δ Dm−r v for some δ with |δ| ≤ M . In the remaining part of the proof, this allows us to consider only the β’s such that |β| > M − r + 1 neglecting all the terms with |γ| ≥ r in equality (2.27). In doing so, we need the following lemma. Lemma 2.17. We have % $ akj (Dt − λj )−k , 1 ≤ ≤ mj , ord akj = h (i) a, (Dt − λj ) = 1≤k≤

for every operator a of order h; $ % (ii) (Dt − λ1 ) , (Dt − λ2 )d =

aij (Dt − λ1 )−i (Dt − λ2 )d−j ,

1≤i≤,1≤j≤d

1 ≤ ≤ m1 , 1 ≤ d ≤ m2 , ord aij = i + j − 1; (iii)

(Dt − λ1 )m1 − (Dt − λ2 )m2 −d =

ai (Dt − λ1 )m1 −i (Dt − λ2 )m2

1≤i≤

+

bi (Dt − λ2 )m2 −i (Dt − λ1 )m1

1≤i≤d

+

cij (Dt − λ1 )i (Dt − λ2 )j ,

0≤i≤m1 −1 0≤j≤m2 −1

0 ≤ < m1 , 0 ≤ d < m2 , + d > 1, ord (ai , bi ) = − − d + i, ord cij = m − − d − i − j − 1. Proof of the lemma. Let P, Q, R be three operators. We have the identity [P, QR] = [P, Q]R + Q[P, R]

(2.28)

provided that all the compositions are well deﬁned. Using this, one can prove (i) by induction on . As it concerns the equality (ii), at ﬁrst one can prove it with = 1 by induction on d and then complete the induction on with a ﬁxed d. In both steps one uses (i) and (2.28). Finally, from the representation of the identity operator 1 = q(Dt − λ1 ) − q(Dt − λ2 ) + a, where q has the symbol (λ2 − λ1 )−1 and a is another suitable pseudodiﬀerential operator of order −1, one can prove (iii) by induction on + d using (i), (ii) and

On the Nonlinear Cauchy Problem

405

(2.28) in (Dt − λ1 )m1 − (Dt − λ2 )m2 −d = (Dt − λ1 )m1 − (q(Dt − λ1 ) − q(Dt − λ2 ) + a)(Dt − λ2 )m2 −d .

Continuation of the proof of Proposition 2.16. Let β be such that M − r + 1 < |β| ≤ M (r > 1) and let us consider the factorization P = P2 P1 + R given by (2.10). Terms of order larger than M + m − r in [∂ β , P ] can only appear from [∂ β , P2 P1 ] = [∂ β , P2 ]P P1 + P2 [∂ β , P1 ]. Using again (2.28) and (2.10), we have ⎧ β [∂ , P2 ]P P1 = [∂ β , (Dt − λ2 )m2 ](Dt − λ1 )m1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2) (1) ⎪ ⎪ + [∂ β , aj ](Dt − λ2 )m2 −j ai (Dt − λ1 )m1 −i ⎪ ⎪ ⎪ ⎨ i,j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+

(2.29) aj [∂ β , (Dt − λ2 )m2 −j ]ai (Dt − λ1 )m1 −i , (2)

(1)

i,j

0 ≤ i ≤ m1 , 0 ≤ j ≤ m2 , i + j > 0.

A repeated use of (2.28) gives ⎧ β ⎪ [∂ , (Dt − λj )mj −d ] = ⎪ ⎨ ⎪ (Dt − λj )mj −d−h [∂ β , (Dt − λj )](Dt − λj )h−1 , ⎪ ⎩

(2.30)

1≤h≤mj −d

so all the terms in the right-hand side of equality (2.29) are compositions of operators a, [∂ β , q], (Dt − λ1 )m1 −i , (Dt − λ2 )m2 −j with a of order 0, q of order less or equal to 1. We have to consider only the truncated expansion aγ ∂ β−γ , ord aγ ≤ 0, 0≤γ<β |γ|
of [∂ β , q] since the remainder leads to terms of order less or equal to |β| + m − r in (2.29). On the other hand, from (2.30) and (i) of Lemma 2.17, in all compositions of order larger than |β| + m − r in (2.29), we can commute and move all the operators

406

Massimo Cicognani and Luisa Zanghirati

of order 0 towards left and all the derivatives ∂ β−γ towards right until we obtain a sum of operators ⎧ ⎪ ai,j,γ (Dt − λ2 )m2 −j (Dt − λ1 )m1 −i ∂ β−γ , ⎪ ⎪ ⎪ ⎪ ⎪ i,j,γ ⎪ ⎪ ⎪ ⎪ ⎨ 0 ≤ i ≤ m1 , 0 ≤ j ≤ m2 , 0 < i + j < r, (2.31) ⎪ ⎪ ⎪ ⎪ 0 ≤ γ < β, |γ| ≤ r − i − j, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ord ai,j,γ ≤ 0. Now, for i + j ≥ 2, we apply (iii) of Lemma 2.17 to (Dt − λ2 )m2 −j (Dt − λ1 )m1 −i until, in at most r − 1 steps, the sum in (2.31) becomes equal to ⎧ ⎪ b,γ1 (Dt − λ1 )m1 − (Dt − λ2 )m2 ∂ β−γ1 + ⎪ ⎪ ⎪ ⎪ ⎪ ,γ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ bd,γ2 (Dt − λ2 )m2 −d (Dt − λ1 )m1 ∂ β−γ2 , (2.32) d,γ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ , d ≤ r − 1, 0 ≤ γ1 , γ2 < β, |γ1 | < r − , |γ2 | < r − d, ⎪ ⎪ ⎪ ⎪ ⎩ ord (b,γ1 , bd,γ2 ) ≤ 0, modulo a negligible operator of order |β| + m − r. Applying to the function v the operator in (2.32), from (2.24) and (2.13), we obtain (β) (δ) (β) qδ,j vj , ord qδ,j ≤ 0. δ≤β 0≤j≤2m−1

In a similar way we can deal with P2 [∂ β , P1 ], so obtaining (2.25).

In (2.25), let us write F (t, x, DM+m−r u) = F (t, x, 0) + F˜ (t, x, DM+m−r u). The foregoing Proposition 2.16, system (2.14) and (2.16) show that, for M ≥ M6 , the equations (2.23) are equivalent to the system ⎧ ˜ V˜ + F˜ Q ˜ V˜ = F˜0 , ⎨ ∂t V˜ − K (2.33) ⎩ ˜ ˜ V (0) = G, with ˜ = K(t, ˜ x, DM6 +m−r u, Dx ), Q ˜ = Q(t, ˜ x, DM6 +m−r u, Dx ), K F˜ = F˜ (t, x, DM+m−r u), F˜ (t, x, 0) = 0, F˜0 = F˜0 (t, x). ˜ is symmetric hyperbolic, K ˜ = i∆ ˜ +R ˜ 0 with a real diagonal The operator ∂t − K ˜ of order 1, F (t, x, 0)Q ˜ among the terms R ˜0 matrix pseudodiﬀerential operator ∆ of order 0.

On the Nonlinear Cauchy Problem

407

˜ and K ˜ +K ˜ ∗ (t) in H µ are locally bounded funcThe continuity norms of Q j tion of the norms ∂ ∂t u(t, ·)H µ+M +m−r−j , 0 ≤ j ≤ k0 , provided that M is taken suﬃciently larger than M6 . Furthermore, for µ > n/2, we have ˜ V˜ (t)H µ ≤ cµ F˜ H µ Q ˜ V˜ (t)H µ ≤ Cu,µ V˜ (t) F˜ Q ∂tj u(t, ·)H µ+M +m−r−j , with Cu,µ is another locally bounded function of the norms ∂ 0 ≤ j ≤ k0 . So, now Proposition 2.4 yields: Proposition 2.18. For every µ > n/2 there are k7 and M7 , such that for all M ≥ M7 , if k0 ≥ k7 and µ0 = µ + M , then the Cauchy problem (2.33) has a unique solution V˜ ∈ C ([0, T ]; H µ ) ˜ ∈ H µ and F˜0 ∈ C ([0, T ]; H µ ). The solution V˜ satisﬁes the estimate to any given G t 2 ˜ 2H µ + ˜ Cu ) G F˜0 (τ ))2H µ dτ , t ∈ [0, T ], V (t)H µ ≤ (1 + tC 0

where Cu is a positive locally bounded function of the norms ∂ ∂tj u(t, ·)H µ0 +m−r−j , 0 ≤ j ≤ k0 . Now, the derivatives of u that we need to assume in C ([0, T ]; H µ) in order to solve (2.33) in H µ , are exactly the same DM+m−r u that appears in the symbol ˜ + F˜ Q. ˜ of ∂t − K 2.6. The equivalent quasilinear system For |β| ≤ M , β0 ≤ k0 , M > M7 , k0 > k7 , let us denote (β)

U (β) = (uj ; 0 ≤ j ≤ 2m − 1)

(2.34)

(β)

the vector of components uj deﬁned by (2.13) taking there ∂ β u in place of v, and, for k ≤ M , let us denote ˆk = (U (β) ; |β| ≤ k), U ˆ =U ˆM . U

(2.35)

ˆk ⊂ U ˆ , |γ| ≤ M − k. ∂xγ U

(2.36)

By deﬁnition, we have

ˆ = Q(t, ˆ x, U ˆM4 +m−r , Dx ) such that From (2.26), there is Q ⎧ M+m−r ˆU ˆ, u=Q ⎨ D ⎩ ˆ M4 0 Q ∈ C [0, T ]; S 0 + C [0, T ]; H µ+M−M S ,

(2.37)

408

Massimo Cicognani and Luisa Zanghirati

so, by (2.33) and (2.37), we have that the quasilinear scalar problem (2.3) can be reduced to a quasilinear symmetric hyperbolic system ⎧ ˆ − K(t, ˆ x, U ˆ , Dx )U ˆ = Fˆ (t, x, Q ˆU ˆ ), ⎨ ∂t U (2.38) ⎩ ˆ ˆ U(0) = G, where ˆ = K(t, ˜ x, U ˆM6 +m−r , Dx ) + F˜ (t, x, Q ˆU ˆ )Q(t, ˜ x, U ˆM6 +m−r , Dx ), K

(2.39)

˜ F˜ , Q ˜ as in (2.33), and Fˆ is a smooth function of its arguments such that K, Fˆ (t, x, 0) has compact support. We prove Theorem 2.5 after ﬁxing µ > n/2 + 1 and showing that for every M > M7 there is a unique solution ˆ ∈ C ([0, T ∗]; H µ ) U ˆ H µ . From Proposition 2.18, for every ˆ ∈ H µ . Let us take R > G for any given G µ ˆ ˆ U ∈ C ([0, T ]; H ) with U(t) ≤ R, 0 ≤ t ≤ T , there is a unique solution Vˆ ∈ C ([0, T ]; H µ ) of the linear problem ⎧ ˆ x, U ˆ , Dx )Vˆ = Fˆ (t, x, Q ˆ ), ˆU ⎨ ∂t Vˆ − K(t, ⎩ ˆ ˆ V (0) = G. This solution satisﬁes

t 2 ˆ ˆ 2H µ + V (t)H µ ≤ (1 + tCR ) G Fˆ (τ ))2H µ dτ , t ∈ [0, T ],

(2.40)

0

so there is a T ∗ < T such that Vˆ (t) ≤ R, 0 ≤ t ≤ T ∗ . ˆ , Uˆ ∈ C ([0, T ∗ ]; H µ ) with U ˆ (t), Inequality (2.40) implies also that for every U ∗ ˆ ˆ ˆ U (t) ≤ R, 0 ≤ t ≤ T , the corresponding solutions V , V satisfy t 2 ˆ ˆ V (t) − V (t)H µ−1 ≤ LR Vˆ (t) − Vˆ (τ )2H µ−1 dτ, t ∈ [0, T ∗]. 0

ˆ ). So, the usual approximation sequence deﬁned by Let us denote Vˆ = S(U ˆ0 = G, ˆ U ˆk+1 = S(U ˆk ), k = 0, 1, . . . U

(2.41)

ˆk ∈ C ([0, T ∗ ]; H µ ), U ˆk (t) ≤ R, 0 ≤ t ≤ T ∗ , for every k, and, is such that U ∗ ˆ of (2.38). taking a smaller T , it converges in C [0, T ∗ ]; H µ−1 to a solution U This solution satisﬁes t ˆ 2 µ−1 + ˆ (t)2 µ−1 ≤ (1 + tCR ) G Fˆ (τ ))2 µ−1 dτ , (2.42) U H

H

H

0

t ∈ [0, T ∗ ], hence it is unique. The proof of Theorem 2.5 is complete. The estimate (2.7), with s = µ + M , s0 = µ + M7 , follows from (2.37) and (2.42).

On the Nonlinear Cauchy Problem

409

2.7. Local C ∞ solutions Let Ω ⊂ Rn+1 be an open set where the variable is denoted by y and let S = {ϕ(y) = 0} be an hypersurface with a real-valued C ∞ function ϕ, ∇ϕ = 0 in S, ν = ∇ϕ/∇ϕ is the normal vector. Let us consider the Cauchy problem ⎧ m m ⎪ ⎪ ⎨ P (y, D u, Dy )u = f (y, D u), (2.43) ⎪ ∂j u ⎪ ⎩ = g in S, 0 ≤ j < m, j ∂ν j gj ∈ C ∞ (S), j = 1, . . . , m − 1, f ∈ C ∞ (Ω × W ), W is a neighborhood of the Cauchy data, for an operator P (y, Dm u, Dy ) = aα (y, Dm u)Dyα (2.44) |α|≤m

∞

with m < m, aα ∈ C (Ω × W ), |α| ≤ m. Given y0 , ν0 with y0 ∈ S and ν0 the normal vector to S at y0 , we say that the operator P (y, Dm u, Dy ) satisﬁes the condition of hyperbolicity (2.4) and the Levi conditions (2.5), (2.6) at (y0 , ν0 ) if there is a neighborhood of y0 in Rn+1 where these conditions are fulﬁlled taking local coordinates y = (t, x) with dt = ν0 . It is well known that, in this case, the same holds true for (y0 , −ν0 ). From Theorem 2.5 and Remark 2.7 we have the following result of local existence and uniqueness of C ∞ solutions. Theorem 2.19. Let the condition of hyperbolicity (2.4) and the Levi conditions (2.5), (2.6) be satisﬁed at (y, ν) for every y ∈ S. Then, for every gj ∈ C ∞ (S), j = 1, . . . , m − 1, there is a neighborhood A of S in Ω such that problem (2.43) has a unique solution u ∈ C ∞ (A). In particular, uniqueness can be proved in domains of inﬂuence. Let D ⊂ Ω be a compact domain ) D= S¯µ , 0≤µ≤1

where {Sµ ; 0 ≤ µ ≤ 1} is a continuous family of surfaces satisfying for local coordinates z = (z0 , z1 , . . . , zn ) = (z0 , z ): (i) S¯0 is the closure of a bounded open set S0 ⊂ {z0 = 0} with C ∞ boundary; (ii) S¯µ = {(z0 , z ); z0 = ϕµ (z ), z ∈ S¯0 }, where ϕ0 = 0 and {ϕµ } is a family of C ∞ functions deﬁned on S¯0 such that ϕµ = 0, ϕµ (z ) < ϕµ (z ), z ∈ S0 , 0 ≤ µ < µ ≤ 1. ∂S0

We say that D is a domain of inﬂuence based on S¯0 if for all µ ∈ [0, 1] the condition of hyperbolicity (2.4) and the Levi conditions (2.5), (2.6) are satisﬁed at every point (y, ν), y ∈ S¯µ , ν is a normal vector to S¯µ at y.

410

Massimo Cicognani and Luisa Zanghirati

For such a domain D and solutions u1 , u2 of

P (y, Dm u, Dy )u = f (y, Dm u)

(2.45)

in a neighborhood of D, we have ∂ j u1 ∂ j u2 = in S0 , j = 0, . . . , m − 1 = =⇒ u1 = u2 in D. j ∂ν ∂ν j 2.8. Analytic regularity Now, let us consider a ﬁxed C ∞ real solution u(t, x) of ⎧ P (t, x, Dm u, Dt,x )u = f (t, x, Dm u), ⎪ ⎪ ⎨ m α ⎪ P (t, x, D u, D ) = aα (t, x, Dm u)Dt,x , t,x ⎪ ⎩

(2.46)

(2.47)

|α|≤m

for (t, x) ∈ [0, T ] × M, M is a real analytic compact manifold of dimension n. Let us assume that aα (t, x, w), |α| ≤ m, and f (t, x, w) are real analytic functions of their arguments in a neighborhood of (2.48) the set {(t, x, Dm u(t, x)); t ∈ [0, T ], x ∈ M}. We prove the following result of propagation of analytic regularity of the solution u. Theorem 2.20. Let the condition (2.48) be fulﬁlled and let the operator P (t, x, Dm−r u, Dt,x ) satisfy the conditions (2.4), (2.5) and (2.6) for the ﬁxed function u. If the traces ∂tj u(0, x), j = 0, . . . , m−1 are analytic, then u is analytic in [0, T ]×M. Remark 2.21. If the functions f and aα , |α| ≤ m, are respectively continuous and C k with respect to t with k large enough and if we assume for them analytic regularity only with respect to (x, w), then we can prove that the solution u(t, x) is analytic with respect to x for all t ∈ [0, T ]. From the Cauchy-Kovalevsky Theorem, there is an analytic solution v in a right-sided neighborhood of {t = 0}, say for 0 < t < T0 , with the same traces of u and there we have u = v. The above theorem means that any C ∞ solution with T0 , T ]. lifespan T > T0 cannot develop analytic singularities at t ∈ [T We use the reduction of the scalar equation (2.47) to the equivalent system ˆ hence its ﬁrst (2.38) in the cylinder [0, T ] × M and prove that the vector U, β component u, is analytic. In doing so, we take derivatives ∂ = ∂xβ and estimate ˆβ = ∂ β U by induction on |β| using the energy inequality (2.42). U As in [2], one can localize this kind of proof using cutoﬀ functions χN (t, x), N = |β| + N0 , such that α |∂ ∂t,x χN (t, x)| ≤ (CN )|α| , |α| ≤ N,

On the Nonlinear Cauchy Problem

411

N0 is a suitable ﬁxed positive integer depending on m, r, n. So, in the same way, in the equation (2.45), one can prove the propagation of analytic regularity in domains of inﬂuence for a ﬁxed C ∞ solution u(y) in an open set A ⊂ Ω, provided that the coeﬃcients satisfy aα (y, w), |α| ≤ m, f (y, w) are analytic functions of their arguments in a neighborhood of {(y, Dm u(y)); y ∈ A}.

(2.49)

Theorem 2.22. Let the condition (2.49) be fulﬁlled, S be an analytic hypersurface, u ∈ C ∞ (A) be a ﬁxed solution in A of equation (2.45) and let D ⊂ A be a domain of inﬂuence determined by u and based on S¯0 , S0 is a bounded open set in S with C ∞ boundary. ∂j u , 0 ≤ j ≤ m − 1, are analytic in S¯0 , this means, analytic in an If the traces ∂ν j S open neighborhood of S¯0 , then u is analytic in D. The remaining part of the section is devoted to the proof of Theorem 2.20. We are going to deal with symbols p(t, x, w, ξ) in S m (M × Rn ) depending on parameters (t, w) ∈ [0, T ]×W , W is a neighborhood of the set {Dm u(t, x); (t, x) ∈ [0, T ] × M}. From (2.48), these symbols are analytic functions of (t, x, w) ∈ [0, T ] × M × W , so they satisfy ⎧ j γ ⎪ p(t, x, w, ξ) ≤ Cα Aj+|β|+|γ| j!β!γ!ξm−|α| , ⎨ ∂t ∂ξα ∂xβ ∂w (2.50) ⎪ ⎩ (t, x, w) ∈ [0, T ] × M × W. From (2.38), (2.37) and (2.39), the equation

P u = f (t, x, Dm u), (t, x) ∈ [0, T ] × M, is equivalent to the system ⎧ ˆ −K ˜U ˆ = Φ, ⎪ ∂t U ⎪ ⎪ ⎪ ⎨ ˆ )Q ˜U ˆ + Fˆ (t, x, QU ˆ ), Φ = F˜ (t, x, QU ⎪ ⎪ ⎪ ⎪ ⎩ (t, x) ∈ [0, T ] × M,

(2.51)

(2.52)

where now: • the matrix F˜ and the vectors Fˆ are analytic functions of the variable (t, x, w) ∈ C¯ × W ; • the symbols of ˜ = K(t, ˜ x, DM6 +m u, Dx ) K (of order 1) and of ˆ = Q(t, ˆ x, DM6 +m u, Dx ), Q ˜ = Q(t, ˜ x, DM6 +m u, Dx ) Q

412

Massimo Cicognani and Luisa Zanghirati

(of order 0) satisfy (2.50); • we can choose M as larger than M6 as we need, in particular for any ﬁxed M∗ we may assume ˆ. DM6 +m +M∗ +1 u ⊂ U (2.53) ˆj = Applying derivatives ∂j = ∂xj , j = 1, . . . , n, we obtain equations for U ˆ ∂j U , ⎧ ˜U ˆj + * Aij U ˆj − K ˆ i = Φj , ⎪ ∂t U i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ∂j ] = * Aij ∂i , ⎨ [K, i (2.54) ⎪ ⎪ M +m +1 6 ⎪ Ai,j = Ai,j (t, x, D , Dx ), ord Aij = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Φj = ∂j Φ, 1 ≤ i, j ≤ n. Our aim now is to take further derivatives ∂ β = ∂xβ and to prove that the function u is analytic with respect to x ∈ M estimating ˆβ,j = ∂ β Uj , U hence its ﬁrst component ∂ β ∂j u, by induction on |β|. We obtain equations ⎧ * ˆβ,j − K ˆβ,i = Φβ,j , ˜U ˆβ,j + Aij U ⎨ ∂t U i

⎩

˜ U ˆj + Φβ,j = −[∂ β , K]

(2.55)

*

β β ˆ i [∂ , Aij ]Ui + ∂ ∂j Φ,

for which the energy inequality gives ⎧ ˆβ,j (t)2 µ ⎪ U ⎪ H (M) ≤ ⎪ ⎪ ⎪ j ⎨ t ⎪ ⎪ ⎪ ˆβ,j (0)2 µ ⎪ C {U + Φβ,j (τ )2H µ (M) dτ }. ⎪ H (M) ⎩ 0

(2.56)

j

ˆβ,j (t)H µ (M) by Gronwall’s method. This will allow us to estimate U We need to introduce some notation. Let us consider the sequence mp = p! , where the constant c is chosen in order to satisfy c (p + 1)2 α α m|β| m|α−β| ≤ m|α| , m|β| m|α−β|+1 ≤ |α|m|α| . β β 0≤β≤α

0<β≤α

Let µ > (n/2) be a ﬁxed integer. For ε > 0, p ≥ 1, w ∈ C ∞ (M), let us denote wµ = wH µ (M) , Mp = ε1−p mp and |w|q . 0
|w|p = sup ∂ ∂xα wµ , [w]p = sup |α|=p

We use the following lemma which is proved in [2].

On the Nonlinear Cauchy Problem

413

Lemma 2.23. Let w = (w1 , . . . , wν ), wj ∈ C ∞ (M), be real-valued functions, and let φ be an analytic function in a neighborhood of M × w(M). Then there exist positive constants ε0 , δ, L such that for p ≥ 1, ε ∈]0, ε0 ] the condition ε[w]p ≤ δ implies

⎧ ⎨ (i) ⎩

(2.57)

[φ(·, w)]p ≤ L(1 + [w]p ), (2.58)

(ii)

|φ(·, w)|p+1 ≤ L{|w|p+1 + Mp+1 (1 + [w]p )}.

We apply Lemma 2.23 to w(t, x), wj ∈ C ∞ ([0, T ] × M), φ(t, x, w(t, x)) is analytic in a neighborhood of [0, T ] × M × w([0, T ] × M), where t is considered as a parameter. In this case, there exist positive constants ε0 , δ, L such that for every integer p ≥ 1, ε ∈]0, ε0 ] if ε[w(t, ·)]p ≤ δ for every t ∈ [0, T ], then (i) and (ii) hold uniformly with respect to t in this interval. Let us denote Mpt = (εe−λt )1−p mp = e−λt(1−p) Mp , where λ ≥ 0 is a parameter to be chosen later. For a function v(t, x), v ∈ C ∞ ([0, T ] × M), 0 < ε ≤ ε0 , 0 ≤ t ≤ T , p ≥ 1 we deﬁne |v|tq , p ≥ 1, 0
|v|tp = max ∂ ∂xβ v(t, ·)µ , [v]tp = max |β|=p

(2.59)

ˆ we write then, for the vector U, ˆ |t |U q , p ≥ 2. t 0≤t≤T 1≤q≤p Mq−1

ˆ ) = sup max Ψp (U

(2.60)

Lemma 2.24. There exist positive constants ε0 , δ, λ1 and L such that for 0 < ε ≤ ε0 , λ ≥ λ1 , the condition ˆ ))2 ≤ δ ε(1 + Ψp (U (2.61) implies ˆ t + M t (1 + Ψp (U ˆ ))2 } Φβ,j (t, ·)µ ≤ Lp{|U| p+1 p

(2.62)

for |β| = p, p ≥ 2, t ∈ [0, T ].

Proof. Let Q = Q(t, x, DM6 +m u, Dx ) be an operator of order 0 which satisﬁes ˆ |tp under assumption (2.61). (2.50) and let us estimate |QU γ γ For |γ| = p, ∂ = ∂x , we have γ γ ˆ ˆ ∂ QU(t)µ ≤ Q(γ ) ∂ γ−γ U(t) µ γ γ ≤γ

ˆ |tp + ≤ C|U

γ ˆ ), Mγt −γ Ψp−1 (U Q(γ ) (t)M γ

0<γ ≤γ

414

Massimo Cicognani and Luisa Zanghirati

where Q(γ ) (t) has the symbol ∂ γ Q(t), so its continuity norm Q(γ ) (t) as an op erator in H µ (M) is an analytic function of (x, DM6 +M∗ +m +|γ | u), M∗ depending only on µ and n. The inclusion (2.53) gives

ˆ )) [DM6 +M∗ +m u]tp ≤ C(1 + Ψp−1 (U so, from Lemma 2.23, condition (2.61) implies t ˆ |Q(γ ) (t) ≤ CM|γ 0 ≤ t ≤ T, | (1 + Ψp−1 (U )),

then ˆ t ≤ C{|U ˆ |t + M t (1 + Ψp−1 (U ˆ ))2 }. |QU| p p p

(2.63)

ˆ |t , q ≤ p, we obtain also This holds true for arbitrary p, hence, applying it to |QU q ˆ tp ≤ C(1 + Ψp (U ˆ ))2 . [QU]

(2.64)

Consider now the term ∂ β ∂j Φ in (2.55). Since Φ is an analytic function of all ˆU ˆ, Q ˜U ˆ ), from (ii) in Lemma 2.23, (2.64) and (2.63), condition its arguments (t, x, Q (2.61) implies ˆ tp+1 + Mpt+1 (1 + Ψp (U ˆ ))2 }, |β| = p, ∂ β ∂j Φ(t)µ ≤ C{|U| that is the desired estimate (2.62) for this term in Φβ,j . ˆ, ˜ U ˆj + * [∂ β , Aij ]U ˆi , U ˆj = ∂j U It remains to prove the same for −[∂ β , K] i |β| = p. ˜ = K(t, ˜ x, DM6 +m u, Dx ) of order 1, we have For the operator K β β ˜ ˆ ˆj (t)µ ˜ (γ) ∂ β−γ U [∂ , K]Uj (t)µ ≤ K γ 0<γ≤β

ˆ |t + CM ˆ) + ≤ Cp|U Mp Ψp (U p+1

γ≤β,|γ|≥2

β ˆ )), ˜ (γ) (t)M|β|−|γ|+1(1 + Ψp (U K γ

˜ (γ) (t) : H µ+1 (M) → H µ (M) is bounded ˜ (γ) (t) of K where the continuity norm K by ˜ (γ) (t) ≤ CM t (1 + Ψp (U ˆ )) K |γ| provided that condition (2.61) is fulﬁlled. This gives ˜ U ˆj (t)µ ≤ Cp{|U| ˆ tp+1 + Mp (1 + Ψp (U ˆ ))2 }. [∂ β , K] ˆi completing Since the operators Ai,j are of order 0, this holds also for [∂ β , Aij ]U the proof of (2.62). Now, we can apply Gronwall’s inequality in (2.56).

On the Nonlinear Cauchy Problem

415

Lemma 2.25. There exist positive constants ε0 , δ, λ0 , L1 such that for p ≥ 2, 0 < ε ≤ ε0 , λ ≥ λ0 , condition (2.61) implies ˆ |tp+1 ≤ L1 Mpt |U

ˆ ))2 (1 + Ψp (U , t ∈ [0, T ]. λ

(2.65)

Proof. Let us assume (2.61) with ε ≤ ε0 , λ ≥ λ1 . From inequalities (2.62) and (2.56), we obtain t ˆ |tp+1 ≤ C0 |U ˆ |0p+1 + C0 Lp ˆ τp+1 + Mpτ (1 + Ψp (U ˆ ))2 }dτ, 0 ≤ t ≤ T. |U {|U| 0

ˆ (0) is analytic, we have |U ˆ |0 ≤ C1 ε1−p mp , thus, by Gronwall’s inequality Since U p+1 0 t t pL t 1−p 2 ˆ ˆ |U |p+1 ≤ C2 e ε0 mp + C2 (1 + Ψp (U )) p epL (t−τ ) Mpτ dτ, L = C0 L. 0

Recalling the deﬁnition of Mpτ , we get for λ ≥ 6L , ε ≤ ε0 , p ≥ 2 t 3M Mpt , C2 epL t ε1−p p epL (t−τ ) Mpτ dτ ≤ mp ≤ C3 Mpt . 0 λ 0 It is now suﬃcient to take λ ≥ λ0 = max{λ1 , 6L } in order to obtain (2.65).

Proof of Theorem 2.20. We can now prove that there exist positive ε, λ, H such that ˆ ) ≤ H, p ≥ 2. (2.66) Ψp (U Let us choose H = max{L1 , Ψ2 (U )} and ﬁx ε ∈]0, ε0 [ such that ε(1 + H)2 < δ. Taking λ = max{λ0 , (1 + H)2 } and assuming ˆ ) ≤ Hp ≥ 2, Ψp (U Lemma 2.25 gives ˆ |t ≤ HM |U Mpt . p+1 Since ' |U |tp+1 ( ˆ ) = max Ψp (U ˆ ), sup , Ψp+1 (U Mpt 0≤t≤T we have Ψp+1 (U ) ≤ H, so (2.66) holds for all p ≥ 2 by induction on p. ˆ (t, ·), This gives the analyticity with respect to the space variable x of the vector U hence of the solution u(t, ·) of the equation (2.47), for all t ∈ [0, T ]. So far, we have not used analytic regularity with respect to t in (2.48). Taking derivatives ∂tj in (2.47), (see for example [38] and [2]), one can prove that u is an analytic function of (t, x) in [0, T ] × M completing the proof of Theorem 2.20.

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Massimo Cicognani and Luisa Zanghirati

3. Well-posedness in Gevrey classes 3.1. The linear problem s s For s > 1 and A > 0 let us denote by γA = γA (Rn ) the space of all functions f satisfying f γAs := sup |Dα f (x)|A−|α| α!−s < ∞ x∈Rn α∈Zn +

s,1 s,1 and by γA = γA (Rn × W ), W is a neighborhood of the origin in RN , the space of all functions f (x, w) such that

f γ s,1 := A

So γ s :=

)

sup (x,w)∈Rn ×W (α,β)∈Zn ×ZN + +

s γA and γ s,1 :=

A>0

β |Dxα Dw f (x, w)|A−|α|−|β| |α!|−s |β!|−1 < ∞.

)

s,1 γA , endowed with the limit topology as A → ∞,

A>0

are Gevrey spaces. For s = 1 one obtains classes of analytic functions. m,s In a corresponding way, we denote by S,A the space of all symbols a(x, ξ) of order m such that m,s = sup sup |∂ ∂xα ∂ξβ a(x, ξ)|ξ|β|−m A−|α| α!−s < ∞, aS,A α∈Zn + |β|≤

R2n

and deﬁne m,s S,A , S m,s := lim Sm,s . Sm,s := lim → ← A→+∞

→+∞

We also use symbols depending on a parameter w ∈ W . In fact, we denote m,s,1 (Rn × W × Rn ) the space of all symbols a(x, w, ξ) of order m such that by S,A β γ aS m,s,1 = sup sup |∂ ∂xα ∂w ∂ξ a(x, w, ξ)|ξ|γ|−m A−|α|−|β|−|γ|α!−s β!−1 < ∞, ,A

α,β |γ|≤

x,w,ξ

and we deﬁne m,s,1 S,A , S m,s,1 := lim Sm,s,1 . Sm,s,1 := lim → ← A→+∞

→+∞

Next we introduce Gevrey-Sobolev spaces. For τ, µ ≥ 0, s > 1, we denote by H τ,s,µ (Rn ) the space of all functions u such that: uH τ,s,µ := eτ Dx

1/s

uH µ < ∞,

H µ = H µ (Rn ) is the usual Sobolev space. From the Paley-Wiener theorem, it follows that s ∩ C0∞ , 0 ≤ τ ≤ τ0 , uH τ,s,µ ≤ CuγAs , u ∈ γA

(3.1)

with τ0 = τ0 (A, s, n) and C = C(A, s, µ, n) are positive constants. Conversely, we have: s , τ > 0, A > Cs τ −s , µ > n/2, H τ,s,µ ⊂ γA

with continuous embedding.

(3.2)

On the Nonlinear Cauchy Problem

417

Moreover, for µ > n/2, H τ,s,µ is a Banach algebra, it holds uvH τ,s,µ ≤ Cµ uH τ,s,µ vH τ,s,µ .

(3.3)

Taking τ < 0 we have dual spaces of ultradistributions. m,s For a symbol a(x, ξ) in S,A and Λ = τ Dx 1/s , τ ∈ R, let us denote by aΛ (x, Dx ) the operator aΛ = eΛ ae−Λ . The continuity of a(x, Dx ) in GevreySobolev spaces is of course equivalent to the continuity of aΛ (x, Dx ) in usual Sobolev spaces. In fact, from [31] we have the following result that we state for later use directly for operators and spaces depending on a parameter t ∈ [0, T ]. m,s ), Λ = λ(T − t)Dx 1/s , λ ∈ R. Proposition 3.1. Let a(t, x, ξ) ∈ C k ([O, T ]; S,A Then there are T0 = T0 (A, s, n) > 0 and for every a positive 0 = 0 ( , s, n) such that: m,σ , |λT | ≤ T , ≥ , aΛ C k ([0,T ];S m ) ≤ CaC k ([0,T ];S,A 0 0 )

(3.4)

with C = C(A, , s, k, n) > 0 is independent of a(t, x, ξ). Furthermore, we have the expansion m−1+1/s

aΛ = a + r(Λ) , r(Λ) ∈ C k ([0, T ]; S

), |λT | ≤ T0 , ≥ 0 .

