Nonlinear Evolution Equations

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CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

NONLINEAR EVOLUTION EQUATIONS

133

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgässner, Universität Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

NONLINEAR EVOLUTION EQUATIONS

133

Library of Congress Cataloging-in-Publication Data Zheng, Songmu. Nonlinear evolution equations / by Song-Mu Zheng. p. cm. — (Chapman & Hall/CRC monographs and surveys in pure and applied mathematics) Includes bibliographical references and index. ISBN 1-58488-452-5 (alk. paper) 1. Evolution equations, Nonlinear. I. Title. II. Series. QA377 .Z435 2004 515'.353--dc22

2004049388

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2004 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-452-5 Library of Congress Card Number 2004049388 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Dedicated to Weixi, Leijun, Lingzhou and Eric

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Motivations from Other Branches of Science . . . . . . . . . . . . . . 1 1.2. Local Solutions and Global Solutions . . . . . . . . . . . . . . . . . . . . . 4 1.3. Some Basic Knowledge on Sobolev Spaces and PDE . . . . . . 8 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6

Sobolev Spaces W m, p (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Gagliardo–Nirenberg and Poincar´e Inequalities 10 Abstract Functions Valued in Banach Spaces . . . . . . 11 Linear Elliptic Boundary Value Problems . . . . . . . . . . 12 Interpolation Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Notation and Some Useful Inequalities . . . . . . . . . . . . 17

1.4. Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2. Semigroup Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1. Semigroups of Linear Contraction Operators . . . . . . . . . . . . 21 2.1.1 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 The Infinitesimal Generator . . . . . . . . . . . . . . . . . . . . . . . 23 2.2. Hille-Yosida Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3. Regularities of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4. Non-homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5. Semilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.6. Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.7. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.8. Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3. Compactness Method and Monotone Operator Method . . . . . . . . . 95 3.1. Compactness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1.1. Some Results on Convergence and Compactness . . 96 vii

viii

Contents 3.1.2. Method and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2. Monotone Operator Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.2.1. General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.2.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.2.3. Generalizations and Supplements . . . . . . . . . . . . . . . . 128 3.3. Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4. Monotone Iterative Method and Invariant Regions. . . . . . . . . . . . .133 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.2. Monotone Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.2.1. Monotone Iterative Method for Nonlinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.2.2. Monotone Iterative Method for Nonlinear Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.3. Invariant Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.3.2. Methods and Applications . . . . . . . . . . . . . . . . . . . . . . . 155 4.4. Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5. Global Solutions with Small Initial Data . . . . . . . . . . . . . . . . . . . . . . 175 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2. IVP for Fully Nonlinear Parabolic Equations . . . . . . . . . . 181 5.2.1. Decay Rate of the Solution to the Cauchy Problem for Linear Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.2.2 Global Solution to the Cauchy Problem for Fully Nonlinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3. IBVP for Fully Nonlinear Parabolic Equations . . . . . . . . .200 5.3.1. Decay Rate of the Solution to IBVP for Linear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.3.2. Global Solution to IBVP for Fully Nonlinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.4. Nonexistence of Global Solutions with Small Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 5.5. Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6. Asymptotic Behavior of Solutions and Global Attractors . . . . . . 221

Contents

ix

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.2. A Lemma in Analysis and Its Applications . . . . . . . . . . . . . 225 6.3. Convergence to Stationary Solutions . . . . . . . . . . . . . . . . . . . 236 6.3.1 6.3.2 6.3.3 6.3.4

Some Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Gradient Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 The Lojasiewicz–Simon Inequality . . . . . . . . . . . . . . . . 245 Proof of Theorem 6.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . .250

6.4. Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.5. Bibliographic Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287

Preface This book is designed to introduce some important methods for nonlinear evolution equations. The material in this book has been used in recent years as lecture notes for the graduate students at Fudan University. Nonlinear evolution equations, i.e., partial differential equations with time t as one of the independent variables, arise not only from many fields of mathematics, but also from other branches of science such as physics, mechanics and material science. For example, Navier-Stokes and Euler equations from fluid mechanics, nonlinear reaction-diffusion equations from heat transfers and biological sciences, nonlinear KleinGorden equations and nonlinear Schr¨ odinger equations from quantum mechanics and Cahn-Hilliard equations from material science, to name just a few, are special examples of nonlinear evolution equations. Complexity of nonlinear evolution equations and challenges in their theoretical study have attracted a lot of interest from many mathematicians and scientists in nonlinear sciences. The first question to ask in the theoretical study is whether for a nonlinear evolution equation with given initial data there is a solution at least locally in time, and whether it is unique in the considered class. Generally speaking, this problem has been solved for a wide class of nonlinear evolution equations by two powerful methods in nonlinear analysis, i.e., the contraction mapping theorem and the Leray-Schauder fixed-point theorem. Roughly speaking, since the 1960s, much more attention has been paid to the global existence and uniqueness of a solution, i.e., when a local solution can be extended to become a global one in time. Furthermore, if there is a global solution for a given nonlinear evolution equation, one also wants to know about the asymptotic behavior of the solution as time goes to infinity. A series of useful methods and theories have been developed, especially since the 1960s. Basic knowledge about the central issues of this subject, i.e., global existence and uniqueness, and long time behavior of a solution as time goes to infinity is extremely important for graduate xi

xii

Preface

students and researchers in mathematics, as well as in other branches of science. This book aims to concisely explain in a clear and easily readable manner a wide range of relevant theories and methods for nonlinear evolution equations such as the semigroup method, compactness and monotone operator method, monotone iterative method and invariant regions, global existence and uniqueness theory for small initial data as well as results on convergence to a stationary solution and the existence of global attractors. Moreover, some material in the book consists of the research results of the author, while results on convergence to a stationary solution in higher-space dimensions are updated. Most are only available in various journals, and have not yet been seen in book form. In addition, there are bibliographic comments in each chapter that provide the reader with references and further reading materials, though this book is not intended to be exhaustive. The book consists of six chapters. Chapter 1 is a preliminary chapter in which we not only describe motivations from other branches of science, but also introduce the concepts of local well-posedness and global well-posedness, and outline the theory of global solutions, which is the main concern of this book. In this chapter we also collect some basic material on PDE and Sobolev spaces for the convenience of the reader. In the next five chapters, we introduce some important methods for nonlinear evolution equations which have been developed from the 1960s to the present day. Chapter 2 is concerned with the semigroup method. This method was developed from the end of the 1940s to the 1960s, and it is still very useful nowadays. Based on the Hille-Yosida theorem and a theorem by I. Segal, which are introduced in detail, global solvability of linear and semilinear evolution equations as well as regularities are discussed. Examples of applications to nonlinear parabolic equations and nonlinear hyperbolic equations are presented. Chapter 3 is devoted to the compactness method and monotone operator method. These two methods were developed in the 1960s. Combining the Faedo-Galerkin method with the compactness argument yields a powerful method which allows us to deal with some nonlinear evolution equations. The monotone operator method is very useful to deal with some quasilinear evolution equations. Examples illustrating the applications to concrete nonlinear evolution equations are also shown in detail.

Preface

xiii

Chapter 4 is concerned with the monotone iterative method and invariant regions. The monotone iterative method was introduced by H. Amann and D. Sattinger around 1971 to deal with nonlinear parabolic equations as well as nonlinear elliptic equations. Based on the comparison principle, one can draw a conclusion about the existence of a solution to these nonlinear PDEs provided that a suitable pair of an upper solution and a lower solution can be found. This method has been significantly developed to deal with certain nonlinear parabolic systems nowadays. It is important to concisely introduce these methods and ideas. A related method, i.e., the invariant regions, was initiated by H. Amann and K. Chueh, C. Conley and J. Smoller around 1977. In this book we mainly follow the line of descriptions by K. Chueh, C. Conley and J. Smoller with some modifications of proof. Applications to not only nonlinear parabolic equations but also some nonlinear hyperbolic equations are shown in this chapter. During the 1970s, a systematic method of dealing with highly nonlinear evolution equations with small initial data emerged. Chapter 5 of this book introduces this important method by illustrating how it works for the fully nonlinear parabolic equations to which the author of this book is one of the main contributors. Since the 1980s, convergence of solutions of nonlinear evolution equations to stationary solutions as time goes to infinity and the study of the related infinite-dimensional dynamical system become two of the main concerns in the field of nonlinear evolution equations. The final chapter of this book is devoted to these topics. A lemma in analysis, which was established by the author and his collaborator in 1993, is introduced and the applications to the study of long time behavior of solutions are also shown. Convergence of solution of nonlinear evolution equations to a corresponding stationary solution as time goes to infinity has been a problem of great interest and importance for a long time. For the one-dimension case, significant progress has been made, as can be seen from the work by H. Matano in 1978 and other works later on. For the higher space dimension case, it has been a focus of many researchers in the last twenty years. In a paper published in 1983 L. Simon extended a lemma by S. Lojasiewicz on analytic functions to the infinite dimensional case and developed a method for the study of convergence to the stationary solution when the nonlinear term is analytic. Since then, there have been many research papers on the gen-

xiv

Preface

eralizations, although they have not yet been seen in book form. We will introduce these ideas and methods in a clear and detailed manner in the final chapter of this book. Another aspect of the study of the long time behavior of solutions to nonlinear evolution equations is to investigate whether there exists a global attractor, etc. when initial data vary in any given bounded set. Since the 1980s, this has become a hot topic in research, as three books by R. Temam, J.K. Hale and A.V. Babin & M.I. Vishik indicated. In the final chapter we will also concisely introduce some results on this topic especially for the gradient system. It is expected that after reading this book, the reader will know the ideas and essences of several important methods in nonlinear evolution equations and their applications. This together with the references and further reading materials provided in the bibliographic comments will enable the reader to prepare for further study and research. I appreciate the stimulating interactions with graduate students at Fudan University where I have taught this course in recent years. Special thanks go to Hao Wu and Yuming Qin for their careful reading and many suggestions. I would also like to acknowledge the NSF of China for their continued support. Currently, this book project is being supported by the grants No. 19331040 and No. 10371022 from NSF of China. Finally, my deepest gratitude goes to my wife, Weixi Shen, also a mathematician and my collaborator at Fudan University, for her constant encouragement, advice and support in my career. Professor Songmu Zheng Institute of Mathematics Fudan University Shanghai 200433, China Email: [email protected] [email protected]

Chapter 1 Preliminaries

In this chapter we first present some examples to show how nonlinear evolution equations naturally arise from other branches of science. We will also give definitions of local and global solutions, and outline the theory of global solution. Since this book is designated for graduate students as well as researchers, some basic materials on partial differential equations and Sobolev’s spaces, which will be needed in the remainder of the book, are introduced for convenience. Most results are just recalled without proofs, but the relevant references are given. In the final section of this chapter, some references for further reading are also given.