(3.5)

In particular, at every ﬁxed t, the operator a(t, x, Dx ) is continuous from H λ(T −t),s,µ+m to H λ(T −t),s,µ provided that ≥ 1 (µ, s, n) and |λT | ≤ T0 . Together with the spaces H τ,s,µ we consider the following classes of functions and symbols depending on t ∈ [0, T ]: ⎧ 1/s ⎨ CTk (H λ,s,µ ) = {u(t, x, ); t → eλ(T −t)Dx Dtj u(t, ·) ∈ (3.6) ⎩ C([0, T ]; H µ−j (Rn )), j = 0, . . . , k} with norm uCTk (H λ,s,µ ) = sup

sup Dtj u(t, ·)H λ(T −t),s,µ−j ;

0≤j≤k 0≤t≤T

⎧ k m λ,s,µ S , H ) = {a(t, x, ξ) ∈ C k ([0, T ]; S m); ∂xα ∂ξβ a(·, ·, ξ) ∈ ⎨ CT (S (3.7)

⎩

CTk (H λ,s,µ ), |α| + |β| ≤ , ξ ∈ Rn }

with norm aCTk (Sm ,H λ,s,µ ) =

sup |α|+|β|≤

sup ∂ ∂xα ∂ξβ a(·, ·, ξ)CTk (H λ,s,µ ) . ξ

In both deﬁnitions (3.6) and (3.7) we take k ∈ Z+ , λ > 0, µ > n/2. Together with Proposition 3.1, we have the following statement proved in [32] without any restriction on λT . Sm , H λ,s,µ ) and Λ = Λ(t) = λ(T − t)Dx 1/s . Proposition 3.2. Let a ∈ CTk (S Then for every ∈ Z+ there is 0 = 0 ( , s, n) ∈ Z+ such that a±Λ C k ([0,T ];S m ) ≤ CaCTk (Sm ;H λ,s,µ ) , ≥ 0 ,

(3.8)

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Massimo Cicognani and Luisa Zanghirati

with a constant C = C(µ, , s, k, n) > 0 which is independent of a(t, x, ξ). Furthermore, we have the expansion m−1+1/s

a±Λ = a + r(Λ) , r(Λ) ∈ C k ([0, T ]; S

), ≥ 0 .

(3.9)

Concerning the composition of u ∈ CTk (H λ,s,µ ) with maps, we have [32]: s,1 ) with f (t, x, 0) = 0 and let u ∈ Proposition 3.3. Let f (t, x, w) ∈ C k ([0, T ]; γA k λ,s,µ CT (H ) with uCTk (H λ,s,µ ) ≤ r, r > 0. Then there are positive µ0 = µ0 (s, µ), T0 = T0 (s, µ), r0 = r0 (k, µ, n), C = C(f, s, µ) such that ⎧ ⎨ f (t, x, u(t, x))CTk (H λ,s,µ ) ≤ CuCTk (H λ,s,µ ) , (3.10) ⎩ µ ≥ µ0 , λT A1/s < T0 , rA ≤ r0 .

In particular, for a(t, x, w, ξ) ∈ C k ([0, T ]; S m,s,1) and u ∈ CTk (H λ,s,µ0 + ), we have a(t, x, u(t, x), Dx ) = a0 (t, x, Dx ) + a1 (t, x, u(t, x), Dx ) with a0 = a(t, x, 0, ξ) ∈ C k ([0, T ]; S m,s), a1 ∈ CTk (S Sm ; H λ,s,µ0 ) provided that λT and uCTk (H λ,s,µ0 + ) are small. Then, one can use Proposition 3.1 and Proposition 3.2 to have a precise estimate of the norm of a(t, x, u(t, x), Dx ) as a continuous operator from CTk (H λ,s,µ ) to CTk (H λ,s,µ−m ) in view of the well-known continuity of aΛ ∈ Sm in usual Sobolev spaces. So, for a linear hyperbolic system in the unknown V , ⎧ ⎨ ∂t V (t, x) = K(t, x, U, Dx )V (t, x) + F (t, x), (3.11) ⎩ V (0, x) = G(x), with a given vector U (t, x), by the energy method one gets: Proposition 3.4. Let K ∈ C([0, T ]; S 1,s,1 ) and K + K ∗ ∈ C([0, T ]; S 1/s,s,1 ). Given any µ there are µ0 > µ, R0 > 0, λ0 > 0 and T0 > 0 such that for any U ∈ CT0 (H λ,s,µ0 ) with U CT0 (H λ,s,µ0 ) ≤ R0 , λ ≥ λ0 , the problem (3.11) has a unique solution V ∈ CT0 (H λ,s,µ ) for every given G ∈ H λT,s,µ and F ∈ CT0 (H λ,s,µ ), provided that T ≤ T0 /λ. Such a solution satisﬁes the estimate ⎧ V (t)2H λ(T −t),s,µ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t ⎨ (3.12) ≤ (1 + tC) G2H λT ,s,µ + F (τ )2H λ(T −τ ),s,µ dτ , ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ t ∈ [0, T ], λT ≤ T0 .

On the Nonlinear Cauchy Problem

419

The constants R0 , λ0 and C depend on µ, s, n and K(t, x, w, ξ); T0 depends on s, n and K(t, x, 0, ξ). The Sobolev exponent µ0 depends only on µ, s, n. Proof. Let us denote S = ∂t − K and SΛ = eλ(T −t)Dx have

1/s

Se−λ(T −t)Dx

1/s

; we

SΛ = ∂t + λDx 1/s − KΛ . It is suﬃcient to prove that one can choose λ ≥ λ0 such that the inequality t V (t)2H µ ≤ C V (0)2H µ + SΛ V (τ )2H µ dτ (3.13) 0

holds for every V ∈ C ([0, T ]; H ) ∩ C ([0, T ]; H µ ), t ∈ [0, T ]. From Propositions 3.1, 3.2 and 3.3, there are µ0 > µ, R0 > 0 and T0 > 0 such that KΛ = KΛ (t, x, U, Dx ) satisﬁes 0

µ+1

1

2$ (K KΛ V (t), V (t))H µ ≤ λ0 (Dx 1/s V (t), V (t))H µ , t ∈ [0, T ], for every V as in (3.13) and for any given U ∈ CT0 (H λ,s,µ0 ) with U CT0 (H λ,s,µ0 ) ≤ R0 , provided that T ≤ T0 /λ. The constant λ0 remains bounded for such U and λT , so, for λ ≥ λ0 , one has d V (t)2H µ ≤ C(V (t)2H µ + SΛ V (t)2H µ dt which implies (3.13) by Gronwall’s method.

In Proposition 3.3, we assume analytic regularity of f in the variable w. We refer to [6] for results of this type under almost the same γ s regularity of f with respect to both variables x and w. 3.2. Gevrey-Levi conditions Let us consider the quasilinear Cauchy problem ⎧ α ⎪ aα (t, x, Dm u)Dt,x u = f (t, x, Dm u), ⎪ ⎨ ⎪ ⎪ ⎩

|α|≤m

(3.14)

Dtj u(0, x) = gj (x), 0 ≤ j < m,

where t ∈ [0, T ], x ∈ Rn , m < m, gj ∈ γ s (Rn ), j = 1, . . . , m − 1, f, aα ∈ B ∞ ([0, T ]; γ s,1 (Rn × W )), |α| ≤ m, W is a neighborhood in RN of the Cauchy data. With a change of the variable u, one can reduce to null Cauchy data, so without loss of generality we may assume that 0 ∈ W and that f (t, x, 0), gj , j = 1, . . . , m − 1, have compact supports with small norms of the gj ’s in Sobolev-Gevrey spaces (compensated by a larger norm of f (t, x, 0)).

420

Massimo Cicognani and Luisa Zanghirati For the principal symbol Pm of the linear operator m P (t, x, w, Dt , Dx ) = Pj (t, x, w, Dt , Dx ), j=0

Pj (t, x, w, Dt , Dx ) =

α aα (t, x, w)Dt,x ,

|α|=j

we still assume the hyperbolicity condition (2.4) Pm (t, x, w, τ, ξ) =

d 1

(τ − λj (t, x, w, ξ))mj ,

j=1

λj ∈ R, j = 1, . . . , d, λh = λk for every (t, x, w, ξ), ξ = 0, h = k, still denoting by r the largest multiplicity. As shown in [32], this is suﬃcient to have a well-posed problem (3.14) in Gevrey classes γ s for s ≤ r/(r−1), m = m−1, but it is not suﬃcient for s > r/(r − 1) already in the linear case. So we have to assume Levi conditions. The nonlinear one is (3.15) m ≤ m − r(1 − 1/s),

then, for any u ∈ B ∞ ([0, T ]; γ s ) such that Dm u takes values in W , we require that the linear problem in the unknown v ⎧ ⎨ P (t, x, Dm u, Dt , Dx )v(t, x) = f0 (t, x), ⎩

Dtj v(0, x) = gj (x), 0 ≤ j < m,

satisﬁes the suﬃcient Gevrey-Levi condition for well-posedness in γ s stated in [30], that is, for every real solution ϕ(t, x) of

∂t ϕ = λj (t, x, Dm u, ∇x ϕ), for every real C ∞ function ψ(t, x) and every C ∞ function h(t, x) with compact support we assume P (heiϕ+i

1/s

ψ

) = O(m−mj (1−1/s) ), → +∞, j = 1, . . . , d.

(3.16)

The necessity has been proved for operators with analytic coeﬃcients [30]. We refer also to [34] for a complete study of the linear problem in Gevrey classes. We use a consequence of (3.16) in (3.19) whereas in the examples after Theorem 3.5 we deal with an equivalent condition that shows that to require (3.16) for any given u leads to analytic relations for the coeﬃcients aα (t, x, w), |α| > m − r(1 − 1/s). The terms Pj of P with j ≤ m − r(1 − 1/s) are not involved in (3.16), in particular this is an empty condition for s ≤ r/(r − 1). In this case, also (3.15) becomes simply m ≤ m − 1, that is, one can consider any quasilinear equation.

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The main result of this section is the following: Theorem 3.5. Assume that conditions (2.4), (3.15) and (3.16) are fulﬁlled. Then, for every gj ∈ γ s , j = 1, . . . , m − 1, there is T ∗ ≤ T such that the Cauchy problem (3.14) has a unique solution u ∈ C ∞ ([0, T ∗]; γ s ) . Remark 3.6. In fact we prove that there are positive µ0 , λ0 , T0 such that, for any ∗ gj ∈ H λT ,s,µ+m−1−j , j = 1, . . . , m − 1, with µ > µ0 , λ > λ0 , the problem (3.14) has a unique solution (H λ,s,µ+m−1−(r−1)(1−1/s) ), λT ∗ ≤ T0 . u ∈ CTm−1 ∗ The solution u satisﬁes the estimate ⎧ m−1 j ⎪ ⎪ ⎪ ∂ ∂t u(t, ·)2H λ(T ∗ −t),s,µ+m−1−j−(r−1)(1−1/s) ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎡ ⎤ t m−1 ⎪ ⎪ gj 2H λT ∗ ,s,µ+m−1−j + f (τ, ·, Dm u(τ, ·))2H λ(T ∗ −τ ),s,µ dτ ⎦ , ⎪ ≤C⎣ ⎪ ⎪ 0 ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t ∈ [0, T ∗]. (3.17) Remark 3.7. Using the condition (2.4) and the estimate (3.17), one can still prove that the equation has cones of dependence. Remark 3.8. For s ≤ r/(r − 1), (3.16) and (3.15) are empty conditions, so by Theorem 3.5 we obtain as in [32] the well-posedness in such Gevrey classes without any Levi condition. In the other direction, taking s = +∞ we reobtain formally the well-posedness in C ∞ but it is well known that the union of all Gevrey classes is strictly contained in C ∞ . Example 3.9. Let P satisfy condition (2.4), so r

P = Ar11 . . . Add + Rm−1 , with Aj , j = 1, . . . , d , are strictly hyperbolic diﬀerential operators, Rm−1 is an operator of order m − 1. Denoting (r −k)+

Qk = A1 1

(r

. . . Ad d

−k)+

, k = 0, 1, . . . , (z)+ = max{z, 0}, A0 = I,

P fulﬁlls the Gevrey-Levi condition (3.16) if (and only if in case of analytic coefﬁcients) m Bj Qkj , kj = [js/(s − 1)], (3.18) P = Q0 + j=1

where Bj denotes an arbitrary diﬀerential operator of order m − j − ord Qkj .

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Notice that for j ≥ r(1 − 1/s) one has kj ≥ r so Qkj = I. This means that the terms of order less or equal to m − r(1 − 1/s) can be chosen arbitrarily, in particular, (3.18) becomes trivial for s ≤ r/(r − 1). Let us consider the rational numbers sh = j/k, j, k ∈ Z, 1 ≤ k < j ≤ r, r/(r − 1) = s1 < s2 < . . . < sν = r. The integers kj , so the structure (3.18) of P , remain the same for all s in a ﬁxed interval ]sh , sh+1 ], h = 0, . . . , ν, (s0 = 1, sν+1 = +∞) without any further condition besides (2.4) in the ﬁrst of such intervals, ]1, r/(r − 1)]. In this sense, we have only ν distinct Gevrey-Levi conditions, including the C ∞ condition (2.6) which persists for all s > r (there is only this one for operators with r = 2, that is with at most double roots). So, taking strictly hyperbolic operators A1 and A2 of respective orders µ1 and µ2 , operators P of order m = 3µ1 + 2µ2 which satisfy all the assumptions of Theorem 3.5 are given by P = A31 A22 + Rm−2µ1 −µ2 −1 A21 A2 + Rm−µ1 −2 A1 + Rm−3 , s > 3, P = A31 A22 + Rm−2µ1 −µ2 −1 A21 A2 + Rm−2 , 2 < s ≤ 3, P = A31 A22 + Rm−µ1 −1 A1 + Rm−2 , 3/2 < s ≤ 2, where Rj denotes an arbitrary operator of order j, A1 = A1 (t, x, Dµ1 −1 , Dt , Dx ), and A2 = A2 (t, x, Dµ2 −3 , Dt , Dx ), Rj = Rj (t, x, Dm−3 , Dt , Dx ), s > 3, A2 = A2 (t, x, Dµ2 −2 , Dt , Dx ), Rj = Rj (t, x, Dm−2 , Dt , Dx ), s ≤ 3. The orders of the nonlinear terms, µ2 − 2 or µ2 − 3, m − 2 or m − 3, are determined by condition (3.15). We have proved Theorem 3.5 in [15]. We recall the main steps in the proof through the following Subsections 3.3, 3.4 and 3.5. 3.3. Factorization under Gevrey-Levi conditions Let u ∈ CTk0 +m H λ,s,µ0 +m , be a given function with Dm u(t, x) ∈ W , W is a neighborhood of the origin in RN , µ0 so large that Propositions 3.3 can be applied, k0 large enough to allow all the ﬁnitely many derivatives with respect to t we need to take into consideration. Let the linear operator P α P (t, x, Dm u, Dt , Dx ) = aα (t, x, Dm u)Dt,x , |α|≤m

aα ∈ B k0 [0, T ]; γ s,1 , |α| ≤ m,

On the Nonlinear Cauchy Problem

423

satisfy the hyperbolicity condition (2.4) and the linear Gevrey-Levi condition (3.16). Following [41], one can take k0 large enough in order to have ⎧ m ⎪ ⎪ ⎪ P = P ◦ · · · ◦ P ◦ P + R, R = r Dtm− , d 2 1 ⎪ ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (j) (j) mj mj −1 ⎪ ⎪ + · · · + a mj , ⎪ Pj = (Dt − λj ) + a1 (Dt − λj ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ λj = λj (t, x, D u, Dx ), ⎪ ⎪ ⎪ ⎪ ⎨ λj (t, x, w, ξ) ∈ C k0 [0, T ]; S 1,s,1 , j = 1, . . . , d; (3.19) ⎪ ⎪ ⎪ ⎪ (j) M1 +m ⎪ ⎪ a(j) u, Dx ), ⎪ k = ak (t, x, D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a(j) (t, x, w, ξ) ∈ C m−1 [0, T ]; S k/s,s,1 , k = 1, . . . , mj ; ⎪ k ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r = r (t, x, DM1 +m u, Dx ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r (t, x, w, ξ) ∈ C [0, T ]; S −m,s,1 , = 1, . . . , m. Remark 3.10. In the factorization allowed by the condition (2.4), the Gevrey-Levi condition (3.16) ensures (j)

ord ak ≤ k/s. All the symbols in (3.19) are analytic functions of ﬁnitely many derivatives of the symbol of P , so they still have analytic regularity in the variable w. 3.4. Linear systems From the factorization (3.19) we can reduce the linear scalar equation in the unknown function v,

P (t, x, Dm u, Dt , Dx )v = f,

(3.20)

to an equivalent system. Now, the related Cauchy problem is well posed in Gevrey classes of exponent s. Also here, just to have a simpler notation, let us consider the case d = 2 of an operator with two multiple characteristic roots together with its two factorizations ˜ P = P2 ◦ P1 + R, P = P˜1 ◦ P˜2 + R. Let us deﬁne the functions v0 , . . . , vm−1 , vm , . . . , v2m−1

424

Massimo Cicognani and Luisa Zanghirati

as in (2.13). Now, we have ⎧ j ⎪ (j) ⎪ j ⎪ D v = c (t, x, DM2 +m u, Dx )vj− + vj , ⎪ t ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎨

j (j) j ⎪ ⎪ Dt v = c˜ (t, x, DM2 +m u, Dx )vm+j− + vm+j , ⎪ ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎩ j = 0, . . . , m − 1,

with

(3.21)

(j) (j) c (t, x, w, ξ), c˜ (t, x, w, ξ) ∈ C [0, T ]; S ,s,1 ,

˜ as follows: so we can represent the remainders R, R ⎧ m−1 ⎪ ⎪ ⎪ Rv = bj (t, x, DM3 +m u, Dx )vj , ⎪ ⎪ ⎪ ⎪ j=0 ⎨ * ⎪ Rv ˜ = m−1 ˜bj (t, x, DM3 +m u, Dx )vm+j , ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎩ bj (t, x, w, ξ), ˜bj (t, x, w, ξ) ∈ C [0, T ]; S 0,s,1 .

(3.22)

Here we do not take V as the vector of components vj , j = 0, . . . , 2m − 1, but we need to introduce a weight, deﬁning ⎧ V = (˜ v0 , . . . , v˜m−1 , v˜m , v˜m+1 , . . . , v˜2m−1 ), ⎪ ⎪ ⎪ ⎪ ⎨ (3.23) v˜ = Dx (m−1−j)/s vj , v˜m+j = Dx (m−1−j)/s vm+j , ⎪ j ⎪ ⎪ ⎪ ⎩ j = 0, . . . , m − 1. From (2.14), (3.22) and (3.23), we have that the scalar Cauchy problem ⎧ ⎨ P (t, x, Dm u, Dt , Dx )v(t, x) = f0 (t, x), (3.24) ⎩ j Dt v(0, x) = gj (x), 0 ≤ j < m, is equivalent to

⎧ ⎨ ∂t V − K(t, x, DM3 +m u, Dx )V = F0 , ⎩

(3.25) V (0) = G,

F0 = (0, . . . , 0, f0 , 0, . . . , 0, f0 ), for a hyperbolic 2m × 2m system ∂t − K which satisﬁes all the assumptions of Proposition 3.4, K = i∆ + R0 , ∆ is a real diagonal matrix pseudodiﬀerential operator of order 1, R0 is a matrix pseudodiﬀerential operator of order 1/s.

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425

But, in order to solve (3.24) in CTm−1 H λ,s,µ , we have to take u ∈ CTk0 +m H λ,s,µ0 +m with µ0 > µ. So we need again to deal with derivatives ∂ β v and commutators [∂ β , P ]v. Following now the proofs of Proposition 2.14 and Proposition 2.16 we obtain the following statement: Proposition 3.11. Let the vector V be deﬁned by (3.23). Then there is a matrix Q = Q(t, x, DM4 +m u, Dx ) such that Dm v = QV, Q(t, x, w, ξ) ∈ C [0, T ]; S m −m+r(1−1/s)+1/s,s,1 .

(3.26)

Remark 3.12. The nonlinear Gevrey-Levi condition m ≤ m − r(1 − 1/s) ensures that Q has an order less or equal to 1/s. Proposition 3.13. Let M ≥ k0 be any ﬁxed integer and let (β) V˜ = (˜ vj ; 0 ≤ j ≤ 2m − 1, β0 ≤ k0 , |β| ≤ M )

(3.27)

(β)

be the vector of components vj deﬁned by (2.13) and (3.23) taking there v (β) = ∂ β v = ∂tk0 ∂xβ v in place of v. ˜ = Then there are a matrix of functions F = F (t, x, DM+m u) and a matrix Q M5 +m ˜ Q(t, x, D u, Dx ) of operators such that ⎧ β ˜ V˜ , ⎨ [∂ , P ]v; β0 ≤ k0 , |β| ≤ M = F Q (3.28) ⎩ ˜ x, w, ξ) ∈ C [0, T ]; S 1/s,s,1 . F (t, x, w) ∈ C [0, T ]; γ s,1 , Q(t, In (3.28), let us write F (t, x, DM+m u) = F (t, x, 0)+F˜ (t, x, DM+m u). Taking derivatives ∂ β , the two above propositions, (2.14), (3.22), (3.23) and (3.27) show that, for M ≥ M5 , the problem (3.24) is equivalent to a system ⎧ ˜ V˜ + F˜ Q ˜ V˜ = F˜0 , ⎨ ∂t V˜ − K (3.29) ⎩ ˜ ˜ V (0) = G,

with

˜ = K(t, ˜ x, DM5 +m u, Dx ), Q ˜ = Q(t, ˜ x, DM5 +m u, Dx ), K

F˜ = F˜ (t, x, DM+m u), F˜ (t, x, 0) = 0, F˜0 = F˜0 (t, x). ˜ is hyperbolic, K ˜ = i∆ ˜ +R ˜ 0 with ∆ ˜ is a real diagonal matrix The operator ∂t − K ˜ among the terms R ˜ 0 of order pseudodiﬀerential operator of order 1, F (t, x, 0)Q 1/s. Let us ﬁx µ such that Proposition 3.3 can be applied and consider µ0 = µ+M for any suﬃciently large M .

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Massimo Cicognani and Luisa Zanghirati

Taking Proposition 3.11 into account, the following proposition shows that now, assuming u ∈ CTk0 +m H λ,s,µ0 +m of small norm, we ﬁnd a unique solution v ∈ CTk0 +m H λ,s,µ0 +m−r(1−1/s)−1/s of (3.24). This means that we loose only δ = m − m + r(1 − 1/s) + 1/s derivatives and, imposing condition (3.15), we have δ = 1/s < 1. Proposition 3.14. There are M0 , R0 > 0, λ0 > 0 and T0 > 0 such that, for any u ∈ CTk0 +m H λ,s,µ+M+m with λ ≥ λ0 , M ≥ M0 , uC k0 +m T

(H λ,s,µ+M +m )

≤ R0 ,

the Cauchy problem (3.29) has a unique solution V˜ ∈ CT0 (H λ,s,µ ) for every given ˜ ∈ H λT,s,µ and F˜0 ∈ C 0 (H λ,s,µ ), provided that T ≤ T0 /λ. G T Such a solution satisﬁes the estimate ⎧ 2 ˜ ⎪ ⎪ V (t)H λ(T −t),s,µ ⎪ ⎪ ⎪ ⎪ t ⎨ ˜ 2 λT ,s,µ + (3.30) ≤ (1 + tC) G F˜0 (τ )2H λ(T −τ ),s,µ dτ , H ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ t ∈ [0, T ], λT ≤ T0 . Proof. We can follow the proof of Proposition 3.4 if we can control the term ˜ x, DM5 +m u, Dx )V˜ F˜ (t, x, DM+m u)Q(t,

which contains the derivatives of highest order of u. ˜ −λ(T −t)Dx 1/s ; ˜ + F˜ Q ˜ and S˜Λ = eλ(T −t)Dx 1/s Se Let us denote S˜ = ∂t − K we have ˜ Λ + (F˜ Q) ˜ Λ. S˜Λ = ∂t + λDx 1/s − K As in the proof of Proposition 3.4, it is suﬃcient to prove that one can choose λ ≥ λ0 such that the inequality t 2 SΛ V (τ )2H µ dτ (3.31) V (t)H µ ≤ C V (0)2H µ + 0

holds for every V ∈ C ([0, T ]; H ) ∩ C ([0, T ]; H µ ), t ∈ [0, T ]. 1/s Let us denote V˜ (t) = e−λ(T −t)Dx V (t). Since M >> M5 , the continuity norm of ˜ : H λ(T −t),s,µ+1/2s → H λ(T −t),s,µ−1/2s , Q(t) 0

µ+1

1

λT ≤ T0 , is bounded by C(1 + R0 ) with a positive C depending only on µ, s, n.

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427

Using also the Banach algebra property of H λ(T −t),s,µ−1/2s and Proposition 3.3, we have ˜˜ (F Q)Λ V (t), V (t) Hµ

˜ V˜ (t), V˜ (t) = F˜ (t)Q ˜ V˜ (t) ≤ F˜ (t)Q

H

λ(T −t),s,µ

H λ(T −t),s,µ−1/2s

≤ C(1 + R0 ) F˜ (t)

˜ V (t)

H λ(T −t),s,µ−1/2s

H λ(T −t),s,µ+1/2s

˜ 2 V (t)

≤ C1 R0 (1 + R0 ) Dx 1/s V˜ (t), V˜ (t)

H λ(T −t),s,µ+1/2s

H λ(T −t),s,µ

= C1 R0 (1 + R0 ) Dx 1/s V (t), V (t) H µ . Since in the proof of Proposition 3.4 we have already checked such an estimate for ˜ Λ , we can choose also here λ0 such that for all λ ≥ λ0 it holds K d V (t)2H µ ≤ C(V (t)2H µ + S˜Λ V (t)2H µ ) dt which implies (3.31) by Gronwall’s inequality. 3.5. The equivalent quasilinear system in Gevrey spaces For |β| ≤ M , β0 ≤ k0 , M > M0 , let us denote by (β)

U (β) = (uj ; 0 ≤ j ≤ 2m − 1)

(3.32)

(β)

the vector of components uj deﬁned by (3.23) taking there ∂ β u in place of v, and, for k ≤ M , let us denote by ˆk = (U (β) ; |β| ≤ k), U ˆ =U ˆM . U (3.33) By deﬁnition, we have ˆk ⊂ U ˆ , |γ| ≤ M − k. ∂xγ U ˆ = Q(t, ˆ x, U ˆM3 +m , Dx ) such that From (3.11), there is a Q ⎧ M+m ˆU ˆ, ⎪ u=Q ⎨ D ⎪ ⎩ Q(t, ˆ x, w, ξ) ∈ C [0, T ]; S m −m+r(1−1/s)+1/s,s,1 ,

(3.34)

(3.35)

so, by (3.29) and (3.35), we have that the quasilinear scalar Cauchy problem (3.14) can be reduced to a quasilinear system ⎧ ˆ − K(t, ˆ x, U ˆ , Dx )U ˆ = Fˆ (t, x, Q ˆU ˆ ), ⎨ ∂t U (3.36) ⎩ ˆ ˆ U(0) = G,

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Massimo Cicognani and Luisa Zanghirati

where ˆ = K(t, ˜ x, U ˆM5 +m , Dx ) + F˜ (t, x, Q ˆU ˆ )Q(t, ˜ x, U ˆM5 +m , Dx ), K (3.37) s,1 ˜ ˜ ˜ ˆ ˆ K, F , Q as in (3.29), and F (t, x, w) ∈ C([0, T ]; γ ) such that F (t, x, 0) has compact support. We prove Theorem 3.5 and the inequality (3.17) after ﬁxing µ and showing that for every M > M0 there is a unique solution ˆ ∈ C 0 ∗ (H λ,s,µ ) U T

ˆ in a small neighborhood of the origin in H λT ∗ ,s,µ , λ ≥ λ0 , λT ∗ ≤ for any given G T0 . ˆ is of order 0, that is, if m ≤ m − r(1 − 1/s) − 1/s, Proposition 3.14 If Q implies the convergence of the usual approximation sequence deﬁned by (2.41) to such a unique solution. ˆ is positive, we can linearize the terms Fˆ (t, x, Q ˆU ˆ ) and If the order of Q ˜ ˆ ˆ F (t, x, QU ) applying derivatives ∂j = ∂xj , j = 1, . . . , n, so obtaining other n ˆ: ˆ, U ˆj = ∂j U ˆ0 = U equations, besides (3.36), for U ⎧ ˆj − K ˆU ˆj = Fˆxj + Fˆw [∂ ˆU ˆj − [∂ ˆ U ˆ0 − Fˆw Q ˆU ˆ0 , ∂j , K] ∂j , Q] ⎨ ∂t U ⎩ ˆ ˆj , Uj (0) = G where we can write ˆ0 = U

n

ˆi , ai ∈ S −1 . ai (Dx )U

i=0

By means of Proposition 3.14, we obtain again the convergence of the usual apˆ is less proximation sequence to a unique solution provided that the order of Q or equal to 1/s. This is exactly condition (3.15) which turns out to be the right Gevrey-Levi nonlinear condition of exponent s. Now, the proof of Theorem 3.5 is complete. 3.6. Local Gevrey solutions and propagation of the analytic regularity Let Ω ⊂ Rn+1 be an open set, where the variable is denoted by y. We introduce the class Gs (Ω) of local Gevrey functions deﬁned as the space of all functions f in C ∞ (Ω) such that for every compact set K ⊂ Ω |∂ ∂yα f (y)| ≤ CA|α| |α|!s , y ∈ K, α ∈ Zn+1 + with positive constants C = C(K), A = A(K). Let S = {ϕ(y) = 0} be an hypersurface with ϕ ∈ Gs (Ω) is a real-valued function, ∇ϕ = 0 in S, ν = ∇ϕ/∇ϕ is the unit normal vector, and let us consider the Cauchy problem ⎧ ⎪ P (y, Dm u, Dy )u = f (y, Dm u), ⎪ ⎨ (3.38) ⎪ ∂j u ⎪ ⎩ = g in S, 0 ≤ j < m, j ∂ν j

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429

gj ∈ Gs (S), j = 1, . . . , m − 1, f ∈ Gs (Ω; G1 (W )), W is a neighborhood of the Cauchy data, for an operator P (y, Dm u, Dy ) = aα (y, Dm u)Dyα (3.39) |α|≤m

with m < m, aα ∈ Gs (Ω; G1 (W )), |α| ≤ m. Given y0 , ν0 with y0 ∈ S and ν0 the normal vector to S at y0 , we say that the operator P (y, Dm u, Dy ) satisﬁes the condition of hyperbolicity (2.4) and the Levi conditions (3.15), (3.16) at (y0 , ν0 ) if there is a neighborhood of y0 in Rn+1 where these conditions are fulﬁlled taking local coordinates y = (t, x) with dt = ν0 . From Theorem 3.5 and Remark 3.7 we have the following result of local existence and uniqueness of Gs solutions. Theorem 3.15. Let the condition of hyperbolicity (2.4) and the Gevrey-Levi conditions (3.15), (3.16) be satisﬁed at (y, ν) for every y ∈ S. Then, for every gj ∈ Gs (S), j = 1, . . . , m − 1, there is a neighborhood Ω of S in Ω such that the Cauchy problem (3.38) has a unique solution u ∈ Gs (Ω ). In particular, uniqueness can be proved in domains of inﬂuence ) S¯µ D= 0≤µ≤1

with {Sµ ; 0 ≤ µ ≤ 1} is a continuous family of surfaces such that for all µ ∈ [0, 1] the condition of hyperbolicity (2.4) and the Gevrey-Levi conditions (3.15), (3.16) are satisﬁed at every point (y, ν), y ∈ S¯µ , ν is the normal vector to S¯µ at y. For a ﬁxed solution u ∈ Gs (A) of

P (y, Dm u, Dy )u = f (y, Dm u)

(3.40)

in an open set A ⊂ Ω, one can prove the propagation of analytic regularity in domains of inﬂuence with analytic boundary, provided that the coeﬃcients satisfy aα (y, w), |α| ≤ m, f (y, w) are analytic functions of their arguments in a neighborhood of {(y, Dm u(y)); y ∈ A}.

(3.41)

Theorem 3.16. Let the condition (3.41) be fulﬁlled, S be an analytic hypersurface, u ∈ Gs (A) be a ﬁxed solution in A of equation (3.40) and let D ⊂ A be a domain of inﬂuence determined by u and based on S¯0 , S0 is a bounded open set in S with C ∞ boundary. ∂j u , 0 ≤ j ≤ m − 1, are analytic in S¯0 , then u is analytic in D. If the traces ∂ν j S Starting now from the equivalent system (3.36) and using inequality (3.30), we can prove Theorem 3.16 in the same way as Theorem 2.20, taking here norms in Gevrey-Sobolev spaces H λ,s,µ instead of usual Sobolev spaces H µ .

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4. Strictly hyperbolic equations with non-Lipschitz coeﬃcients and C ∞ solutions 4.1. Log-Lipschitz coeﬃcients or unbounded derivatives Let us consider now the Cauchy problem ⎧ ⎨ P (t, x, Dm−1 u, Dt , Dx )u(t, x) = f (t, x, Dm−1 u), ⎩

(4.1) ∂tj u(t0 , x) = gj ,

0 ≤ j < m,

with Cauchy data at t0 ∈ [0, T [ for ⎧ m−1 ⎪ ⎪ m ⎪ P = D + aj (t, x, Dm−1 u, Dx )Dtj , ⎪ t ⎪ ⎨ j=0 ⎪ ⎪ ⎪ ⎪ ⎪ aj = ⎩

ajα (t, x, D

m−1

(4.2)

u)Dxα ,

|α|≤m−j

with principal symbol Pm (t, x, w, τ, ξ) = τ m +

ajα (t, x, w)ξ α τ j .

j+|α|=m

The variable w belongs to an open neighborhood W of the origin in RN . Let us assume ⎧ m 1 ⎪ ⎪ ⎪ (τ − λj (t, x, w, ξ)), λj ∈ R, ⎨ Pm = j=1 (4.3) ⎪ ⎪ ⎪ ⎩ λ1 (t, x, w, ξ) < λ2 (t, x, w, ξ) < . . . < λm (t, x, w, ξ), ξ = 0, so we are now dealing with strictly hyperbolic equations. In this section we are concerned with the problem of determining the sharp regularity of the coeﬃcients ajα with respect to the time variable t in order to have well-posedness for the Cauchy problem (4.1) in C ∞ . Only the ajα ’s for j + |α| = m are important since we do not need much regularity for lower order terms: we take continuous bounded coeﬃcients in t there but, for instance, L1 (0, T ) in t is suﬃcient. We use and mix two diﬀerent scales of regularity of global and local type: the modulus of continuity and/or the behaviour with respect to |t − t1 |−q , q ≥ 1, of the ﬁrst derivative as t tends to a point t1 ∈ [t0 , T ], say t1 = T. Both are ways to weaken the Lipschitz regularity in t. Precisely, the coeﬃcients of the principal part are either supposed to be LogLipschitz continuous, i.e., |ajα (t + τ, x, w) − ajα (t, x, w)| ≤ C|τ || log |τ ||, C > 0, 0 < |τ | ≤ 1/4, x ∈ Rn , w ∈ W,

On the Nonlinear Cauchy Problem

431

or to satisfy the following condition: ajα ∈ C 1 ([0, T [; B ∞(Rn × W )) ∩ C 0 ([0, T ]; B ∞(Rn × W )), |∂ ∂t ajα (t, x, w)| ≤ C(T − t)−1 , C > 0, x ∈ Rn , w ∈ W. We use the notation a ∈ LL([0, T ]) for a Log-Lipschitz function a in [0, T ]. Theorem 4.1. Let the operator P in (4.2) be strictly hyperbolic with ajα ∈ C 0 ([0, T ]; B ∞(Rn × W )), j + |α| ≤ m, and let the coeﬃcients of the principal part satisfy either ajα ∈ LL([0, T ]; B ∞(Rn × W )), j + |α| = m,

(4.4)

or

(4.5) (T − t)∂ ∂t ajα ∈ B 0 ([0, T [; B ∞ (Rn × W )), j + |α| = m. 0 ∞ n Let f ∈ C ([0, T ]; B (R ×W )) with f (t, x, 0) of compact support. Then, for every gj ∈ H ∞ (Rn ), j = 0, . . . , m − 1, there is a T ∗ > 0 such that if T − t0 ≤ T ∗ , then the Cauchy problem (4.1) has a unique solution u in C m−1 ([t0 , T ]; H ∞ (Rn )). Remark 4.2. We have well-posedness in H ∞ with a loss of derivatives, that is, there is a positive δ such that for every Cauchy data gj ∈ H µ+δ+m−1−j , the solution u belongs to m−1 2 C j ([t0 , T ]; H µ+m−1−j ). j=0

We are able to describe the loss of derivatives as a function of t ﬁnding a decreasing weight function q(t, ξ) such that q(T, ξ) = 1 ≤ q(t, ξ) ≤ cξδ , |∂ ∂ξα q(t, ξ)| ≤ cα ξ−|α| q(t, ξ), and such that m−1 q(t, Dx )∂ ∂tj u(t)µ+m−1−j j=0

⎛

m−1

≤C⎝

⎞ t

q(t0 , Dx )gj µ+m−1−j +

j=0

q(s, Dx )f (s)µ ds⎠ .

t0

This estimate of strictly hyperbolic type (see [46]) in the weighted Sobolev spaces µ Hq(t) (Rn ) := {g ∈ H µ (Rn ); q(t, Dx )g ∈ H µ (Rn )}, gµ,q(t) := q(t, Dx )gµ

allows us to prove existence and uniqueness of smooth solutions by a standard ﬁxed point argument. Remark 4.3. The existence of cones of dependence can be proved as in [29]. This implies well-posedness in C ∞ (Rn ).

432

Massimo Cicognani and Luisa Zanghirati

In the case m = 2 of second order operators, Theorem 4.1 has been proved in [35] under the assumption of unbounded derivatives and, following our procedure, in [3] assuming Log-Lipschitz coeﬃcients. The proof of the general case that we are going to give is the same as in [13], see also [12]. 4.2. The linear problem with non-regular coeﬃcients Let us ﬁrst prove Theorem 4.1 for linear equations ⎧ ⎨ P (t, x, Dt , Dx )u(t, x) = f (t, x), ⎩

(4.6) ∂tj u(t0 , x) = gj (x),

0 ≤ j < m.

In the linear case, we have T ∗ = T , that is, global in time solutions. In our scheme, this time the scalar equation P u = f will be reduced to an equivalent m × m system SU = F (4.7) with ⎧ S = ∂t + i∆(t, x, Dx ) + R(t, x, Dx ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎞ ⎛ ˜ 0 λ1 (4.8) ⎪ ⎟ ⎜ .. ⎪ ⎪ ∆=⎝ ⎠, . ⎪ ⎪ ⎩ ˜m 0 λ ˜ j , j = 1, . . . , m, are real-valued molliﬁed roots of P and λ ∆, R ∈ C 0 ([0, T ]; S 1 ).

(4.9)

Here the matrix R(t) is pointwise of order 1 but there will be a function ϕ(t, ξ) such that ⎧ |R(t, x, ξ)| ≤ ϕ(t, ξ) for all t, x, ξ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ∈ C 0 ([0, T ]; S 1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T (4.10) ϕ(t, ξ)dt ≤ c0 + δ logξ, c0 , δ ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ ⎩ ∂ξα ϕ(s, ξ)ds ≤ cα ξ−|α| . t

The function

q(t, ξ) = exp

T

ϕ(s, ξ)ds

(4.11)

t

satisﬁes

⎧ ⎨ q(T, ξ) = 1 ≤ q(t, ξ) ≤ cξδ , δ ≥ 0, ⎩

|∂ ∂ξα q(t, ξ)| ≤ cα ξ−|α| q(t, ξ). Then we shall apply the following statement:

(4.12)

On the Nonlinear Cauchy Problem Theorem 4.4. Let S be as in (4.8), (4.9) and let us assume (4.10). For every U ∈ C 0 ([0, T ]; H µ+1+δ (Rn )) ∩ C 1 ([0, T ]; H µ+δ (Rn )) we have ⎧ 2 ⎪ q(t, Dx )U (t)µ ⎪ ⎨ t ⎪ ⎪ Cµ ) q(t0 , Dx )U (t0 )2µ + q(s, Dx )SU (s)2µ ds , ⎩ ≤ (1 + (t − t0 )C

433

(4.13)

t0

t0 ∈ [0, T [, t0 ≤ t ≤ T, Cµ > 0, with q(t, ξ) deﬁned by (4.11). Proof. It is suﬃcient to prove (4.13) for µ = 0 since Dx µ SDx −µ satisﬁes the same hypotheses as S for every µ. Let us deﬁne Sq = q(t, Dx )Sq −1 (t, Dx ). We have to prove U (t)20

≤

C0 U (t0 )20

t

+

Sq U (s)20 ds , t0 ≤ t ≤ T, C0 > 0,

(4.14)

t0

for every U ∈ C 0 ([0, T ]; H 1 (Rn )) ∩ C 1 ([0, T ]; H 0 (Rn )). One has Sq = ∂t + i∆(t, x, Dx ) + R(t, x, Dx ) + ϕ(t, Dx )I + R0 (t, x, Dx ) with R0 ∈ C 0 ([0, T ]; S 0 ). Since ∆ is a real diagonal matrix pseudodiﬀerential operator, also the symbol a(t, x, ξ) of i∆ + (i∆)∗ belongs to C 0 ([0, T ]; S 0 ). Furthermore, the inequality |R(t, x, ξ)| ≤ ϕ(t, ξ) allows us to apply the sharp G˚ ˚ arding inequality to the operator R + ϕI obtaining: 2Re RU (t) + ϕU (t), U (t) ≥ −CU (t)20 . So we have d U (t)20 ≤ CU (t)20 + Sq U (t)20 dt which implies (4.14) by Gronwall’s inequality. The proof is complete.

Since (4.13) implies

t U (t)2µ ≤ (1 + (t − t0 )C Cµ ) U (t0 )2µ+δ + SU (s)2µ+δ ds t0

the Cauchy problem for the system S is H ∞ well posed with a loss of δ derivatives. Now we can apply Theorem 4.4 to the Cauchy problem for a scalar linear operator P . As usual in our scheme, the reduction to an equivalent system is based on a factorization of P . Here the operator is strictly hyperbolic, so it is natural to deal with a factorization only of the principal part. On the other hand, we can bring the algebraic factorization (4.3) to the operators level only after having molliﬁed the symbols λj which are not diﬀerentiable in t, see (4.16) below.

434

Massimo Cicognani and Luisa Zanghirati

Theorem 4.5. Let the linear operator P satisfy all the assumptions of Theorem 4.1 (with coeﬃcients independent of w). Then there is a function q(t, ξ) as in (4.11), m 2 C j ([0, T ]; H µ+δ+m−j ) we have: (4.12) such that for every u ∈ j=0

⎧ m−1 ⎪ ⎪ ⎪ q(t, Dx )∂ ∂tj u(t)µ+m−1−j ⎪ ⎪ ⎪ ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎛ ⎞ t m−1 ⎪ ⎪ q(t0 , Dx )∂ ∂tj u(t0 )µ+m−1−j + q(s, Dx )P u(s)µ ds⎠ , ⎪ ≤ Cµ ⎝ ⎪ ⎪ t 0 ⎪ j=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t0 ∈ [0, T [, t0 ≤ t ≤ T, Cµ > 0.

(4.15)

Proof. Let λj (t, x, ξ), j = 1, . . . , m, denote the characteristic roots of P . After a modiﬁcation in a neighborhood of ξ = 0 we may assume λj ∈ C 0 ([0, T ]; S 1 ) and either λj ∈ LL([0, T ]; S 1) or (T −t)∂ ∂t λj ∈ B 0 ([0, T [; S 1 ) according to the respective condition (4.4) or (4.5). Let us deﬁne molliﬁed symbols by ˜ x, ξ) = λ(t − sξ−1 , x, ξ)(s)ds (4.16) λ(t, with ∈ C0∞ (R), 0 ≤ ≤ 1, supp ⊂]0, ∞[,

(s)ds = 1, λj (s, x, ξ) = λj (0, x, ξ)

for s ≤ 0. Our aim is to perform a reduction of the scalar equation P u = f to a system SU = F which satisﬁes the assumptions in Theorem 4.4. The ﬁrst step is to check that the symbols Rjk , k ≥ 0, ˜ j , Rjk = ξ−k ∂ k λ ˜ Rj0 = λj − λ t j , k ≥ 1, fulﬁll all the conditions in (4.10). Under the assumption (4.4) we have |∂ ∂ξα ∂xβ Rjk | ≤ Cαβ ξ−|α| (1 + logξ) which gives (4.10) with ϕ = γ(1 + logξ) independent of t, γ > 0, δ = γT. If the coeﬃcients of P satisfy (4.5) we have both Rjk ∈ C 0 ([0, T ]; S 1 ) and (T − t)Rjk ∈ B 0 ([0, T [; S 0), thus we have |Rjk (t, x, ξ)| ≤ ϕ(t, ξ) choosing ϕ(t, ξ) = ψ((T − t)ξ)δξ + (1 − ψ((T − t)ξ))δ/(T − t) ψ(y) = 1 for |y| ≤ 1, ψ(y) = 0 for |y| ≥ 2,

On the Nonlinear Cauchy Problem

435

with a large δ > 0 and a smooth function ψ, 0 ≤ ψ ≤ 1. Also the other conditions in (4.10) are fulﬁlled, in particular, T −2ξ−1 T T ϕ(t, ξ)dt ≤ δ (T − t)−1 dt + δ ξdt ≤ c0 + δ logξ. 0

0

T −2ξ−1

So we can factorize P : ˜ m (t, x, Dx )) · · · (Dt − λ ˜ 1 (t, x, Dx ))+ P (t, x, Dt , Dx ) = (Dt − λ m−1

Rj (t, x, Dx )Dx m−1−j Dtj

j=0

with Rj satisfying (4.10) for j = 0, . . . , m − 1. Given the scalar function u, we deﬁne the vectors Z = (z0 , . . . , zm−1 ), U = M (t, x, Dx )Z, by

(4.17)

⎧ z0 = Dx m−1 u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜1 )u ⎪ z1 = Dx m−2 (Dt − λ ⎪ ⎨ ⎪ .. ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ ˜m−1 ) · · · (Dt − λ ˜1 )u, zm−1 = (Dt − λ

where M is the diagonalizer of the m × m matrix ⎛ ˜ 1 Dx λ 0 0 ˜2 ⎜ 0 λ D 0 0 x ⎜ ⎜ . . . . ⎜ . . ⎜ ⎝ ˜ 0 0 λm−1 Dx ˜m 0 0 λ

(4.18)

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

Notice that also ∂t M and ∂t M −1 fulﬁll (4.10). So the scalar equation P u = f is equivalent to a system SU = F , F = (0, . . . , 0, f ), for the vector U with S that satisﬁes all the assumptions in Theorem 4.4. Furthermore we have U = Q0 (t, x, Dx )(Dx m−1 u, Dx m−2 Dt u, . . . , Dtm−1 u) (Dx m−1 u, Dx m−2 Dt u, . . . , Dtm−1 u) = Q(t, x, Dx )U, Q, Q0 ∈ C 0 ([0, T ]; S 0 ), so inequality (4.13) yields (4.15).

436

Massimo Cicognani and Luisa Zanghirati

Theorem 4.1, in the linear case, follows immediately from Theorem 4.5 since (4.15) implies ⎧ m−1 ⎪ j ⎪ ⎪ ∂ ∂t u(t)µ+m−1−j ⎪ ⎪ ⎪ ⎪ ⎨ j=0 (4.19) ⎛ ⎞ ⎪ t ⎪ m−1 ⎪ ⎪ ⎪ ⎪ ≤ Cµ ⎝ ∂ ∂tj u(t0 )µ+m−1−j+δ + P u(s)µ+δ ds⎠ . ⎪ ⎩ t0 j=0

The Cauchy problem is H ∞ well posed with a loss of δ derivatives. Remark 4.6. • If the coeﬃcients of the principal part of P are Lipschitz continuous in t, then (4.19) holds with δ = 0, that is, the Cauchy problem is well posed in H ∞ without loss of derivatives. • For Log-Lipschitz coeﬃcients we have q(t, ξ) = eγ(T −t) ξγ(T −t) ,

(4.20)

so from (4.15) we get m−1

∂ ∂tj u(t)µ+m−1−j−γt

j=0

⎛

m−1

≤ Cµ ⎝

∂ ∂tj u(t0 )µ+m−1−j−γt0

j=0

⎞ t

+

P u(s)µ−γs ds⎠ ,

t0

that is, the loss of derivatives is a linear function of t. 4.3. The map u → v In this section we prove Theorem 4.1 in the general case. Considering the map u → v given by ⎧ ⎨ P (t, x, Dm−1 u, Dt , Dx )v(t, x) = f (t, x, Dm−1 u), (4.21) ⎩ j 0 ≤ j < m, ∂t v(t0 , x) = gj , µ (Rn ) and to look for a soluinequality (4.15) suggests to take Dm−1 u(t, ·) ∈ Hq(t) µ tion v such that also Dm−1 v(t, ·) ∈ Hq(t) (Rn ), where ⎧ µ n µ n µ n ⎨ Hq(t) (R ) := {g ∈ H (R ); q(t, Dx )g ∈ H (R )}, (4.22) ⎩ gµ,q(t) := q(t, Dx )gµ .

On the Nonlinear Cauchy Problem

437

Proposition 4.7. For every µ1 , µ2 ≥ 0 there is a positive constant C such that ⎧ ⎨ u1 u2 µ,q(t) ≤ Cu1 µ1 ,q(t) u2 µ2 ,q(t) , (4.23) ⎩ µ = min{µ1 , µ2 , µ1 + µ2 − n/2 − ε}, 0 ≤ t ≤ T, µ1 µ2 for any u1 ∈ Hq(t) , u2 ∈ Hq(t) and ε > 0. The constant C is independent of t and of u1 , u2 . µ In particular for µ > n/2 the space Hq(t) is a Banach algebra. Furthermore, there exists a µ0 > n/2 such that for every µ > µ0 µ µ f ∈ B ∞ , f (0) = 0, u ∈ Hq(t) = f (u) ∈ Hq(t) =⇒

(4.24)

with f (u)µ,q(t) ≤ a(uµ,q(t) ) for a polynomial a(y) = aµ,f (y), a(0) = 0, with positive coeﬃcients which are independent of t and u. Proof. There is not anything to prove if q(t, ξ) is given by (4.20) (Log-Lipschitz coeﬃcients). In the other case, we still have q(t, ξ + η) ≤ Kδ q(t, ξ)q(t, η) with Kδ independent of t. So (4.23) can be proved as in Lemma 1.3 and 1.4 in [5], where the same property for usual Sobolev spaces is considered. From this, we obtain also the composition in (4.24). In its proof, it is not restrictive to assume f (0) = f (0) = · · · = f (N ) (0) = 0 with a large N . In fact, one writes the (N ) (0) = · · · = gN (0) = 0, then Taylor expansion f (u) = pN (u) + gN (u), gN (0) = gN one applies the composition property to gN and the Banach algebra inequality to the polynomial pN . The ﬁrst step is to prove that there exists a µ0 > n/2 such that for every µ > µ0 µ f ∈ B ∞ , f (0) = 0, u ∈ Hq(t) = f (u) ∈ Hq0(t) . =⇒

(4.25)

∞

In doing so, let us write f (u) = g(u)u with g ∈ B . We have just to prove that 0 B j is a space of multipliers for Hq(t) provided that j is large enough. Let us take j h ∈ B ; the operator q(t, Dx )h(x)q −1 (t, Dx ) has a symbol of order 0 with ﬁnite regularity in x but with j large, say j ≥ j0 , this is enough for the L2 -boundness. We obtain (4.25) choosing µ0 > j0 + n/2. Next, for a ﬁxed integer k, 0 ≤ k ≤ [µ], we assume that for µ > µ0 µ f ∈ B ∞ , f (0) = · · · = f (k) (0) = 0, u ∈ Hq(t) = f (u) ∈ Hqk(t) =⇒

(4.26)

and prove that for µ > µ0 µ f ∈ B ∞ , f (0) = · · · = f (k+1) (0) = 0, u ∈ Hq(t) = f (u) ∈ Hq(t) =⇒

min{k+1,µ}

In

∂xj f (u) = f (u)uxj

.

(4.27)

438

Massimo Cicognani and Luisa Zanghirati

µ−1 k we have f (u) ∈ Hq(t) by (4.26) and uxj ∈ Hq(t) , so inequality (4.23) gives min{k,µ−1}

min{k+1,µ}

, that is, f (u) ∈ Hq(t) . ∂xj f (u) ∈ Hq(t) The composition property (4.24) is proved by induction on k.

As it concerns dependence on time variable, we deﬁne Cqk (t0 , T ; H µ ) as the space of all functions u(t, x) such that each map t → q(t, Dx )∂ ∂tj u(t, ·), j = 0, . . . , k, is continuous from [t0 , T ] to H µ−j with norm uCqk ,µ =

sup

sup ∂ ∂tj u(t, ·)µ−j,q(t) .

j=0,...,k t∈[t0 ,T ]

Now let us ﬁx µ > µ0 , µ0 as in Proposition 4.7. Since in the proof of the linear case we have performed only a ﬁnite number of operations, we can use also here symbols p ∈ H M S m of Sobolev limited regularity. So Proposition 4.7 and the proof of the linear case yield: Proposition 4.8. Let P satisfy the assumptions of Theorem 4.1. Then there are a function q(t, ξ) as in (4.11), (4.12) and an integer M0 such that for any u ∈ Cqm t0 , T ; H µ+m−1+M0 , Dm−1 u(t, x) ∈ W , the linear Cauchy problem (4.21) has a unique solution v ∈ Cqm t0 , T ; H µ+m−1 , for every given µ+m−1−j data gj ∈ Hq(t , j = 0, . . . , m − 1. 0) Such a solution satisﬁes m−1

q(t, Dx )∂ ∂tj v(t)µ+m−1−j

j=0

⎛ ≤ a(uCqm ,µ+m−1+M0 ) ⎝

m−1

⎞ t

q(t0 , Dx )gj µ+m−1−j +

j=0

q(s, Dx )f (s)µ dss⎠.

t0

(4.28) with a(y) is a polynomial with positive coeﬃcients. Proof. Following the proof of Theorem 4.5 the equation P (t, x, Dm−1 u, Dt , Dx )v(t, x) = f (t, x) is equivalent to a system SV = F with ⎧ ⎨ S = ∂t + i∆(t, x, Dm−1 u, Dx ) + R(t, x, DM1 +m−1 u, Dx ), ⎩

(4.29) DM1 +m−1 u := (∂ ∂xβ Dm−1 u; |β| ≤ M1 ),

where the integer M1 is determined by the number of operations we need to perform. Furthermore, we have Dm−1 v = Q(t, x, DM2 +m−1 u, Dx )V

(4.30)

On the Nonlinear Cauchy Problem

439

with Q of order 0. So inequality (4.28) follows from (4.15) taking an integer M0 > M2 determined by M2 (≥ M1 ), the Sobolev index µ and the dimension n. We have uCqm,µ+m−1+M0 in the right-hand side of (4.28) because the derivative ∂tm u is involved in the estimates of the symbols Rjk (t, x, Dm−1 u(t, x), ξ) in the proof of Theorem 4.5. Of course the solution v of the Cauchy problem (4.21) belongs to Cqm t0 , T ; H µ+m−1 . We are again at a point where we need u ∈ Cqm t0 , T ; H µ+m−1+M0 to solve (4.21) in Cqm t0 , T ; H µ+m−1 . At such a point, in our procedure, we take derivatives ∂xβ in (4.21) for all |β| ≤ M , M ≥ M0 , and look for a reﬁned equivalent system. Denoting v (β) = ∂xβ v one obtains the Cauchy problems ⎧ (β) ∂xβ , P ]v (0) = ∂xβ f, ⎨ P v + [∂ |β| ≤ M. (4.31) ⎩ j (β) (β) ∂t v (t0 , x) = gj , 0 ≤ j < m, Starting from the scalar function v (β) let us deﬁne the vector V (β) in the same way as the vector U starting form the function u in (4.17) and (4.18); then we denote V˜ := (V (β) ; |β| ≤ M ). From (4.30), there are a matrix of functions F = F (t, x, DM+m−1 u) and matrices ˆ = Q(t, ˆ x, DM2 +m u, Dx ), Q ˜ = Q(t, ˜ x, DM2 +m u, Dx ) of operators of order 0 such Q that ⎧ M+m−1 ˆ V˜ , D v=Q ⎪ ⎪ ⎪ ⎪ ⎨ ˜ V˜ , (4.32) [∂ β , P ]v; |β| ≤ M = F Q ⎪ ⎪ ⎪ ⎪ ⎩ F (t, x, w) ∈ C ([0, T ]; B ∞(Rn × W )) . So, for M ≥ M0 , the Cauchy problems (4.31) are equivalent to a system ⎧ ˜ V˜ + F˜ Q ˜ V˜ = Fˆ , ⎨ ∂t V˜ − K ⎩ ˜ ˜ V (t0 ) = G,

(4.33)

with ˜ = K(t, ˜ x, DM2 +m−1 u, Dx ), Q ˜ = Q(t, ˜ x, DM2 +m−1 u, Dx ), K F˜ = F˜ (t, x, DM+m−1 u), F˜ (t, x, 0) = 0, Fˆ = Fˆ (t, x, DM+m−1 u). ˜ = ∂t − i∆ ˜ +R ˜ satisﬁes all the assumptions of Proposition The operator ∂t − K 4.4. Notice that the representation of the commutators becomes very simple when Levi conditions are not needed (cf. Propositions 2.16 and 3.13). Now, as always at this point of our method, the Banach algebra property, µ ˜ V˜ , ensuring that not any here of Hq(t) (Rn ), allows us to control the new term F˜ Q extra derivatives of u comes in.

440

Massimo Cicognani and Luisa Zanghirati

Proposition 4.9. For every V˜ ∈ Cq1 t0 , T ; H µ+1 which satisﬁes (4.33) we have t 2 2 ˜ ˜ Fˆ (s)2µ,q(s) ds , V (t)µ,q(t) ≤ (1 + (t − t0 )a(uCqm ,µ+m−1+M )) Gµ,q(t0 ) + t0

(4.34) t0 ≤ t ≤ T , the weight function q is as in Proposition 4.8, a(y) = aM (y) is a polynomial with positive coeﬃcients. Proof. Following the proof of Theorem 4.4 and denoting by (·, ·)µ,q(t) the scalar µ product in Hq(t) (Rn ), we have only to check ˜ V˜ (t), V˜ (t) 2Re F˜ Q ≥ −a(uCqm,µ+m−1+M )V˜ (t)2µ,q(t) (4.35) µ,q(t) for this new term. From Proposition 4.7, we have F˜ Q ˜ V˜ (t), V˜ (t)

µ,q(t)

˜ V˜ (t)µ,q(t) V˜ (t)µ,q(t) ≤ F˜ (t)µ,q(t) Q ≤ a(uCqm−1 ,µ+m−1+M )V˜ (t)2µ,q(t)

which gives (4.35).

We are now at the ﬁnal step of our procedure. For |β| ≤ M , M > M0 , let us denote by (β) U (β) = (uj ; 0 ≤ j ≤ 2m − 1) (4.36) (β)

the vector of components uj deﬁned by (4.17), (4.18) taking there ∂ β u in place of u, and, for k ≤ M , ˆk = (U (β) ; |β| ≤ k), U ˆ =U ˆM . U (4.37) So, from (4.32) and (4.33), we have proved that for any such M the quasilinear scalar Cauchy problem (4.1) can be reduced to a quasilinear system ⎧ ˆ − K(t, ˆ x, U ˆ , Dx )U ˆ = Fˆ (t, x, Q ˆU ˆ ), ⎨ ∂t U (4.38) ⎩ ˆ ˆ U(t0 ) = G, where ˆ = K(t, ˜ x, U ˆM2 +m−1 , Dx ) + F˜ (t, x, Q ˆU ˆ )Q(t, ˜ x, U ˆM2 +m−1 , Dx ), K (4.39) ˜ F˜ , Q, ˜ Fˆ as in (4.33), and, without loss of generality, we may assume that K, ˆ F (t, x, 0) has compact support. We prove Theorem 4.1 after ﬁxing µ and showing that for every M > M0 there is a unique solution ˆ ∈ Cq0 (t0 , T ; H µ ) U ˆ in H µ (Rn ). for any given G q(t0 )

On the Nonlinear Cauchy Problem

441

By Proposition 4.9, the usual iteration scheme converges to such a unique solution, provided that T − t0 is suﬃciently small. This concludes the proof of Theorem 4.1.

5. Holder ¨ coeﬃcients and Gevrey Solutions 5.1. Gevrey well-posedness In this section we apply our method to quasilinear equations with H¨ ¨older coeﬃcients of exponent χ ∈]0, 1]. Let us consider the quasilinear Cauchy problem ⎧ α ⎪ aα (t, x, Dm−1 u)Dt,x u = f (t, x, Dm−1 u), ⎪ ⎨ |α|≤m (5.1) ⎪ ⎪ ⎩ j Dt u(0, x) = gj (x), 0 ≤ j < m, gj ∈ γ s (Rn ), j = 1, . . . , m − 1, f, aα ∈ B 0 ([0, T ]; γ s,1 (Rn × W )), |α| ≤ m, W is a neighborhood in RN of the Cauchy data. As usual, without loss of generality, we may assume 0 ∈ W , and that f (t, x, 0), gj , j = 1, . . . , m − 1, have compact supports. For the principal part α Pm (t, x, Dm−1 u, Dt , Dx ) = aα (t, x, Dm−1 u)Dt,x |α|=m

of the operator

P (t, x, Dm−1 u, Dt , Dx ) =

α aα (t, x, Dm−1 u)Dt,x ,

|α|≤m

we assume the hyperbolicity condition (2.4) Pm (t, x, w, τ, ξ) =

d 1

(τ − λj (t, x, w, ξ))mj ,

j=1

λj ∈ R, j = 1, . . . , d, λh = λk for every (t, x, w, ξ), ξ = 0, h = k, still denoting by r ≥ 1 the largest multiplicity, and aα ∈ C χ ([0, T ]; γ s,1 (Rn × W )),

|α| = m, χ ∈]0, 1],

(5.2)

where s ≤ r/(r − χ). The result of well-posedness is the following:

(5.3)

Theorem 5.1. Assume that conditions (2.4), (5.2) and (5.3) are fulﬁlled. Then, for every gj ∈ γ s , j = 1, . . . , m − 1, there is T ∗ ≤ T such that the Cauchy problem (5.1) has a unique solution u ∈ C m ([0, T ∗ ]; γ s ) .

442

Massimo Cicognani and Luisa Zanghirati

Remark 5.2. Even in the strictly hyperbolic case, we do not have C ∞ wellposedness but the bound s ≤ 1/(1 − χ) for the Gevrey index which is optimal already for linear equations, [20], [11]. Remark 5.3. In the Lipschitz case χ = 1, we obtain well-posedness for s ≤ r/(r−1) which is the largest possible Gevrey index without any Levi conditions. A nonlinear result of this type is proved in [33]. Taking there regular roots one does not reach the bound r/(r − χ) that we conjecture to be optimal in this case. We have proved Theorem 5.1 in [16] under the technical assumption P = P (t, x, Dm−2 u, Dt , Dx ) for r ≥ 2 that we remove here. The result holds under the more general assumption of regular roots, not necessarily of constant multiplicity. We refer to [13] for linear equations and to [4] for quasilinear ones. 5.2. From the factorization to the quasilinear system Let u ∈ CTm H λ,s,µ0 +m−1 be a given function with Dm−1 u(t, x) ∈ W , W is a neighborhood of the origin in RN , µ0 is so large that Proposition 3.3 can be applied. After a modiﬁcation of the characteristic roots in a neighborhood of ξ = 0, we ˜ j as in (4.16), may assume λj ∈ C χ ([0, T ]; S 1,s,1), so deﬁning molliﬁed symbols λ now for Rjk , k ≥ 0, ˜ ˜ j , Rjk = ξ−k ∂ k λ Rj0 = λj − λ t j , k ≥ 1, we have Rjk ∈ C 0 ([0, T ]; S 1−χ,s,1 ). This allows the following factorization of P , which is the ﬁrst step in our method. Since the Gevrey index s is assumed smaller than r/(r − 1), we can take arbitrary lower order terms (not any Gevrey-Levi condition) and take care of the principal part only. In fact, we have ⎧ m ⎪ ⎪ ⎪ P = P ◦ · · · ◦ P ◦ P + R, R = r Dtm− , d 2 1 ⎪ ⎪ ⎪ ⎪ =1 ⎨ (5.4) ⎪ ˜ j )mj , P = (D − λ ⎪ j t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r = r (t, x, DM1 +m−1 u, Dx), r (t, x, w, ξ) ∈ C [0, T ]; S −χ,s,1 , where now DM1 +m−1 u = DxM1 Dm−1 u, M1 still depends only on m, n and the largest multiplicity r. In the second step of our method, from the factorization (5.4) we reduce the linear scalar equation in the unknown v, P (t, x, Dm−1 u, Dt , Dx )v = f, to an equivalent system.

(5.5)

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443

Also here, just to have a simpler notation, let us consider the case d = 2 of an operator with two multiple characteristic roots together with its two factorizations ˜ P = P2 ◦ P1 + R, P = P˜1 ◦ P˜2 + R. Let us take functions v0 , . . . , vm−1 , vm , . . . , v2m−1 as in (2.13). Then, we introduce a weight Dx , 0 ≤ < 1 to be chosen later, and deﬁne ⎧ V = (˜ v0 , . . . , v˜m−1 , v˜m , v˜m+1 , . . . , v˜2m−1 ), ⎪ ⎪ ⎪ ⎪ ⎨ (5.6) v˜j = Dx (m−1−j) vj , v˜m+j = Dx (m−1−j) vm+j , ⎪ ⎪ ⎪ ⎪ ⎩ j = 0, . . . , m − 1. From Proposition 3.11, for every real positive m there is a matrix Q = Q(t, x, DM2 +m−1 u, Dx ) such that

⎧ ⎪ (Dx m −j Dtj v, j = 0, . . . , m − 1) = QV, ⎨

⎪ ⎩ Q(t, x, w, ξ) ∈ C [0, T ]; S m −m+r(1−)+,s,1 .

(5.7)

With m = m − χ, we have to choose = (r − χ)/r in order to have Q of order . Provided this and (5.3), from (5.4), (5.6) and (5.7), we have that the scalar Cauchy problem ⎧ ⎨ P (t, x, Dm−1 u, Dt , Dx )v(t, x) = f0 (t, x), (5.8) ⎩ j Dt v(0, x) = gj (x), 0 ≤ j < m, is equivalent to ⎧ ⎨ ∂t V − K(t, x, DM2 +m u, Dx )V = F0 , (5.9) ⎩ V (0) = G, F0 = (0, . . . , 0, f0 , 0, . . . , 0, f0 ), for a hyperbolic 2m × 2m system ∂t − K with diagonal principal part which satisﬁes all the assumptions of Proposition 3.4, in particular K = i∆+R0 with ∆ is a real diagonal matrix pseudodiﬀerential operator of order 1 and R0 a matrix pseudodiﬀerential operator of order (r − χ)/r ≤ 1/s. The third step of our procedure is a reﬁnement of this reduction taking derivatives ∂ β ∂xβ , |β| ≤ M , in (5.8) and representing the commutators [∂ β , P ]v. When Levi conditions are not involved, like now, this representation is simple. In fact, taking m = m − 1 in (5.7) and writing (β) V˜ = (˜ vj ; 0 ≤ j ≤ 2m − 1, |β| ≤ M ),

(5.10)

444

Massimo Cicognani and Luisa Zanghirati (β)

the vector of components vj deﬁned by (2.13) and (5.6) taking there v (β) = ∂ β v in place of v, there are a matrix of functions F = F (t, x, DM+m−1 u) and matrices ˜ = Q(t, ˜ x, DM3 +m u, Dx ) of operators such that ˆ = Q(t, ˆ x, DM3 +m u, Dx ), Q Q ⎧ M+m−1 ˆ V˜ , D v=Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [∂ β , P ]v; |β| ≤ M = F Q ˜ V˜ , ⎨ (5.11) ⎪ s,1 ⎪ F (t, x, w) ∈ C [0, T ]; γ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆ ˜ x, w, ξ) ∈ C [0, T ]; S χ(1−1/r),s,1 , Q(t, x, w, ξ), Q(t, where χ(1 − 1/r) ≤ (r − χ)/r ≤ 1/s. So, for M ≥ M3 , the Cauchy problems (5.8) are equivalent to a system ⎧ ˜ V˜ + F˜ Q ˜ V˜ = F˜0 , ⎨ ∂t V˜ − K (5.12) ⎩ ˜ ˜ V (0) = G, with ˜ = K(t, ˜ x, DM3 +m−1 u, Dx ), Q ˜ = Q(t, ˜ x, DM3 +m−1 u, Dx ), K F˜ = F˜ (t, x, DM+m u), F˜ (t, x, 0) = 0, F˜0 = F˜0 (t, x). ˜ is hyperbolic with diagonal principal part and satisﬁes all the The operator ∂t − K assumptions of Proposition 3.4. In our forth step, we establish a sharp (without a large loss of derivatives in the map u → v) energy estimate for the solution of the system (5.12). Here, like in Proposition 3.14, there are M0 , R0 > 0, λ0 > 0 and T0 > 0 such that, for any u ∈ CTm H λ,s,µ+M+m−1 with λ ≥ λ0 , M ≥ M0 ,

uC m−1(H λ,s,µ+M +m−1 ) ≤ R0 , T

the Cauchy problem (5.12) has a unique solution V˜ ∈ CT0 (H λ,s,µ ) for every given ˜ ∈ H λT,s,µ and F˜0 ∈ C 0 (H λ,s,µ ), provided that T ≤ T0 /λ. G T Such a solution satisﬁes the estimate ⎧ t ⎪ 2 2 ⎪ ˜ ˜ F˜0 (τ )2H λ(T −τ ),s,µ dτ , ⎨ V (t)H λ(T −t),s,µ ≤ (1 + tC) GH λT ,s,µ + ⎪ ⎪ ⎩

0

t ∈ [0, T ], λT ≤ T0 .

(5.13) As always at this point of our method, the Banach algebra property allows us to ˜ V˜ , with respect to the proof of Proposition 3.4, ensuring control the new term F˜ Q that not any extra derivative of u comes in. In the ﬁnal step, for |β| ≤ M , M > M0 , we denote by (β)

U (β) = (uj ; 0 ≤ j ≤ 2m − 1)

(5.14)

On the Nonlinear Cauchy Problem

445

(β)

the vector of components uj deﬁned by (5.6) taking there ∂ β u in place of v, and, for k ≤ M , ˆ =U ˆM . ˆk = (U (β) ; |β| ≤ k), U (5.15) U So, from (5.11) and (5.12), we have that the quasilinear scalar Cauchy problem (5.1) can be reduced to a quasilinear system ⎧ ˆ − K(t, ˆ x, U ˆ , Dx )U ˆ = Fˆ (t, x, Q ˆU ˆ ), ⎨ ∂t U (5.16) ⎩ ˆ ˆ U(0) = G, where ˆ = K(t, ˜ x, U ˆM3 +m−1 , Dx ) + F˜ (t, x, Q ˆU ˆ )Q(t, ˜ x, U ˆM3 +m−1 , Dx ), K

(5.17)

˜ F˜ , Q ˜ as in (5.12), and Fˆ (t, x, w) ∈ C([0, T ]; γ s,1 ) such that Fˆ (t, x, 0) has comK, pact support. We prove Theorem 5.1 after ﬁxing µ and showing that for every M > M0 there is a unique solution ˆ ∈ C 0 ∗ (H λ,s,µ ) U T ∗ ˆ for any given G in a small neighborhood of the origin in H λT ,s,µ , λ ≥ λ0 , λT ∗ ≤ T0 . ˆ is of order 0, that is in the hyperbolic case r = 1 or if the coeﬃcients If Q of P depend only on Dm−2 u, the estimate (5.13) implies the convergence of the usual approximation sequence deﬁned by (2.41) to such a unique solution. ˆ is positive, we can linearize the terms Fˆ (t, x, Q ˆU ˆ ) and If the order of Q ˆU ˆ ) applying derivatives ∂j = ∂xj , j = 1, . . . , n, so obtaining other n F˜ (t, x, Q ˆ: ˆ0 = U ˆ, U ˆj = ∂j U equations, besides (5.16), for U ⎧ ˆj − K ˆU ˆj = Fˆxj + Fˆw [∂ ˆU ˆj − [∂ ˆ U ˆ0 − Fˆw Q ˆU ˆ0 , ∂j , K] ∂j , Q] ⎨ ∂t U ⎩ ˆ ˆj , Uj (0) = G where we can write ˆ0 = U

n

ˆi , ai ∈ S −1 . ai (Dx )U

i=0

By means of (3.30), we obtain again the convergence of the usual approximation ˆ is less or equal to 1/s. sequence to a unique solution since the order of Q This concludes the proof of Theorem 5.1.