1.1

Motivations from Other Branches of Science

An evolution equation usually means a partial differential equation with one of the independent variables being time t. The linear wave equation or linear heat equation describing vibration of string or heat conduction are two simple examples of evolution equations. However, there are many nonlinear evolution equations naturally arising from physics, mechanics, biology, chemistry, material science needing to be investigated. In the following text we give some examples. 1. Let us first look at a first-order nonlinear evolution equation ∂u ∂u +u = 0. (1.1.1) ∂t ∂x This equation is derived when one studies the one-dimensional traffic problem. In this situation, u(x, t) represents the density of cars at place x and time t. Equation (1.1.1) is also often taken as a model equation for the study of gas dynamics with conservation law.

1

2

NONLINEAR EVOLUTION EQUATIONS

2. In practice we will often encounter second-order nonlinear evolution equations. For instance, if we investigate the heat transfer process in a body with a heat source that produces heat quantity at the unity of time depending on the instant temperature, then we are led to the following nonlinear heat equation: ∂u = div(k∇u) + f (u). (1.1.2) ∂t Considering vibration of a strain with the external force nonlinearly depending on displacement, one is led to the nonlinear wave equation: 2 ∂2u 2∂ u − a = f (u). (1.1.3) ∂t2 ∂x2 In quantum mechanics one encounters various nonlinear evolution equations such as the nonlinear Sine-Gorden equation:

utt − ∆u + sinu = 0,

(1.1.4)

the Klein-Gorden equation, utt − ∆u + mu + γu3 = 0,

(1.1.5)

and the nonlinear Schr¨ odinger equation, iut − ∆u + γ|u|2 u = 0,

(1.1.6)

or even a coupled system of nonlinear evolution equations. Hereafter we use the subscript t or x to denote differentiation with respect to t or x. 3. In addition to the second-order equations, some even higher-order nonlinear evolution equations naturally arise in other branches of science. For instance, in the study of phase transitions of binary alloys such as polymers, glasses etc., the following Cahn-Hilliard equation is derived: ut + ε∆2 u = ∆φ(u)

(1.1.7)

where ε is a given small positive constant and typically φ(u) = u3 − u. Notice that equation (1.1.7) is a fourth-order nonlinear evolution equation. Other examples of a higher-order evolution equation include the following famous KdV equation: ut + 6uux + uxxx = 0 which is a third-order evolution equation.

(1.1.8)

Preliminaries

3

4. When a complicated physical, mechanical or a biological system is investigated, it is quite often that not only one independent variable is not enough, but also a system of dependent variables are needed to describe the whole system. In these circumstances one is led to study a system of nonlinear evolution equations. There are many examples in this aspect. For instance, the following famous Navier-Stokes equations, which describe the flow of viscous incompressible fluids, is a system of this kind.   u = 0,  div ~

  u~t + (~ u · ∇)~u + ρ1 grad p = µ∆~u

(1.1.9)

where ~u denotes the velocity vector, ρ is density and p is pressure. Other examples include the Euler equations describing the flow of viscous and heat-conductive compressible fluids, and nonlinear thermoelastic systems or nonlinear thermoviscoelastic systems. Some systems arise from biology and ecology. For instance, the following reactiondiffusion system is a good example of this kind: u~t − D∆~u = f~(~u) (1.1.10) where ~u and f~ are vector functions and D is a matrix function. In ecology, prey and predator coexist in an ecological system, and their existence and development is an interactive process that can be described by system (1.1.10). From the mathematical point of view, (1.1.10) is a semilinear parabolic system when D is a positive definite matrix. Another example of this kind is the following Fitzhugh-Nagumo equations, which describes the propagations of pulses in a nerve system:   ut − ∆u = f (u) − z, 

zt = σu − γz.

(1.1.11)

This is a system of a nonlinear parabolic equation coupled with an ordinary differential equation, which can be considered as a degenerate parabolic equation. 5. In practice, one will encounter degenerate nonlinear evolution equations that change the type of equation in some regions. For instance, the following porous media equation is a good example of this kind: ut − ∆(um ) = 0.

(1.1.12)

This equation describes the flow of underground water in porous media. When u = 0, the above equation is degenerate. It is expected that

4

NONLINEAR EVOLUTION EQUATIONS

many properties, including uniqueness will not be the same as other non-degenerate equations. 6. Sometimes one also encounters nonlinear evolution equations defined on a manifold. For the needs of large-scale weather broadcasting, one is led to the study the nonlinear evolution equations defined on a manifold, i.e., the surface of the earth. In differential geometry, to find a harmonic mapping from an n-dimensional Riemannian manifold M to another m-dimensional Riemannian manifold N , J. Eells and J.H. Sampson in 1964 proposed a new approach by studying the corresponding nonlinear parabolic system defined on a Riemannian manifold. The above are just a few examples, but enough to exhibit the variety and complexity of nonlinear evolution equations, and consequently, the challenge of their study.

1.2

Local Solutions and Global Solutions

For a given nonlinear evolution equation, two basic problems posed to study are the initial value problem and the initial boundary value problem. For the preceding problem, while the initial data are given at t = 0, and the space variables (x1 , ·, xn ) vary in Rn or in an ndimensional Riemannian manifold without boundary, one wants to find the solution for later time t > 0. For the later problem, the space variables (x1 , ·, xn ) vary in a domain Ω of Rn or an n-dimensional Riemannian manifold with boundary. Then for a solution to be well defined, in addition to the initial conditions at t = 0, suitable boundary conditions on the boundary Γ of Ω or a Riemannian manifold also need to be given. In addition to the above two basic problems, finding a travelling-wave solution to a given nonlinear evolution equation is also an interesting and important problem. In contrast to the previous two problems, the initial condition is not posed. Finding a travelling-wave solution is often reduced to solving nonlinear elliptic equations subject to boundary conditions, and we will not go into the detail here since it is beyond the scope of the present book. For a given nonlinear evolution equation, under general assumptions on initial data and boundary conditions, we very often find that existence and uniqueness of the solution in a small time interval near the origin t = 0, i.e., the local well-posedness can be obtained by two fun-

Preliminaries

5

damental tools in nonlinear analysis, namely, the contraction mapping theorem and the Leray-Schauder fixed-point theorem. However, as we know from the theory of ordinary differential equations, there is a significant difference between a linear equation and a nonlinear equation: for a linear equation, one can often find a solution defined globally for all t > 0, while for a nonlinear equation it is not always possible. This is also true for more complicated nonlinear evolution equations. Let us first look at the initial value problem for equation (1.1.1): ut + uux = 0, u|t=0 = φ(x).

(1.2.1) (1.2.2)

For a solution u(x, t) to equation (1.2.1), consider the following characteristic equation: dx = u(x, t). (1.2.3) dt Then equation (1.2.1) implies that along the characteristic line x = x(t) in the (x, t) plane, which is the solution to equation (1.2.3), du = 0, (1.2.4) dt i.e., u remains the same constant. It turns out from equation (1.2.3) that the characteristic line is a straight line. For given initial data φ(x), no matter how smooth and small they are, if there exist two points x1 and x2 with x1 < x2 such that φ(x1 ) − φ(x2 ) < 0, x1 − x2

(1.2.5)

then two characteristic lines through (x1 , 0) and (x2 , 0) in the (x, t) plane must intersect in a finite time t∗ . This implies that the solution cannot be uniquely defined at that time, and singularity of solution u to problem (1.2.1), (1.2.2) must occur at least at t = t∗ where the first-order derivatives of u become infinity. In other words, for initial data satisfying the condition (1.2.5), problem (1.2.1), (1.2.2) admits a local solution when t < t∗ , but does not have a global solution for all t > 0 in the classical sense, and the solution blows up in the finite time t∗ . Actually such phenomena can occur for other nonlinear evolution equations. Let us now look at another example. Consider the following initial boundary value problem for a nonlinear heat equation with the Neumann boundary condition:

6

NONLINEAR EVOLUTION EQUATIONS ut − ∆u = u2 , t > 0, x ∈ Ω ⊂ Rn , ¯ ∂u ¯¯ = 0, t > 0, ∂n ¯Γ u|t=0 = u0 (x), x ∈ Ω ⊂ Rn

(1.2.6) (1.2.7) (1.2.8)

where Ω is a bounded domain in Rn with smooth boundary Γ. In what follows we prove that the classical solution u to problems (1.2.6)–(1.2.8) must blow up in a finite time provided that u0 (x) is nonnegative and R u (x)dx > 0. Indeed, integrating equation (1.2.6) with respect to x 0 Ω and using boundary condition (1.2.7), we get Z

Z

d udx = u2 dx. dt Ω Ω Using the following H¨older inequality, µZ

¶2

Ω

udx

≤C

Z

Ω

(1.2.9)

u2 dx,

(1.2.10)

where C = |Ω| denotes the measure of Ω, we infer from (1.2.9) that µZ

Z

1 d udx ≥ udx dt Ω C Ω Solving this differential inequality yields Z

Ω

udx ≥

where C0 = Thus, when t →

C C0 ,

Z Ω

CC0 , C − C0 t

Z Ω

u0 dx.

¶2

.

(1.2.11)

(1.2.12)

(1.2.13)

u(x, t)dx → +∞,

i.e., the solution of problem (1.2.6)–(1.2.8) must blow up in the finite ¤ time t∗ = CC0 . Thus for a given nonlinear evolution equation it does not always have a global solution defined for all t > 0. Research on local well-posedness made much progress around the 1960s, though it is still a topic of research for new problems nowadays. Since the 1960s, investigating whether there will be a global solution in the classical sense or in the generalized sense has become one of the central issues in the study of nonlinear evolution equations.