References [1] R. Agliardi and M. Cicognani, The Cauchy problem for a class of Kovalevskian pseudo-diﬀerential operators, Proc. Amer. Math. Soc. 132 (2004), 841–845. [2] S. Alinhac and G. Metivier, Propagation de l’analycit´ ´e des solutions de syst´mes hyperboliques non-lin´eaires, Invent. Math. 75 (1984), 189–203.

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[3] A. Ascanelli, Quasilinear hyperbolic operators with log-Lipschitz coeﬃcients, J. Math. Anal. Appl. 295 (2004), 70–79. [4] A. Ascanelli, The Cauchy problem for quasilinear operators with non-absolutely continuous coeﬃcients, Preprint Math. Dep. Ferrara Univ. (2004). [5] M. Beals, Propagation and interaction of singularities in nonlinear hyperbolic problems, Birkh¨ ¨ auser, Boston, 1989. [6] G. Bourdaud, M. Reissig and W. Sickel, Hyperbolic equations, function spaces with exponential weights and Nemytskij operators, Ann. Mat. Pura Appl. 182 (2003), 409– 455. [7] M. D. Bronˇ ˇste˘ın, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moskov. Mat. Obshch. 41 (1980), 83–99. [8] J. Chazarain, Op´ ´erateurs hyperboliques a caract´ ´eristiques de multiplicit´ ´ constante, Ann. Inst. Fourier 24 (1974), 173–202. [9] J. Chazarain, Propagation des singularit´ es pour une classe d’op´rateurs ´ a caracteristiques ´ multiples et r´ ´esolubilit´ ´ locale, Ann. Inst. Fourier 24 (1974), 203–223. [10] M. Cicognani, Weakly hyperbolic equations with Lipschitz or H H¨ o ¨lder continuous coefﬁcients with respect to time, Ann. Univ. Ferrara Sez. VII (N.S.) 38 (1992), 193–215. [11] M. Cicognani, On the strictly hyperbolic equations which are H¨lder ¨ continuous with respect to time, Italian J. Pure Appl. Math. 4 (1998), 73–82. [12] M. Cicognani, The Cauchy problem for strictly hyperbolic operators with nonabsolutely continuous coeﬃcients, Tsukuba J. Math. 27 (2003), 1–12. [13] M. Cicognani, Coeﬃcients with unbounded derivatives in hyperbolic equations, Math. Nachr. 276, (2004), 31–46. [14] M. Cicognani and L. Zanghirati, The Cauchy problem for nonlinear equations with Levi conditions, Bull. Sci. Math. 123 (1999), 413–435. [15] M. Cicognani and L. Zanghirati, Nonlinear weakly hyperbolic equations with Levi conditions in Gevrey classes, Tsukuba J. Math. 25 (2001), 85–102. [16] M. Cicognani and L. Zanghirati, Nonlinear hyperbolic Cauchy problems in Gevrey classes, Chinese Ann. Math. Ser. B 22 (2001), 417–426. [17] M. Cicognani and L. Zanghirati, Analytic regularity for solutions of nonlinear weakly hyperbolic equations, Boll. Un. Mat. Ital. (B) 11 (1997), 643–679. [18] M. Cicognani and L. Zanghirati, Analytic regularity for solutions to semilinear weakly hyperbolic equations, Rend. Sem. Mat. Univ. Politec. Torino 51 (1993), 387–396. [19] M. Cicognani and L. Zanghirati, Propagation of Gevrey and analytic regularity for a class of semilinear weakly hyperbolic equations, Rend. Sem. Mat. Univ. Padova 94 (1995), 99–111. [20] F. Colombini, E. De Giorgi and S. Spagnolo, Sur les ´equations hyperboliques avec des coeﬃcients qui ne d´pendent ´ que du temps. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 6 (1979), 511–559. [21] F. Colombini, D. Del Santo and T. Kinoshita, Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coeﬃcients, Ann. Sc. Norm. Sup. Pisa Cl. Sci. 1 (2002), 327–358. [22] F. Colombini, D. Del Santo and M. Reissig, On the optimal regularity coeﬃcients in hyperbolic Cauchy problem, Bull. Sci. Math. 127 (2003), 328–347.

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[23] F. Colombini, E. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coeﬃcients depending on time, Ann. Sc. Norm. Sup. Pisa Cl. Sci. 10 (1983), 291–312. [24] F. Colombini and N. Lerner, Hyperbolic operators with non-Lipschitz coeﬃcients, Duke Math. J. 77 (1995), 657–698. [25] P. D’Ancona and S. Spagnolo, Quasi-symmetrization of hyperbolic systems and propagation of analytic regularity, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. 1 (1998), 169–185. [26] J.C. De Paris, Probl` eme de Cauchy oscillatoire pour un op´ ´erateur diﬀ´ ﬀ rentiel a ﬀ´ ` caracteristiques ´ multiples; lien avec l’hyperbolicit´, J. Math. Pures Appl. 51 (1972), 231–256. [27] H. Flascka and G. Strang, The correctness of the Cauchy problem, Adv. Math. 6 (1971), 349–379. [28] D. Gourdin, Une classe d’operateurs faiblement hyperboliques non lin´aires ´ , Bull. Sci. Math. 113 (1989), 349–379. [29] F. Hirosawa and M. Reissig, Well-posedness in Sobolev spaces for second order strictly hyperbolic equations with non-diﬀerentiable oscillating coeﬃcients, Ann. Global Anal. Geom. 25 (2004), 99–119. [30] V. Ya. Ivrii, Conditions for correctness in Gevrey classes of the Cauchy problem for weakly hyperbolic equations, Siberian Math. J. 17 (1976), 422–435. [31] K. Kajitani, Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes, J. Math. Kyoto Univ. 23 (1983), 599–616. [32] K. Kajitani, Local solution of Cauchy problem for nonlinear hyperbolic systems in Gevrey spaces, Hokkaido Math. J. 12 (1983), 436–460. [33] K. Kajitani, The Cauchy problem for nonlinear hyperbolic systems, Bull. Sci. Math. 110 (1986), 3–48. [34] H. Komatsu, Linear hyperbolic equations with Gevrey coeﬃcients, J. Math. Pures Appl. 59 (1980), 145–185. [35] A. Kubo and M. Reissig, C ∞ -well-posedness of the Cauchy problem for quasilinear hyperbolic equations with coeﬃcients non-Lipschitz in t and smooth in x, Banach Center Publ. 60 (2003), 131–150. [36] A. Kubo and M. Reissig, Construction of parametrix for hyperbolic equations with fast oscillations in non-Lipschitz coeﬃcients, Comm. Partial Diﬀerential Equations 28 (2003), 1741–1502. [37] H. Kumano-Go, Pseudo-diﬀerential operators, The MIT Press, Cambridge London, 1982. ´ [38] J. Leray and Y. Ohya, Equations et syst` emes non-lin´ ´eaires, hyperboliques nonstricts, Math. Ann. 170 (1967) 167–205. [39] E. E. Levi, Caratteristiche multiple e problema di Cauchy, Annali di Mat. 16 (1909), 161–201. [40] S. Matsuura, On non-strict hyperbolicity, Proc. Internat. Conf. on Funct. Anal. and Related Topics, Univ. of Tokyo Press, Tokyo, 1970, 171–176. [41] S. Mizohata, On the Cauchy problem, Notes and Reports in Mathematics in Science and Engineering, 3, Academic Press, Inc., Orlando, FL; Science Press, Beijing, 1985.

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[42] S. Mizohata and Y. Ohya, Sur la condition de E.E. Levi concernant des ´equations hyperboliques, Publ. Res. Inst. Math. Sci. 4 (1968), 511–526. [43] A. Montanari, Equivalent forms of Levi Condition, Ann. Univ. Ferrara, Sez. VII (N.S.) 45 (1999), 191–203. [44] T. Nishitani, Sur les ´equations hyperboliques a ` coeﬃcients h¨ld´ ¨ ´riens en t et de classe de Gevrey en x, Bull. Sci. Math. 107 (1983), 113–138. [45] Y. Ohya and S. Tarama, Le probl`eme de Cauchy a ` caract´ ´eristiques multiples dans la classe de Gevrey I. Coeﬃcients h¨lderiens ¨ en t, Hyperbolic equations and related topics (Katata/Kyoto, 1984), 273–306, Academic Press, Boston, MA, 1986. [46] M. Reissig, Weakly hyperbolic equations with time degeneracy in Sobolev spaces, Abstr. Appl. Anal. 2 (1997), 239–256. [47] S. Spagnolo, Some results of analytic regularity for the semi-linear weakly hyperbolic equations of the second order, Nonlinear hyperbolic equations in applied sciences, Rend. Sem. Mat. Univ. Politec. Torino 1988, Special Issue, 203–229. [48] M. E. Taylor, Pseudodiﬀerential operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [49] J. Vaillant, Donnees ´ de Cauchy port´ ´es par une caract´ ´ristique double, dans le cas d’un syst` eme lin´ ´eaire d’´equations aux d´ ´ eriv´ ´ees partielles, roˆle des bicaract´ ´eristiques, J. Math. Pures Appl. 47 (1968), 1–40. [50] S. Wakabayashi, The Lax-Mizohata theorem for nonlinear Cauchy problems, Comm. Partial Diﬀerential Equations 26 (2001), 1367–1384. [51] K. Yagdjian, The Lax-Mizohata theorem for nonlinear gauge invariant equations, Proceedings of the Second ISAAC Congress, Vol. 2 (Fukuoka, 1999), 1547–1561, Int. Soc. Anal. Appl. Comput., 8, Kluwer Acad. Publ., Dordrecht, 2000. Massimo Cicognani Dipartimento di Matematica Piazza di Porta S. Donato, 5 40127 Bologna Italy and Facolt` a ` di Ingegneria II Via Genova, 181 47023 Cesena Italy e-mail: [email protected] Luisa Zanghirati Dipartimento di Matematica Via Machiavelli, 35 44100 Ferrara Italy e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 159, 449–511 c 2005 Birkhauser ¨ Verlag Basel/Switzerland

Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators Michael Dreher and Ingo Witt Abstract. The intention of this article is twofold: First, we survey our results from [20, 18] about energy estimates for the Cauchy problem for weakly hyperbolic operators with ﬁnite time degeneracy at time t = 0. Then, in a second part, we show that these energy estimates are sharp for a wide range of examples. In particular, for these examples we precisely determine the loss of regularity that occurs in passing from the Cauchy data at t = 0 to the solutions. 2000 Mathematics Subject Classiﬁcation. Primary: 35L80; Secondary: 35L30, 35S10. Keywords. Weakly hyperbolic operators, ﬁnite time degeneracy, well-posedness of the Cauchy problem, sharp energy estimates, loss of regularity.

1. Introduction This article is devoted to the study of the Cauchy problem for certain degenerate hyperbolic operators. These operators, P , are either ﬁrst-order, N × N , pseudodiﬀerential systems, P = Dt − A(t, x, Dx ),

1 A ∈ C ∞ ((0, T ], Op Scl ),

(1.1)

or higher-order, scalar, pseudodiﬀerential equations, P = Dtm +

m

aj (t, x, Dx )Dtm−j ,

j aj ∈ C ∞ ((0, T ], Op Scl ),

(1.2)

j=1 j j where Scl = Scl (Rn × Rn ) is the space of jth-order classical pseudodiﬀerential symbols. The precise assumptions as t → +0 are stated in (1.7), (1.8) below. (Note that the interval (0, T ] is open at t = 0.) In particular, the symbols A(t, x, ξ)

450

Michael Dreher and Ingo Witt

and aj (t, x, ξ), respectively, are smooth up to t = 0. Diﬀerential operators in this class are of the form + ajα (t, x) t(j+(l∗ +1)|α|−m) Dtj Dxα , P = j+|α|≤m

where ajα ∈ Cb∞ ([0, T ] × Rn ) for j + |α| ≤ m, y + = max{y, 0} for y ∈ R. Some examples are discussed, e.g., in Sections 1.2.1, 3.1. 1.1. Well-posedness of the Cauchy problem For most function spaces, X, hyperbolicity is a necessary condition for the Cauchy problem for the operator P to be X well-posed. Thereby, the operator P is said to be hyperbolic if all its characteristic roots, x, ξ) of the equation * i.e., jthe roots τj (t, m−j det(τ 1N − σ 1 (A)(t, x, ξ)) = 0 and τ m + m = 0, respectively, j=1 σ (aj )(t, x, ξ)τ j are real. Here, σ j (a) denotes the principal symbol of a ∈ Scl . It is known that in order to ensure X well-posedness of the Cauchy problem for the operator P , additional assumptions — besides hyperbolicity — have to be made, usually depending on the function space X. If X = A(Rn ), the space of analytic functions, then the Cauchy problem for diﬀerential operators P (not necessarily hyperbolic) is always well posed, for the initial hypersurface t = 0 is non-characteristic for P . If X = Gs (Rn ), the Gevrey space of index s, and 1 < s ≤ m/(m − 1), then hyperbolicity is a necessary and suﬃcient condition for the well-posedness of the ¨ rmander [24], Cauchy problem for scalar operators (1.2), see Bronstein [9], Ho Ivrii [29], Kajitani [37], Komatsu [40], Nishitani [51]. A similar result holds for ﬁrst-order systems. The famous Lax–Mizohata theorem states that hyperbolicity is a necessary condition for the C ∞ well-posedness of the Cauchy problem for diﬀerential operators as above, see Lax [42], Mizohata [49]. Hyperbolicity, however, is not a suﬃcient condition, as will be seen below. Suﬃcient conditions are, e.g., strict hyperbolicity (the characteristic roots τj (t, x, ξ) are distinct) and symmetric hyperbolicity (the matrix σ 1 (A)(t, x, ξ) is Hermitian), see Leray [43], Petrovsky [58]. The situation is delicate for non-strictly hyperbolic operators, so-called weakly hyperbolic operators; and many questions have been remained open until now. Roughly speaking, there are two eﬀects causing ill-posedness of the Cauchy problem: First, oscillations in the coeﬃcients can occur and, secondly, the lower-order terms play a crucial role. In Colombini–Spagnolo [14], e.g., an oscillating function a ∈ C ∞ ([0, T ]; R), a(t) ≥ 0, has been constructed for which there are data u0 , u1 ∈ C ∞ (R) such that the Cauchy problem utt (t, x) − a(t)uxx (t, x) = 0, (t, x) ∈ (0, T ) × R, u(0, t) = u0 (x),

ut (0, x) = u1 (x)

Sharp Energy Estimates

451

has even no distributional solution u. Several examples of ill-posed Cauchy problems for hyperbolic ﬁrst-order systems with oscillating smooth coeﬃcients have been given by Matsumoto [46]. Concerning the inﬂuence of the lower-order terms for hyperbolic ﬁrst-order systems, we mention the results by Nishitani [54], who has studied hyperbolic operators of the form Dt − A(t, x)Dx + B(t, x) with analytic 2 × 2 matrices A and B; and who has given necessary and suﬃcient conditions for the C ∞ wellposedness, formulated in terms of certain Newton polygons. For related results, see Nishitani–Vaillant [55], Vaillant [71], and the references therein. One of the earliest examples of ill-posedness for a second-order operator is due to Gevrey [21]: The non-characteristic Cauchy problem for the side-reversed heat operator ∂t2 − ∂x is not well posed in Gs , s > 2. Moreover, this Cauchy problem is neither well posed in C ∞ nor in the Sobolev spaces H s . Conditions on the lower-order terms that guarantee well-posedness in a given function space, X, are called Levi conditions, see Levi [44, 45]. For large classes of hyperbolic operators, Levi conditions for Gs with s > m/(m−1) and C ∞ have been given by Colombini–Ishida–Orru [11], Colombini–Jannelli–Spagnolo [12], ¨ rmander [25], Ishida–Yagdjian [27], Ivrii [30, 31], Colombini–Orru [13], Ho Ivrii–Petkov [33], and Oleinik [56]. For the model operator P = Dt2 − t2l Dx2 + b(t)tk Dx ,

k, l ∈ N0 ,

where b(0) = 0 and b is suﬃciently smooth, these conditions are as follows: • k < l − 1 : The Cauchy problem is well posed in Gs if 1 < s < (2l − k)/(l − k − 1). This bound on s is sharp. • k = l − 1 : The Cauchy problem is well posed in Gs , C ∞ , and the scale of Sobolev spaces H s , however, with a certain loss of regularity in the latter case. For more about this, see also this article. • k ≥ l : The Cauchy problem is well posed in Gs , C ∞ , and the scale of Sobolev spaces H s , now without any loss of regularity. 1.2. Degenerate diﬀerential operators In this paper, we will be concerned with the case k = l − 1 — in our understanding this is the most interesting one. Henceforth, l will be denoted by l∗ . As noted above, the main example for an operator in the class under consideration is the mth-order, scalar, hyperbolic diﬀerential operator + ajα (t, x) t(j+(l∗ +1)|α|−m) Dtj Dxα , (1.3) P = j+|α|≤m

where ajα ∈

Cb∞ ([0, T ] ×

Rn ), with principal symbol

σ m (P )(t, x, τ, ξ) =

m 1 τ − tl∗ µk (t, x, ξ) . k=1

(1.4)

452

Michael Dreher and Ingo Witt

Here, the µk ∈ C ∞ ([0, T ]; S (1) ) are real-valued, where S (j) = S (j) (Rn × (Rn \ 0)) is the space of pseudodiﬀerential symbols that are positively homogeneous of degree j in ξ = 0. In the notation of (1.2), + am−j,α (t, x) t((l∗ +1)|α|−j) ξ α . (1.5) aj (t, x, ξ) = |α|≤j

The operator P is weakly hyperbolic, for its characteristic roots τk (t, x, ξ) = tl∗ µk (t, x, ξ) coincide at t = 0. Operators of such structure — given by (1.3), (1.4) — will be investigated in detail in this paper. They exhibit two phenomena attracting our attention: The loss of regularity and a non-standard propagation of the singularities under favourable circumstances. Generically, a singularity coming along one null bicharacteristic from, say, t < 0, continues to propagate along all its m connecting null bicharacteristics in t > 0 after it has passed over t = 0. It is a discrete phenomenon when this complete branching does not occur, i.e., when at least one of the m branches in t > 0 is missing in the propagation picture. 1.2.1. A ﬁrst example. Both phenomena have been observed in the following example by Qi [59]. It will be generalized in Section 3.1 below: utt − t2 uxx − (4k + 1)ux = 0, (t, x) ∈ (0, T ) × R, (1.6) u(0, x) = u0 (x), ut (0, x) = 0, for some k ∈ N0 , with explicit solution representation k t2 2j (j) cjk t u0 u(t, x) = x+ , ckk = 0. 2 j=0 One sees that the solution u has k derivatives less as compared with the initial data u0 , and that the family of characteristics x − t2 /2 = const traveling to the right has disappeared. Note that the number k in (1.6) can indeed be any real number, or even be a smooth complex-valued function k = k(t, x). In these two cases, however, the singularities generically do not propagate in the exceptional manner just described; but we still have a loss of regularity of |$k(0, x) + 1/4|−1/4 derivatives at time t = 0 and spatial point x, as our calculus clearly reveals. In particular, the loss of regularity is a Lipschitz function of x, but may fail to be C 1 . 1.2.2. Main tools. Our approach in treating the operator P from (1.3), (1.4) consists in converting it into an equivalent ﬁrst-order, m×m, pseudodiﬀerential system and then to diagonalize the latter as far as it is needed. This is why we consider systems of the form (1.1). The system resulting from converting P has necessarily to be pseudodiﬀerential, since for ﬁrst-order diﬀerential systems in the class under consideration it can be shown that no loss of regularity occurs — hence such a system cannot be equivalent to P . Therefore, we establish a calculus for a class S m,η;λ of pseudodiﬀerential symbols a(t, x, ξ) on [0, T ] × R2n , where m, η ∈ R are the parameters involved and

Sharp Energy Estimates

453

the function λ(t) = tl∗ is to ﬁx the kind of degeneracy at t = 0 under consideration. In case m = η, this class of pseudodiﬀerential symbols a(t, x, ξ) expresses the degeneracy at t = 0 of the principal part of the operator from (1.1) and (1.2), respectively, as well as Levi conditions on the lower-order terms in a very precise manner, see (1.7), (1.8). The case m = η is needed to formulate the hyperbolicity assumption, see (2.11). The classes S m,η;λ are introduced with the help of two weight functions g(t, ξ), h(t, ξ), see (2.3) and Deﬁnition 2.1. The diagonalization procedure requires two steps: In fact, after a ﬁrst step the principal part of the operator A(t, x, Dx ) from (1.1) has become diagonal. Then a second step that up to lower-order terms eﬀects the operator A(t, x, Dx ) only at t = 0 proves to be necessary in order to read oﬀ the precise loss of regularity. Accordingly, we single out a subclass S˜m,η;λ ⊂ S m,η;λ of pseudodiﬀerential symbols a(t, x, ξ) possessing a principal symbol σ m (a) as usual and, in addition, a subor˜ 0,1 (A)(x, ξ) in dinated secondary symbol σ ˜ m−1,η (a). Both symbols σ 1 (A)(t, x, ξ), σ case of (1.1) parallel the diagonalization procedure. For more details, see Deﬁnition 2.6 and thereafter. We complete our assumptions as t → +0 in (1.1) and (1.2), respectively: In (1.1), we shall assume that A(t, x, Dx ) ∈ Op S 1,1;λ ,

(1.7)

while, in (1.2), we shall assume that aj (t, x, Dx ) ∈ Op S j,j;λ ,

1 ≤ j ≤ m. (1.8) * It is important to note that the diﬀerential symbol |α|≤j a ˜α (t, x)ξ α , where a ˜α ∈ ∞ n j,j;λ if and only if Cb ([0, T ] × R ) for |α| ≤ j, belongs to the symbol class S +

a ˜α (t, x) = t((l∗ +1)|α|−j) aα (t, x)

(1.9)

for certain aα (t, x) ∈ Cb∞ ([0, T ] × Rn ). In this sense, (1.7), (1.8) express sharp Levi conditions on the lower-order terms of (1.1) and (1.2), respectively. 1.2.3. Other approaches and further results. Some authors have constructed parametrices for the hyperbolic operators P from (1.3), (1.4), see Kumano-go [41], Nakamura–Uryu [50], Taniguchi–Tozaki [69], Yoshikawa [74]; see also Aleksandrian [1], Yagdjian [73] for related results. In Amano–Nakamura [4], these parametrices have been exploited to classify the exceptional cases for the propagation of singularities, with an explicit description for m = 2. The energy method for operators in the class has been developed by Kumano-go [41], Nishitani [52, 53], among others. A relation to Stokes phenomena and hypoellipticity of certain associated operators has been established by Amano–Nakamura [3], Reissig–Yagdjian [60], Shinkai [66]. The case m = 2, l∗ = 1 is of particular interest. The Cauchy problem for the operator P + Q is then C ∞ well posed for any ﬁrst-order diﬀerential operator Q

454

Michael Dreher and Ingo Witt

with smooth coeﬃcients, see (1.3); one says that the operator P is regularly hyperbolic. Regular hyperbolicity on the level of the principal symbol has been characterized by Ivrii–Petkov [33] (necessary conditions) and by Ivrii [32], Iwasaki [34], Melrose [48] (suﬃcient conditions). Pseudodiﬀerential calculi for a treatment of such operators have been introduced by Boutet de Monvel [6], Joshi ¨ strand [67], Witt [72], and others. Operators with non-involutive char[36], Sjo acteristics and propagation phenomena have been further studied among others by Alinhac [2], Boutet de Monvel–Tr` eves [7], Bove–Lewis–Parenti [8], Ivrii [28], Kajitani–Nishitani [38], Melrose [47]. The question on the propagation of singularities in this special case has ﬁnally been settled by Hanges [23], who has given a symplectically invariant condition for a complete branching of singularities to do not occur. Semilinear problems connected with the operator P from (1.3), (1.4) have been investigated by Dreher–Reissig [17], Dreher–Witt [20], Iwasaki [35]. 1.3. Notation Here, we list notation that will be used in the sequel. Note that the positive integer l∗ ∈ N+ is ﬁxed throughout this paper: λ(t) = tl∗ — characterizes the kind of degeneracy under consideration at time t = 0 β∗ = 1/(l∗ + 1) — constant 0t Λ(t) = 0 λ(t ) dt — primitive of λ(t) ξ = (1 + |ξ|2 )1/2 ξK = (K + |ξ|2 )1/2 H s(x) = H s(x) (Rn ) — Sobolev space of variable order for s ∈ Cb∞ (Rn ; R), see (2.1) s,δ(x);λ s,δ(x);λ n =H ((0, T ) × R ) — function space of Sobolev type for H s ∈ R, δ ∈ Cb∞ (Rn ; R), see Deﬁnitions 2.2 and 3.8 j j = Scl (Rn × Rn ) — space of jth-order classical pseudoScl diﬀerential symbols j j n n — space of jth-order pseudodiﬀerentS1,δ = S1,δ (R × R ) ial symbols of type 1, δ, where 0 ≤ δ<1 — space of pseudodiﬀerential symbols S (j) = S (j) (Rn × (Rn \ 0)) which are positively homogeneous of order j in ξ = 0 S m,η;λ — symbol class, see Deﬁnition 2.1 S˜m,η;λ — symbol class, see Deﬁnition 2.6 m,η;λ — symbol class, see Deﬁnition 4.7 S+ — principal symbol of a ∈ S˜m,η;λ σ m (a) = σ m (a)(t, x, ξ)

Sharp Energy Estimates ˜ m−1,η (a)(x, ξ) σ ˜ m−1,η (a) = σ χ(t)

χ+ (t, ξ) χ+ K (t, ξ) χ− (t, ξ) χ− K (t, ξ) g h $Q

= = = = = = =

χ(Λ(t)ξ) χ(Λ(t)ξK ) 1 − χ+ (t, ξ) 1 − χ+ K (t, ξ) g(t, ξ) h(t, ξ) (Q + Q∗ )/2

%Q = (Q − Q∗ )/(2i)

MN ×N (C) 1N

455

— secondary symbol of a ∈ S˜m,η;λ — cut-oﬀ function at t = ∞, i.e., χ ∈ C ∞ (R+ ), 0 ≤ χ ≤ 1, χ(t) = 0 if t ≤ 1/2, and χ(t) = 1 if t ≥ 1 — cuts into the hyperbolic zone — cuts into the pseudodiﬀerential zone — see (2.3) — see (2.3) and (4.5) — real part of the matrix Q, selfadjoint part of the operator Q — imaginary part of the matrix Q, anti-self-adjoint part of the operator Q — space of N × N matrices with complex entries — N × N identity matrix

2. Formulation of the results In this section, our main results are stated. Beforehand, however, we provide further motivation. 2.1. Motivation and plan of the paper 2.1.1. Qi’s example reviewed. We now come back to Qi’s example (1.6) and utilize it to explain typical features and diﬃculties connected with our approach. This approach consists of two components: A calculus for a class of pseudodiﬀerential operators generalizing (1.5) and an adapted scale of Sobolev-type function spaces. These function spaces are most appropriate for the hyperbolic operators under consideration in so far as they allow energy estimates including a sharp loss of regularity. In case of problem (1.6) with k ∈ N0 , we already know that u0 ∈ H s+k (R) s implies u ∈ Hloc ((0, T ) × R). Therefore, we are looking for Sobolev-type function spaces whose elements exhibit H s regularity for t > 0, but (essentially) H s+k regularity at t = 0 via a trace theorem. Moreover, we can consider the Cauchy problem (1.6) also with initial data u(0, x) = u0 (x), ut (0, x) = u1 (x). A diﬀerent representation of the solution (to be discussed in Section 3.1 below) then tells us that u0 ∈ H s+k (R), u1 ∈ H s+k−1/2 (R) s provides a solution u ∈ Hloc ((0, T )×R). This is unexpected inasmuch as one would expect that the orders of regularity of u0 and u1 diﬀer by 1, as is the case for the wave equation case. Of course, we wish our ffunction spaces to reﬂect this particular feature.

456

Michael Dreher and Ingo Witt

There is more about the loss of regularity: Consider (1.6) again, but now with a smooth function k = k(t, x) satisfying k(0, x) ≥ 0, x ∈ Rn , that takes the diﬀerent integer values k1 = k2 in a neighbourhood V1 of (0, x1 ) and a neighbourhood V2 of (0, x2 ), respectively. By virtue of the local uniqueness of the solution u, we have kp t2 2j (j) u(t, x) = cjk,p t u0 x+ , (t, x) ∈ Vp , p = 1, 2, 2 j=0 on certain smaller neighbourhoods Vp ⊂ Vp . One can see that the loss of regularity at the diﬀerent points (0, x1 ) = (0, x2 ) diﬀers. One then guesses (in fact, we are going to prove this below) that the Cauchy problem (1.6) with initial data u0 belonging to the Sobolev space H s+k(0,x) (R) of variable order s + k(0, x) has a s ((0, T ) × R). Here, solution u ∈ Hloc s+k(0,x) −1 2 L (Rn ), (2.1) H s+k(0,x) (R) := Dx K s+k(0,x)

where the parameter K > 0 is chosen large to ensure the operator Dx K with symbol (K + |ξ|2 )(s+k(0,x))/2 be invertible on S (R). Our main results are stated in Theorems 2.5, 2.7, 2.8, 2.9, and 2.10. To let the reader to get acquainted with them, we now describe these results as applied to the operator P from Qi’s example (1.6), where k = k(t, x) is a smooth function satisfying k(0, x) ≥ 0, x ∈ Rn . To begin with, we postulate function spaces H s.δ(x);λ ((0, T )×R), where s ∈ R is Sobolev regularity for t > 0 with respect to space-time, δ(x) = 2k(0, x) is related to the loss of regularity at the point (0, x), and λ(t) = t is as above (l∗ = 1), where these function spaces possess the following properties: = H s ((T , T ) × R) for all 0 < T < T , • H s,δ(x);λ ((0, T ) × R) (T ,T )×R s

((0, T ) × R) ⊆ H ((0, T ) × R) provided that δ(x) ≥ 0, • H • For 0 ≤ j < s − 1/2, the trace map τj : u → Dtj u|t=0 maps the function space H s,δ(x);λ ((0, T ) × R) onto H s+δ(x)−j/2−1/4 (R). Further properties of these spaces as well as details of their construction will be discussed in Section 3.4 (for the special case that δ ∈ R is independent of x) and Section 4.4 (for general δ ∈ Cb∞ (Rn ; R)). The Cauchy problem utt − t2 uxx − (4k(t, x) + 1)ux = f (t, x), (2.2) u(0, x) = u0 (x), ut (0, x) = u1 (x) . / is well posed in the scale H s,δ(x);λ ((0, T ) × R) s ≥ 0 in the following sense: s,δ(x);λ

• For uj ∈ H s+δ(x)/2−j/2 (R), j = 0, 1, and f ∈ H s−1,δ(x)+1;λ ((0, T ) × R), the Cauchy problem (2.2) possesses a unique solution u ∈ H s,δ(x);λ ((0, T ) × R). • The choice δ(x) = 2k(0, x) ≥ 0 is optimal; the statement in the previous item becomes false if δ(x0 ) < 2k(0, x0 ) for some x0 ∈ R. • The solution u is locally unique in H 1,δ(x);λ ((0, T ) × R).

Sharp Energy Estimates

457

2.1.2. Plan of the paper. In the next section, Section 2.2, we state our main results. To formulate these results, we need to introduce the basic function spaces H s,δ(x);λ ((0, T )× Rn ) as well as the symbol classes S m,η;λ and S˜m,η;λ , respectively. In Section 3, we discuss the example of a second-order, scalar operator P with coeﬃcients that are independent of x ∈ Rn (m = 2). This case is treated by Fourier transformation with respect to x, followed by dealing with the resulting family of ordinary diﬀerential equations on the half-space R+ (with variable t) depending on the parameter ξ = 0. The symbol classes S m,η;λ , S˜m,η;λ and the function spaces H s,δ(x);λ ((0, T ) × Rn ) mentioned above are thoroughly studied in Section 4. Note that a speciﬁc role is played by the “shift operator” Θ, that is introduced in (4.4). m,η;λ , which A treatment of Θ enforces us to enlarge the symbol class S m,η;λ to S+ is done in Section 4.4. Our main results are then proved in Section 5. Compared to Dreher–Witt [20, 18], the representation is now enhanced in several respects. Furthermore, Theorems 2.9 and 2.10 stating local uniqueness of the solutions and sharpness of the energy estimates, respectively, are new results not contained in previous publications. In Appendix A, we collect and prove some auxiliary material, while, in Appendix B, some open problems are listed. 2.2. Main results Already here we formulate our main results, although part of the motivation, in particular, for the introduction of the symbol classes S 1,1;λ in Deﬁnition 2.1 and S˜1,1;λ in Deﬁnition 2.6 will be given only in Section 3. In Bourdaud–Reissig–Sickel [5], Colombini–Ishida [10], Ishida–Yagdjian [27], Kajitani–Wakabayashi–Yagdjian [39], Reissig–Yagdjian [61], Yagdjian [73], the approach to weakly hyperbolic operators with time degeneracy at time t = 0 is based on dividing the (t, ξ) strip [0, T ] × Rn into two zones: The pseudodiﬀerential (or inner) zone Zpd given by Λ(t)ξ ≤ 1 and the hyperbolic (or outer) zone Zhyp given by Λ(t)ξ ≥ 1. This reﬂects the fact that the microlocal properties of the operators under consideration are diﬀerent in these two zones. Our approach is based on weight functions. A careful analysis, e.g., in Section 3.1, shows that one should employ the following two weight functions: g¯(t, ξ) = ξβ∗ + λ(t)ξ, ¯h(t, ξ) = (t + ξ−β∗ )−1 . ¯ ∈ S 0,1;λ , but g¯ ∈ ¯∈ We have g¯ ∈ S 1,1;λ , h / S˜1,1;λ , h / S˜0,1;λ . In order to stay within ¯ ξ): the smaller symbol classes, we will change g(t, ξ) for g¯(t, ξ) and h(t, ξ) for h(t, g(t, ξ) = χ− (t, ξ)ξβ∗ + χ+ (t, ξ)λ(t)ξ, (2.3) h(t, ξ) = χ− (t, ξ)ξβ∗ + χ+ (t, ξ) t−1 , which does not eﬀect the symbol estimates in Deﬁnition 2.1.

458

Michael Dreher and Ingo Witt We then consider the Cauchy problems for ﬁrst-order pseudodiﬀerential sys-

tems

Dt U (t, x) = A(t, x, Dx )U (t, x) + F (t, x),

(t, x) ∈ (0, T ) × Rn ,

U (0, x) = U0 (x) and for mth–order, scalar, pseudodiﬀerential equations ⎧ m ⎪ ⎪ ⎨ Dtm u(t, x) + aj (t, x, Dx )Dtm−j u(t, x) = f (t, x), (t, x) ∈ (0, T ) × Rn , j=1 ⎪ ⎪ ⎩ Dj u(0, x) = u (x), t

j

(2.4)

(2.5)

0 ≤ j ≤ m − 1.

Here, U , U0 , and F are N vectors and A(t, x, ξ) is an N × N matrix symbol belonging to the symbol class S 1,1;λ , as in (1.7), and u, uj , and f are scalar functions and the aj (t, x, ξ) are scalar symbols from the symbol class S j,j;λ , as in (1.8). These symbol classes are deﬁned as follows: Deﬁnition 2.1. For m, η ∈ R, the symbol class S m,η;λ consists of all functions a ∈ C ∞ ([0, T ] × R2n ; MN ×N (C)) such that, for each multi-index (j, α, β) ∈ N1+2n , there is a constant Cjαβ > 0 with the property that j α β ∂t ∂x ∂ a(t, x, ξ) ≤ Cjαβ g(t, ξ)m h(t, ξ)η−m+j ξ−|β| (2.6) ξ for all (t, x, ξ) ∈ [0, T ] × R2n . The parameter m counts powers of gh−1 ∼ 1 + Λ(t)ξ, while the parameter η counts powers of (t + ξ−β∗ )−1 . In particular, m ), S m,η;λ ⊂ C ∞ ((0, T ]; S1,0

while, for j = 0, 1, 2, . . . , (η+j)β∗

∂tj a(0, x, ξ) ∈ S1,0

when a ∈ S m,η;λ .

m Note also that C ∞ ([0, T ], S1,0 ) ⊂ S m,m(l∗ +1);λ . Next, we introduce the function spaces H s,δ;λ ((0, T ) × Rn ):

Deﬁnition 2.2. For s ∈ N0 , δ ∈ Cb∞ (Rn ; R), and T > 0, we deﬁne the function space H s,δ;λ ((0, T ) × Rn ) by the ﬁniteness of the norm s 1/2 T 2 2l−1 l Θsl (t)Dt u(t, ·) 2 n dt uH s,δ(x);λ ((0,T )×Rn ) = T , (2.7) L (R ) l=0

0

where Θsl (t) = g s−l h(s+δ)l∗ (t, x, Dx ), Cb∞ (Rn ; R),

0 ≤ l ≤ s.

the function spaces H s,δ;λ ((0, T ) × Rn ) are For general s ∈ R, δ ∈ deﬁned by interpolation and duality.

Sharp Energy Estimates

459

Details of this construction can be found in Sections 3.4 and 4.4. We now discuss the well-posedness of the Cauchy problems (2.4) and (2.5) in the scale of function spaces H s,δ(x);λ ((0, T ) × Rn ): For the mth-order, N × N matrix, pseudodiﬀerential operator m P = Dtm + Aj (t, x, Dx )Dtm−j , j=1

where Aj (t, x, Dx ) ∈ Op S

j,j;λ

for 1 ≤ j ≤ m, we consider the Cauchy problem

P U = F (t, x),

Dtj U (0, x) = Uj (x),

(t, x) ∈ (0, T ) × Rn ,

(2.8)

0 ≤ j ≤ m − 1.

Deﬁnition 2.3. (a) For s ∈ N0 , δ ∈ Cb∞ (Rn ; R), the Cauchy problem for the operator P is said to be (s, δ(x))-well-posed if, for all Uj ∈ H s+m+β∗ δ(x)l∗ −β∗ j−1 (Rn ), 0 ≤ j ≤ m − 1, and F ∈ H s,δ(x)+m−1;λ ((0, T ) × Rn ), problem (2.8) possesses a unique solution U ∈ H s+m−1,δ(x);λ ((0, T ) × Rn ). Moreover, this solution U is subject to the estimate s+m−1 l=0

2 t2l Θs+m−1,l (t)Dtl U (t, ·)L2 (Rn ) ⎛

≤C⎝

m−1

⎞ 2 U Uj H s+m+β∗ δ(x)l∗ −β∗ j−1 (Rn )

+

2 t2 F H s,δ(x)+m−1;λ ((0,t)×Rn ) ⎠

(2.9)

j=0

for all 0 ≤ t ≤ T , where the constant C = C(s, δ, T ) > 0 is independent of Uj , F . (b) For δ ∈ Cb∞ (Rn ; R), the Cauchy problem for the operator P is said to be δ(x)-well-posed if it is (s, δ(x))-well posed for all s ∈ N0 . We obviously have the following result: Lemma 2.4. (a) If the Cauchy problem for the operator P is (s, δ(x))-well posed, then we have the estimate U 2H s+m−1,δ(x);λ ((0,T )×Rn ) ⎞ ⎛ m−1 2 2 ≤C⎝ U Uj H s+m+β∗ δ(x)l∗ −β∗ j−1 (Rn ) + T 2 F H s,δ(x)+m−1;λ ((0,T )×Rn ) ⎠ . (2.10) j=0

(b) If the Cauchy problem for the operator P is δ(x)-well posed, then estimate (2.10) holds for all s ≥ 0, with suitable constants C = C(s, δ, T ) > 0. Assuming symmetrizable hyperbolicity for (2.4), we have: Theorem 2.5. Assume the symbol A(t, x, ξ) ∈ S 1,1;λ in (2.4) is symmetrizablehyperbolic in the sense that there is an N × N matrix M ∈ S 0,0;λ such that | det M (t, x, ξ)| ≥ c for |ξ| ≥ C and some C, c > 0 and χ(|ξ|/2C) %(M AM −1 ) ∈ S 0,1;λ ,

(2.11)

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for the cut-oﬀ function χ(t) see Section 1.3. Then, for each s ≥ 0, there is a function δ ∈ Cb∞ (Rn ; R) such that (2.4) is (s, δ(x))-well posed. This statement can be reﬁned to δ(x)-well-posedness — including a sharp upper bound on δ — if we assume the symbol A(t, x, ξ) is composed of two homogeneous components and a lower-order remainder: Deﬁnition 2.6. For m, η ∈ R, the symbol class S˜m,η;λ consists of all functions a ∈ C ∞ ([0, T ] × R2n ) of the form (2.12) a(t, x, ξ) = χ+ (t, ξ) t−η a0 (t, x, tl∗ +1 ξ) + a1 (t, x, tl∗ +1 ξ) + a2 (t, x, ξ), where a0 ∈ C ∞ ([0, T ]; S (m)),

a1 ∈ C ∞ ([0, T ]; S (m−1)),

and a2 ∈ S m−2,η;λ + S m−1,η−1;λ . With a(t, x, ξ) as in (2.12), we associate two homogeneous symbol components, σ m (a)(t, x, ξ) = t−η a0 (t, x, tl∗ +1 ξ) ∈ tm(l∗ +1)−η C ∞ ([0, T ]; S (m) ), σ ˜ m−1,η (a)(x, ξ) = a1 (0, x, ξ) ∈ S (m−1) . Note that each symbol a(t, x, ξ) of the form (2.12) does belong to the symbol class S m,η;λ , i.e., we have S˜m,η;λ ⊂ S m,η;λ . Example. Consider Eq. (1.6) with k = k(t, x) and introduce the vector U (t, x) = (g(t, Dx )u(t, x), Dt u(t, x))t . Then U solves the 2 × 2 ﬁrst-order system Dt U (t, x) = A(t, x, Dx )U (t, x) ˜1,1;λ

for a certain A ∈ S

, where 0 1 σ 1 (A)(t, x, ξ) = λ(t)|ξ| , 1 0

σ ˜ 0,1 (A)(x, ξ) = −i

1 ξ k(0, x) |ξ|

0 . 0

Theorem 2.7 (Dreher–Witt [18, Theorem 1.1]). Let A ∈ S˜1,1;λ , where σ 1 (A)(t, x, ξ) = λ(t)|ξ|A0 (t, x, ξ),

σ ˜ 0,1 (A)(x, ξ) = −il∗ A1 (0, x, ξ);

A0 ∈ C ∞ ([0, T ]; S (0)), A1 ∈ S (0) . Assume A(t, x, ξ) symmetrizable–hyperbolic in the sense that there is a symbol M0 ∈ C ∞ ([0, T ]; S (0) ) satisfying | det M0 (t, x, ξ)| ≥ c for ξ = 0 and some c > 0 such that the matrix (M M0 A0 M0−1 )(t, x, ξ) is Hermitian

(2.13)

for all (t, x, ξ) ∈ [0, T ] × Rn × (Rn \ 0). Let M1 ∈ S (0) be an arbitrary N × N matrix and δ ∈ Cb∞ (Rn ; R) satisfy $ % $ M0 A1 M0−1 + M1 M0−1 , M0 A0 M0−1 (0, x, ξ) ≤ δ(x)1N , (2.14) for all (x, ξ) ∈ Rn × (Rn \ 0), where [·, ·] denotes the commutator of matrices. Then the Cauchy problem (2.4) is δ(x)-well posed.

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461

Under these assumptions, Theorem 2.5 is applicable, where the symmetrizer M can be chosen to belong to Op S˜0,0;λ and satisfy σ 0 (M )(t, x, ξ) = M0 (t, x, ξ),

σ ˜ −1,0 (M )(x, ξ) = −il∗ |ξ|−1 M1 (t, x, ξ).

When applied to the mth-order, scalar, diﬀerential operator P from (1.3), (1.4), we infer from Theorem 2.7: Theorem 2.8 (Dreher–Witt [18, Proposition 5.6]). Let δ ∈ Cb∞ (Rn ; R) satisfy τ ∂2p 2 ∂τ 2 + $q (2.15) (0, x, µh (0, x, ξ), ξ), δ(x) ≥ sup sup − ∂p 1≤h≤m ξ=0

∂τ

where p(τ ) = p(t, x, τ, ξ) is the compressed principal symbol of P , p(t, x, τ, ξ) = ajα (t, x)τ j ξ α

(2.16)

j+|α|=m

q(τ ) = q(x, τ, ξ) is (up to the factor il∗−1 ) the secondary symbol of P , q(x, τ, ξ) = il∗−1 ajα (0, x)τ j ξ α , j+|α|=m−1, |α|>0

and the τh = tl∗ µh for 1 ≤ h ≤ m are the characteristic roots of P . Then the Cauchy problem (2.5) is δ(x)-well posed. In order to study the local uniqueness, we introduce local versions of the function spaces H s,δ(x);λ ((0, T ) × Rn ): For Ω ⊆ (0, T ) × Rn being an open set, the function space H s,δ(x);λ (Ω) is deﬁned as the space of restrictions of functions from H s,δ(x);λ ((0, T ) × Rn ) to Ω; and it is equipped with the inﬁmum norm. Theorem 2.9. Let P be the mth-order partial diﬀerential operator from (1.3), (1.4) with characteristic roots τj = tl∗ µj , where the µj are real and distinct, |µj (t, x, ξ) − µk (t, x, ξ)| ≥ c |ξ|,

j = k,

c > 0.

Let the function δ ∈ Cb∞ (Rn ; R) satisfy condition (2.15) of Theorem 2.8. Further let Ω ⊆ (0, T ) × Rn be open such that its closure Ω is a neighbourhood of (0, 0) in [0, T ] × Rn . Set Ω0 := Ω ∩ {t = 0}. Under these assumptions we have that if the function u ∈ H m−1,δ(x);λ (Ω) is an energy solution to P u(t, x) = 0, (t, x) ∈ Ω, Dtj u(0, x) = 0,

x ∈ Ω0 ,

0 ≤ j ≤ m − 1,

then u ≡ 0 in a certain open set Ω ⊆ Ω, where Ω is a neighbourhood of (0, 0) in [0, T ] × Rn .

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Upon a suitable choice of the matrix M1 ∈ S (0) , (2.14) provides an optimal lower bound for δ(x) in a number of cases. Here, this is exempliﬁed for the scalar operator P from (1.3), (1.4), where we assume strict hyperbolicity for t > 0. For a discussion of other cases, see Dreher–Witt [18, Section 5]. (Note that, in general, the lower bound for δ(x) is a Lipschitz function in x, but may fail to be C 1 , while δ ∈ Cb∞ (Rn ; R).) Theorem 2.10. Suppose that Dt −A is strictly hyperbolic for t > 0 in the sense that A ∈ S˜1,1;λ and the characteristic roots τj (t, x, ξ) = tl∗ µj (t, x, ξ) of the principal part τ 1N − σ 1 (A)(t, x, ξ) are real-valued and satisfy |µj (t, x, ξ) − µk (t, x, ξ)| ≥ c|ξ|,

j = k,

c > 0.

(2.17)

Then there are symbols νj ∈ S˜1,1;λ , j = 1, . . . , N , which coincide with the eigenvalues of ˜ 0,1 (A)(x, ξ), Λ(t)ξ ≥ C, σ 1 (A)(t, x, ξ) + t−1 σ for some large C > 0 and which possess the following properties: (a) If a function δ ∈ Cb∞ (Rn ; R) satisﬁes $(i˜ σ 0,1 (ννj ))(x, ξ) ≤ δ(x)l∗ ,

(x, ξ) ∈ Rn × (Rn \ 0),

for all 1 ≤ j ≤ N , then the Cauchy problem (2.4) is δ(x)-well posed. (b) If δ ∈ Cb∞ (Rn ; R) and $(i˜ σ 0,1 (ννj ))(x0 , ξ0 ) > δ(x0 )l∗ for some j and some (x0 , ξ0 ) ∈ Rn × (Rn \ 0), then the Cauchy problem (2.4) is not (0, δ(x))-well posed.

3. A model case To motivate our considerations later on, we ﬁrst consider the Cauchy problem for the operator P from (1.3), (1.4) in the special case that P is of the second order and its coeﬃcients are independent of x ∈ Rn : P u(t, x) = f (t, x), (t, x) ∈ (0, T ) × Rn , (3.1) u(0, x) = u0 (x), Dt u(0, x) = u1 (x), where P = Dt2 + 2

n j=1

λ(t)cj (t)Dt Dj −

n

λ2 (t)ajk (t)Dj Dk

j,k=1

−i

n j=1

λ (t)bj (t)Dj + c0 (t)Dt , (3.2)

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463

ajk , cj ∈ C ∞ ([0, T ]; R), and bj ∈ C ∞ ([0, T ]). We shall assume hyperbolicity for P : ⎛ ⎞2 n n ⎝ cj (t)ξξj ⎠ + ajk (t)ξξj ξk ≥ α0 |ξ|2 , (t, ξ) ∈ [0, T ] × Rn , j=1

j,k=1

for some α0 > 0. This special case is comparatively easy to analyze, since Fourier transform with respect to x turns (3.1) into a family of ordinary diﬀerential equations with parameter ξ ∈ Rn . For the complete derivation, see Dreher–Witt [20]. 3.1. Taniguchi–Tozaki’s example The following example by Taniguchi–Tozaki [69], with n = 1, is particularly instructive: P u = (Dt2 − λ2 (t)Dx2 − iλ (t)bDx )u = 0, (t, x) ∈ (0, T ) × R, (3.3) u(0, x) = u0 (x), ut (0, x) = u1 (x), . / where b ∈ R. The Fourier transform u ˆ(t, ξ) = Fx→ξ u(t, x) of the solution u is given by β∗ (1 − b)l∗ u ˆ(t, ξ) = e−iΛ(t)ξ 1 F1 , β∗ l∗ ; 2iΛ(t)ξ u ˆ0 (ξ) 2 β∗ (1 − b)l∗ + te−iΛ(t)ξ 1 F1 + β∗ , β∗ (l∗ + 2); 2iΛ(t)ξ uˆ1 (ξ), 2 where 1 F1 (α, γ; z) is the conﬂuent hypergeometric function. It behaves asymptotically like 1 F1 (α, γ, z)

=

Γ(γ) z α−γ Γ(γ) e±iπα z −α + e z + O(|z|−α−1 + |z|α−γ−1 ) as |z| → ∞, Γ(γ − α) Γ(α)

with the upper sign being taken if −π/2 < arg z < 3π/2 and the lower sign being taken if −3π/2 < arg z ≤ −π/2. From this representation we can now easily read ˆ(t, ξ)|. oﬀ the asymptotic behaviour of |ˆ u(t, ξ)| and |Dt u First, one of the exponents −α and α − γ is negative, since γ > 0. Therefore, one of the terms z −α and z α−γ is negligible for large |z|. Then we check that λ(t)ξ|ˆ u(t, ξ)| ≤ λ(t)ξ (Λ(t)ξ)β∗ (−1+|b|)l∗ /2 c1 |ˆ u0 (ξ)| + c2 ξ−β∗ |ˆ u1 (ξ)| , β (−1+|b|)l∗ /2 c3 |ˆ |Dt uˆ(t, ξ)| ≤ λ(t)ξ (Λ(t)ξ) ∗ u0 (ξ)| + c4 ξ−β∗ |ˆ u1 (ξ)| , for large values of Λ(t)ξ and certain cj > 0. Moreover, we can replace “≤” by “∼” if one of the initial data u ˆ0 (ξ), uˆ1 (ξ) vanishes. Hence, it is natural to assume that Dx β∗ u0 and u1 obey the same Sobolev regularity. Combining the two cases Λ(t)ξ → ∞ and t = 0, |ξ| → ∞, we ﬁnd that β∗ u(t, ξ)| and |Dt u ˆ(t, ξ)| ξ + λ(t)ξ |ˆ

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Michael Dreher and Ingo Witt

exhibit the same asymptotic behaviour as |ξ| → ∞ when 0 ≤ t ≤ T . This hints at the importance of the weight function g(t, ξ) ∼ ξβ∗ + λ(t)ξ. As a side remark, we note that one of the two characteristic curves emanating from a point on the initial line t = 0 cannot transport any singularities at all if α ∈ −N0 or γ − α ∈ −N0 , since the Gamma function has poles at the non-positive integers. 3.2. Conversion into a 2 × 2 system This observation in case of Eq. (3.3) hints at the conversion of the general case (3.1) into a ﬁrst-order pseudodiﬀerential system: Utilizing the weight function g(t, ξ) from (2.3), we introduce the vector g(t, Dx )u(t, x) U (t, x) = Dt u(t, x) and obtain the Cauchy problem Dt U (t, x) = A(t, Dx )U (t, x) + F (t, x), U (0, x) = U0 (x), where A(t, ξ) = A˜0 (t, ξ) + A˜1 (t, ξ) (3.4) Dt g(t,ξ) 0 g(t, ξ) 0 g(t,ξ) ( = λ(t)2 |ξ|2 + Dt λ(t)| ξ| g(t,ξ) a(t, ξ) −2λ(t)|ξ|c(t, ξ) g(t,ξ) b(t, ξ) −c0 (t) and n ξj ξk ξj , b(t, ξ) = − bj (t) , |ξ|2 |ξ| j=1 j,k=1 β∗ Dx u0 (x) 0 U0 (x) = . , F (t, x) = f (t, x) u1 (x)

a(t, ξ) =

n

ajk (t)

c(t, ξ) =

n j=1

cj (t)

ξj , |ξ|

The ﬁrst matrix in the deﬁnition of A is the ﬁrst-order principal part, while β∗ )∩ the second matrix constitutes a lower-order term belonging to L∞ ((0, T ), S1,0 0 t−1 L∞ ((0, T ), S1,0 ). The imaginary part of this second term plays a decisive role in determining the loss of regularity. 3.3. Estimation of the fundamental matrix ˆ (t, ξ) of U (t, x) can be represented as The partial Fourier transform U t ˆ (t, ξ) = X(t, 0; ξ)Uˆ0 (ξ) + i U X(t, t ; ξ)Fˆ (t , ξ) dt ,

(3.5)

0

where X(t, t ; ξ), (t, t ; ξ) ∈ [0, T ]2 × Rn , is the fundamental matrix of the system Dt − A(t, ξ): Dt X(t, t ; ξ) = A(t, ξ)X(t, t ; ξ), X(t , t ; ξ) = 12 .

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465

An estimation of the matrix norm X(t, t ; ξ) can be found via a diagonalization approach, see Dreher–Reissig [17], Dreher–Witt [20], and also Kajitani– Wakabayashi–Yagdjian [39]: Proposition 3.1. We have X(t, t ; ξ) ≤ C, δ λ(t) 0 X(t, t ; ξ) ≤ C , λ(t ) δ0 λ(t) X(t, t ; ξ) ≤ C , λ(ξ−β∗ )

0 ≤ t ≤ t ≤ ξ−β∗ ,

(3.6)

ξ−β∗ ≤ t ≤ t ≤ T,

(3.7)

0 ≤ t ≤ ξ−β∗ ≤ t ≤ T,

(3.8)

where the number

|$b(0, ξ) + c(0, ξ)| 1 + sup 2 ξ∈Rn 2 c(0, ξ)2 + a(0, ξ) is related to the loss of regularity β∗ δ0 l∗ . δ0 =

3.4. Function spaces: An approach via edge Sobolev spaces The departing point is the observation that P = t−m P˜ (t, x, tDt , tl∗ +1 Dx )

(3.9)

˜ is a polynofor the mth-order diﬀerential operator P from (1.3), where P˜ (t, x, τ˜, ξ) l∗ +1 ˜ mial of degree m in the compressed covariables τ˜ = tτ , ξ = t ξ that is smooth up to t = 0. (Note that σ m (P˜ ) = p, with p taken from (2.16).) This representation hints at P as some kind of “generalized” edge-degenerate diﬀerential operators (with respect to the hypersurface t = 0). Edge-degeneracy is encountered when l∗ = 0 in (3.9). Introducing the function spaces H s,δ;λ ((0, T ) × Rn ) for s, δ ∈ R, here we adopt Schulze’s approach to edge-degenerate problems, see Schulze [63, 64]. In particular, one separates the action of the operator P in direction of t from its action in the directions of the spatial variables xj for 1 ≤ j ≤ n. This is accompanied by corresponding function spaces: The function spaces H s,δ;λ ((0, T ) × Rn ) — which should “somehow” be related to the kind of degeneracy at t = 0 — are obtained by restricting ffrom the open “model wedge” R+ × Rn , H s,δ;λ ((0, T ) × Rn ) = H s,δ;λ (R+ × Rn ) n, (0,T )×R

where the function spaces H (R+ × R ) appear as realizations of the abstract concept of an edge Sobolev space with respect to the “edge” {0}×Rn of the “model wedge” R+ × Rn , see in 3.4.2 below. s,δ;λ

n

3.4.1. Geometric content of relation (3.9). Before we proceed, we look at (3.9). Since [t∂ ∂t , tl∗ +1 ∂xj ] = (l∗ + 1)tl∗ +1 ∂xj , [tl∗ +1 ∂xj , tl∗ +1 ∂xk ] = 0 for 1 ≤ j, k ≤ n, where [ , ] is the commutator on vector ﬁelds, we have: • The vector ﬁelds t∂ ∂t , tl∗ +1 ∂x1 , . . . , tl∗ +1 ∂xn form a basis (over Cb∞ ([0, T ] × n R )) of the Lie algebra generated by them,

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• The operator tm P belongs to the envelope of this Lie algebra. The local belonging of a vector ﬁeld to this Lie algebra is clariﬁed by the next result: Lemma 3.2. A vector ﬁeld X on R × Rn belongs to the Lie algebra generated by the vector ﬁelds t∂ ∂t , tl∗ +1 ∂x1 , . . . , tl∗ +1 ∂xn (over C ∞ (R × Rn )) if and only if, for all functions a ∈ Cc∞ (Rn ), b ∈ Cc∞ (R × Rn ), (3.10) X a(x) + tl∗ +1 b(t, x) vanishes to the (l∗ + 1)th order at t = 0. Remark 3.3. A function c ∈ Cc∞ (R × Rn ) is of the form a(x) + tl∗ +1 b(t, x) for some a ∈ Cc∞ (Rn ), b ∈ Cc∞ (R × Rn ) if and only if ∂tj c(0, x) ≡ 0 for 1 ≤ j ≤ l∗ . In fact, (3.10) need to be checked only when a = xj , b = 0 for some 1 ≤ j ≤ n and when a = 0, b = 1. Moreover, the latter can be replaced by checking that X(t) vanishes at t = 0. Condition (3.10), however, has the advantage of being coordinate invariant as is seen that coordinate changes which keep the geometric situation under consideration are (locally near t = 0) of the form t˜ = tφ(t, x), (3.11) x˜ = κ(x) + tl∗ +1 ψ(t, x), φ(0, x) > 0, with suitable C ∞ functions κ, φ, ψ. This characterizes the situation under consideration as being a cuspidal one. 3.4.2. Function spaces on the half-space R+ ×Rn . The concept of an abstract edge Sobolev space requires the introduction of a certain function space H s,δ;λ (R+ ) on (δ) the half-axis R+ as well as of a strongly continuous group {κν }ν>0 acting on it. (δ) (δ) (δ) (δ) In particular, κν κν = κνν for all ν, ν > 0, κ1 = 1H s,δ;λ (R+ ) . s (R+ ) Deﬁnition 3.4. For s, δ ∈ R, the space H s,δ;λ (R+ ) consists of all u ∈ Hloc being of the form (3.12) u(t) = λ(1 + t)1/2+δ v(Λ(1 + t)) for some v ∈ H s (R+ ).

Note that the behaviour of a function u ∈ H s,δ;λ (R+ ) as t → +0 and t → ∞, respectively, is diﬀerent: we have (1 − χ(t))u ∈ H s (R+ ), while χ(t)u belongs to a certain weighted H s space. Lemma 3.5. (a) For s ∈ N0 , δ ∈ R, u ∈ H s,δ;λ (R+ ) if and only if λ(1 + t)−(j+δ) Dtj u ∈ L2 (R+ ),

0 ≤ j ≤ s.

(3.13)

(δ) {κν }ν>0

(b) For s, δ ∈ R, deﬁned by (δ) β∗ /2−β∗ δl∗ u(ν β∗ t), κν u (t) = ν

t ∈ R+ ,

acts as strongly continuous group on H s,δ;λ (R+ ).

ν > 0,

(3.14)

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467

Proof. (a) Note that

. / L2 (R+ ) = λ(1 + t)1/2 v(Λ(1 + t)) v ∈ L2 (R+ ) .

(3.15)

(b) For u represented as in (3.12), (δ) κν u (t) = ν β∗ /2−β∗ δl∗ λ(1 + ν β∗ t)1/2+δ v(Λ(1 + ν β∗ t)) = ν 1/2 λ(ν −β∗ + t)1/2+δ v(ν Λ(ν −β∗ + t)),

ν > 0,

(δ)

which obviously belongs to H s,δ;λ (R+ ). We conclude that κν ∈ L(H s,δ;λ (R+ )) (δ) for each ν > 0 as well as the map R+ & ν → κν ∈ L(H s,δ;λ (R+ )) is strongly continuous. (δ)

Note that the group {κν }ν>0 for δ ∈ R has been chosen in such way that (i) It reﬂects the considered kind of degeneracy at t = 0, (ii) It acts as group of isometries on {λ(t)1/2+δ v(Λ(t)) | v ∈ L2 (R+ )}, where the latter is the L2 space on R+ with the speciﬁc weighting of H 0,δ;λ (R+ ) as t → ∞ prolongated to all of R+ (i.e., when λ(1 + t) is replaced with λ(t)). Proof of (ii). The natural norm on {λ(t)1/2+δ v(Λ(t)) | v ∈ L2 (R+ )} is 1/2 ∞ 2 −2δ |u(t)| λ(t) dt . u → 0

Then (ii) follows by changing variables under the integral sign.

We proceed to abstract edge Sobolev spaces: Deﬁnition 3.6. For s ∈ R and a Hilbert space E equipped with a strongly ncontin s R ;E = } acting on it, the abstract edge Sobolev space W uous group {κ ν ν>0 ˆ ∈ L2loc (Rn ; E) and W s Rn ; E, {κν }ν>0 consists of all u ∈ S (Rn ; E) such that u 1/2 2 uW s (Rn ;E) := ξ2s κ(ξ)−1 u ˆ(ξ)E dξ < ∞, (3.16) Rn

where κ(ξ) := κξ for ξ ∈ Rn .

The abstract edge Sobolev space W s Rn ; E equipped with the norm (3.16) is a Hilbert space.

Example. The basic example is provided by the standard Sobolev spaces H s (R+ × Rn ): For s ≥ 0, H s (R+ × Rn ) = W s (Rn ; H s (R+ )) κν u)(t) = ν 1/2 u(νt) for ν > 0, see with respect to the group {¯ κν }ν>0 given by (¯ Schulze [64, Example 1.3.23]. For the next result, see Seiler [65]:

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Michael Dreher and Ingo Witt

˜ {˜ Proposition 3.7. Let (E, {κν }ν>0 ), (E, κν }ν>0 ) be Hilbert spaces equipped with ˜ such strongly continuous group actions. Further let a ∈ C ∞ (Rn × Rn ; L(E, E)) that m−β κ ˜ (ξ)−1 (∂ ∂xα ∂ β a)(x, ξ)κ(ξ) , (x, ξ) ∈ Rn × Rn , ˜ ≤ Cαβ ξ L(E,E)

ξ

for some m ∈ R and certain constants Cαβ > 0. Then, for each s ∈ R, ˜ a(x, D) : W s (Rn ; E) → W s−m (Rn ; E) . / continuously, where a(x, D)u = Fξ−1 u(ξ) as usual. →x a(x, ξ)ˆ After this short digression to the abstract theory, we now deﬁne: Deﬁnition 3.8. For s, δ ∈ R, we set

H s,δ;λ (R+ × Rn ) := W s Rn ; H s,δ;λ (R+ ), {κ(δ) ν }ν>0 .

Moreover, we set

H s,δ;λ ((0, T ) × Rn ) := H s,δ;λ (R+ × Rn ) (0,T )×Rn .

For ﬁxed T > 0, the Hilbert norm on the function space H s,δ;λ ((0, T ) × Rn ) following from this deﬁnition is equivalent to the Hilbert norm given by (2.7). We summarize properties of the spaces H s,δ;λ ((0, T ) × Rn ): Proposition 3.9. (a) {H s,δ;λ ((0, T ) × Rn ) s ∈ R} forms an interpolation scale of Hilbert spaces with respect to the complex interpolation method. (b) H 0,0;λ ((0, T ) × Rn )) = L2 ((0, T ) × Rn )), and H −s,−δ;λ ((0, T ) × Rn ) is the dual to H s,δ;λ ((0, T ) × Rn ) with respect to the L2 -scalar product. (c) H s,δ;λ (R+ × Rn ) (T ,T )×Rn = H s ((T , T ) × Rn ) for any 0 < T < T . (d) The space Cc∞ ([0, T ] × Rn ) ⊂ H s,δ;λ ((0, T ) × Rn ) is dense. (e) For s > 1/2, the map [s−1/2]−

H

s,δ;λ

((0, T ) × R ) → n

1

H s+β∗ δl∗ −β∗ j−β∗ /2 (Rn ),

j=0

u → Dtj u t=0 0≤j≤[s−1/2]− , −

is surjective. Here, [s − 1/2] is the largest integer strictly less than s − 1/2. (f) H s,δ;λ ((0, T ) × Rn ) ⊂ H s ,δ ;λ ((0, T ) × Rn ) if and only if s ≥ s , s + β∗ δl∗ ≥ s + β∗ δ l∗ . Moreover, this embedding is locally compact if and only if both inequalities are strict. (g) If, formally, l∗ = 0, then H s,δ;λ ((0, T )×Rn ) is independent of δ and coincides with the standard Sobolev space H s ((0, T ) × Rn ). Proof. Properties (a) to (g) have been shown in Dreher–Witt [20]. For instance, in [20, Lemma 2.5], it has been proved that . / H s,δ;λ (R+ × Rn ) (T ,∞)×Rn = λ(t)1/2+δ v(Λ(t), x) v ∈ H s (R+ × Rn ) n (T ,∞)×R

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for any T > 0, and (c) follows. To obtain (e), we argue as follows: We write u ∈ H s,δ;λ (R+ × Rn ) as . (δ) / (ξ)w(t, ˆ ξ) , u(t, x) = Fξ−1 →x κ (δ)

where κ(δ) (ξ) = κξ and w ∈ H s (Rn ; H s,δ;λ (R+ )). The function w can be rewritten as w(t, x) = (1 − χ(t))

[s−1/2]− j j=0

∂tj w(0, x)

where wj (x) = We conclude that u(t, x) =

Fξ−1 →x +

.

∈ H (R ) and w ¯ ∈ H s (Rn ; H s (R)), w(t, ¯ x) = 0 for t < 0. s

n

/ 1 − χ(ξ t)

Fξ−1 →x

β∗

.

t wj (x) + w(t, ¯ x), j!

κ

(δ)

[s−1/2]− j

/ ¯ ξ) (ξ)w(t,

j=0

t Dx −β∗ δl∗ +β∗ j+β∗ /2 wj (x) j!

and ∂tj u(0, x) = Dx −β∗ δl∗ +β∗ j+β∗ /2 wj (x) ∈ H s+β∗ δl∗ −β∗ j−β∗ /2 (Rn ) for 0 ≤ j ≤ [s − 1/2]− .

Proposition 3.10. For the mth-order diﬀerential operator P from (1.3), P : H s+m,δ;λ ((0, T ) × Rn ) → H s,δ+m;λ ((0, T ) × Rn ) for all s, δ ∈ R. Proof. This is a direct consequence of Proposition 3.7 from inspecting all the respective causes. For instance, we obtain Dt : H s+1,δ;λ (R+ × Rn ) → H s,δ+1;λ (R+ × Rn ) because of Dt ∈ L(H s+1,δ;λ (R+ ), H s,δ+1;λ (R+ )) and κ(δ+1) (ξ)−1 Dt κ(δ) (ξ) = ξDt . Similarly, tl : H s,δ;λ (R+ × Rn ) → H s,δ+l/l∗ ;λ (R+ × Rn ), Dx j : H

s+1,δ;λ

(R+ × R ) → H n

s,δ;λ

(R+ × R ), n

a(t, x) : H s,δ;λ (R+ × Rn ) → H s,δ;λ (R+ × Rn ), The proof is complete.

l = 0, 1, 2 . . . , 1 ≤ j ≤ n, a ∈ Cb∞ (R+ × Rn ).

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3.5. Establishing energy estimates For the estimation of U (t, x), we deﬁne a weight by the aid of the symbol ϑ00 , ϑ00 (t, ξ) = χ− (t, ξ)λ(ξ−β∗ )−δ0 + χ+ (t, ξ)λ(t)−δ0 , see Dreher [16], Dreher–Witt [20]. Then (3.5) yields t ˆ (t, ξ)| ≤ C |ϑ00 (0, ξ)U ˆ0 (ξ)| + |ϑ00 (t , ξ)Fˆ (t , ξ)| dt . |ϑ00 (ξ)U 0

Squaring this inequality and integration over (0, T ) × Rn gives the estimate 2 2 2 U H 0,δ0 ;λ ((0,T )×Rn ) ≤ C U U0 H β∗ δ0 l∗ (Rn ) + T 2 F H 0,δ0 ;λ ((0,T )×Rn ) , 2

where V H 0,δ0 ;λ ((0,T )×Rn ) deﬁned by V 2H 0,δ0 ;λ ((0,T )×Rn ) =

1 T

T

Rn

0

|ϑ00 (t, ξ)V (t, ξ)|2 dξ dt

is an equivalent norm on the space H 0,δ0 ;λ ((0, T ) × Rn ), see (2.7). To estimate higher-order derivatives of U as well, we choose some s ∈ N0 , set ϑsl (t, ξ) = χ− (t, ξ)ξs−l λ(ξ−β∗ )−δ0 −1−l + χ+ (t, ξ)ξs−l λ(t)−δ0 −1−l for 0 ≤ l ≤ s, and deﬁne the norm s T 2l−1 V 2H s,δ0 ;λ ((0,T )×Rn ) = l=0

0

T

Rn

|ϑsl (t, ξ)Dtl Vˆ (t, ξ)|2 dξ dt,

see (2.7) again. Diﬀerentiating (3.5) with respect to t and induction on s then implies the estimate 2 2 2 U0 H s+β∗ δ0 l∗ (Rn ) + T 2 F H s,δ0 ;λ ((0,T )×Rn ) . U H s,δ0 ;λ ((0,T )×Rn ) ≤ C U This estimate is the main ingredient in the proof of Theorem 2.8 in the model case (3.1). We see that the loss of regularity — as predicted by this estimate — is at most β∗ δ0 l∗ . It turns out that this result is sharp, see Theorem 2.10 and the examples by Qi and Taniguchi–Tozaki. 3.6. Summary of Section 3 Starting from second-order operators P from (3.2) with coeﬃcients independent of x, we have been led via partial Fourier transformation with respect to x to certain estimates on the solutions to the Cauchy problem. These estimates have been brought to function spaces H s,δ;λ ((0, T ) × Rn ) building upon the machinery of abstract edge Sobolev spaces. Among others, this approach enables a precise (δ) control of the degeneracy as t → +0, e.g., by the choice of the group {κν }ν>0 . These considerations will guide us in Sections 4 and 5 — where we will be treating operators with coeﬃcients depending on x — where, however, we need to replace the partial Fourier transformation with respect to x by pseudodiﬀerential techniques relying on certain weight functions. Moreover, the variable loss of regularity will require function spaces H s,δ(x);λ ((0, T ) × Rn ), where δ = δ(x) is

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a function of x (instead of being a constant), such that the technique of abstract edge Sobolev spaces is not longer applicable. It will be replaced by an approach also based on the weight functions just mentioned.