Preliminaries

7

This issue can be addressed from a different angle: for a given nonlinear evolution equation, under what circumstances will the solution to that nonlinear evolution equation blow up in a finite time? The previous examples show some hints on the techniques that may be useful. However, we should point out that verification of blow-up in a finite time for a given nonlinear evolution equation in general is not easy. There are some general methods in this direction (see, e.g., [122] by L.E. Payne). But they normally work only for nonlinear evolution equations in the simple form. If blow-up does occur in a finite time, one may further ask: What is the set of the blow-up points? What is the blow-up rate of the solution when time approaches the blow-up time? These problems, which are also important from the point of view of applications, have become one of interesting topics of research since the middle of the 1980s. Quite often solutions to nonlinear evolution equations are sought in the generalized sense or in the weak sense. Enlarging the scope of search not only increases the chance of the existence of solutions, but also allows one to use many powerful methods in functional analysis. However, when a solution to a given nonlinear evolution equation is sought in the generalized sense or the weak sense, a natural question is about the regularity of weak solutions. Will solution be more regular if initial data and boundary data are more regular? On the other hand, once global existence (and uniqueness) is concluded, one may ask further questions on asymptotic behavior of the solution: as time goes to infinity, what is the behavior of solutions? In particular, for a given initial datum, does the solution converge to an equilibrium, or a stationary solution? More generally, what is the long time behavior of solutions when initial data vary in any bounded set in a Sobolev space associated with the given problem? In particular, do solutions converge to a compact invariant set, namely, a global attractor as time goes to infinity? Clearly, these questions are not easy to answer, and so far much progress has mainly been made on dissipative systems. Roughly speaking, research on the above aspects constitutes the major parts of the theory of global solutions. Starting with the next chapter of this book, we will introduce some important methods and theories.

8

1.3

NONLINEAR EVOLUTION EQUATIONS

Some Basic Knowledge on Sobolev Spaces and PDE

For the convenience of the reader, in this section we recall some basic facts on Sobolev spaces and partial differential equations. Most materials in this section are just given without proofs, but the relevant references are indicated.

1.3.1

Sobolev Spaces W m, p (Ω)

Let Ω be a bounded or an unbounded domain of Rn with smooth boundary Γ. For m ∈ N, 1 ≤ p ≤ ∞, W m, p (Ω) is defined to be the space of functions u in Lp (Ω) whose distribution derivatives of order up to m are also in Lp (Ω). Then, it is known (see, e.g., R.A. Adams [1], J.L. Lions and E. Magenes [95]) that W m, p (Ω) is a Banach space for the norm 

kukW m, p (Ω) = 

X |α|≤m

1

kDα ukpLp (Ω) 

p

(1.3.1)

where α = {α1 , · · · , αn } ∈ N n , |α| = α1 + · · · + αn , and Dα u =

∂ α1 +···+αn u . ∂xα1 1 · · · ∂xαnn

When p = 2, we usually denote W m, p (Ω) by H m (Ω) and this is a Hilbert space for the induced inner product. Let C k (Ω) (k ∈ N or k = ∞) be the space of k times continuously differentiable functions on Ω. We denote by C0k (Ω) the space of C k (Ω) functions on Ω with compact support in Ω. The closure of C0∞ (Ω) in W m, p (Ω) is denoted by W0m, p (Ω), which is a subspace of W m, p (Ω). We now recall some important properties of the Sobolev spaces W m, p (Ω) (see, e.g., R.A. Adams [1]). THEOREM 1.3.1 (Density Theorem) If Ω is a C m domain, m ≥ 1, 1 ≤ p < ∞, then ¯ is dense in W m, p (Ω). C m (Ω) THEOREM 1.3.2 (Imbedding and Compactness Theorem) Assume that Ω is a bounded

Preliminaries

9

domain of class C m . Then we have (i) If mp < n, then W m, p (Ω) is continuously imbedded in Lq (Ω) with ∗

1 = 1 − m: p n q∗

W m, p (Ω) ,→ Lq (Ω). ∗

(1.3.2)

Moreover, the imbedding operator is compact for any q, 1 ≤ q < q ∗ . (ii) If mp = n, then W m, p (Ω) is continuously imbedded in Lq , ∀q, 1 ≤

q<∞:

W m, p (Ω) ,→ Lq (Ω).

(1.3.3)

Moreover, the imbedding operator is compact, ∀q, 1 ≤ q < ∞. If p = 1, m =

n, then the above still holds for q = ∞.

n (iii) If k + 1 > m − n p > k, k ∈ N, then writing m − p = k + α, k ∈ ¯ : N, 0 < α < 1, W m, p (Ω) is continuously imbedded in C k, α (Ω) ¯ W m, p (Ω) ,→ C k, α (Ω), ¯ C k, α (Ω)

(1.3.4)

¯ C k (Ω)

where is the space of functions in whose derivatives of order k are H¨older continuous with exponent α. Moreover, if n = m − k − 1, and α = 1, p = 1, then (1.3.4) holds for α = 1, and the imbedding operator is ¯ ∀ 0 ≤ β < α. compact from W m, p (Ω) to C k, β (Ω),

REMARK 1.3.1 The imbedding properties (i)–(iii) are still valid for smooth unbounded domains or Rn provided that Lq (Ω) in (1.3.3) and ¯ in (1.3.4) are replaced by Lq (Ω) and C k, α (B) for any bounded C k, α (Ω) loc domain B ⊂ Ω, respectively. REMARK 1.3.2 The regularity assumption on Ω can be weakened. When u ∈ W0m, p (Ω), the above imbedding properties are valid without any regularity assumptions on Ω.

Let Ω be a smooth bounded domain of class C m and u ∈ W m, p (Ω). Then we can define the trace of u on Γ which coincides with the value ¯ of u on Γ when u is a smooth function of C m (Ω). THEOREM 1.3.3 (Trace Theorem) Let ν = (ν1 , · · · , νn ) be the unit outward normal on Γ

10

NONLINEAR EVOLUTION EQUATIONS

and

¯

∂ j u ¯¯ ¯ j = 0, · · · , m − 1. γj u = ¯ , ∀u ∈ C m (Ω), ∂ν j ¯Γ

(1.3.5)

Then the trace operator γ = {γ0 , · · · , γm−1 } can be uniquely extended to a m−1 Y 1 continuous operator from W m, p (Ω) to W m−j− p , p (Γ) : j=0

γ:

u ∈ W m, p (Ω) 7→ γu = {γ0 u, · · · , γm−1 u} ∈

m−1 Y

1

W m−j− p , p (Γ).

j=0

(1.3.6)

Moreover, it is a surjective mapping. 1

Notice that W m−j− p , p (Γ) are spaces with fractional-order derivatives. Refer to J.L. Lions and E. Magenes [95] for the definition and more about that.

1.3.2

The Gagliardo–Nirenberg and Poincar´ e Inequalities

Throughout this book the following Gagliardo–Nirenberg interpolation inequalities (see L. Nirenberg [115] and A. Friedman [54]) will be frequently used. First we introduce some notation. For p > 0, |u|p,Ω = kukLp (Ω) . For i h n p < 0, set h = − n p , −α = h + p and define |u|p,Ω = sup |Dh u| ≡ Ω

|u|p,Ω = [Dh u]α,Ω ≡ ≡

X

X

sup |Dβ u|, if α = 0,

|β|=h Ω

X

(1.3.7)

sup[Dβ u]α

|β|=h Ω

|Dβ u(x) − Dβ u(y)| , if α > 0. |x − y|α |β|=h x,y∈Ω sup

(1.3.8)

If Ω = Rn , we simply write |u|p instead of |u|p,Ω . THEOREM 1.3.4 Let j, m be any integers satisfying 0 ≤ j < m, and let 1 ≤ q, r ≤ ∞, and

j ≤ a ≤ 1 such that p ∈ R, m 1 j 1 m 1 − = a( − ) + (1 − a) . p n r n q

(1.3.9)

Preliminaries

11

Then, T (i) For any u ∈ W m, r (Rn ) Lq (Rn ), there is a positive constant C depending only on n, m, j, q, r, a such that the following inequality holds:

|Dj u|p ≤ C|Dm u|ar |u|1−a q

(1.3.10)

with the following exception: if 1 < r < ∞ and m − j − n r is a nonnegative

j

integer, then (1.3.10) holds only for a satisfying m ≤ a < 1.

T (ii) For any u ∈ W m, r (Ω) Lq (Ω) where Ω is a bounded domain with smooth boundary, there are two positive constants C1 , C2 such that the following inequality holds: 1−a |Dj u|p,Ω ≤ C1 |Dm u|ar,Ω |u|q,Ω + C2 |u|q,Ω

(1.3.11)

with the same exception as in (i). T m, r In particular, for any u ∈ W0 (Ω) Lq (Ω), the constant C2 in (1.3.11) can be taken as zero.

The following two theorems are concerned with the useful Poincar´e inequalities. THEOREM 1.3.5 Let Ω be a bounded domain in Rn and u ∈ H01 (Ω). Then there is a positive constant C depending only on Ω and n such that

kukL2 (Ω) ≤ Ck∇ukL2 (Ω) , ∀ u ∈ H01 (Ω).

(1.3.12)

THEOREM 1.3.6 Let Ω be a bounded domain of C 1 in Rn . There is a positive constant C depending only on Ω, n such that for any u ∈ H 1 (Ω), ¯Z ¯¶ µ ¯ ¯ ¯ kukL2 (Ω) ≤ C k∇ukL2 (Ω) + ¯ udx¯¯ . (1.3.13) Ω

1.3.3

Abstract Functions Valued in Banach Spaces

For the study of evolution equations it is convenient to introduce abstract functions valued in Banach spaces. Let X be a Banach space, 1 ≤ p < ∞, −∞ ≤ a < b ≤ ∞. Then Lp ((a, b); X) denotes the space of Lp functions from (a, b) into X. It is

12

NONLINEAR EVOLUTION EQUATIONS

a Banach space for the norm kf kLp ((a,b); X) =

ÃZ

b

a

!1

kf (t)kpX

dt

p

(1.3.14)

where the integral is understood in the Bochner sense. For p = ∞, L∞ ((a, b); X) is the space of measurable functions from (a, b) into X being essentially bounded. It is a Banach space for the norm kf kL∞ ((a,b); X) = sup esskf (t)kX . t∈(a,b)

(1.3.15)

Similarly, when −∞ < a < b < ∞ we can define Banach spaces C k ([a, b]; X) for the norm kf kC k ([a,b]; X) =

k X i=0

max k

t∈[a,b]

di f (t)kX . dti

(1.3.16)

The following result (see, e.g., R. Temam [151]) is needed in the study of linear or nonlinear evolution equations. LEMMA 1.3.1 Let V, H, V 0 be three Hilbert spaces with V 0 being the dual space of V and each space included and dense in the following one:

V ,→ H ∼ = H 0 ,→ V 0 . If an abstract function u belongs to L2 ([0, T ], V ) and its derivative ut in the distribution sense belongs to L2 ([0, T ], V 0 ), then u is almost everywhere equal to a function continuous from [0, T ] into H and we have the following equality, which holds in the scalar distribution sense on (0, T ):

d kuk2H = 2hu0 , ui, dt where h·, ·i denotes the dual product between V and V 0 .