4. Symbol classes and function spaces We refer the reader to Dreher–Witt [18] for proofs and further details. 4.1. The symbol classes S m,η;λ The relevant symbol classes in case η is constant are the symbol classes S m,η;λ that have been introduced in Deﬁnition 2.1. We start with some examples: Example. (a) λ(t)ξ ∈ S 1,1;λ , (t + ξ−β∗ )−1 ∈ S 0,1;λ , Λ(t)ξ ∈ S 1,0;λ . m (b) For a ∈ C ∞ ([0, T ], S1,0 ), we have a ∈ S m,m(l∗ +1)−l;λ if and only if Dtj a(0, x, ξ) = 0, (c) χ+ ∈ S 0,0;λ , χ− ∈ S −∞,0;λ , where S −∞,η;λ =

0 ≤ j ≤ l − 1. 2

S m,η;λ .

m∈R

Remark 4.1. We have the following equivalent descriptions of the symbol classes S m,η;λ : A function a ∈ C ∞ ([0, T ] × R2n ) belongs to S m,η;λ if and only if, for each multi-index (j, α, β) ∈ N1+2n , one of the following inequalities hold: j α β ∂t ∂x ∂ a(t, x, ξ) ≤ Cjαβ (1 + Λ(t)ξ)m h(t, ξ)η+j ξ−|β| , ξ j α β ∂t ∂x ∂ a(t, x, ξ) ≤ Cjαβ g(t, ξ)m−|β| h(t, ξ)η−m−|β|l∗ +j ξ for all (t, x, ξ) ∈ [0, T ] × R2n . To see this, note that gh−1 ∼ 1 + Λ(t)ξ,

ghl∗ ∼ ξ.

We conclude this section with some properties of these symbol classes, which are easily derived:

Proposition 4.2. (a) S m,η;λ ⊆ S m ,η ;λ ⇐⇒ m ≤ m , η ≤ η . (b) Let a ∈ S m,η;λ . Then χ+ (t, ξ)a ∈ S m ,η;λ for some m < m implies a ∈ S m ,η;λ . In particular, if a(t, x, ξ) = 0 for Λ(t)ξ ≥ C and some C > 0, then a ∈ S −∞,η;λ . (c) If a ∈ S m,η;λ , then ∂tj ∂xα ∂ξβ a ∈ S m−|β|,η+j−|β|(l∗+1);λ . (d) If a ∈ S m,η;λ , a ∈ S m ,η ;λ , then a ◦ a ∈ S m+m ,η+η ;λ and a ◦ a = aa

mod S m+m −1,η+η −(l∗ +1);λ ,

where ◦ denotes the Leibniz product with respect to x.

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Michael Dreher and Ingo Witt

(e) If a ∈ S m,η;λ , then a∗ ∈ S m,η;λ and a∗ (t, x, ξ) = a(t, x, ξ)∗

mod S m−1,η−(l∗ +1);λ ,

where a∗ is the (complete) symbol of the adjoint to a(t, x, Dx ) with respect to L2 . (f) If a ∈ S m,η;λ ([0, T ] × R2n ; MN ×N (C)) is elliptic in the sense that N | det a(t, x, ξ)| ≥ c g m (t, ξ) hη−m (t, ξ) , (t, x, ξ) ∈ [0, T ] × R2n , |ξ| ≥ C for some C, c > 0, then there is a symbol a ∈ S −m,−η;λ with the property that a ◦ a − 1, a ◦ a − 1 ∈ C ∞ ([0, T ]; S −∞ ). Moreover, (g)

; m,η

a = a−1

mod S −m−1,−η−(l∗+1);λ .

S m,η;λ = C ∞ ([0, T ]; S −∞ ).

4.2. The symbol classes S˜m,η;λ These symbol classes have been introduced in Deﬁnition 2.6. Again, we consider some examples ﬁrst: Example.

(a) For m, η ∈ R, we have g m hη−m ∈ S˜m,η;λ , m σ m (g m hη−m ) = t−η tl∗ +1 |ξ| , σ ˜ m−1,η (g m hη−m ) = 0,

see (2.3). * + (b) For a(t, x, ξ) = |α|≤j aα (t, x) t(|α|(l∗ +1)−j) ξ α , where aα ∈ Cb∞ ([0, T ] × Rn ) for |α| ≤ j, we have a ∈ S˜j,j;λ , aα (t, x) (tl∗ ξ)α , σ j (a) = |α|=j

* σ ˜ j−1,j (a) =

|α|=j−1

aα (0, x) ξ α

0

if j > 1, if j = 0, 1,

see (1.9). For a ∈ S˜m,η;λ , the principal symbol σ m (a) as well as the secondary symbol σ ˜ (a) are uniquely determined. This follows from the next lemma, whose proof can be found in [18]: m−1,η

Lemma 4.3. (a) The symbols σ m (a), σ ˜ m−1,η (a) are well deﬁned. (b) The short sequence m

m−1,η

σ ) (σ ,˜ 0 −→ S m−2,η;λ + S m−1,η−1;λ −→ S˜m,η;λ −−−−−−−−→ ΣS˜m,η;λ −→ 0

is split exact, where ΣS˜m,η;λ = t(l∗ +1)m−η C ∞ ([0, T ]; S (m)) × S (m−1) comprises the principal and secondary symbol spaces.

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The calculus for S˜m,η;λ requires an additional notation: Let a ∈ S˜m,η;λ be of the form a(t, x, ξ) = χ+ (t, ξ) t−η a0 (t, x, tl∗ +1 ξ) + a1 (t, x, tl∗ +1 ξ) + a2 (t, x, ξ), where a0 ∈ C ∞ ([0, T ]; S (m)), a1 ∈ C ∞ ([0, T ]; S (m−1)), and a2 ∈ S m−2,η;λ + S m−1,η−1;λ . Then we set σ ˜ m,η (a)(x, ξ) = a0 (0, x, ξ). Note that this symbol is not of independent interest, but it is directly derived from σ m (a). Extending Proposition 4.2 we have:

Proposition 4.4. (a) If a ∈ S˜m,η;λ , a ∈ S˜m ,η ;λ , then a ◦ a ∈ S˜m+m ,η+η ;λ and

σ m+m (a ◦ a ) = σ m (a) σ m (a ),

σ ˜ m+m −1,η+η (a ◦ a ) = σ ˜ m,η (a) σ ˜ m −1,η (a ) + σ ˜ m−1,η (a) σ ˜ m ,η (a ). (b) If a ∈ S˜m,η;λ , then a∗ ∈ S˜m,η;λ and σ m (a∗ ) = σ m (a)∗ , σ ˜ m−1,η (a∗ ) = σ ˜ m−1,η (a)∗ . (c) If the symbol a ∈ S˜m,η;λ ([0, T ] × R2n ; MN ×N (C)) is elliptic in the sense of Proposition 4.2 (f), then N | det σ m (a)| ≥ c t(l∗ +1)m−η |ξ|m for some c > 0 and the symbol a from Proposition 4.2 (f) belongs to S˜−m,−η;λ . Moreover, σ −m (a ) = σ m (a)−1 , σ m,η (a)−1 σ ˜ m−1,η (a) σ ˜ m,η (a)−1 . σ ˜ −m−1,−η (a ) = −˜ Proposition 4.5. (a) If q(t, x, Dx ) ∈ Op S˜0,0;λ is invertible on H s,δ;λ for some s ∈ R, δ ∈ Cb∞ (Rn ; R), then q(t, x, Dx ) is invertible on H s,δ;λ for all s ∈ R, δ ∈ Cb∞ (Rn ; R) and q(t, x, Dx )−1 ∈ Op S˜0,0;λ . (b) Conversely, if symbols q0 ∈ C ∞ ([0, T ]; S (0) ), q1 ∈ S (−1) are given, where |det q0 (t, x, ξ)| ≥ c for all (t, x, ξ) ∈ [0, T ] × Rn × (Rn \ 0) and a certain c > 0, then there is an invertible operator q(t, x, Dx ) ∈ Op S˜0,0;λ in the sense of (a) such that ˜ −1,0 (q) = q1 . σ 0 (q) = q0 , σ If a ∈ S˜m,η;λ , then in general ∂t a ∈ S˜m,η+1;λ . But in a special case, an improvement is possible: Lemma 4.6. Let a ∈ S˜m,η;λ and η = (l∗ + 1)m. Then ∂t a ∈ S m−1,η+1;λ + S m,η;λ .

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Proof. We have ∂t a ∈ S˜m,η+1;λ and σ ˜ m,η+1;λ (∂ ∂t a) = (m(l∗ + 1) − η) σ ˜ m,η;λ (a). ∂t a) = 0 in case η = (l∗ + 1)m. The latter implies that ∂t a ∈ Therefore, σ ˜ m,η+1;λ (∂ m−1,η+1;λ + S m,η;λ . S For the reader’s convenience, we summarize what the vanishing of the single symbol components for a ∈ S˜m,η;λ means: ˜ m−1,η (a) = 0 ⇐⇒ a ∈ S m−2,η;λ + S m−1,η−1;λ . • σ m (a) = 0, σ m • σ (a) = 0 ⇐⇒ a ∈ S m−1,η;λ . • σ ˜ m,η (a) = 0 ⇐⇒ a ∈ S m−1,η;λ + S m,η−1;λ . m,η;λ 4.3. The symbol classes S+ for η ∈ Cb∞ (Rn ; R) To get a priori estimates of the solutions of (2.4), we will symmetrize the operator A(t, x, Dx ) up to a certain remainder and then “shift the spectrum” of the new operator A(t, x, Dx ), so preparing for the application of G˚ ˚ arding’s inequality. However, the symbol Θ(t, x, ξ) of the “shift operator” does not belong to S m,η;λ with constant η. Therefore, we need to enlarge our symbol classes: m,η;λ Deﬁnition 4.7. For m ∈ R, η ∈ Cb∞ (Rn ; R), and ∈ N0 , the symbol class S() consists of all a ∈ C ∞ ([0, T ] × R2n ; MN ×N (C)) such that, for each multi-index (j, α, β) ∈ N1+2n , there is a constant Cjαβ > 0 with the property that j α β ∂ ∂ ∂ a(t, x, ξ) (4.1) t x ξ +|α| ≤ Cjαβ g(t, ξ)m h(t, ξ)η(x)−m+j 1 + | log h(t, ξ)| ξ−|β|

for all (t, x, ξ) ∈ [0, T ] × R2n . Moreover, we set ) m,η;λ m,η;λ S+ = S() . ∈N 0,δ(x)l∗ ;λ

Example. A typical example is given by h(t, ξ)δ(x)l∗ ∈ S(0)

.

m,η;λ m ,η ;λ Proposition 4.8. (a) S() ⊆ S( ⇐⇒ m ≤ m , η ≤ η , and ≤ if η = η . ) ; m,η;λ (b) S m,η;λ S+ >0 S m,η+;λ . m−|β|,η−|β|(l∗+1)+j;λ m,η;λ . (c) If a ∈ S() , then ∂tj ∂xα ∂ξβ a ∈ S(+|α|)

m,η;λ m ,η ;λ m+m ,η+η ;λ , a ∈ S( , then a ◦ a ∈ S(+ and (d) If a ∈ S() ) )

a ◦ a = aa

m+m −1,η+η −(l∗ +1);λ

mod S(+ +1)

.

m,η;λ m,η;λ , then a∗ ∈ S() and (e) If a ∈ S()

a∗ (t, x, ξ) = a(t, x, ξ)∗

m−1,η−(l∗ +1);λ

mod S(+1)

0,0;λ 0 ⊂ L∞ ((0, T ); S1,δ ) for any 0 < δ < 1. (f) S(0)

From Proposition 4.8 (f) we conclude:

.

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475

0,0;λ Corollary 4.9. Op S 0,0;λ ⊂ Op S(0) ⊂ L(L2 ).

4.4. Function spaces: An approach via weight functions For δ ∈ Cb∞ (Rn ; R), we employ the weight functions g, h from (2.3) to introduce the function spaces H s,δ(x);λ ((0, T ) × Rn ): Deﬁnition 4.10. For s ∈ N0 , δ ∈ Cb∞ (Rn ; R), the space H s,δ(x);λ ((0, T ) × Rn ) consists of all functions u = u(t, x) satisfying (g s−j h(s+δ)l∗ )(t, x, Dx ) Dtj u ∈ L2 ((0, T ) × Rn ), Cb∞ (Rn ; R),

For general s ∈ R, δ ∈ the space H by means of duality and interpolation.

s,δ(x);λ

0 ≤ j ≤ s.

(4.2)

((0, T ) × R ) is then deﬁned n

Proposition 4.11. For s ∈ N0 , δ ∈ R, Deﬁnitions 3.8 and 4.10 coincide. Proof. Since ghl∗ ∼ ξ by choice of the weight functions, (4.2) is equivalent to (ξs−j h(j+δ)l∗ )(t, x, Dx )Dtj u ∈ L2 ((0, T ) × Rn ), Now, u ∈ H

(R+ × R ) means ξs κ(δ) (ξ)−1 uˆ(t, ξ)

s,δ;λ

0 ≤ j ≤ s.

(4.3)

n

H s,δ;λ (R+ )

∈ L2 ((0, T ) × Rnξ ).

The latter is equivalent to

−β∗ t, ξ ∈ L2 ((0, T ) × Rnξ ), ˆ ξ ξs+β∗ δl∗ −β∗ /2 λ(1 + t)−(j+δ) ∂tj u

0 ≤ j ≤ s,

i.e., equivalent to

ˆ t, ξ ∈ L2 ((0, T ) × Rnξ ), ξs+β∗ δl∗ −jβ∗ λ(1 + ξβ∗ t)−(j+δ) ∂tj u −(j+δ)

Writing λ(1 + ξ t) (4.3). β∗

−β∗ (j+δ)l∗

= ξ

(j+δ)l∗

h(t, ξ)

0 ≤ j ≤ s.

, we see that this is exactly

Remark 4.12. Below we shall make use of Deﬁnition 4.10 as follows: (i) For s ∈ N0 , δ ∈ Cb∞ (Rn ; R), u ∈ H s,δ(x);λ if and only if g s−j (t, Dx )Dtj u ∈ 0,s+δ(x);λ for 0 ≤ j ≤ s. H (ii) For δ ∈ Cb∞ (Rn ; R), a function u belongs to H 0,δ(x);λ if and only if hδl∗ (t, x, Dx )u ∈ L2 ((0, T ) × Rn ). For K > 0, δ ∈ Cb∞ (Rn ; R), let ξK := (K+|ξ|2 )1/2 , χ+ K (t, ξ) := χ(Λ(t)ξK ), := 1 − χ+ K (t, ξ), and

χ− K (t, ξ)

∗ Θ(t, x, ξ) = ΘK,δ (t, x, ξ) := χ− K (t, ξ) ξK

β δ(x)l∗

0,δ(x)l∗ ;λ

Note that Θ(t, x, Dx ) ∈ Op S(0)

−δ(x)l∗ + χ+ . K (t, ξ) t

(4.4)

.

Remark 4.13. Below it is convenient to write Θ(t, x, ξ) = h(t, ξ)δ(x)l∗ , where we have deﬁned β∗ + −1 h(t, ξ) = χ− . (4.5) K (t, ξ)ξK + χK (t, ξ)t m,η;λ m,η;λ ˜m,η;λ Of course, this choice leads to the same symbol classes S ,S and S+ as the choice in (2.3).

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Because of their importance, the proofs of the following two results are repeated from [18]. Lemma 4.14. Given δ ∈ Cb∞ (Rn ; R), there is a K0 > 0 such that the operator

Θ(t, x, Dx ) : H s,δ (x);λ → H s,δ (x)−δ(x);λ

(4.6)

is invertible for all s ∈ R, δ ∈ Cb∞ (Rn ; R), and K ≥ K0 . Moreover, Θ−1 ∈ 0,−δ(x)l∗ ;λ Op S(0) . Proof. Here, we will prove invertibility of the hypoelliptic operator Θ(t, x, Dx ), for 0,−δ(x)l∗ ;λ large K > 0, and also the fact that Θ(t, x, Dx )−1 ∈ Op S(0) . The proof is then completed with the help of the next proposition. 0,δ(x)l∗ ;λ The symbol ΘK,δ (t, x, ξ) belongs to the symbol class S+ , but with pa√ rameter K ≥ 1. Similarly for ΘK,−δ (t, x, ξ). If RK := ΘK,δ ◦ΘK,−δ −ΘK,δ ΘK,−δ , then, for all α, β ∈ Nn0 and certain constants Cαβ > 0, −1 ∗ −l∗ (t, x, ξ)| ≤ Cαβ ξβK∗ + λ(t)ξK (t + ξ−β |∂ ∂xα ∂ξβ RK K ) 1+|α| −|β| ∗ × 1 + | log(t + ξ−β ξK , (t, x, ξ) ∈ [0, T ] × R2n , K ≥ 1 K )| (i.e., we have estimates (2.6), but with ξ replaced by ξK ). From the latter 0 (t, x, ξ) → 0 in L∞ ((0, T ); S1,0 ) as K → ∞, i.e., relation, it is seen that RK 2 RK (t, x, Dx ) → 0 in L(L ) as K → ∞. Now, let RK := ΘK,δ ◦ ΘK,−δ − 1, i.e., RK = RK + ΘK,δ ΘK,−δ − 1. Since 2 (ΘK,δ ΘK,−δ )(t, x, Dx ) → 1 in L(L ) as K → ∞, it follows that RK (t, x, Dx ) → 0 in L(L2 ) as K → ∞. Thus, ΘK,−δ ◦ (1 + RK )−1 is a right inverse to ΘK,δ , for large K > 0. In a similar fashion, a left inverse to ΘK,δ is constructed. −∞,−δ(x)l∗ −(l∗ +1);λ Moreover, Θ−1 = ΘK,−δ mod Op S+ , as is seen from the constructions. Proposition 4.15. We have L(H s,δ(x);λ , H s−m,δ(x)+m+(m−η)/l∗ ;λ ) m,η;λ Op S(0) ⊂ L(H s,δ(x);λ , H s,δ(x)+(m−η)/l∗ ;λ )

if m ≥ 0, if m < 0.

(4.7)

Proof. We prove (4.7) in case m ≥ 0; the proof in case m < 0 is similar. By interpolation and duality, we may assume that s − m ∈ N0 . Then we have m,η;λ to show that, for A ∈ Op S(0) and 0 ≤ k ≤ j ≤ s − m, h(s+δ)l∗ +m−η g s−m−j (Dtj−k A)Dtk u ∈ L2 ((0, T ) × Rn ) provided u ∈ H s,δ(x);λ ((0, T ) × Rn ). We have h(s+δ)l∗ +m−η g s−m−j (Dtj−k A)Dtk u =h

m−η −m−j+k

g

(Dtj−k A)h(s+δ)l∗ g s−k Dtk u

(4.8) +

RDtk u,

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−j+k,0;λ with hm−η g −m−j+k (Dtj−k A) ∈ Op S(j−k) and a certain remainder term R ∈ s−j−1,(s−1)(l +1)+δl −k;λ

−j+k,0;λ 0,0;λ ∗ ∗ . Now Op S(j−k) ⊂ Op S(0) and h(s+δ)l∗ g s−k Dtk u Op S+ 2 n belongs to L ((0, T ) × R ) by assumption, i.e., the ﬁrst summand on the righthand side of (4.8) belongs to L2 ((0, T )×Rn ) by virtue of Corollary 4.9. The second summand is rewritten as

RDtk u = Rg −s+k (ΘK,s+δ )−1 ΘK,s+δ g s−k Dtk u −j+k−1,−(l +1);λ

∗ ⊂ Op S 0,0;λ for large K > 1, where Rg −s+k (ΘK,s+δ )−1 ∈ Op S+ s−k k 2 and again ΘK,s+δ g Dt u ∈ L , i.e., also the second summand on the right-handside of (4.8) belongs to L2 ((0, T ) × Rn ).

The next result extends Proposition 3.9 to the case of variable δ = δ(x). Proposition 4.16. Let s ∈ R, δ ∈ Cb∞ (Rn ; R). Then: / . (a) H s,δ(x);λ ((0, T )× Rn ) s ∈ R forms an interpolation scale of Hilbert spaces with respect to the complex interpolation method. (b) H 0,0;λ ((0, T ) × Rn )) = L2 ((0, T ) × Rn )), and H −s,−δ(x);λ ((0, T ) × Rn ) is the dual to H s,δ(x);λ ((0, T ) × Rn ) with respect to the L2 –scalar product. (c) H s,δ(x);λ (R+ × Rn ) (T ,T )×Rn = H s ((T , T ) × Rn ) for any 0 < T < T . (d) The space Cc∞ ([0, T ] × Rn ) ⊂ H s,δ(x);λ ((0, T ) × Rn ) is dense. (e) For s > 1/2, the map [s−1/2]−

H

s,δ(x);λ

((0, T ) × R ) → n

1

H s+β∗ δ(x)l∗ −β∗ j−β∗ /2 (Rn ),

(4.9)

j=0

u → Dtj u t=0 0≤j≤[s−1/2]− , is surjective. (f) H s,δ(x);λ ⊂ H s ,δ (x);λ if and only if s ≥ s , s + β∗ δ(x)l∗ ≥ s + β∗ δ (x)l∗ . Moreover, the embedding {u ∈ H s,δ(x);λ supp u ⊆ K} ⊂ H s ,δ (x);λ for some K [0, T ] × Rn is compact if and only if s > s and s + β∗ δ(x)l∗ > s + β∗ δ (x)l∗ for all x satisfying (0, x) ∈ K. Proof. We exemplary verify (a), (d): We write H s,δ(x);λ = Θ−1 H s,0;λ for s ∈ R, with Θ being the operator from Lemma 4.14. (a) Since {H s,0;λ | s ∈ R} is an interpolation scale, {H s,δ(x);λ | s ∈ R} is also an interpolation scale with interpolation method. respect to the complex j s−β∗ j−β∗ /2 (Rn ) for 0 ≤ j ≤ j0 , since (d) Let γj u := Dt u t=0 . Then γj Θu ∈ H (4.9) holds if δ = 0. :j0 H s+β∗ δ(x)l∗ −β∗ j−β∗ /2 (Rn ), u → γj u 0≤j≤j0 follows Now, H s,δ(x);λ → j=0 from β δ(x)l∗ −1 γj u = Dx K∗ γj Θu, while the surjectivity of this map is implied by the reverse relation β δ(x)l∗

γj Θu = Dx K∗

γj u

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Michael Dreher and Ingo Witt

and the surjectivity of (4.9) in case δ = 0. 4.5. Summary of Section 4

Our analysis is based on the two weight functions g(t, ξ), h(t, ξ) introduced in (2.3). These weight functions have been designed to reﬂect the kind of degeneracy as t → +0 under consideration. Thereby, the weight function g plays the predominant part, while the weight function h is to control the ﬁne structure. One major achievement has been the reformulation of the results of Section 3 in terms of g, h. More precisely, the symbol classes S m,η;λ come into being. Here, the basic case occurs when m = η, e.g., among others the belonging of A(t, x, Dx ) in (2.4) to Op S 1,1;λ expresses sharp Levi conditions on the lower-order terms. Symbol classes S m,η;λ with m = η are utilized to formulate hyperbolicity assumptions, e.g., A(t, x, Dx ) − A(t, x, Dx )∗ ∈ Op S 0,1;λ in case of symmetric-hyperbolic systems. The symbol classes S m,η;λ are then further reﬁned to S˜m,η;λ , where the elements a(t, x, ξ) of the latter admit two homogeneous symbol components σ m (a), σ ˜ m−1,η (a). These homogeneous symbol components will be used to determine the loss of regularity on a symbolic level. m,η;λ We have also introduced the symbol classes S+ . The only place, where these symbol classes will be of use in this article, is the proof of Theorem 2.7 in Section 5.3, below, where they play an auxiliary role. Therefore, they need not be considered further here. However, these symbol classes are expected to play a role in the parametrix construction. Finally, the properties of the function spaces H s,δ;λ ((0, T )×Rn) in case s, δ ∈ R carry over to the case s ∈ R, δ ∈ Cb∞ (Rn ; R) with the help of the operator Θ considered in Lemma 4.14.

5. The Cauchy problem In this section, we prove Theorems 2.5, 2.7, 2.8, 2.9, and 2.10. Our main tools are a priori estimates, which all are variations of the following simple result, see, e.g., ¨ rmander [26, Chapter 23]: Ho 1 ) be a pseudodiﬀerential symbol Lemma 5.1. Let A = A(t, x, ξ) ∈ L∞ ([0, T ], S1,0 with

$(iA)(t, x, ξ) ≤ C0 ,

(t, x, ξ) ∈ [0, T ] × R2n .

Then the Cauchy problem Dt U (t, x) = A(t, x, Dx )U (t, x) + F (t, x), U (0, x) = U0 (x) is well posed in L2 (Rn ).

(t, x) ∈ (0, T ) × Rn ,

(5.1)

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479

Proof. By the sharp G˚ ˚ arding inequality, 2

∂t U (t, ·)L2 (Rn ) = 2$ (∂ ∂t U (t, ·), U (t, ·)) = 2$ (i(AU )(t, ·) + iF (t, ·), U (t, ·)) 2

2

≤ C U (t, ·)L2 (Rn ) + F (t, ·)L2 (Rn ) . Gronwall’s lemma now yields an a priori estimate, and the L2 (Rn ) well-posedness follows by standard arguments. In Section 5.1, we observe that the real number C0 from (5.1) can be replaced 1 by a scalar symbol q = q(t, ξ) ∈ L∞ ([0, T ], S1,0 ) whose primitive p = p(t, ξ) = 0t ∞ 0 q(t , ξ) dt belongs to L ([0, T ], S ). 1,0 0 The key example for such a symbol q is q(t, ξ) = C0 (g(t, ξ)−1 h(t, ξ)2 + 1), which appears naturally in estimates of symbols from the class S −1,1;λ + S 0,0;λ . The consequences for the case that A ∈ S˜1,1;λ are obvious, as A has the structure A(t, x, ξ) = χ+ (t, ξ)t−1 A0 (t, x, tl∗ +1 ξ) + A1 (x, ξ) + A2 (t, x, ξ), (5.2) A0 ∈ C ∞ ([0, T ], S (1)),

A1 ∈ S (0) ,

A2 ∈ S −1,1;λ + S 0,0;λ .

From the above reasoning we ﬁnd that the Cauchy problem for the operator Dt −A is well posed in L2 (Rn ) (without loss of regularity) provided that $(i(A0 +A1 )) ≤ 0, which can be achieved in two steps as follows: • First, we diagonalize A0 (which is possible by assumption of symmetrizability). This way, the real eigenvalues tl∗ µj (t, x, ξ) appear on the diagonal of A0 , hence $(iA0 ) = 0. • Secondly, we “shift the spectrum” of $(iA1 ) by means of a “shift operator” Θ with symbol Θ(t, x, ξ) ∼ h(t, ξ)δ(x)l∗ . If we choose the parameter function δ = δ(x) suitably, we can arrange that the symbol of the new A1 satisﬁes $(iA1 ) ≤ 0. The predicted loss of regularity is proportional to δ(x). Since we want to describe the loss precisely, we wish to choose δ as small as possible. It turns out that an optimal δ can be chosen if A1 can be diagonalized, which is certainly possible if the µj satisfy (2.17). The details of this reduction are presented in Section 5.3. As application, we consider higher order diﬀerential equations in Section 5.4, and we prove the local uniqueness (and, consequently, the ﬁnite propagation speed) for higher order diﬀerential equations in Section 5.5. The optimality of this choice of δ is proved in Section 5.6, using an a priori estimate from below. See Section 5.6.1 for a detailed exposition. The situation is not so nice if we merely assume that A ∈ S 1,1;λ instead of A ∈ S˜1,1;λ . In that case we cannot longer assume that A can be split into two homogeneous components and a remainder as in (5.2). But we still can show that

480

Michael Dreher and Ingo Witt

the Cauchy problem to Dt − A is well posed with a certain loss of derivatives, see Section 5.2. 5.1. Improvement of G˚ ˚ arding’s inequality The proofs of Theorems 2.5 and 2.7 rely on the following estimate for matrix pseudodiﬀerential initial-value problems. We suppose that the operator Dt − A(t, x, Dx ) possesses a forward fundamental solution X(t, t ) that maps the Sobolev space H ∞ (Rn ) into itself: (Dt − A(t, x, Dx ))X(t, t ) = 0, 0 ≤ t ≤ t ≤ T, X(t , t ) = I,

0 ≤ t ≤ T.

Our assumptions on A(t, x, ξ) are as follows: 1 (Rn × Rn )), (A) A ∈ L∞ ((0, T ), S1,0 (B) $(iA(t, x, ξ)) ≤ q(t, ξ)1N for (t, x, ξ) ∈ (0, T ) × Rn × (Rn \ 0), 1 (C) The real-valued scalar function q(t, ξ) belongs to L∞ ((0, T ), S1,0 (Rn )), while 0t 0 (Rn )). its primitive p(t, ξ) := 0 q(t , ξ) dt belongs to L∞ ((0, T ), S1,0 Lemma 5.2. Under the assumptions (A), (B), (C), each solution U = U (t, x) ∈ C([0, T ], L2 (Rn )) to the Cauchy problem Dt U (t, x) = A(t, x, Dx )U (t, x) + F (t, x), U (0, x) = U0 (x), where U0 ∈ L (R ), F ∈ L2 ((0, T ), L2 (Rn )), such that Dt U ∈ L2 ((0, T ); L2 (Rn )) satisﬁes the a priori estimate 1 t 2 2 U (t, ·)L2 (Rn ) + U (t , ·)L2 (Rn ) dt t 0 t 2 2 ≤ C U U0 L2 (Rn ) + t F (t , ·)L2 (Rn ) dt (5.3) 2

n

0

for all 0 ≤ t ≤ T and some C = C(T ). Proof. Representing the solution U = U (t, x) in terms of the fundamental matrix X(t, t ), t X(t, t )F (t , x) dt , U (t, x) = X(t, 0)U U0 (x) + i 0

we see that it suﬃces to establish the uniform estimate X(t, t )V L2 (Rn ) ≤ C0 V L2 (Rn ) , ∞

0 ≤ t ≤ t ≤ T,

(5.4)

for all V ∈ H (R ), since we then obtain the estimate t U0 L2 (Rn ) + C0 F (t , ·)L2 (Rn ) dt U (t, ·)L2 (Rn ) ≤ C0 U n

0

√

≤ C0 U U0 L2 (Rn ) + C0 t

t

F (t 0

2 , ·)L2 (Rn )

dt

1/2 ,

Sharp Energy Estimates

481

from which the assertion (5.3) follows by squaring and integrating over t. For 0 ≤ t ≤ t ≤ T , we deﬁne a map Y (t, t ) : H ∞ (Rn ) → H ∞ (Rn ) by Y (t, t ) = exp(−p(t, Dx )) exp(p(t , Dx ))X(t, t ). Observe that the zeroth-order pseudodiﬀerential operators exp(±p(t, Dx )) are invertible. Moreover, Y (t , t ) = I and ∂t Y (t, t ) = −q(t, Dx )Y (t, t ) + i exp(−p(t, Dx )) exp(p(t , Dx ))A(t, x, Dx )X(t, t ) = (iA − q1N + [exp(−p(t, Dx )) exp(p(t , Dx )), iA] × exp(−p(t , Dx )) exp(p(t, Dx ))) Y (t, t ) = B(t, x, Dx )Y (t, t ) 1 (Rn × Rn )) that satisﬁes $ B(t, x, ξ) ≤ C a.e. for for some B ∈ L∞ ((0, T ), S1,0 2n ˚ arding’s inequality gives (t, x, ξ) ∈ (0, T ) × R . Then G˚

∂t Y (t, t )V , Y (t, t )V ) ∂t Y (t, t )V L2 (Rn ) = 2$ (∂ 2

= 2 (($ B)Y (t, t )V , Y (t, t )V ) ≤ C Y (t, t )V L2 (Rn ) . 2

Upon applying Gronwall’s inequality, we obtain Y (t, t )V L2 (Rn ) ≤ C Y (t , t )V L2 (Rn ) = C V L2 (Rn ) , 2

2

2

0 ≤ t ≤ t ≤ T,

which gives (5.4), since the factors exp(±p(t, Dx )) are continuous isomorphisms on L2 (Rn ). 5.2. Symmetric-hyperbolic systems Lemma 5.2 enables us to establish estimates on the solutions to (2.4) provided that A ∈ S 1,1;λ has Hermitian principal part and the eigenvalues of $(iA(t, x, ξ)) lie on the negative real axis modulo perturbations from S −1,1;λ + S 0,0;λ : Proposition 5.3. Let A = A(t, x, ξ) ∈ S 1,1;λ satisfy $(iA) ∈ S 0,1;λ , where $(iA(t, x, ξ)) ≤ C0 (g(t, ξ)−1 h(t, ξ)2 + 1)1N ,

(t, x, ξ) ∈ [0, T ] × R2n .

Then the Cauchy problem (2.4) is (0, 0)-well posed in the sense of Deﬁnition 2.3. Proof. We approximate A(t, x, ξ) by Aε (t, x, ξ) = $ A(t, x, ξ) + i

t + ξ−β∗ % A(t, x, ξ), t + ξ−β∗ + ε

for 0 < ε ≤ 1. It is then clear that Aε ∈ S 1,1;λ , $(iAε ) ∈ S 0,1;λ with uniform symbol estimates, where $(iAε (t, x, ξ)) ≤ C0 (g(t, ξ)−1 h(t, ξ)2 + 1),

(ε, t, x, ξ) ∈ (0, 1] × [0, T ] × R2n .

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Michael Dreher and Ingo Witt

The operator Dt − Aε is hyperbolic with Hermitian principal part $ A and a 0 (Rn × Rn )). Consequently, the lower-order term i % Aε belonging to L∞ ((0, T ), S1,0 Cauchy problem Uε (t, x) + F (t, x), Dt Uε (t, x) = Aε (t, x, Dx )U Uε (0, x) = U0 (x) has a unique solution Uε ∈ L∞ ((0, T ), H ∞ (Rn )) for all U0 ∈ H ∞ (Rn ), F ∈ L∞ ((0, T ), H ∞ (Rn )). We now apply Lemma 5.2 with weight q(t, ξ) = C0 (g(t, ξ)−1 h(t, ξ)2 + 1) to obtain the estimate 2 2 2 U Uε H 0,0;λ ((0,T )×Rn ) ≤ C(T ) U U0 L2 (Rn ) + F H 0,0;λ ((0,T )×Rn ) uniformly in 0 < ε ≤ 1. It remains to show that the Uε converges in H 0,0;λ ((0, T )× Rn )) as ε → +0 to a solution U = U (t, x) to (2.4). To this end, we consider Dx M Uε for M > 0, which solves the problem $ % Dt Dx M Uε = Aε Dx M Uε + Dx M , A Uε + Dx M F, Dx M Uε (0, x) = Dx M U0 (x), which together with [Dx M , A]Dx −M ∈ S 0,−l∗ ;λ ⊂ S 0,0;λ and Lemma 5.2 yields the estimate Dx M Uε 2 0,0;λ H ((0,T )×Rn ) 2 ≤ C U U0 2H M (Rn ) + Dx M F H 0,0;λ ((0,T )×Rn ) . (5.5) The diﬀerence Uε − Uε solves Dt (U Uε − Uε ) = Aε (U Uε − Uε ) + (Aε − Aε )U Uε , (U Uε − Uε )(0, x) = 0. Since the set {(Aε − Aε )/(ε − ε ) : 0 < ε < ε ≤ 1} is bounded in S 0,2;λ , we conclude from Proposition 4.15 that 2

U Uε − Uε H 0,0;λ ((0,T )×Rn ) ≤ C|ε − ε | U Uε H 0,2/l∗ ;λ ((0,T )×Rn ) ≤ C|ε − ε | Dx U Uε H 0,0;λ ((0,T )×Rn ) . 2

2

The uniform estimate (5.5) implies the convergence Uε → U in H 0,0;λ as ε → +0. By interpolation, Dx M Uε converges to Dx M U . A density argument then completes the proof. The estimate of the previous proposition can be reﬁned if one has more information about the structure of the symbol A(t, x, ξ): Proposition 5.4. Let A ∈ S˜1,1;λ satisfy the assumptions of Proposition 5.3, i.e., A(t, x, ξ) = χ+ (t, ξ) λ(t)|ξ|A0 (t, x, ξ) − il∗ t−1 A1 (x, ξ) + A2 (t, x, ξ),

Sharp Energy Estimates

483

where A0 ∈ C ∞ ([0, T ], S (0) ), A1 ∈ S (0) , A2 ∈ S −1,1;λ + S 0,0;λ , and A0 = A∗0 ,

$ A1 (x, ξ) ≤ 0.

Then the Cauchy problem (2.4) is 0-well posed. Proof. We need to show that, for any s ∈ N0 , the Cauchy problem (2.4) is (s, 0)well posed. We proceed by induction on s. The (0, 0)-well-posedness follows from Proposition 5.3. Now suppose that (s, 0)-well-posedness has already been proved and consider (s + 1, 0)-well-posedness. By deﬁnition, W ∈ H s+1,0;λ if and only if (ghl∗ )(t, Dx )W, hl∗ (t, Dx )Dt W ∈ s,0;λ H . For ξ ∼ (ghl∗ )(t, ξ), we rephrase this as Dx W , g(t, Dx )−1 Dx Dt W ∈ H s,0;λ . The 2N -vector Dx U (t, x) V (t, x) = g(t, Dx )−1 Dx Dt U (t, x) is a solution to the Cauchy problem ⎧ (00) ⎪ A 0 Dx F ⎪ ⎪ Dt V = V + , ⎨ Dt (g −1 Dx F ) A(10) A(11) ⎪ Dx U U0 (x) ⎪ V (0, x) = V (x) = ⎪ , ⎩ 0 Dx 1−β∗ (A(0, x, Dx )U U0 (x) + F (0, x))

(5.6)

where A(00) (t, x, ξ) = ξ ◦ A(t, x, ξ)ξ−1 ∈ S˜1,1;λ , A(10) (t, x, ξ) = Dt (g(t, ξ)−1 ξ ◦ A(t, x, ξ)) g(t, ξ)ξ−1 ∈ S −1,1;λ , A(11) (t, x, ξ) = g(t, ξ)−1 ξ ◦ A(t, x, ξ)g(t, ξ)ξ−1 ∈ S˜1,1;λ . By direct computation, we ﬁnd (00) 1 A σ (A) 0 0 σ1 = , 0 σ 1 (A) A(10) A(11) (00) 0,1 0 0 σ ˜ (A) A σ ˜ 0,1 = . 0 σ ˜ 0,1 (A) A(10) A(11) Moreover, V0 ∈ H s (Rn ) and Dx F , Dt (g −1 Dx F ) ∈ H s,0;λ ((0, T )×Rn) assuming U0 ∈ H s+1 (Rn ) and F ∈ H s+1,0;λ ((0, T )×Rn ). This brings us in a position to apply the supposed (s, 0)-well-posedness (but for the 2N × 2N system (5.6)), completing the proof this way. 5.3. Symmetrizable-hyperbolic systems Now we are able to prove Theorems 2.5 and 2.7. We bring system (2.4) into a form that allows to apply Propositions 5.3 and 5.4. We proceed as follows: • First, we symmetrize the principal part of A by constructing a suitable symmetrizer M0 .

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Michael Dreher and Ingo Witt

• Secondly, we diagonalize (if possible) the secondary part σ ˜ 0,1 (A) with the help of some matrix M1 . • Thirdly, we shift the spectrum of $i˜ σ 0,1 (A) by utilizing the shift operator Θ from (4.6). Proof of Theorem 2.5. By assumption, there is a matrix M ∈ S 0,0;λ satisfying | det M (t, x, ξ)| ≥ c > 0 for |ξ| ≥ C > 0 and χ(|ξ|/2C)%(M AM −1 ) ∈ S 0,1;λ . By virtue of Lemma A.3, we can assume that the operators M (t, x, Dx ), t ∈ [0, T ], are invertible on L2 (Rn ). We set U (1) (t, x) = M (t, x, Dx )U (t, x), and obtain the system Dt U (1) = M AM −1 + (Dt M )M −1 U (1) + M F = A(1) U (1) + F (1) , (1)

U (1) (0, x) = M (0, x, Dx )U U0 (x) = U0 (x).

(5.7)

The operator A(1) has Hermitian principal part $ A(1) ∈ S 1,1;λ and lower-order part i % A(1) ∈ S 0,1;λ . However, we cannot hope to symmetrize i % A(1) because of lack of information on the structure of A. But there is surely a constant δ0 ∈ R such that $ iA(1) (t, x, ξ) ≤ δ0 h(t, ξ)l∗ 1N ,

(t, x, ξ) ∈ [0, T ] × R2n ,

|ξ| ≥ C.

Therefore, setting Θ(t, ξ) = h(t, ξ)δ0 l∗ ∈ S 0,δ0 l∗ ;λ , U (2) (t, x) = Θ(t, Dx )U (1) (t, x), we arrive at the system ⎧ (2) (1) −1 −1 ⎪ D U (2) + ΘF (1) U = ΘA Θ + (D Θ)Θ ⎪ t t ⎪ ⎨ = A(2) U (2) + F (2) , ⎪ ⎪ ⎪ (2) ⎩ U (2) (0, x) = Θ(0, D )M (0, x, D )U x x U0 (x) = U0 (x).

(5.8)

Since Θ is scalar, we have ΘA(1) Θ−1 = A(1) mod S 0,−l∗ ;λ ⊂ S 0,0;λ . Clearly, the symbol Θ satisﬁes (Dt Θ)Θ−1 = δ0 l∗

Dt h , h

$ i(Dt Θ)Θ−1 = δ0 l∗

ht ≤ −δ0 hl∗ h

mod S −∞,1;λ .

Therefore, the term A(2) satisﬁes the conditions of Proposition 5.3. It follows that (2) U 0,0;λ U (2) 2 n + F (2) 0,0;λ ≤ C n n 0 H ((0,T )×R ) L (R ) H ((0,T )×R ) or, equivalently,

U H 0,δ0 ;λ ((0,T )×Rn ) ≤ C U U0 H β∗ δ0 l∗ (Rn ) + F H 0,δ0 ;λ ((0,T )×Rn ) .

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485

Well-posedness in the spaces H s,δ;λ for s ∈ N0 can be shown in a similar way. Exemplary, we demonstrate this in the case s = 1. As in the proof of Proposition 5.4, we introduce Dx U (1) (t, x) (1) V (t, x) = , g(t, Dx )−1 Dx Dt U (1) (t, x) which is a solution to

(1,00) A 0 Dx F (1) (1) , V = + Dt V Dt (g −1 Dx F (1) ) A(1,10) A(1,11) (1,00) 0 ∈ S 0,1;λ and where $ i A (1,10) (1,11) A A (1,00) A 0 $i (t, x, ξ) ≤ δ1 h(t, ξ)l∗ 12N , A(1,10) A(1,11) (1)

for (t, x, ξ) ∈ [0, T ] × Rn × (Rn \ 0) and some δ1 ∈ R. We set Θ(t, ξ) = h(t, ξ)δ1 l∗ and proceed as above to obtain U H 1,δ1 ;λ ((0,T )×Rn ) ≤ C U U0 H 1+β∗ δ1 l∗ (Rn ) + F H 1,δ1 ;λ ((0,T )×Rn ) , completing the proof in the case s = 1. The parameter functions δ ∈ Cb∞ (Rn ; R) turn out to be constants. The following reﬁned a priori estimate will be useful in the proof of the local uniqueness. Corollary 5.5. Let A and M be as in Theorem 2.5, and δ ∈ Cb∞ (Rn ; R) be a function with $ iA(1) (t, x, ξ) ≤ δ(x)h(t, ξ)l∗ + C(g(t, ξ)−1 h(t, ξ)2 + 1) 1N , for all (t, x, ξ) ∈ [0, T ] × R2n , |ξ| ≥ C, where A(1) = M AM −1 + (Dt M )M −1 . Then the fundamental solution X(t, t ) of the system Dt − A satisﬁes the a priori estimate U0 (·) 2 n ≤ C h(t , Dx )δ(x)l∗ U0 (·) 2 n , h(t, Dx )δ(x)l∗ X(t, t )U L (R )

∞

L (R )

for all U0 ∈ H (R ), 0 ≤ t ≤ t ≤ T , and some constant C = C(T ) > 0. n

U0 (x). Then, by deﬁnition, U is the solution to Proof. Put U (t, x) = X(t, t )U Dt U (t, x) = A(t, x, Dx )U (t, x), (t, x) ∈ (t , T ) × Rn , U (t , x) = U0 (x). Setting U (2) (t, x) = Θ(t, x, Dx )M (t, x, Dx )U (t, x) with Θ(t, x, ξ) = h(t, ξ)δ(x)l∗ we get, as in the proof of Theorem 2.5, Dt U (2) (t, x) = A(2) (t, x, Dx )U (2) (t, x), (t, x) ∈ (t , T ) × Rn , U0 (x), U 2 (t , x) = Θ(t , x, Dx )M (t , x, Dx )U

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Michael Dreher and Ingo Witt

with A(2) satisfying the conditions of Lemma 5.2. Then it suﬃces to exploit (5.4) of Lemma 5.2. Proof of Theorem 2.7. By assumption, there is a matrix M0 ∈ C ∞ ([0, T ], S (0)) such that M0 A0 M0−1 is Hermitian. Choose an arbitrary M1 ∈ S (0) . According to Proposition 4.5, there is an invertible operator M (t, x, Dx ) ∈ Op S˜0,0;λ such that M −1 ∈ Op S˜0,0;λ and σ 0 (M ) = M0 ,

σ ˜ −1,0 (M ) = −il∗ |ξ|−1 M1 .

The inverse operator M −1 has principal symbols σ 0 (M −1 ) = M0−1 ,

σ ˜ −1,0 (M −1 ) = il∗ |ξ|−1 M0 (0, x, ξ)−1 M1 (x, ξ)M M0 (0, x, ξ)−1 ,

see Proposition 4.4. Similarly as in the proof of Theorem 2.5, we set U (1) = M U , leading to the Cauchy problem (5.7). We compute the principal symbols of A(1) ∈ S˜1,1;λ : M0 A0 M0−1 )(t, x, ξ), σ 1 (A(1) ) = σ 0 (M )σ 1 (A)σ 0 (M −1 ) = λ(t)|ξ|(M which is Hermitian, by choice of M0 . Due to Lemma 4.6, (Dt M )M −1 ∈ S −1,1;λ , so we can regard this term as remainder. The secondary symbol of A(1) is, according to Proposition 4.4, σ ˜ 0,1 (A(1) ) = σ ˜ 0,0 (M )˜ σ 1,1 (A)˜ σ −1,0 (M −1 ) + σ ˜ 0,0 (M )˜ σ 0,1 (A)˜ σ 0,0 (M −1 ) σ 1,1 (A)˜ σ 0,0 (M −1 ) +σ ˜ −1,0 (M )˜ $ % = −il∗ M0 A1 M0−1 + M1 M0−1 , M0 A0 M0−1 (0, x, ξ). By assumption, the function δ ∈ Cb∞ (Rn ; R) satisﬁes σ 0,1 (A(1) )(x, ξ) ≤ δ(x)l∗ 1N , $ i˜

(x, ξ) ∈ Rn × (Rn \ 0).

As in the proof of Theorem 2.5, we set Θ(t, x, ξ) = h(t, ξ)δ(x)l∗ , and choose U (2) (t, x) = Θ(t, x, Dx )U (1) (t, x), resulting in the Cauchy problem (5.8). We want to apply Proposition 5.4 to this system. Therefore, we compute the principal symbols of A(2) . By Proposition 4.8, Θ ◦ A(1) ◦ Θ−1 = A(1)

0,−(l∗ +1);λ

mod S(2)

(Dt Θ) ◦ Θ−1 = (Dt Θ)Θ−1 (Dt Θ)Θ−1 = i−1 δ(x)

ht l∗ h

⊂ S 0,0;λ ,

−1,−l∗ ;λ mod S(1) ⊂ S 0,0;λ , −∞,1;λ mod S(0) ,

0,1;λ according to the rules of Proposition 4.8. since (Dt Θ)Θ−1 ∈ S(0)

Hence, we conclude that σ 1 (A(2) ) = σ 1 (A(2) )∗ and $ i˜ σ 0,1 (A(2) ) ≥ 0. Then Proposition 5.4 provides us with the (s, 0)-well-posedness of (5.8), which, in turn, implies the (s, δ(x))-well-posedness of (2.4) for any s ∈ N0 . This completes the proof.

Sharp Energy Estimates

487

5.4. Higher-order scalar equations Proof of Theorem 2.8. We transform problem (2.5) to an m×m system of the ﬁrst order. Then it is equivalent to the Cauchy problem Dt U (t, x) = A(t, x, Dx )U (t, x) + F (t, x), (t, x) ∈ (0, T ) × Rn , U (0, x) = U0 (x), where

⎛

⎞ g m−1 u ⎜g m−2 Dt u⎟ ⎜ ⎟ ⎜ ⎟ .. U =⎜ ⎟ ∈ H s,δ(x)+m−1;λ . ⎜ ⎟ ⎝ gDtm−2 u ⎠ Dtm−1 u

(this holding if and only if u ∈ H s+m−1,δ(x);λ ), ⎞ Dx β∗ (m−1) u0 ⎜Dx β∗ (m−2) u1 ⎟ ⎟ ⎜ ⎟ ⎜ .. U0 = ⎜ ⎟ ∈ H s+β∗ (δ(x)+m−1)l∗ , . ⎟ ⎜ ⎝ Dx β∗ um−2 ⎠ um−1 ⎛

⎛ ⎜ ⎜ ⎜ F =⎜ ⎜ ⎝

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟ ∈ H s,δ(x)+m−1;λ , ⎟ 0 ⎠ f (t, x)

and ⎛

(m − 1) Dgt g ⎜ 0 ⎜ ⎜ 0 ⎜ A(t, x, ξ) = ⎜ .. ⎜ ⎜ . ⎜ ⎝ 0 a0 − gm−1 where aj (t, x, ξ) =

g (m − 2) Dgt g 0 .. . 0 a1 − gm−2

*

... 0 ... 0 ... 0 .. .. . . Dt g ... g . . . − am−2 g

+

|α|≤m−j ˜1,1;λ 1

We have A ∈ S

0 g (m − 3) Dgt g .. . 0 a2 − gm−3

ajα (t, x) t(j+(l∗ +1)|α|−m) ξ α .

, σ (A)(t, x, ξ) = λ(t)|ξ|A0 (t, x, ξ), where ⎛

0 0 0 .. .

⎜ ⎜ ⎜ ⎜ A0 (t, x, ξ) = ⎜ ⎜ ⎜ ⎝ 0 −p0

1 0 0 .. .

0 1 0 .. .

0 −p1

0 −p2

... ... ... .. .

0 0 0 .. .

... 0 . . . −pm−2

0 0 0 .. . 1 −pm−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ , . ⎟ ⎟ g ⎠ −am−1

488

Michael Dreher and Ingo Witt *

ajα (t, x)(ξ/|ξ|)α , and σ ˜ 0,1 (A)(x, ξ) = −il∗ A1 (x, ξ), where ⎞ ⎛ m−1 0 0 ... 0 0 ⎜ 0 m−2 0 ... 0 0⎟ ⎟ ⎜ ⎜ 0 0 m− 3 ... 0 0⎟ ⎟ ⎜ A1 (x, ξ) = ⎜ . .. .. .. .. ⎟ , .. ⎜ .. . . . . .⎟ ⎟ ⎜ ⎝ 0 0 0 ... 1 0⎠ −q1 −q2 . . . −qm−2 0 −q0 * qj (x, ξ) = il∗−1 |α|=m−j−1 ajα (0, x)(ξ/|ξ|)α . Now, it is easy to provide a symmetrizer M0 for A0 , namely with µk from (1.3), (1.4) we choose ⎞ ⎛ 1 1 ... 1 ⎜ µ1 µ2 ... µm ⎟ ⎟ ⎜ M0 (t, x, ξ)−1 = ⎜ . . .. ⎟ . .. .. ⎝ .. . . ⎠

pj (t, x, ξ) =

|α|=m−j

Note that det M0−1 =

µm−1 1

:

h>h (µh

(M M0 (t, x, ξ))hj =

µm−1 2

. . . µm−1 m

− µh ) and, for 1 ≤ h, j ≤ m,

+ pm−1 µm−j−1 + · · · + pj+1 µh + pj µm−j h h ∂p ∂τ (µh )

.

(5.9)

According to our general scheme, to read oﬀ the loss of regularity we have to calculate M0 A1 M0−1 hh = (M M0 )hj (A1 )jk (M M0−1 )kh j, k

=

m−1

(m − j) (M M0 )hj (M M0−1 )jh −

j=1

=m−

m−1

qj −1 (M M0 )hm (M M0−1 )jh

j=1 m

j (M M0 )hj (M M0−1 )jh −

j=1

m−1

qj −1 (M M0 )hm (M M0−1 )jh .

j=1

By virtue of (5.9), m

j(M M0 )hj (M M0−1 )jh

j=1

=

=

1

m m−j−1 µj−1 j µm−j + p µ + · · · + p µ + p m−1 j+1 h j h h h

∂p ∂τ (µh ) j=1

*m j+1 j−1 j=1 2 pj µh ∂p ∂τ (µh )

=

∂p ∂τ

τ ∂2p 2 ∂τ 2 ∂p ∂τ

+

(0, x, µh , ξ)

Sharp Energy Estimates and

m−1

489

*m−1 qj −1 (M M0 )hm (M M0−1 )jh

=

j=1

j−1 j=1 qj −1 µh ∂p ∂τ (µh )

=

q(x, µh , ξ) . ∂p ∂τ (0, x, µh , ξ)

Hence, the assertion follows.

5.5. Local uniqueness Proof of Theorem 2.9. We follow an approach of Kumano-go [41]. Choose a cut-oﬀ function ϕ ∈ Cc∞ (Rn ; R) with supp ϕ Ω0 , ϕ ≡ 1 in a neighbourhood Ω1 of 0 ∈ Rn . Set v(t, x) = ϕ(x)u(t, x) ∈ H m−1,δ(x);λ ((0, T ) × Rn ) (shrink T if necessary). Then v solves the Cauchy problem P v(t, x) = [P, ϕ] u(t, x) =: f (t, x), (t, x) ∈ (0, T ) × Rn , Dtj v(0, x) = 0,

0 ≤ j ≤ m − 1.

We compare v with the solution vε to P vε (t, x) = f (t, x), Dtj vε (ε, x) = 0,

(t, x) ∈ (ε, T ) × Rn , 0 ≤ j ≤ m − 1,

for 0 < ε < T . Observe that v ≡ u in (0, T )×Ω1 and f ≡ 0 in (0, T )×Ω1 . Since the Cauchy problem for vε is strictly hyperbolic and, therefore, has ﬁnite propagation speed, there is a neighbourhood Ω2 Ω1 of 0 such that vε ≡ 0 in (ε, T ) × Ω2 for all 0 < ε < T (shrink T again if necessary). It suﬃces to show that lim v(t, ·) − vε (t, ·)L2 (Rn ) = 0,

ε→+0

(5.10)

for 0 < t < T a.e., because this implies limε→+0 v(t, ·) − vε (t, ·)L2 (Ω2 ) = 0. Writing P in the form (1.2) with aj ∈ S j,j;λ , we ﬁnd [P, ϕ] =

m

[aj , ϕ] Dtm−j ,

j=1

where [aj , ϕ] ∈ S j−1,j−l∗ −1;λ , since ϕ ∈ S 0,0;λ . According to Proposition 4.15, [aj , ϕ] ∈ L(H j−1,δ+m−j−1;λ , H 0,δ+m−1;λ ). From Proposition 3.10, we get Dtm−j ∈ L(H m−1,δ−1;λ ((0, T ) × Rn ), H j−1,δ+m−j−1;λ ((0, T ) × Rn )). Thus, [P, ϕ] ∈ L(H m−1,δ−1;λ ((0, T ) × Rn ), H 0,δ+m−1;λ ((0, T ) × Rn )).

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Michael Dreher and Ingo Witt

We now introduce the vectors ⎛ m−1 ⎞ v g ⎜g m−2 Dt v ⎟ ⎜ ⎟ ⎜ ⎟ .. V =⎜ ⎟, . ⎜ ⎟ ⎝ gDtm−2 v ⎠ Dtm−1 v

⎞ g m−1 vε ⎜g m−2 Dt vε ⎟ ⎟ ⎜ ⎟ ⎜ .. Vε = ⎜ ⎟. . ⎟ ⎜ ⎝ gDtm−2 vε ⎠ Dtm−1 vε ⎛

These vectors solve the Cauchy problems Dt V (t, x) = A(t, x, Dx )V (t, x) + F (t, x),

(t, x) ∈ (0, T ) × Rn ,

V (0, x) = 0, Dt Vε (t, x) = A(t, x, Dx )V Vε (t, x) + F (t, x),

(t, x) ∈ (ε, T ) × Rn ,

Vε (ε, x) = 0. See the proof of Theorem 2.8 for the deﬁnition of A and F . According to Corollary 5.5 and the proof of Theorem 2.8, the fundamental solution X(t, t ) to the ﬁrst-order system Dt − A(t, x, Dx ) satisﬁes the estimate h(t, Dx )(δ+m−1)l∗ X(t, t )U U0 (·)L2 (Rn ) ≤ C h(t , Dx )(δ+m−1)l∗ U0 (·)L2 (Rn ) . Obviously,

t

V (t, x) = i

X(t, t )F (t , x) dt ,

0 < t < T,

X(t, t )F (t , x) dt ,

ε < t < T,

0

Vε (t, x) = i

t

ε

and, therefore,

V (t, x) − Vε (t, x) = i

ε

X(t, t )F (t , x) dt .

0

We have the following estimates: h(t, Dx )(δ+m−1)l∗ (V (t, ·) − Vε (t, ·)) 2 n L (R ) ε h(t , Dx )(δ+m−1)l∗ F (t , ·) 2 n dt , ≤C L (R ) 0

and h(t, Dx )(δ+m−1)l∗ (V (t, ·) − Vε (t, ·))2 2 n L (R ) ε h(t , Dx )(δ+m−1)l∗ F (t , ·)2 2 n dt , ≤ Cε L (R ) 0

2

≤ Cε F H 0,δ+m−1;λ ((0,T )×Rn ) 2

≤ Cε uH m−1,δ−1;λ (Ω) . This implies (5.10) ﬁnishing the proof.

Sharp Energy Estimates

491

5.6. Sharpness of energy estimates We ﬁnally come to the proof of Theorem 2.10. For technical reasons, it is quite long; however, the main ideas are borrowed from the proof of a standard result on the instability of ODE systems. For the reader’s convenience, we recall that result from stability theory ﬁrst, and present the proof of Theorem 2.10 then. Compare also [73]. 5.6.1. Digression to stability theory. Let A ∈ MN ×N (C) be a constant matrix, G : CN → CN be a smooth mapping with G(W ) ≤ C W 2 in a neighbourhood of 0 ∈ CN , and consider the ODE system Dt W (t) = AW (t) + G(W (t)), 0 ≤ t < ∞, (5.11) W (0) = W0 , where W ∈ C 1 ([0, ∞); C) is an unknown N -vector. Deﬁnition 5.6. We say that the zero solution to (5.11) is stable if for every δ > 0 there is a γ > 0 such that W W0 < γ implies global existence of the solution W to (5.11), and W (t) < δ for all 0 ≤ t < ∞. The zero solution is called asymptotically stable if it is stable and there is a γ∗ > 0 such that W W0 < γ∗ implies limt→∞ W (t) = 0. Denote the eigenvalues of A by ν1 ,. . . ,νN . It is well-known that the imaginary parts %ννj determine the stability behaviour: Proposition 5.7. (a) If the values $(iννj ), j = 1, . . . , N , are all negative, then the zero solution to (5.11) is asymptotically stable. (b) If one value $(iννj ) is positive, then the zero solution to (5.11) is not stable. Sketch of proof of (b). For simplicity, assume that A is symmetrizable, and no eigenvalue νj has vanishing imaginary part. Then there is a symmetrizer M with M AM −1 = diag(ν1 , . . . , νN ). Replacing W with M W , we can suppose that A is already diagonalized. Be reordering we may additionally assume that ε ≤ $(iνk ), $(iνk ) ≤ −ε,

k = 1, . . . , d,

k = d + 1, . . . , N,

for some ε > 0 and some 1 ≤ d ≤ N . We divide A into blocks, (00) A 0 A= , 0 A(11) where A(00) , A(11) are d × d, (N − d) × (N − d) matrices, respectively, with $(iA(00) ) ≥ ε1d , $(iA(11) ) ≤ −ε1N −d .

492

Michael Dreher and Ingo Witt

We choose a special initial vector W0 , W0 = γ0 (1, . . . , 1, 0, . . . , 0 )T , 4 566 7 4 566 7 d times

γ0 > 0 small,

N − d times

and deﬁne W = W (t) as the (at least local) solution to (5.11). Our goal is to show that W (t) grows up to a certain value, independent of W W0 . To this end, we deﬁne the Lyapunov functional S(t) =

d

2

|W Wk (t)| −

k=1

N

2 2 2 |W Wk (t)| = W (0) (t) − W (1) (t) .

k=d+1 2

This functional is always bounded by the energy W (t) , |S(t)| ≤ W (t)2 ; and in our case S(t) > 0 for small t, since S(0) > 0. We deduce that ∂t S(t) = 2$ ∂t W (0) (t), W (0) (t) − 2$ ∂t W (1) (t), W (1) (t) = 2$ iA(00) W (0) (t), W (0) (t) + 2$ −iA(11) W (1) (t), W (1) (t) ˜ (t)), W (t) + 2$ G(W 2

≥ 2ε W (t) − C0 W (t)

3

≥ (2ε − C0 W (t)) S(t), assuming 2ε − C0 W (t) > 0 and S(t) > 0, which is true for small γ0 and small t. It follows that S(t) keeps growing until W (t) reaches the value C2ε0 . Therefore, the zero solution is not stable. Now, we compare two evolution equations: Case 1: Dt W = AW + G(W ) as in (5.11), Case 2: Dt U = AU as in Theorem 2.10, and list their similarities: • In both cases, the operators on the right-hand side have a diagonalizable principal part Apr : Apr = A in Case 1 and ˜ 0,1 (A) for some large C > 0, Apr = Op χ(Λ(t)ξ/C) σ 1 (A) + t−1 σ in Case 2. • Both operators on the right-hand side contain a perturbation term Apert , which does not respect the eigenspaces of the diagonalized Apr , but turns out to be negligible: Apert (W ) = G(W ) in Case 1 and Apert ∈ S −1,1;λ + S 0,0;λ in Case 2.

Sharp Energy Estimates

493

• At least after some transformations, the spectrum of $(iApr ) contains a positive part. In the second case, these transformations are the following: – diagonalize Apr , – possibly shift δ, see Lemma 5.8, – cut-oﬀ a subset of a conic neighbourhood of (x0 , ξ0 ), – shift the spectrum using an operator Θ, see (5.24), – restrict the time interval from [0, T ] to some subset [tj , T ]. • In order to handle a (possibly empty) negative part of the spectrum of $(iApr ), we introduce a Lyapunov functional as diﬀerence of L2 norms of components of the solution vector. • The proofs of instability and ill-posedness, respectively, are based on an a priori estimate from below for the Lyapunov functional for suitably chosen initial values. But there are also some diﬀerences: • In Case 2, we have to bring several additional terms under control, which arise from cut-oﬀ operators and commutators. • We consider a whole family Uj = Uj (t, x) of approximate solutions to Dt U = AU . They are supported in a neighbourhood of the line [0, T ]×{x0 }, and their ˆj (t, ξ) are concentrated near [0, T ] × {2j ξ0 }, |ξ0 | = 1. We Fourier transforms U will compare the values of the Lyapunov functional to Uj evaluated at t = tt and t = Tj , obtaining (for large j) a contradiction to the a priori estimate that follows from the assumed (0, δ(x))-well-posedness of the Cauchy problem for the operator Dt − A. Here, tj is deﬁned by Λ(tj )ξξj = N1 for large N1 , √ and Tj = tj . 5.6.2. Proof of Theorem 2.10. Before proving Theorem 2.10, we state a technical result whose proof is postponed to the appendices: Lemma 5.8. Let A ∈ Op S 1,1;λ , δ ∈ Cb∞ (Rn ; R). Assume the Cauchy problem for the operator Dt − A be (0, δ(x))-well-posed. Then, for any ε > 0, the Cauchy problem for this operator is also (0, δ(x) + ε)-well-posed. As announced in Section 5.6.1, we diagonalize A modulo a remainder Apert with symbol from S −1,1;λ + S 0,0;λ : Lemma 5.9. Under the assumptions of Theorem 2.10, there is an invertible operator M ∈ Op S˜0,0;λ with M −1 ∈ Op S˜0,0;λ such that the non-diagonal part of M AM −1 belongs to Op(S −1,1;λ + S 0,0;λ ). Moreover, there are functions νj ∈ S˜1,1;λ for 1 ≤ j ≤ N which coincide with the eigenvalues of σ 1 (A)(t, x, ξ) + t−1 σ ˜ 0,1 (A)(x, ξ) for large values of Λ(t)ξ. Proof. Fix M0 , A0 as in the proof of Theorem 2.7, where the matrix M0 A0 M0−1 is real diagonal. According to the computations there, it suﬃces to ﬁnd an M1 ∈ S (0) with the property that the matrix $ % M0 A1 M0−1 + M1 M0−1 , M0 A0 M0−1

494

Michael Dreher and Ingo Witt

is diagonal. But the existence of such an M1 follows from Lemma A.1. Then ˜ 0,1 (M AM −1 ) are diagonal, and the ﬁrst claim is proved. σ 1 (M AM −1 ) and σ Concerning the second claim, we have to investigate the eigenvalues of λ(t)|ξ|Aε (t, x, ξ) = λ(t)|ξ| (A0 (t, x, ξ) − iεA1 (x, ξ)) for small values of ε = l∗ (tλ(t)|ξ|)−1 , where A0 ∈ C ∞ ([0, T ], S (0) ) and A1 ∈ S (0) . The eigenvalues of A0 (t, x, ξ) are µj (t, x, ξ)|ξ|−1 , hence distinct in the sense of (2.17). Lemma A.2 gives us the desired symbol estimates of the eigenvalues µj,ε of λ(t)|ξ|Aε (t, x, ξ) for large Λ(t)ξ. Then νj (t, x, ξ) = χ(Λ(t)ξ/C)µj,ε (t, x, ξ)

are as desired.

Proof of Theorem 2.10. Part (a) follows from Theorem 2.7 and Lemma 5.9. To prove part (b), we may suppose that the complete symbol A(t, x, ξ) is diagonalized modulo S −1,1;λ + S 0,0;λ . Without loss of regularity, we may suppose that |ξ0 | = 1. Denote by Uγ for γ > 0 a truncated conic neighbourhood of (x0 , ξ0 ) ∈ Rn × (Rn \ 0), &

ξ ξ0 n n < γ, |ξ| ≥ 1 . Uγ := (x, ξ) ∈ R × (R \ 0) : |x − x0 | < γ, − |ξ| |ξ0 | By renumbering, if necessary, we can achieve that $(i˜ σ 0,1 (ν1 ))(x0 , ξ0 ) ≥ $(i˜ σ 0,1 (ν2 ))(x0 , ξ0 ) ≥ . . . ≥ $(i˜ σ 0,1 (νN ))(x0 , ξ0 ). Lemma 5.8 allows us to assume that σ 0,1 (νk ))(x, ξ) ≤ (δ(x) + 2ε)l∗ , (δ(x) + ε)l∗ ≤ $(i˜ $(i˜ σ

0,1

(νk ))(x, ξ) ≤ (δ(x) − ε)l∗ ,

k = 1, . . . , d, k = d + 1, . . . , N,

(5.12)

for (x, ξ) ∈ Uγ , certain small positive ε and γ, and some 1 ≤ d ≤ N . Choose a cut-oﬀ function v0 ∈ Cc∞ (Rn ; R) supported in {|x − x0 | < γ/4}, and put v0,j (x) = v0 (x) exp ix · 2j ξ0 . Then the Fourier transform vˆ0,j is concentrated near 2j ξ0 . Lemma A.6 gives us the crucial estimate (k) (5.13) C −1 v0,j H s(x) (Rn ) ≤ ϕj (x, Dx )v0,j s(x) n ≤ C v0,j H s(x) (Rn ) , H

for j large, k = 1, 2, s ∈ follows. Fix

(2) ϕj

Cb∞ (Rn ; R)

(R )

(1)

(2)

and cut-oﬀ operators ϕj , ϕj

deﬁned as

∈ S with 0

(2)

⊂ Uγ ∩ {(x, ξ) : 2j−1 < |ξ| < 2j+1 },

& 3 j+1 5 j−1 (2) < |ξ| < · 2 ϕj (x, ξ) = 1 on Uγ/2 ∩ (x, ξ) : · 2 . 4 4 supp ϕj

(1)

(2)

Deﬁne ϕj similarly, with ϕj subset of S 0,0;λ .