1.3.4

Linear Elliptic Boundary Value Problems

In this section we introduce some basic results on linear elliptic boundary value problems (refer to L. Nirenberg [116], A. Friedman [54] and J.L. Lions and E. Magenes [95]). Let Ω be a domain with smooth boundary Γ. Any linear partial differ¯ has the ential operator with, for simplicity, C ∞ coefficients aα (x) in Ω

Preliminaries form P (x, D) =

X

aα (x)Dα .

13

(1.3.17)

|α|≤µ

¯ if the leading homogeneous part The operator P is called elliptic in Ω ¯ of P (x, ξ) does not vanish for ξ 6= 0 and x ∈ Ω: Pµ (x, ξ) =

X

¯ ξ ∈ Rn \{0}. aα (x)ξ α 6= 0, ∀ x ∈ Ω,

(1.3.18)

|α|=µ

The Laplace operator ∆ and the biharmonic operator ∆2 are the most familiar elliptic operators. It follows that for an elliptic operator P with real coefficients aα , µ must be an even number 2m, m ≥ 0, m ∈ N . Usually, one studies boundary value problems for elliptic operators:  Pu = 

X

aα (x)Dα u = f , x ∈ Ω,

|α|≤2m

Bj u|Γ = gj , j = 0, · · · m − 1,

(1.3.19)

¯ and Γ, respectively, and Bj where f and gj are given functions in Ω are certain partial differential operators defined on Γ and the order of each Bj is less than 2m. Well-posedness When gj ≡ 0, the problem (1.3.19) is said to be well-posed if (i) Ker P belongs to C ∞ and dim(Ker P ) = ν < ∞. (ii) In suitable function spaces X, Y the operator P : X 7→ Y is continuous and has closed range in Y of finite codimension ν ∗ , i.e., P is Fredholm. The index of operator P is defined as follows: ind P = ν − ν ∗ .

(1.3.20)

The boundary operators Bj (j = 0, · · · , m − 1) for which the general boundary problems (1.3.19) are well-posed have been characterized and are known as the Lopatinsky boundary conditions. As far as spaces ¯ C k+α (Ω) ¯ (k ∈ N, 0 < α < 1) and X, Y are concerned, C 2m+k+α (Ω), 2m+k, p k, p W (Ω), W (Ω) (1 < p < ∞) are two suitable choices of Sobolev spaces.

14

NONLINEAR EVOLUTION EQUATIONS

Basic Results in C k+α (i) Fredholm The mapping P :

n

o

k+α ¯ ¯ u ∈ C 2m+k+α (Ω)|B (Ω) j u|Γ = 0, j = 0, · · · , m − 1 7→ C

(1.3.21)

is Fredholm and its index is independent of k. (ii) Regularity ¯ and u is a weak solution of problem (1.3.19) If gj = 0, f ∈ C k+α (Ω) ¯ Thus all functions in in the distribution sense, then u ∈ C 2m+k+α (Ω). ∞ Ker P are in C . (iii) A Priori Estimate There exist two positive constants C1 , C2 independent of u such that ¯ satisfying Bj u|Γ = 0, (j = 0, · · · , m − 1), for any u ∈ C 2m+k+α (Ω) kukC 2m+k+α ≤ C1 kP ukC k+α + C2 kukC .

(1.3.22)

Moreover, if Ker P = 0, i.e., the uniqueness of problem (1.3.19) holds, then there is a positive constant C3 independent of u such that kukC 2m+k+α ≤ C3 kP ukC k+α .

Basic Results in W k,p (Ω) (k ∈ N, 1 < p < ∞) (i) Fredholm The mapping P :

n

(1.3.23)

o

u ∈ W 2m+k, p (Ω)| Bj u|Γ = 0, j = 0, · · · , m − 1 7→ W k, p (Ω) (1.3.24)

is Fredholm and its index is independent of k. (ii) Regularity If gj ≡ 0, f ∈ W k, p (Ω) and u is a weak solution of problem (1.3.19), then u ∈ W 2m+k, p (Ω). Thus all functions in Ker P are in C ∞ . (iii)A Priori Estimate There exist two positive constants C1 , C2 independent of u such that for any u ∈ W 2m+k, p (Ω) satisfying Bj u|Γ = 0, (j = 0, · · · , m − 1) in the trace sense, kukW k+2m, p ≤ C1 kP ukW k, p + C2 kukLp .

(1.3.25)

Moreover, if Ker P = 0, i.e., the uniqueness of problem (1.3.19) holds, then there is a positive constant C3 independent of u such that kukW k+2m, p ≤ C3 kP ukW k, p .

(1.3.26)

Preliminaries

15

REMARK 1.3.3 For the nonhomogeneous boundary value problem (1.3.19) with gj 6= 0, (j = 0, · · · , m−1) being in Sobolev spaces of fractional order, the similar wellposedness holds (see J.L. Lions and E. Magenes [95]).

The following Lax–Milgram theorem (e.g., see K. Yosida [155]) plays a very important role in the study of existence of solutions to linear elliptic boundary value problems. THEOREM 1.3.7 Let H be a Hilbert space, and B(x, y) be a functional mapping from H × H into a complex number. Suppose that the following conditions are satisfied: (i) (sesquilinearity) for any complex numbers αi , βi , (i = 1, 2),

B(α1 x1 + α2 x2 , y) = α1 B(x1 , y) + α2 B(x2 , y), B(x, β1 y1 + β2 y2 ) = β¯1 B(x, y1 ) + β¯2 B(x, y2 ); (ii) (boundedness) there is a positive constant K such that for all x, y ∈ H ,

|B(x, y)| ≤ KkxkH kykH ; (iii) (coerciveness) there is a positive constant δ such that for all x ∈ H ,

B(x, x) ≥ δkxk2H . Then there is a unique bounded linear operator S from H into H that has a bounded inverse operator S −1 such that for all x, y ∈ H ,

(x, y) = B(x, Sy). Moreover,

kSk ≤ δ −1 , kS −1 k ≤ K.

1.3.5

Interpolation Spaces

We recall a few facts about linear operators associated with a bilinear form and interpolation spaces associated with a positive definite operator in Hilbert spaces. Let V, H be separable Hilbert spaces such that V is dense in H and the injection V ,→ H is continuous and compact. Thus, by the Riesz representation theorem we can write V ,→ H ∼ (1.3.27) = H 0 ,→ V 0 . The dual product between V and V 0 is denoted by h , i and the inner product in H by ( , ).

16

NONLINEAR EVOLUTION EQUATIONS

Let A be a linear continuous operator from V to V 0 . We can associate it with a bilinear form a on V in such a way that a(u, v) = hAu, vi, ∀u, v ∈ V.

(1.3.28)

Suppose that a is symmetric: a(u, v) = a(v, u)

(1.3.29)

and a is coercive, i.e., there exists a positive constant α > 0 such that a(u, u) ≥ αkuk2V , ∀u ∈ V.

(1.3.30)

D(A) = {u| u ∈ V, Au ∈ H}.

(1.3.31)

Let Thus by the Lax–Milgram theorem, a(u, v) can be considered as an equivalent inner product. Furthermore, A is a strictly positive selfadjoint operator in H and the spectral theorem allows us to define the powers As of A for s ∈ R. Since we assume that the injection V ,→ H is compact, there exists (see [155] by K. Yosida) a complete orthonormal basis {wj } of H and a sequence {λj } such that wj ∈ D(A) and  Awj = λj wj ,          

j = 1, 2, · · · ,

0 < λ1 ≤ λ2 ≤ · · · ,

λj → ∞, as j → ∞,

a(wi , wj ) = λi δij ,

∀i, j ∈ N.

(1.3.32)

Thus, for s > 0, we define

¯  ¯∞  X ¯ D(As ) = u ∈ H ¯¯ λ2s |(u, wj )|2 < ∞ , j   ¯j=1  1 2 ∞ X 2s 2  kukD(As ) = λ |(u, wj )|  

j

j=1

(1.3.33)

(1.3.34)

and for negative s, D(As ) is the completion of H for the norm 

1

2 λ2s j |(u, wj )| j=1

kukD(A ) =  s

∞ X

1

2

.

In particular, we have V = D(A 2 ). Let X and Y be two Banach spaces, with X ⊂ Y, X dense in Y , and the injection X ,→ Y being continuous. In general, there are several different methods to define the intermediate spaces between X and Y .

Preliminaries

17

The framework described above gives a sort of definition of intermediate spaces, namely, the interpolation spaces. Let 1

X = V = D(A 2 ), Y = H = D(A0 ). a(u, v) = (u, v)X .

(1.3.35) (1.3.36)

Then interpolation spaces [X, Y ]θ , (0 ≤ θ ≤ 1) are given by [X, Y ]θ = D(A

1−θ 2 ),

∀θ ∈ [0, 1].

(1.3.37)

The interpolation inequality 1−θ kuk[X,Y ]θ ≤ C(θ)kukX kukθY , ∀u ∈ X, θ ∈ [0, 1]

(1.3.38)

also follows from (1.3.37). A typical example of the above framework is V = H01 (Ω), H = L2 (Ω), A = −∆

(1.3.39)

D(A) = H 2 (Ω) ∩ H01 (Ω)

(1.3.40)

and with Ω being a bounded domain of C 2 .

1.3.6

Notation and Some Useful Inequalities

Throughout this book we use the following common notation. 1. In addition to the notation ∂k , ∂tk

∂xα1 1

∂α , · · · ∂xαnn

we also use Dtk , Dα to denote the corresponding partial derivatives, i.e., Dtk =

∂k ∂α α , D = . ∂xα1 1 · · · ∂xαnn ∂tk

The subscripts t and x are often used to denote the partial derivatives with respect to t and x, respectively, i.e., utt =

∂2u ∂2v , vxx = , 2 ∂t ∂x2

etc. 2. We often use C, Ci (i ∈ N ) to denote a universal constant that may vary in different places.

18

NONLINEAR EVOLUTION EQUATIONS

In addition to the Gagliardo–Nirenberg and Poincar´e inequalities, the following elementary inequalities are very useful and will be frequently referred to in the remainder of the book (refer to [25] by E.F. Beckenbach and R. Bellman for the proofs): 1. The Young inequality 1 + 1 = 1. Then Let a, b and ε be positive constants and p, q ≥ 1, p q ab ≤

bq εp ap + q. p qε

(1.3.41)

2. The Jensen inequality Let ϕ(u) : u ∈ [α, β] 7→ R be a convex function. Suppose that f : t ∈ [a, b] 7→ [α, β], and P (t) are continuous functions with P (t) ≥ 0, P (t) 6≡ 0. Then the following inequality holds: ϕ

ÃR b a

f (t)P (t) dt

Rb a

P (t) dt

!

≤

Rb a

ϕ(f (t))P (t) dt Rb a

P (t) dt

.

(1.3.42)

3. The Gronwall inequality Suppose that a, b are nonnegative constants and u(t) is a nonnegative integrable function. Suppose that the following inequality holds for 0 ≤ t ≤ T: u(t) ≤ a + b

Z t 0

u(s) ds.

(1.3.43)

Then for 0 ≤ t ≤ T , u(t) ≤ aebt .

(1.3.44)

The following uniform Gronwall inequality plays an important role in the study of the asymptotic behavior of solutions to nonlinear evolution equations. 4. The Uniform Gronwall inequality Let g, h, y be three positive locally integrable functions on (t0 , +∞) such that y 0 is locally integrable on (t0 , +∞) and the following inequalities are satisfied: dy ≤ gy + h dt Z t+r t

g(s)ds ≤ a1 ,

Z t+r t

∀t ≥ t0 ,

h(s)ds ≤ a2 ,

Z t+r t

(1.3.45)

y(s)ds ≤ a3 ∀t ≥ t0 (1.3.46)

Preliminaries where r, ai , (i = 1, 2, 3) are positive constants. Then µ ¶ a3 y(t + r) ≤ + a2 ea1 ∀t ≥ t0 . r

1.4

19

(1.3.47)

Bibliographic Comments

We refer to the books [1] by R.A. Adams and [95] by J.L. Lions and E. Magenes for the detailed descriptions on Sobolev spaces and their properties. The book [95] contains comprehensive study on elliptic boundary value problems as well as evolution equations with extensive references. We refer to the paper [115] by L. Nirenberg for the proof of the the Gagliardo–Nirenberg inequality; see also the book [54] by A. Friedman. We refer to the papers [72]–[77] by T. Kato for nonlinear evolution equations with applications to nonlinear parabolic equations as well as nonlinear hyperbolic equations. For the recent development in the nonlinear semigroup theory with applications to nonlinear parabolic systems with nonlinear boundary conditions, see the papers [8]–[17] by H. Amann. The books [52] by A. Friedman and [89] by N.V. Krylov contain thorough treatment of the linear parabolic equations. We also refer to the book [90] by O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Uralceva for the study of both linear and nonlinear parabolic equations. See also the book [164] by S. Zheng for a lot of results on global existence and uniqueness for nonlinear parabolic equations and hyperbolicparabolic coupled systems. We refer to the book [122] by L.E. Payne for the several useful methods to study blow-up of solutions. In the literature, there are many papers regarding blow-up results. See, e.g., the papers [91], [92] by H.A. Levine; the papers [57], [58] by R.T. Glassey; the papers [52], [56] by H. Fujita; the papers [21], [22] by J.M. Ball for easy access. Concerning the study of asymptotic behavior of solution to nonlinear evolution equations, we refer to the paper [135] by W. Shen and S. Zheng for a useful lemma. See also the papers [87] by P. Krejci and J. Sprekels and [165] by S. Zheng in this aspect. Concerning convergence of solution to a stationary solution as time goes to infinity, we refer to the paper [141] by L. Simon and more recent developments stated in

20

NONLINEAR EVOLUTION EQUATIONS

the bibliographical comments section in Chapter 6. Concerning the study of infinite-dimensional dynamical systems, we refer to the books [151] by R. Temam; [59] by J.K. Hale; [20] by A.V. Babin and M.I. Vishik for the comprehensive study of global attractors, inertial manifolds, etc. The book [94] by J.L. Lions gives detailed descriptions on several important methods in the study of nonlinear evolution equations. These include the compactness method and monotone operator method as well as the penalty method, which is also useful for the study of evolution variational inequality. There are several other important methods, which we will not discuss in this book, for the study of nonlinear evolution equations. These include the compensated compactness method (see, e.g., the papers [149] by T. Tartar and [113] by F. Murat, and the book [103] by Y. Lu); the concentrated compactness method (see, e.g., the papers [96], [97] by P.L. Lions) and other weak convergence methods (see e.g., the book [49] by L.C. Evans); and the viscosity solution method (see, e.g., the paper [36] by M.G. Crandall, H. Ishii and P.L. Lions).

Chapter 2 Semigroup Method

The semigroup method is a powerful tool for solving evolution equations. It can be used to deal with many initial value problems or initial boundary value problems for both linear and nonlinear evolution equations, and we will introduce it in this chapter in detail. Sections 2.1–2.4 are mainly concerned with the C0 -semigroups of linear contractions, and in Section 2.5 we are devoted to semilinear problems. In order to apply the semigroup method to second-order parabolic equations in the general form, in the sixth section, we briefly introduce the results on the C0 -semigroups of bounded operators and analytic semigroups. Some applications are given in Section 2.7. Some references for further reading are given in the last section on bibliographic comments.

2.1 2.1.1

Semigroups of Linear Contraction Operators Semigroups

To introduce the concept of semigroups of linear contraction operators, let us first look at a simple example. Consider the following initial boundary value problem for the heat equation:   ut − uxx = 0,           

u|x=0 = u|x=π = 0,

(2.1.1)

u|t=0 = u0 (x) ∈ L2 (0, π).

Applying the method of separate variables, it is easy to see that the solution to the above problem can be expressed as a Fourier series: u(x, t) =

∞ X

ak e−k t sin kx 2

(2.1.2)

k=1

21

22

NONLINEAR EVOLUTION EQUATIONS

where ak =

2 π

Since it is assumed that u0 ∈ ∞ X

Z π 0 L2 ,

a2k =

k=1

u0 (x) sin kxdx.

2 π

(2.1.3)

by the Parseval equality, Z π 0

u20 (x)dx.

(2.1.4)

Since the series given by (2.1.2) has a fast decay factor e−k t for t > 0, it can be easily proved that u(x, t) given by this series is infinitely time differentiable with respect to x and t for t > 0, and it satisfies the equation and boundary conditions for (2.1.1). Moreover, we have 2

Z π 0

≤

(u(x, t) − u0 (x))2 dx =

∞ πX

Z π X ∞ 0

|

ak (e−k t − 1) sin kx|2 dx 2

k=1

a2k (e−k t − 1)2 → 0, 2

(2.1.5)

2 k=1

as t → 0. In other words, as t → 0, u(x, t) converges to u0 (x) in L2 (0, π) sense. It can be easily seen from the energy method that solution u(x, t) is uniquely determined by u0 . Thus we can view u(x, t) as the image of u0 (x) under a mapping S(t), i.e., u(x, t) = S(t)u0 (x) =

∞ X

ak e−k t sin kx. 2

(2.1.6)

k=1

We infer from this definition that for any t ∈ [0, +∞), S(t) is a linear operator from L2 (0, π) to L2 (0, π). Now we investigate the properties of S(t). By definition (2.1.6), (i) S(0) = I. Moreover, for any t1 , t2 ≥ 0, S(t2 )S(t1 )u0 (x) = =

∞ X

∞ X

(ak e−k

2

t1

)e−k

2

t2

sin kx

k=1

ak e−k

2

(t1 +t2 )

sin kx = S(t1 + t2 )u0 (x).

k=1

Since u0 is any function in L2 (0, π), we have (ii) S(t1 + t2 ) = S(t2 )S(t1 ) = S(t1 )S(t2 ).

(2.1.7)

Semigroup Method

23

By Parseval’s inequality, we have ∞ πX 2 ku(·, t)k = a2k e−2k t 2 k=1 2

≤ e−2t

∞ πX a2 = e−2t ku0 k2 . 2 k=1 k

(2.1.8)

Therefore, for all t ≥ 0, (iii) kS(t)k ≤ 1. As in (2.1.5) we can show that (iv) For any u0 ∈ L2 (0, π), S(t)u0 ∈ C([0, +∞), L2 (0, π)), i.e., for t ∈ [0, ∞), u(x, t) can be viewed as an abstract continuous function valued in L2 (0, π). We usually call one-parameter operators {S(t); t ≥ 0} satisfying the above four properties as a one-parameter, strongly continuous semigroup of contractions, in short, a linear semigroup of contractions, or simply say that it is a C0 -semigroup in a Banach space. It follows from the previous argument that solving problem (2.1.1) alternatively is to find a linear semigroup of operators S(t). In general, in the definition of a semigroup of operators S(t), L2 should be replaced by a general Banach space B. Concerning the basic question on how to find S(t) for a given problem, in the following section we will see that actually S(t) can be generated from an infinitesimal generator A, which can be easily identified for a given problem.

2.1.2

The Infinitesimal Generator

Suppose {S(t); t ≥ 0} is a linear semigroup of contractions defined on Banach space B. We denote by D a subset of B such that for x ∈ D, S(t)x is differentiable at t = 0 from the right, i.e., ¯ ¾ ¯ S(h)x − x ¯ exists . x ∈ B ¯ lim h→+0 h

½

D=

(2.1.9)

For x ∈ D, we denote −Ax = lim

h→+0

S(h)x − x . h

(2.1.10)

24

NONLINEAR EVOLUTION EQUATIONS

It is clear that A is a linear operator defined on D, and it is usually called the infinitesimal generator of S(t). We will see later on that A has a very close relationship with S(t). By definition, once S(t) is known, then A is derived from (2.1.10). On the other hand, once A is given, then we will see later on that S(t) can be generated by A. REMARK 2.1.1 In some other books and papers in the literature, instead of A, the operator −A defined by (2.1.10) is called the infinitesimal generator of S(t).

First we prove some lemmas. LEMMA 2.1.1 For any x ∈ D, S(t)x ∈ C 1 ([0, +∞), B). Moreover, for t ≥ 0, Z t Z t

x − S(t)x =

0

AS(τ )xdτ =

0

S(τ )Axdτ,

(2.1.11)

and

d(S(t)x) + A(S(t)x) = 0. dt

(2.1.12)

Proof. For x ∈ D, and h > 0, we have S(t + h)x − S(t)x (S(h) − I)S(t)x S(h) − I = = S(t) x. (2.1.13) h h h Since as h → +0, the limit of the third term in the above exists, it turns out that the second term in the above has a limit, which implies that S(t)x ∈ D, and −AS(t)x = −S(t)Ax.