(1)

≡ 1 on supp ϕj . These symbols form a bounded

Sharp Energy Estimates

495

With d from (5.12), we introduce a vector V0,j (x) = v0,j (x)(1, . . . , 1, 0, . . . , 0 )T . 4 566 7 4 566 7 d times

(5.14)

N − d times

We need an auxiliary vector function Vj , which is deﬁned as the solution to (2) Dt Vj (t, x) = A(t, x, Dx )ϕj (x, Dx )V Vj (t, x), Vj (tj , x) = V0,j (x), where tj is given by the relation Λ(tj )2j = N1 , and N1 will be chosen later. To estimate Vj in terms of V0,j , we note that (5.12) implies $ iAϕj (t, x, ξ) ≤ ((δ(x) + 2ε)h(t, ξ)l∗ + Cg(t, ξ)−1 h(t, ξ)2 + C)1N (2)

for (t, x, ξ) ∈ [0, T ] × Rn × (Rn \ 0). Then the proof of Theorem 2.5 gives us ≤ C V Vj (0, ·)H β∗ (δ(x)+2ε)l∗ (Rn ) , 0 ≤ t ≤ T. h(t, Dx )(δ(x)+2ε)l∗ Vj (t, ·) L2 (Rn )

(5.15) belongs to the To relate Vj (0, ·) with V0,j , we note that the symbol A ◦ 0 Hormander ¨ class S1,0 , for 0 ≤ t ≤ tj , with symbol seminorms O(2β∗ j ). By choice of tj , it holds 2β∗j tj ≤ C. Then it can be concluded that (2) ϕj

V0,j H s(x) (Rn ) ≤ V Vj (t, ·)H s(x) (Rn ) ≤ C V V0,j H s(x) (Rn ) C −1 V

(5.16)

Cb∞ (Rn ; R).

for 0 ≤ t ≤ tj , and all s ∈ We will need a reﬁnement of these estimates. Denote by {ψj }j≥0 the standard Littlewood-Paley decomposition (see the proof of Lemma 5.8 in the Appendix for a precise deﬁnition), and set ζj (ξ) = ψj−1 (ξ) + ψj (ξ) + ψj+1 (ξ), (2)

which is identical 1 on the support of ϕj . Then (2) (2) Dt (ζζj Vj ) = Aϕj (ζζj Vj ) + ζj , Aϕj Vj , where the commutator on the right belongs to Op S −∞ . By the same reasoning as before, Vj (t, ·) (5.17) h(t, Dx )(δ(x)+2ε)l∗ ζj (Dx )V L2 (Rn )

≤ C ζζj V0,j H β∗ (δ(x)+2ε)l∗ (Rn ) + Ck 2−jk V V0,j H −k (Rn ) , for any k ∈ R, reﬁning (5.15). Now we are ready to deﬁne Uj , (1)

Vj (t, x), Uj (t, x) = ϕj (x, Dx )V which is a solution to

(2) (1) (1) (2) Dt Uj = AU Uj + A(ϕj − 1)ϕj Vj + ϕj , Aϕj Vj = AU Uj + Fj .

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Michael Dreher and Ingo Witt

Now we bring the assumption of (0, δ(x))-well-posedness into play. According to Deﬁnition 2.3, the a priori estimate 2 Tj , Dx )δ(x)l∗ Uj (T Tj , ·) (5.18) h(T L2 (Rn ) ≤ C U Uj (0, ·)2H β∗ δ(x)l∗ (Rn ) + Tj F Fj 2H 0,δ(x);λ ((0,TTj )×Rn ) holds for 0 ≤ Tj ≤ T . By the estimates (5.13) and (5.16), Vj (0, ·)H β∗ δ(x)l∗ (Rn ) U Uj (0, ·)H β∗ δ(x)l∗ (Rn ) ≤ C V

(5.19)

≤ C V V0,j H β∗ δ(x)l∗ (Rn ) ≤ C U Uj (tj , ·)H β∗ δ(x)l∗ (Rn ) . Choosing the number k suﬃciently large, we can conclude that V V0,j H −k (Rn ) ≤ C V V0,j H β∗ δ(x)l∗ (Rn ) ≤ C U Uj (tj , ·)H β∗ δ(x)l∗ (Rn ) ≤ C W Wj (tj , ·)L2 (Rn ) , (5.20) where we have introduced Wj = ΘU Uj and Θ(t, x, ξ) = h(t, ξ)δ(x)l∗ . Then we can rewrite (5.18) as 2 2 2 W Wj (T Tj , ·)L2 (Rn ) ≤ C0 W Wj (tj , ·)L2 (Rn ) + Tj ΘF Fj L2 ((0,TTj )×Rn ) . (5.21) To derive an estimate from below, we deﬁne the Lyapunov functional Sj (t) =

d

2

W Wj,k (t, ·)L2 (Rn ) −

k=1

N

2

W Wj,k (t, ·)L2 (Rn ) ,

k=d+1

where Wj,k is the kth component of the vector Wj . Observe that 2

Wj (t, ·)L2 (Rn ) , |S Sj (t)| ≤ W Sj (tj ) =

2 W Wj (tj , ·)L2 (Rn )

0 ≤ t ≤ T, ,

by choice of V0,j , see (5.14). The vector Wj solves Dt Wj = ΘAΘ−1 + (Dt Θ)Θ−1 Wj + ΘF Fj = AΘ Wj + ΘF Fj , (00) (01) AΘ AΘ AΘ = (10) (11) , AΘ AΘ

(5.22) (5.23)

(5.24)

(00) (11) where AΘ , AΘ ∈ S˜1,1;λ are d × d, (N − d) × (N − d) matrices, respectively, and (01) (10) AΘ , AΘ ∈ S −1,1;λ + S 0,0;λ . By (5.12), (00) $ iAΘ (t, x, ξ) ≥ εh(t, ξ)l∗ − C(g(t, ξ)−1 h(t, ξ)2 + 1) 1d , (11) $ iAΘ (t, x, ξ) ≤ −εh(t, ξ)l∗ + C(g(t, ξ)−1 h(t, ξ)2 + 1) 1N −d , (1)

for (t, x, ξ) ∈ [0, T ] × Uγ . Remember that ϕj ≡ 0 outside [0, T ] × Uγ . Then it follows that ∂t Sj ≥ 2$ (εhl∗ − Cg −1 h2 − C)W Fj 2L2 (Rn ) −C2−2jk V Wj , Wj −C ΘF V0,j 2H −k (Rn ) .

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497

If t ≥ tj and the constant N1 in the deﬁnition of tj is suﬃciently large, then εh(t, ξ)l∗ − Cg(t, ξ)−1 h(t, ξ)2 − C ≥

ε εl∗ h(t, ξ)l∗ = , 2 2t

shrinking T if necessary. As a consequence, εl∗ 2 Fj )(t, ·)L2 (Rn ) + 2−2jk Sj (tj ) , Sj (t) − C1 (ΘF ∂t Sj (t) ≥ t for tj ≤ t ≤ T , exploiting (5.20), (5.22), and (5.23). By Gronwall’s Lemma, εl∗ Tj Sj (T Tj ) ≥ Sj (tj )− tj Tj −εl∗ t 2 (ΘF Fj )(t, ·)L2 (Rn ) + 2−2jk Sj (tj ) dt . − C1 tj tj Combining this with (5.21) and (5.22), we ﬁnd, for large j, k, and small ε, 2 (5.25) Fj L2 ((0,TTj )×Rn ) C0 Sj (tj ) + Tj ΘF εl∗ Tj −εl∗ Tj t 1 ≥ Sj (tj ) − C1 (ΘF Fj )(t, ·)2L2 (Rn ) dt . tj 2 tj tj T

This will be a contradiction for large values of tjj provided that we get control on the term ΘF Fj . Recall that (1) (1) (2) Fj = A(ϕ(2) − 1)ϕj Vj + ϕj , Aϕj Vj (1) (1) (2) = A(ϕ(2) − 1)ϕj ζj Vj + ϕj , Aϕj ζj Vj , / . (2) (1) (2) (1) since ζj = ζj (ξ) ≡ 1 on supp ϕj supp ϕj . Clearly, (ϕj − 1)ϕj j ∈ N0 ⊂ / . (1) (2) S −∞ and [ϕj , Aϕj ] j ∈ N0 ⊂ S 0,−l∗ ;λ are bounded subsets. Therefore, we get 2 2 (ΘF Fj )(t, ·)L2 (Rn ) ≤ C h(t, Dx )−l∗ (Θζζj Vj )(t, ·)L2 (Rn ) 2 ≤ C h(t, Dx )−(1+2ε)l∗ h(t, Dx )(δ(x)+2ε)l∗ ζj (Dx )V Vj (t, ·) 2 n . L (R )

For 0 ≤ t ≤ tj , we have h(t, ξ)−(1+2ε)l∗ ζj (ξ) ∼ 2−β∗ j(1+2ε)l∗ , hence (ΘF Fj )(t, ·)L2 (Rn ) ≤ C2−β∗ j(1+2ε)l∗ ζζj V0,j H β∗ (δ(x)+2ε)l∗ (Rn ) + Ck 2−jk V V0,j H −k (Rn ) ≤ C2−β∗ jl∗ ζζj V0,j H β∗ δ(x)l∗ (Rn ) + Ck 2−jk W Wj (tj , ·)L2 (Rn ) ≤ C2−β∗ jl∗ W Wj (tj , ·)L2 (Rn ) ,

498

Michael Dreher and Ingo Witt

due to (5.17), (5.19), and (5.20). And in the case of tj ≤ t ≤ T , we know that h(t, ξ)−(1+2ε)l∗ ζj (ξ) ∼ t(1+2ε)l∗ , from which follows that (ΘF Fj )(t, ·)L2 (Rn ) ≤ Ct(1+2ε)l∗ ζζj V0,j H β∗ (δ(x)+2ε)l∗ (Rn ) + Ck 2−jk V V0,j H −k (Rn ) 2εl∗ ≤ Ctl∗ 2β∗ j t W Wj (tj , ·)L2 (Rn ) . Inserting these estimates into (5.25), we obtain 4εl∗ C0 Sj (tj ) 1 + CT Tj tj 2−2β∗ jl∗ + CT Tj2+2l∗ 2β∗ j Tj εl∗ 4εl∗ Tj 1 − CT Tj2+2l∗ 2β∗ j Tj ≥ Sj (tj ) . tj 2 √ We recall that tj = C2−β∗ j . Choosing Tj = tj , we ﬁnd that 4εl∗ = 0, lim Tj tj 2−2β∗ jl∗ + Tj2+2l∗ 2β∗ j Tj j→∞

for small ε > 0, which gives a contradiction as j → ∞.

Appendix A. Supplements Proof of Lemma 5.8. We have to show that 2 h(t, Dx )(δ(x)+ε)l∗ U (t, ·)

L2 (Rn )

2 2 ≤ C U U0 H β∗ (δ(x)+ε)l∗ (Rn ) + t2 F H 0,δ(x)+ε;λ ((0,t)×Rn )

for any solutions U = U (t, x) to Dt U = A(t, x, Dx )U + F (t, x), U (0, x) = U0 (x), where the constant C does not dependent on t, 0 ≤ t ≤ T . By density arguments, we can assume that U0 ∈ H ∞ (Rn ), F ∈ H ∞ ([0, T ) × Rn ). Then U ∈ H ∞ ((0, T ) × Rn ). We use the well-known Littlewood-Paley decomposition: ψj ∈ Cc∞ (Rn ; R), 0 ≤ ψj (ξ) ≤ 1,

j = 0, 1, 2, . . . , ξ ∈ Rn ,

j = 0, 1, 2, . . . ,

supp ψ0 ⊂ {ξ ∈ R | |ξ| < 2}, n

supp ψj ⊂ {ξ ∈ Rn | 2j−1 < |ξ| < 2j+1 }, ψj (ξ) = ψ1 (21−j ξ), ∞ j=0

ψj (ξ) = 1,

j = 1, 2, . . . ,

ξ ∈ Rn .

j = 1, 2, . . . ,

Sharp Energy Estimates

499 (j)

The set {ψj | j = 0, 1, . . . } ⊂ S 0,0;λ is bounded. Denote U (j) = ψj (Dx )U , U0 U0 , and F (j) = ψj (Dx )F . Then ψj (Dx )U ⎧ ∞ ⎪ ⎪ ⎨ D U (j) = AU (j) + [ψ , A] ψ U + F (j) , (t, x) ∈ (0, T ) × Rn , t

⎪ ⎪ ⎩

j

=

l

l=0 (j)

U (j) (0, x) = U0 (x).

. / From Proposition 4.2, we have that the set* [ψj , A] j = 0, 1, . . . ⊂ S 0,−l∗ ;λ is bounded. Moreover, the operator [ψj , A] |l−j|≥2 ψl is regularizing, where, for any k ∈ N0 , the set ( ' 2jk [ψj , A] ψl j = 0, 1, . . . ⊂ S 0,0;λ |l−j|≥2

is bounded. By virtue of the (0, δ(x))-well-posedness, 2 h(t, Dx )δ(x)l∗ U (j) (t, ·) 2 n L (R ) ⎛ (j) 2 ≤ C ⎝U0 β δ(x)l H

∗

∗ (Rn )

+ t2

(l) 2 U

H 0,δ(x);λ ((0,t)×Rn )

|l−j|≤1

2 2 +2−2jk t2 U H 0,δ(x);λ ((0,t)×Rn ) + t2 F (j)

H 0,δ(x);λ ((0,t)×Rn )

.

If Λ(t)2j ≤ 1, then h(t, ξ)εl∗ ∼ 2β∗ jεl∗ on supp hδl∗ ψj , hence 2 h(t, Dx )(δ(x)+ε)l∗ U (j) (t, ·) 2 n L (R ) ⎛ (j) 2 (l) 2 U ≤ C ⎝U0 + t2 β (δ(x)+ε)l n H

∗

∗ (R

)

|l−j|≤1

H 0,δ(x)+ε;λ ((0,t)×Rn )

2 +2−2jk+2β∗ jεl∗ t2 U 2H 0,δ(x);λ ((0,t)×Rn ) + t2 F (j)

H 0,δ(x)+ε;λ ((0,t)×Rn )

. (A.1)

If Λ(t)2j ≥ 1, then h(t, ξ)εl∗ ∼ t−εl∗ on supp hδl∗ ψj . Moreover, t−εl∗ ≤ Ch(t , ξ)εl∗ for 0 ≤ t ≤ t, ξ ∈ supp ψj . Thus, we have (A.1) also in this case. For the further treatment of all the terms of (A.1) (except the third on the right), we have to commute h(δ+ε)l∗ and ψj : h(δ+ε)l∗ ◦ ψj = ψj + h(δ+ε)l∗ , ψj ◦ (h(δ+ε)l∗ )−1 ◦ h(δ+ε)l∗ ,

500

Michael Dreher and Ingo Witt −1,−(l∗ +1);λ

and [h(δ+ε)l∗ , ψj ] ◦ (h(δ+ε)l∗ )−1 ∈ S(1)

0,−εl∗ ;λ with S(0) symbol seminorms

O(2−j/2 ), for small ε, whence 2 h(t, Dx )(δ(x)+ε)l∗ U (j) (t, ·) 2 n L (R ) 2 2 (δ(x)+ε)l∗ ≤ ψj h(t, Dx ) U (t, ·) 2 n + C2−j h(t, Dx )δ(x)l∗ U (t, ·) 2 n , L (R ) L (R ) 2 (j) U0 β (δ(x)+ε)l n ∗ (R ) H ∗ 2 2 ≤ ψj h(0, Dx )(δ(x)+ε)l∗ U0 + C2−j U U0 H β∗ δ(x)l∗ (Rn ) , L2 (Rn ) (l) 2 U 0,δ(x)+ε;λ ((0,t)×Rn ) H t 2 ≤ dt + C2−j U 2H 0,δ(x);λ ((0,t)×Rn ) . ψl h(t , Dx )(δ(x)+ε)l∗ U (t , ·) L2 (Rn )

0

2 The term F (j) H 0,δ(x)+ε;λ ((0,t)×Rn ) from (A.1) can be treated in a similar way as (l) 2 U 0,δ(x)+ε;λ . Inserting these inequalities into (A.1), we ﬁnd H ((0,t)×Rn ) 2 ψj h(t, Dx )(δ(x)+ε)l∗ U (t, ·) 2 n L (R ) 2 ≤ C ψj h(0, Dx )(δ(x)+ε)l∗ U0 2 n L (R ) 2 t + t2 dt ψl h(t , Dx )(δ(x)+ε)l∗ U (t , ·) |l−j|≤1

0

L2 (Rn )

t 2 ψj h(t , Dx )(δ(x)+ε)l∗ F (t , ·) 2 n dt L (R ) 0 2 2 + 2−j t2 U H 0,δ(x);λ ((0,t)×Rn ) + 2−j h(t, Dx )δ(x)l∗ U (t, ·) 2 n L (R ) 2 2 −j −j 2 +2 U U0 H β∗ δ(x)l∗ (Rn ) + 2 t F H 0,δ(x);λ ((0,t)×Rn ) . + t2

Summing over j and exploiting the embeddings from Proposition 4.16, we obtain the estimate 2 h(t, Dx )(δ(x)+ε)l∗ U (t, ·) 2 n L (R ) 2 2 ≤ C U U0 H β∗ (δ(x)+ε)l∗ (Rn ) + t2 F H 0,δ(x)+ε;λ ((0,t)×Rn ) t 2 + t2 h(t , Dx )(δ(x)+ε)l∗ U (t , ·) 2 n dt L (R ) 0 2 2 2 δ(x)l∗ +t U H 0,δ(x);λ ((0,t)×Rn ) + h(t, Dx ) U (t, ·) 2 n . L (R )

Sharp Energy Estimates

501

Finally, we apply Gronwall’s lemma and the assumed (0, δ(x))-well-posedness, and deduce the a priori estimate for the (0, δ(x) + ε)-well-posedness. From Taylor [70, Chap. IX, Lemma 1.1], we quote the following result: Lemma A.1. For E ∈ MM ×M (C), F ∈ MN ×N (C), the map MN ×M (C) → MN ×M (C),

T → T F − ET,

is bijective if and only if E and F have disjoint spectra. We also need: Lemma A.2. Let A ∈ MN ×N (C) have distinct eigenvalues µj , |µj − µk | ≥ c0 > 0,

j = k.

Then, for each B ∈ MN ×N (C), there is a constant ε0 > 0 with the property that the eigenvalues µjε of A + εB for |ε| ≤ ε0 are distinct, c0 |µjε − µkε | ≥ , j = k, |ε| ≤ ε0 , 2 and depend analytically on ε and the entries of A and B. The bound ε0 depends analytically on c0 and the norms A, B, and M −1 , where M with M = 1 is a diagonalizer of A. Here, denotes the row-sum matrix norm. Proof. By deﬁnition of M , we have M AM −1 = diag(µ1 , . . . , µN ), and the eigenvalues µj as well as the entries of M depend analytically on the entries of A. Therefore, we can assume that A = diag(µ1 , . . . , µN ) is diagonal. By Gerschgorin’s theorem, each of the N balls ' ( |Bjk | Ωjε = z ∈ C |z − (µj + εBjj )| ≤ |ε| k= j

contains exactly one eigenvalue µjε of A + εB provided that these balls do not intersect. This happens when |ε| ≤ ε0 and 2ε0 B ≤ c0 /2. The eigenvalues µjε of A + εB are solutions to the polynomial equation 0 = φε (µ) = det (A + εB − µ1N ) , and are given by the integral µjε =

1 2πi

@ ∂ Ωjε

µφε (µ) dµ, φε (µ)

which completes the proof.

We likewise need the following results: 0 Lemma A.3. For each N × N matrix symbol q ∈ S1,0 (Rn × Rn ) that satisﬁes 2n | det q(x, ξ)| ≥ c for all (x, ξ) ∈ R , |ξ| ≥ C, and some constants C, c > 0, there 0 (Rn ) such that is an invertible operator Q ∈ Op S1,0 −1 (Rn ). Q − q(x, Dx ) ∈ Op S1,0

502

Michael Dreher and Ingo Witt

Proof. We construct two invertible operators Q1 = q1 (x, Dx ), Q2 = q2 (x, Dx ) ∈ 0 (Rn ) such that Op S1,0 −1 (Rn × Rn ), q1 (x, ξ) ≡ χ(|ξ|/(2C))q(x, ξ)q(x0 , ξ)−1 mod S1,0 −1 mod S1,0 (Rn × Rn ).

q2 (x, ξ) ≡ q(x0 , ξ)

Here, the point x0 ∈ Rn is chosen arbitrarily. Then the composition Q1 Q2 possesses the desired properties. Construction of Q1 . We employ the parameter calculus of Grubb [22]. Rename (1 − χ(|ξ|/(2C)))1N + χ(|ξ|/(2C))q(x, ξ)q(x0 , ξ)−1 to q(x, ξ). Then | det q(x, ξ)| ≥ c for |ξ| ≥ 2C and some c > 0 and q(x0 , ξ) = 1N for all ξ ∈ Rn . By a standard application of the mapping degree and homotopy theory, we obtain 0 (Rn × Rn ) such that h(x, ξ) = q(x, ξ) for an N × N matrix function h ∈ S1,0 |ξ| ≥ 2C and | det h(x, ξ)| ≥ c /2 for all (x, ξ) ∈ R2n , by changing q(x, ξ) for |ξ| ≤ 2C if necessary. We then further get an N × N matrix function p0 (x, ξ, µ) ∈ 0 S1,0 (Rn × Rn × R+ ), where µ ≥ 0 enters as additional covariable, such that | det p0 (x, ξ, µ)| ≥ c /2, p0 (x, ξ, 0) = q(x, ξ),

(x, ξ, µ) ∈ R2n × R+ ,

|ξ, µ| ≥ 2C,

(x, ξ) ∈ R , 2n

For that it suﬃces to set |ξ, µ| ≥ 2C, µ ≥ |ξ|, √ √ √ and then to connect p0 (x, ξ, 0) = q(x, ξ) to p0 (x, ξ/ √2, |ξ|/ 2) = h(x, ξ/( 2|ξ|)) for |ξ| ≥ 2C along√ the curve [0, 1] & σ → (x, (1−κσ)ξ, 2κσ − κ2 σ 2 |ξ|) ∈ R2n ×R+ , where κ = 1 − 1/ 2, appropriately: ξ ξ +σ p˜0 (x, ξ, µ) = h x, (1 − σ) , |ξ, µ| ≥ 2C, µ < |ξ|, 1 − κσ |ξ, µ| p˜0 (x, ξ, µ) = h(x, ξ/|ξ, µ|),

where σ = κ−1 (1 − |ξ|/|ξ, µ|), i.e., ξ/(1−κσ) = |ξ, µ| ξ/|ξ|. The symbol ˜0 thus obtained (appropriately continued into |ξ, µ| < 2C) is continuous, but only piecewise C ∞ ; smoothing ˜0 along |ξ, µ| ≥ 2C, µ = |ξ|, whilst keeping the symbol estimates and invertibility, yields the symbol p0 . We now set p(x, ξ, µ) = χ(|ξ, µ|) p1 (x, ξ, µ) + χ(|ξ|) p0 (x, ξ, µ) − p1 (x, ξ, µ) , * where p1 (x, ξ, µ) = |α| 0, see Grubb [22, Remark 2.1.13]. According to Grubb [22, Theorem 3.2.11], there is a µ0 ≥ 0 such that, for all µ ≥ µ0 , the operator p(x, Dx , µ) : L2 (Rn ) → L2 (Rn ) is invertible. Eventually, it suﬃces to set Q1 = p(x, Dx , µ), where µ ≥ µ0 . Construction of Q2 . Rename q(x0 , ξ) to q(ξ). The task to construct a symbol 0 q2 ∈ S1,0 such that q2 (x, Dx ) is invertible and q(Dx ) − q2 (x, Dx ) ∈ Op S −∞ can be fulﬁlled within the framework of SG-calculus, where one has symbols which fulﬁl

Sharp Energy Estimates

503

independently symbol estimates in both the x- and the ξ-variables. Recall that S m;η = S m;η (Rn × Rn ) is the class of all a ∈ C ∞ (R2n ) such that |∂ ∂xα ∂ξβ a(x, ξ)| ≤ Cαβ xη−|α| ξm−|β| ,

(x, ξ) ∈ R2n .

Recall also that a(x, Dx ) : xr H s (Rn ) → xr+η H s−m (Rn ) for a ∈ S m;η is a Fredholm operator if and only if, for some R > 0, det a(x, ξ) = 0 (x, ξ) ∈ R2n ,

|x| + |ξ| ≥ R,

(A.2)

and xm ξη a−1 (x, ξ) ≤ c,

(x, ξ) ∈ R2n ,

|x| + |ξ| ≥ R.

(A.3)

Now we choose a symbol q2 ∈ S 0;0 that is elliptic in the sense of (A.2), (A.3) such that (A.4) q2 (x, ξ) ≡ q(ξ) mod S −1;0 . The choice of q2 (x, ξ) relies on the split exactness of the short sequence (σm ,ση )

0 −−−−→ S m−1;η−1 −−−−→ S m;η −−−−−e→ ΣS m;η −−−−→ 0 where σ m (a) = a + S m−1;η is the principal symbol of a ∈ S m;η , σeη (a) = a + S m;η−1 is its principal exit symbol, and both symbols are. subject to the condition σ m (a)+S m;η−1 = σeη (a)+S m−1;η . Accordingly,/ΣS m;η = (a, ae ) ∈ S m;η /S m−1;η × S m;η /S m;η−1 a ≡ ae mod S m−1;η + S m;η−1 . (A.4) says that σ 0 (q2 ) = σ 0 (q), while the choice of σe0 (q2 ) is restricted by the requirement σe0 (h) ≡ σ 0 (q) mod S −1;0 + S 0;−1 and is free otherwise (except that σe0 (q2 ) needs to be elliptic). Then q2 (x, Dx ) : L2 (Rn ) → L2 (Rn ) is a Fredholm operator. It follows from standard SG–calculus that, upon an appropriate choice of σe0 (q2 ), one can achieve each integer as index of this operator. We choose q2 (x, ξ) in such a way that q2 (x, Dx ) : L2 (Rn ) → L2 (Rn ) has index 0. Then, by adding a contribution from Op S −∞;−∞ (Rn × Rn ) = Op S(R2n ) if necessary, we ﬁnally arrive at an operator Q2 = q2 (x, Dx ) that is invertible as operator on L2 (Rn ). For more on SG-calculus we refer to Cordes [15], Parenti [57], Schrohe [62], and Schulze [64]. Proposition A.4. For each N × N matrix symbol q ∈ S 0,0;λ that satisﬁes | det q(t, x, ξ)| ≥ c,

(t, x, ξ) ∈ [0, T ] × R2n , |ξ| ≥ C,

for some C, c > 0, there exists an invertible operator Q ∈ Op S 0,0;λ such that Q − q(t, x, Dx ) ∈ Op S −1,−(l∗ +1);λ . Proof. A parameter version of Lemma A.3 yields the existence of an invertible 0 operator Q ∈ C ∞ ([0, T ]; Op S1,0 ) ⊂ Op S 0,0;λ such that −1 Q − q(t, x, Dx ) ∈ C ∞ ([0, T ]; Op S1,0 ) ⊂ Op S −1,−(l∗ +1);λ .

504

Michael Dreher and Ingo Witt There is the following improvement of Proposition A.4:

Proposition A.5. Given N ×N matrix symbols q0 ∈ C ∞ ([0, T ]; S (0) ) and q1 ∈ S (−1) such that |det q0 (t, x, ξ)| ≥ c for all (t, x, ξ) ∈ [0, T ] × R2n and some c > 0, there is an invertible operator Q ∈ Op S˜0,0;λ that satisﬁes σ 0 (Q) = q0 ,

σ ˜ −1,0 (Q) = q1 .

Proof. See Dreher–Witt [18, Proposition 3.6 (b)].

Finally, we show that, for special functions, certain cut-oﬀ operators can be estimated from below: Lemma A.6. Deﬁne v0,j ∈ Cc∞ (Rn ; R) and ϕj , ϕj for j = 1, 2, . . . as in the proof of Theorem 2.10. Then, for each s ∈ Cb∞ (Rn ; R), there are constants C and j0 such that (k) C −1 v0,j H s(x) (Rn ) ≤ ϕj (x, Dx )v0,j s(x) n ≤ C v0,j H s(x) (Rn ) , (1)

(2)

(R )

H

for k = 1, 2 and j ≥ j0 . Proof. The Fourier transform vˆ0,j is given by vˆ0,j (ξ) = vˆ0 (ξ − 2j ξ0 ),

|ξ0 | = 1,

and decays rapidly, |ˆ v0,j (2j ξ0 + η)| ≤ CN η−N ,

N ∈ N.

For s ∈ R, we split the norm v0,j H s (R) : 2

v0,j H s (Rn ) = I1,s + I2,s =

ξ |ˆ v0,j (ξ)| dξ + 2s

|ξ −2j ξ0 |≤2j−1

Since vˆ0 decays rapidly, we have 2js I1,s ∼ 2

|η |<2j−1

2

|ξ −2j ξ0 |≥2j−1

ξ2s |ˆ v0,j (ξ)|2 dξ.

|vˆ0 (η)|2 dη ∼ 22js v0 2L2 (Rn ) .

To show that I1,s is the main part of v0,j H s (Rn ) , we estimate I2,s = 2j ξ0 + η2s |ˆ v0,j (2j ξ0 + η)|2 dη |η |≥2j−1 2 ≤ CN 2j ξ0 + η2s η−2N dη |η |≥2j−1

≤ CM 2

−2jM

,

Sharp Energy Estimates

505

for all M and j. Hence we conclude that C −1 2js ≤ v0,j H s (Rn ) ≤ C2js ,

j ∈ N,

s ∈ R.

Next, let s = s(x) be a non-constant function from Cb∞ (Rn ; R). Then we have, with s− = inf s(x) < s+ = sup s(x), x∈R

x∈R

the estimate s(x) C −1 f H s− (Rn ) ≤ Dx K f (x)

L2 (Rn )

≤ C f H s+ (Rn ) ,

f ∈ S(Rn ).

The proof is complete if we can show that s(x) (k) Dx K (1 − ϕj (x, Dx ))v0,j (x)

L2 (Rn )

≤

1 s(x) Dx K v0,j (x) 2 n , 2 L (R )

j ≥ j0 .

Choose a cut-oﬀ function w0 ∈ Cc∞ (Rn ; R) with w0 (x) ≡ 1 on supp v0,j and supp w0 ⊂ {|x − x0 | ≤ γ/3}. Then w0 (x)v0,j (x) = v0,j (x). Since (1 − w0 )(1 − ϕj(k) ) acts as regularizing operator on functions with support contained in supp v0,j , it follows that s(x) (k) (k) Dx K (1 − ϕj (x, Dx ))v0,j (x) 2 n ≤ C (1 − ϕj (x, Dx ))v0,j (x) s+ n L (R ) H (R ) (k) ≤ C w0 (x)(1 − ϕj (x, Dx ))v0,j (x) s H + (Rn ) (k) + C (1 − w0 (x))(1 − ϕj (x, Dx ))v0,j (x) s n H + (R ) (k) ≤ C w0 (x)(1 − ϕj (x, Dx ))v0,j (x) s n + CM 2−jM v0,j (x)H s+ −M (Rn ) . H

+ (R

)

We may choose M = s+ − s− , leading to s(x) 2−jM v0,j (x)H s+ −M (Rn ) ≤ C2−j(s+ −s− ) Dx K v0,j (x)

L2 (Rn )

.

(k)

If w0 (x)(1 − ϕj (x, ξ)) = 0, then |ξ − 2j ξ0 | ≥ ε2j for some positive ε. Hence (k)

(k)

w0 (x)(1 − ϕj (x, ξ)) = w0 (x)(1 − ϕj (x, ξ)) ◦ χ((ξ − 2j ξ0 )/(ε2j )),

506

Michael Dreher and Ingo Witt (k)

0 and {w0 (1 − ϕj )}j is a bounded subset of S1,0 . Therefore 2 (k) w0 (x)(1 − ϕj (x, Dx ))v0,j (x) s H

+ (Rn )

2 ≤ C χ((Dx − 2j ξ0 )/(ε2j ))v0,j (x)H s+ (Rn ) =C ξ2s+ χ((ξ − 2j ξ0 )/(ε2j ))2 |ˆ v0,j (ξ)|2 dξ ≤C

Rn ξ

|η |≥ε 2j

2j ξ0 + η2s+ |ˆ v0,j (2j ξ0 + η)|2 dη

≤ CM 2−2jM ≤ CM 2−2j(M+s− ) v0,j (x)2H s− (Rn ) 2 s(x) −2j(M+s− ) ≤ CM 2 Dx K v0,j (x)

L2 (Rn )

,

for all M and j. We select M = s+ − 2s− , and deduce that s(x) (k) s(x) Dx K (1 − ϕj (x, Dx ))v0,j (x) 2 n ≤ C2−j(s+ −s− ) Dx K v0,j (x) L (R )

completing the proof.

L2 (Rn )

,

Appendix B. Open problems We list some open problems: 1. For A(t, x, Dx ) ∈ Op S 1,1;λ satisfying (2.11), is it true that the (0, δ(x))well-posedness of the Cauchy problem for the operator Dt − A(t, x, Dx ) already implies its δ(x)-well-posedness? By virtue of Theorem 2.7, this is true if A(t, x, Dx ) ∈ Op S˜1,1;λ and the symmetrizer M (t, x, Dx ) is found in the class Op S˜0,0;λ . 2. Prove (or disprove) that for A(t, x, Dx ) ∈ Op S˜1,1;λ — assuming symmetrizable hyperbolicity in the sense of (2.13) — the loss of regularity only depends on σ 1 (A), σ ˜ 0,1 (A). In particular, this is true if the conditions of Theorem 2.10 are met. 3. More generally, assuming an answer in the aﬃrmative to Problem 1 prove (or disprove) that the operators A(t, x, Dx ) and A(t, x, Dx ) + A2 (t, x, Dx ), where A(t, x, Dx ) ∈ Op S 1,1;λ is symmetrizable-hyperbolic in the sense of (2.11) and A2 (t, x, Dx ) belongs to the reminder class Op S 0,0;λ + Op S −1,1;λ , always admit the same loss of regularity. 4. In case the right-hand side of (2.15) fails to be C ∞ as a function of x (it is always globally Lipschitz), our result is sharp in the class of functions δ ∈ Cb∞ (Rn ; R), but it is surely not sharp in general. Improve this result. 5. Address degeneracies at time t = 0 other than the one characterized by λ(t) = tl∗ , l∗ ∈ N+ . For admissible λ(t), see Yagdjian [73].

Sharp Energy Estimates

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6. In the previous item, allow also mixed types of degeneracies, in which one has, e.g., a degeneracy like e−1/t in direction of x1 , a degeneracy like t3 in direction of x2 , and a degeneracy like t2 in all other directions. Some results can be found in Kajitani–Wakabayashi–Yagdjian [39], Tahara [68]. 7. Establish an upper bound for the microlocal loss of regularity. The result is conjectured to be the same as in (2.15), but with the two supremums skipped. 8. Show that the microlocal estimates from the previous item are generically sharp. In the remaining cases, we expect exceptional propagation of singularities, with the complete branching of singularities does not occur. 9. Discuss the invariance of the operator classes Op S m,η;λ , Op S˜m,η;λ for m, η ∈ R under coordinate changes of the form (3.11). See the authors’ article [19] for the case t˜ = t, x ˜ = κ(x). 10. Solve semilinear problems related to the operator P from (1.3), (1.4), P u = F (t, x, Qu), (t, x) ∈ (0, T ) × Rn , Dtj u(0, x) = uj (x),

0 ≤ j ≤ m − 1,

where Q is a diﬀerential operator like P , but of order m − 1. Among others, this requires to discuss the superposition operator F (u) for u ∈ H s,δ;λ (or u ∈ H s,δ;λ ∩ L∞ ). For the case F is analytic, see Dreher–Reissig [17], Dreher–Witt [20]. It can be shown that H s,δ;λ ((0, T ) × Rn ) ⊂ L∞ ((0, T ) × Rn ) if and only if s > (n + 1)/2, s + β∗ δ(x)l∗ ≥ (n + 1)/2.

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Acknowledgment We would like to thank Prof. Reissig und Prof. Schulze for the invitation of our contribution to this volume. Michael Dreher Fachbereich Mathematik und Statistik Universit¨ ¨ at Konstanz D-78457 Konstanz Germany e-mail: [email protected] Ingo Witt Institut f¨ fur Mathematik Universit¨ ¨ at Potsdam 14415 Potsdam Germany Current address: Department of Mathematics Imperial College 180 Queen’s Gate London SW7 2BZ United Kingdom e-mail: [email protected]