(2.1.14)

On the other hand, from S(t − h)(S(h) − I)x S(t)x − S(t − h)x = (2.1.15) h h and the fact that S(t)x ∈ C([0, +∞), B), we can deduce that as h → 0, (2.1.15) has a limit, which should be −S(t)Ax. Combining the previous discussions yields that S(t)x ∈ C 1 ([0, +∞), B) and d (S(t)x) = −S(t)Ax = −AS(t)x. (2.1.16) dt Integrating with respect to t, we get (2.1.11). Thus the proof is complete. ¤

Semigroup Method

25

REMARK 2.1.2 (2.1.12) indicates that for x ∈ D, u = S(t)x is a classical solution to the following initial value problem for the abstract firstorder evolution equation:  du   + Au = 0,  dt (2.1.17)   

u(0) = x.

In Section 2.7 we will see how an initial boundary value problem for a concrete linear evolution equation can be converted into the initial value problem for the first-order abstract evolution equation. LEMMA 2.1.2

A is a closed operator. Proof. By definition of the closed operator, it suffices to prove that if xn ∈ D, xn → x in B, and Axn → y in B, then x ∈ D and y = Ax. For h > 0, by (2.1.11), we have xn − S(h)xn 1 = h h

Z h 0

AS(τ )xn dτ =

1 h

Z h 0

S(τ )Axn dτ.

(2.1.18)

Letting n → +∞, we get 1 1 (x − S(h)x) = h h

Z h 0

S(τ )ydτ → y,

as h → 0. Thus x ∈ D and Ax = y, and the proof is complete.

(2.1.19) ¤

For x ∈ D(A) = D, we introduce the graph norm: ³

kxkD(A) = kxk2 + kAxk2

´1 2

.

(2.1.20)

Throughout this chapter we simply denote by k·k the norm in B. Since A is a closed operator, it is easy to verify that D(A) equipped with the graph norm defined by (2.1.20) is a Banach space, continuously imbedded in B.

26

NONLINEAR EVOLUTION EQUATIONS

LEMMA 2.1.3

D is dense in B . Moreover, for any x ∈ B , t > 0, Z t

S(τ )xdτ ∈ D,

0

and

x − S(t)x = A Proof. Let xt =

Z t 0

Z t 0

S(τ )xdτ,

S(τ )xdτ.

(2.1.21)

∀x ∈ B.

(2.1.22)

Then for h > 0, R

R

t S(τ )xdτ − 0t S(τ + h)xdτ −S(h)xt + xt = 0 h h Rh R t+h S(τ )xdτ − t S(τ )xdτ = 0 → x − S(t)x, h

(2.1.23)

as h → 0. Therefore, xt ∈ D and Axt = x − S(t)x. This implies (2.1.21). On the other hand, for t > 0, we get xt 1 = t t

Z t 0

S(τ )xdτ → x,

(2.1.24)

as t → 0. Thus the density of D in B is proved. ¤ The above lemmas show that an infinitesimal generator of linear semigroup of contractions S(t) in a Banach space B must be a densely defined operator. In next subsection we will show what kind of densely defined operators can be an infinitesimal generator of a linear semigroup of contractions.

2.2

Hille–Yosida Theorem

The following Hille–Yosida theorem gives necessary and sufficient conditions for a densely defined operator to be an infinitesimal generator of a linear semigroup of contractions in a Banach space.

Semigroup Method

27

THEOREM 2.2.1 Let A be a linear operator defined in a Banach space B ,

A : D(A) ⊂ B 7→ B. Then the necessary and sufficient conditions for A being an infinitesimal generator of linear semigroup of contractions are: (i) A is a densely defined operator in B , (ii) for all λ > 0, λI + A is a one-to-one and onto mapping, and

k(λI + A)−1 k ≤

1 . λ

Proof. We first prove the necessity. If A is an infinitesimal generator of a linear semigroup of contractions S(t). Then by Lemma 2.1.2 and Lemma 2.1.3, A is a densely defined operator. For any λ > 0, {e−λt S(t); t ≥ 0} is also a linear semigroup of contractions that clearly has an infinitesimal generator λI + A with the same domain D. Applying (2.1.11) and (2.1.21) to this semigroup, we get for all x ∈ D and y ∈ B, and t ≥ 0: x−e

−λt

S(t)x =

Z t 0

e−λτ S(τ )(λI + A)xdτ,

y − e−λt S(t)y = (λI + A)

Z t

e−λτ S(τ )ydτ.

0

(2.2.1) (2.2.2)

Since ke−λt S(t)yk ≤ e−λt kyk, and the latter is integrable on [0, +∞), it follows that Z ∞ 0

exists and lim

Z t

t→+∞ 0

e

−λτ

e−λτ S(τ )ydτ

S(τ )ydτ =

Z ∞ 0

e−λτ S(τ )ydτ.

(2.2.3)

Let t → +∞ in (2.2.2). Since Z t 0

e−λτ S(τ )ydτ ∈ D,

and the term on the left-hand side of (2.2.2) has the limit y as t → +∞, it follows from λI + A being a closed operator that Z ∞ 0

e−λτ S(τ )ydτ ∈ D

28

NONLINEAR EVOLUTION EQUATIONS

and y = (λI + A)

Z ∞ 0

e−λτ S(τ )ydτ

(2.2.4)

which implies that λI + A is an operator from D onto B. On the other hand, letting t → +∞ in (2.2.1), we obtain x=

Z ∞ 0

e−λτ S(τ )(λI + A)xdτ,

(2.2.5)

which implies that λI + A is a one-to-one operator from D onto B. Thus we infer from (2.2.4) that −1

k(λI + A) ≤

Z ∞ 0

yk ≤ k

e−λτ dτ kyk =

Z ∞ 0

e−λτ S(τ )ydτ k

1 kyk, λ

(2.2.6)

i.e., k(λI + A)−1 k ≤

1 . λ

(2.2.7)

Thus, the proof of necessity is complete. The proof of sufficiency consists of three parts: (i) construct a sequence of bounded operators Aλ to approximate A; (ii) construct linear semigroups of contractions Sλ (t) = e−tAλ corresponding to Aλ ; (iii) prove that the limit lim Sλ (t) = S(t)

λ→0

exists, and S(t) is the desired linear semigroup of contractions. In what follows we give the proof along these lines. (i) For any λ > 0, by the previous result, I +λA is a one-to-one mapping from D(A) ⊂ B onto B. Therefore, Jλ = (I + λA)−1

(2.2.8)

is a linear bounded operator from B into itself. Jλ is usually called a resolvent operator. Let 1 (2.2.9) Aλ = (I − Jλ ), λ which is usually called the Yosida approximation of A. Since Jλ is a linear bounded operator, for any λ > 0, Aλ is also a linear bounded operator. For Aλ and Jλ , we have the following results.

Semigroup Method

29

LEMMA 2.2.1 (1) For any y ∈ B , and λ > 0,

Aλ y = A(Jλ y).

(2.2.10)

(2) For any y ∈ D(A) and λ > 0,

Aλ y = Jλ (Ay).

(2.2.11)

kAλ yk ≤ kAyk.

(2.2.12)

lim Jλ y = y.

(2.2.13)

lim Aλ y = Ay.

(2.2.14)

(3) For any y ∈ D(A), (4) For any y ∈ B , λ→0

(5) For any y ∈ D(A), λ→0

Proof. By (2.2.8) and (2.2.9), 1 1 Aλ y = (I − Jλ )y = ((I + λA) − I)Jλ y = AJλ y. (2.2.15) λ λ Thus (2.2.10) is proved. (2.2.11) can be proved in the same way. It follows from (2.2.7) that µ

¶−1

1 k I+ A λ

k ≤ 1.

(2.2.16)

Thus, kJλ k ≤ 1.

(2.2.17)

Combining (2.2.17) with (2.2.11) yields (2.2.12). To prove (2.2.13), we first observe that for any y ∈ D(A), kJλ y − yk = k(I − (I + λA))Jλ yk = λkA(Jλ y)k = λkAλ yk ≤ λkAyk → 0,

(2.2.18)

as λ → 0. Next, for any given y ∈ B, since D(A) is dense in B, for any ε > 0, there is yA ∈ D(A) such that ε kyA − yk ≤ . (2.2.19) 3 Therefore, kJλ y − yk = kJλ y − Jλ yA − y + yA + Jλ yA − yA k ≤ kJλ kky − yA k + ky − yA k + kJλ yA − yA k ε ε ε ≤ + + =ε (2.2.20) 3 3 3

30

NONLINEAR EVOLUTION EQUATIONS

as long as λ is sufficiently small. Thus, (2.2.13) is proved. Combining (2.2.13) with (2.2.11) yields (2.2.14). ¤ (ii) We now continue the proof of the present theorem. Since Aλ is a linear bounded operator, the one-parameter operators defined as Sλ (t) = e

−tAλ

=

∞ X (−tAλ )n n=0

n!

(2.2.21)

make sense for any given λ > 0 and all t ≥ 0. In the following we prove that Sλ (t) is a C0 -semigroup of contractions on B. It is easy to see from the definition of Sλ (t) given by (2.2.21) that Sλ (0) = I,

Sλ (t1 + t2 ) = Sλ (t1 )Sλ (t2 )

(2.2.22)

for any t1 , t2 ∈ R, which also implies that Sλ (t) is a group. Strong continuity of Sλ (t) also easily follows from the definition. Now it remains to verify that Sλ (t) are linear operators of contractions. Indeed, −t

t

kSλ (t)k = ke−tAλ k = e λ ke λ Jλ k µ ¶ −t −t t t kJλ k ≤ e λ e λ = 1. ≤ e λ exp λ

(2.2.23)

We infer from the definition that Sλ (t) is C ∞ differentiable with respect to t. Moreover, for any x ∈ B, d (Sλ (t)x) = −Aλ Sλ (t)x = −Sλ (t)Aλ x. dt (iii) We now prove that the limit

(2.2.24)

lim Sλ (t)x

λ→0

exists for any x ∈ B. For any y ∈ D(A), and λ > 0, µ > 0, Sλ (t)y − Sµ (t)y = =

Z t 0

Z t d 0

ds

(Sµ (t − s)Sλ (s)y) ds

Sµ (t − s)Sλ (s)(Aµ y − Aλ y)ds.

(2.2.25)

By (2.2.14), for y ∈ D(A), Aλ y → Ay, as λ → 0. Thus it follows from (2.2.25) that Sλ (t)y converges in B. Furthermore, the convergence is uniform with respect to t ∈ [0, T ] for any T > 0. Since Sλ (t) are operators of contraction, we easily deduce that for any x ∈ B, Sλ (t)x converges in B, and the convergence is uniform with respect to t in any finite interval.

Semigroup Method

31

Now we define S(t) as follows. For x ∈ B, S(t)x = lim Sλ (t)x. λ→0

(2.2.26)

It is easy to see from the definition that S(t) are linear operators from B to B. Contraction of S(t) easily follows from that of Sλ (t). Since Sλ (t)x ∈ C([0, +∞), B) and the convergence in (2.2.26) is uniform with respect to t in any finite interval, S(t)x ∈ C([0, +∞), B) follows. Furthermore, since Sλ (t) is a semigroup of contractions and the convergence in (2.2.26) is uniform with respect to t in any finite interval, we can deduce that S(t) is also a semigroup of contractions. Thus, {S(t); t ≥ 0} is a C0 -semigroup of contractions in B. To finish the proof of the present theorem, it remains to prove that A is the infinitesimal generator of S(t). For any y ∈ D(A), it follows from Lemma 2.2.1 and (2.2.26) that Sλ (t)Aλ y → S(t)Ay,

(2.2.27)

as λ → 0. Taking the limit on both sides of the following equality Sλ (h)y − y = − yields S(h)y − y = −

Z h 0

Z h 0

Sλ (t)Aλ ydt

(2.2.28)

S(t)Aydt.

(2.2.29)

This implies that as h → +0, the limit of S(h)y − y h exists, and it equals −Ay. If we denote by Λ the infinitesimal generator of S(t), then the above indicates that D(A) ⊂ D(Λ) and for y ∈ D(A), Ay = Λy. In other words, Λ is an extension of A. By the assumption, I + A is a surjective mapping. On the other hand, it follows from the proof of necessity that I + Λ is injective. Therefore, D(A) = D(Λ) and A = Λ. Thus the proof is complete. ¤ We now introduce the following definition for the sake of convenience. DEFINITION 2.2.1 Let A be a linear operator defined in a Banach space B , A : D(A) ⊂ B 7→ B . If for any x, y ∈ D(A) and any λ > 0,

kx − yk ≤ kx − y + λ(Ax − Ay)k,

(2.2.30)

then A is said to be an accretive operator. Moreover, if A is a densely defined accretive operator, and I + A is surjective, i.e., R(I + A) = B , then A is

32

NONLINEAR EVOLUTION EQUATIONS

said to be a maximal accretive operator, in short, m-accretive. LEMMA 2.2.2 If A is m-accretive, then A is a closed operator, and for all λ > 0, R(I + λA) = B , and

k(I + λA)−1 k ≤ 1.

(2.2.31)

Proof. We first prove that A is a closed operator. Let xn ∈ D(A), xn → x, Axn → y. Then xn + Axn = (I + A)xn → x + y.

(2.2.32)

By (2.2.30), (I + A)−1 is a continuous operator from B into D(A) ⊂ B. Therefore, x = (I + A)−1 (x + y).

(2.2.33)

This shows that x ∈ D(A) and Ax = y, i.e., A is a closed operator. To prove that R(I + λA) = B, it suffices to show that if for λ0 > 0, R(I + λ0 A) = B, then for any λ > λ20 , R(I + λA) = B holds. This amounts to showing that for any y ∈ B, and λ > λ20 , equation x + λAx = y

(2.2.34)

is solvable. Notice that (2.2.34) can be rewritten in the following form: x + λ0 Ax =

λ0 λ0 y + (1 − )x, λ λ

which is equivalent to −1

x = (I + λ0 A)

µ

(2.2.35) ¶

λ0 λ0 y + (1 − )x λ λ

(2.2.36)

because (I +λ0 A) is invertible. By the well-known contraction mapping theorem, (2.2.36) has a unique solution x ∈ D(A) provided |1 − i.e., λ >

λ0 2 .

λ0 | < 1, λ

Thus the proof is complete.

¤

REMARK 2.2.1 By Definition 2.2.1 and Lemma 2.2.2, the necessary and sufficient condition for A being an infinitesimal operator of the linear strongly continuous semigroup of contractions is that A is m-accretive.

Semigroup Method

33

LEMMA 2.2.3 Suppose that H is a Hilbert space. Then the necessary and sufficient conditions for A being m-accretive are: (i) Re(Ax, x) ≥ 0 for all x ∈ D(A), (ii) R(I + A) = H .

Proof. We first prove the necessity. By (2.2.30), (x, x) = kxk2 ≤ kx + λAxk2 = (x, x) + 2λRe(Ax, x) + λ2 kAxk2 .

(2.2.37)

Thus, for all λ > 0, λ Re(Ax, x) ≥ − kAxk2 . (2.2.38) 2 Letting λ → 0, we get (i). Furthermore, (ii) immediately follows from the fact that A is m-accretive. We now prove the sufficiency. It follows from (i) that for all λ > 0, kx − yk2 ≤ Re(x − y, x − y + λA(x − y)) ≤ kx − ykkx − y + λ(Ax − Ay)k,

(2.2.39)

i.e., (2.2.30) holds. Now it remains to prove that A is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element x0 belonging to the orthogonal supplement of D(A) such that for all x ∈ D(A), (x, x0 ) = 0.

(2.2.40)

It follows from (ii) that there is x∗ ∈ D(A) such that x∗ + Ax∗ = x0 . Taking the inner product of (2.2.41) with

x∗ ,

(2.2.41)

we deduce that

(x∗ + Ax∗ , x∗ ) = 0.

(2.2.42)

Taking the real part of (2.2.42), we deduce that x∗ = 0, and by (2.2.41), x0 = 0, a contradiction. Thus the proof is complete. ¤ DEFINITION 2.2.2 Suppose that A is a linear operator from a Hilbert space H into itself. A is said to be a monotone operator if it satisfies

Re(Ax − Ay, x − y) ≥ 0,

∀x, y ∈ D(A).

(2.2.43)

A is said to be a maximal monotone operator if A is monotone and it satisfies R(I + A) = H .

34

NONLINEAR EVOLUTION EQUATIONS

It can be seen from Definition 2.2.2 and Lemma 2.2.3 that in a Hilbert space, an accretive operator or m-accretive operator amounts to a monotone operator or a maximal monotone operator, respectively. Consider the following initial value problem for the abstract firstorder evolution equation:  du   + Au = 0,    

dt

(2.2.44)

u(0) = u0 .

where A is an m-accretive operator in a Banach space B, and u0 ∈ D(A). Then we have the following solvability theorem for problem (2.2.44). THEOREM 2.2.2 Suppose that A is m-accretive in a Banach space B , and u0 ∈ D(A). Then problem (2.2.44) has a unique classical solution u such that

u ∈ C([0, +∞), D(A)) ∩ C 1 ([0, + ∞), B). Moreover, the following estimates hold:

ku(t)k ≤ ku0 k, ∀t ≥ 0, du k (t)k ≤ kAu0 k, ∀t > 0, dt

(2.2.45) (2.2.46)

where D(A) is understood as a Banach space equipped with the graph norm given by (2.1.20).

Proof. By Lemma 2.1.1 and Theorem 2.2.1, u(t) = S(t)u0 is a classical solution in the required class. To prove the uniqueness, we use a contradiction argument. Suppose that there are two solutions u1 and u2 . Then u = u1 − u2 ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), B) and it satisfies

 du   + Au = 0,    

dt

(2.2.47)

u(0) = 0.

For any T > 0, let φ(t) = S(t)u(T − t)

∀t ∈ [0, T ].

(2.2.48)

Semigroup Method

35

Then by Lemma 2.1.1, we have dφ du(T − t) = −S(t) − S(t)Au(T − t) dt dt = S(t)Au(T − t) − S(t)Au(T − t) = 0.

(2.2.49)

u(T ) = φ(0) = φ(T ) = S(T )u(0) = 0,

(2.2.50)

Thus, and the uniqueness follows. Since u = S(t)u0 , (2.2.45) immediately follows. On the other hand, we notice that du = −Au = −AS(t)u0 = −S(t)Au0 . (2.2.51) dt Thus, (2.2.46) follows, and the whole proof is complete. ¤ REMARK 2.2.2 Suppose A is a linear bounded operator from Banach space B into B . Then

S(t) = e−tA =

∞ X (−tA)n n=0

n!

is a C0 -semigroup with A being its infinitesimal generator. Thus for general C0 -semigroup S(t) with A being its infinitesimal generator, in the literature, it is still often formally written as

S(t) = e−tA .

2.3

Regularities of Solutions

For the initial value problem of the abstract first-order evolution equation (2.2.44), if the initial datum u0 is more regular, then the solution u is expected to be more regular. We first introduce the following definition. DEFINITION 2.3.1 k

Let

D(A ) = {u| u ∈ D(Ak−1 ), Au ∈ D(Ak−1 ), k ∈ N }

36

NONLINEAR EVOLUTION EQUATIONS = {u|Aj u ∈ B, j = 0, · · · , k, A0 u = u}

(2.3.1)

equipped with the norm 

kukD(Ak ) = 

k X

1 2

j

2

kA uk

.

(2.3.2)

j=0

Since A is a closed operator, then it is easy to verify that D(Ak ) is a Banach space. THEOREM 2.3.1 Suppose that u0 ∈ D(Ak ) and A is m-accretive. Then the solution u of problem (2.2.44) belongs to k \

C k−j ([0, +∞), D(Aj )).

j=0

Proof. We use the induction argument to prove this theorem. For k = 1, the assertion has been proved by Theorem 2.2.2. Now suppose that the assertion is true for k − 1 for any given k > 1, we want to prove that the assertion holds for k. Consider the following auxiliary problem:    dv + Av = 0,    

dt

(2.3.3)

v(0) = −Au0 ∈ D(Ak−1 ),

where u0 ∈ D(Ak ). By induction, problem (2.3.3) has a unique solution: v(t) = −S(t)Au0 ∈

k−1 \

C k−1−j ([0, +∞), D(Aj )).

(2.3.4)

j=0

Let u(t) = u0 +

Z t 0

v(τ )dτ = u0 −

Z t 0

S(τ )Au0 dτ.

(2.3.5)

Clearly, k−1 \ du = v(t) ∈ C k−1−j ([0, +∞), D(Aj )). dt j=0

(2.3.6)

Semigroup Method

37

On the other hand, we infer from (2.3.5) that u(t) = u0 −

Z t 0

Z t du

dτ = u(t).

(2.3.7)

\ du k−1 ∈ C k−1−j ([0, +∞), D(Aj )). dt j=0

(2.3.8)

AS(τ )u0 dτ = u0 +

0

dτ

Therefore, Au(t) = −

Combining (2.3.6) with (2.3.7) and (2.3.8) yields the desired regularity. Thus the proof is complete. ¤ REMARK 2.3.1 For any u0 ∈ B that does not belong to D(A), in general, S(t)u0 is not necessarily differentiable, and neither belongs to D(A) for t > 0.

The following is such an example. Ex 2.3.1. Let B = {u(x)|u(x) ∈ C[0, 1], u(0) = 0} equipped with the following norm: kukB = max |u(x)|.

(2.3.9)

D(A) = {u|u ∈ C 1 [0, 1], u(0) = u0 (0) = 0}.

(2.3.10)

0≤x≤1

Let It is easy to see that D(A) is a dense set in B. Consider the operator A=

d dx

(2.3.11)

in D(A). To prove that A is m-accretive, it suffices to verify that for any f ∈ B, the problem  du   + u = f, λ   

dx

(2.3.12)

u(0) = 0.

has a unique solution in D(A), and k(I + λA)−1 k ≤ 1. Indeed, (2.3.12) has a unique solution that can be explicitly expressed as Z x ξ 1 1 e λ f (ξ)dξ. (2.3.13) u(x) = e− λ x λ 0

38

NONLINEAR EVOLUTION EQUATIONS

It is easy to see that u ∈ D(A), and 1 1 |u(x)| ≤ e− λ x λ

Z x 0

1

e λ ξ dξkf kB

1

= (1 − e− λ x )kf kB ≤ kf kB

(2.3.14)

which yields kukB ≤ kf kB ,

(2.3.15)

and we conclude that A is m-accretive. The corresponding evolution equation is  ∂u ∂u   + = 0,    

∂t

∂x

(2.3.16)

u|x=0 = 0, u|t=0 = u0 (x)

with u0 ∈ B. It is easy to solve this problem by using the characteristic line method, and the resulting solution is given by u(x, t) = S(t)u0 =

  u0 (x − t), 

0,

x − t ≥ 0, 0 ≤ x < t.

(2.3.17)

Thus if u0 is any continuous, but not differentiable function, then for any t ≥ 0, u(t) = S(t)u0 is not differentiable, and neither belongs to D(A). ¤ When u0 ∈ B, but does not belong to D(A), we still can view u = S(t)u0 as a sort of solution in the generalized sense, and it is usually called the mild solution in the literature. On the other hand, as we saw from the example for the heat equation in Section 2.1 that in some special cases, even when u0 ∈ B, for t > 0, S(t)u0 is still a classical solution. In what follows we consider the case when A is self-adjoint and B is a Hilbert space. Recall that if H is a Hilbert space and A is a densely defined operator from H into itself, then we can define the adjoint operator A∗ of A such that for all u ∈ D(A), v ∈ D(A∗ ): (Au, v) = (u, A∗ v). If D(A) =

D(A∗ ),

and A =

A∗ ,

(2.3.18)

then we say that A is self-adjoint.

THEOREM 2.3.2 Let H be a Hilbert space, and A be self-adjoint and m-accretive in H . Then

Semigroup Method

39

for any u0 ∈ H , problem (2.2.44) admits a unique classical solution u(t) = S(t)u0 such that

u ∈ C([0, +∞), H) ∩ C k ((0, +∞), D(Aj )), ∀k, j = 0, 1, · · · . (2.3.19) Moreover, the following estimates hold:

ku(t)k ≤ ku0 k, ∀t ≥ 0, k

du 1 (t)k = kAu(t)k ≤ ku0 k, ∀t > 0. dt t

(2.3.20) (2.3.21)

Proof. We use the density argument to prove this theorem. Since D(A), and also D(Ak ), k = 1, 2, · · · is dense in H, for the time being we assume that u0 ∈ D(A2 ). Then by Theorem 2.3.1, problem (2.2.44) admits a unique classical solution u = S(t)u0 such that u ∈ C 2 ([0, +∞), H) ∩ C 1 ([0, +∞), D(A)) ∩ C([0, +∞), D(A2 )). Now we would like to derive the estimates (2.3.20), and (2.3.21) for such a classical solution. Once it has been done, then we use the density argument to pass to the limit. Indeed, (2.3.20) has already been derived in Theorem 2.2.2. Therefore, it remains to prove (2.3.21). First we take the inner product of the equation du + Au = 0 dt with u to obtain the following equality: 1d ku(t)k2 + Re(Au, u) = 0. 2 dt

(2.3.22)

(2.3.23)

Since Re(Au, u) ≥ 0, it follows that ku(t)k2 decreases with respect to t. Next, let v=

du . dt

Then v satisfies dv + Av = 0. dt

(2.3.24)

In the same argument as above, kvk2 also decreases with respect to t. Now we take the inner product of the equation (2.3.22) with tv, using the fact that A is self-adjoint, to obtain tk

du 2 1 d 1 k + (tRe(Au, u)) − Re(Au, u) = 0. dt 2 dt 2

(2.3.25)

40

NONLINEAR EVOLUTION EQUATIONS

Adding (2.3.23), (2.3.25) together, then integrating with respect to t yields Z

1 t 1 t Re(Au, u)dτ ku(t)k2 + Re(Au, u) + 2 2 2 0 Z t du 1 τ k (τ )k2 dτ = ku0 k2 . + dt 2 0

(2.3.26)

Since Re(Au, u) ≥ 0 and kvk2 decrease with respect to t, it follows from (2.3.26) that 1 2 du t k (t)k2 ≤ 2 dt

Z t 0

τk

du 1 (τ )k2 dτ ≤ ku0 k2 . dt 2

(2.3.27)

Thus, when u0 ∈ D(A2 ), (2.3.21) holds. (k) When u0 ∈ H, we can find a sequence of u0 ∈ D(A2 ) to approximate u0 . Let u(k) be the corresponding solution. Then it follows from (2.3.20) and (2.3.21) that for any large T > 0 and small δ > 0 such that (k) u(k) is a Cauchy sequence in C([0, T ], H), and ut as well as Au(k) is a Cauchy sequence in C([δ, T ], H). Thus the limit function u belongs to C([0, +∞), H) ∩ C 1 ((0, +∞), H) ∩ C((0, +∞), D(A)), and it is a classical solution to problem (2.2.44). To prove the uniqueness, let u1 , u2 be two solutions, and let u = u1 −u2 . Then we take the inner product of the equation of problem (2.2.44) with u to obtain 1d ku(t)k2 + Re(Au, u) = 0. (2.3.28) 2 dt Integrating with respect to t, and using Re(Au, u) ≥ 0, we obtain ku(t)k2 ≤ 0

∀t ≥ 0,

(2.3.29)

and the uniqueness follows. It remains to prove the regularity of u, i.e., u ∈ C k ((0, +∞), D(Aj )),

∀k, j = 0, 1, · · · .

We use the bootstrap argument. First if u0 ∈ D(A), then as for the proof of Theorem 2.3.1, let v be the solution to problem  dv   + Av = 0,  dt (2.3.30)   

v(0) = −Au0 ∈ H.

By the previous argument, v belongs to C([0, +∞), H) ∩ C 1 ((0, +∞), H) ∩ C((0, +∞), D(A)).

Semigroup Method

41

Therefore, as shown in the proof of Theorem 2.3.1, the solution u of problem (2.2.44), which can be expressed as u = u0 +

Z t 0

v(τ )dτ,

(2.3.31)

belongs to C 2 ((0, +∞), H) ∩ C 1 ((0, +∞), D(A)). Next, for the solution u of problem (2.2.44) and for any δ > 0, as proved before, u(δ) ∈ D(A), and u(t + δ) is a unique classical solution to the following problem:    dv + Av = 0,    

dt

(2.3.32)

v(0) = u(δ) ∈ D(A).

Then it follows from the previous argument that v(t) ∈ C 2 ((0, +∞), H) ∩ C 1 ((0, +∞), D(A)) ∩ C((0, +∞), D(A2 )), and u(t + δ) = v(t). Since δ is arbitrary, we conclude that u(t) ∈ C 2 ((0, +∞), H) ∩ C 1 ((0, +∞), D(A)) ∩ C((0, +∞), D(A2 )). Repeating this process yields u ∈ C k ((0, +∞), D(Aj )) for k, j = 1, 2, · · ·. Thus the proof is complete.

2.4

¤

Nonhomogeneous Equations

In this section we consider the initial value problem for the nonhomogeneous equation:  du   + Au = f (t),    

dt

u(0) = u0 ,

(2.4.1)

42

NONLINEAR EVOLUTION EQUATIONS

where A is still a maximal accretive operator defined in a dense subset D(A) of a Banach space B, and u0 ∈ D(A). First we have the following result. THEOREM 2.4.1 Suppose that

f (t) ∈ C 1 ([0, +∞), B),

u0 ∈ D(A).

Then problem (2.4.1) admits a unique classical solution u such that

u ∈ C 1 ([0, +∞), B) ∩ C([0, +∞), D(A)) which can be expressed as

u(t) = S(t)u0 +

Z t 0

S(t − τ )f (τ )dτ.

(2.4.2)

Proof. Since S(t)u0 satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that w(t) given by w(t) =

Z t 0

S(t − τ )f (τ )dτ

(2.4.3)

belongs to C 1 ([0, +∞), B) ∩ C([0, +∞), D(A)) and satisfies the nonhomogeneous equation. Consider the following quotient of difference w(t + h) − w(t) h 1 = h =

1 h

=

1 h

ÃZ

t+h

0

Z t+h t

Z t+h t

S(t + h − τ )f (τ )dτ −

S(t + h − τ )f (τ )dτ +

1 h

S(z)f (t + h − z)dz +

1 h

Z t 0

!

S(t − τ )f (τ )dτ

Z t 0

Z t 0

(S(t + h − τ ) − S(t − τ ))f (τ )dτ S(z)(f (t + h − z) − f (t − z))dz. (2.4.4)

When h → 0, the terms in the last line of (2.4.4) have limits: S(t)f (0) +

Z t 0

S(z)f 0 (t − z)dz ∈ C([0, +∞), B).

(2.4.5)

Semigroup Method

43

It turns out that w ∈ C 1 ([0, +∞), B) and the terms in the third line of (2.4.4) have limits, too, which should be S(0)f (t) − Aw(t) = f (t) − Aw(t).

(2.4.6)

Thus the proof is complete.

¤

From (2.4.3) we immediately have the following result. COROLLARY 2.4.1 If

f (t) ∈ C([0, +∞), D(A)), u0 ∈ D(A), then the function u(t) given by (2.4.2) is a classical solution.

The following result shows that the condition in Theorem 2.4.1 can be slightly weakened. COROLLARY 2.4.2 If

u0 ∈ D(A), f ∈ C([0, +∞), B) and for any T > 0,

f 0 ∈ L1 ([0, T ], B), then the function u given by (2.4.2) is a classical solution.

Proof. We first prove that for any g ∈ L1 ([0, T ], B), the function w given by the following integral: w(t) =

Z t 0

S(t − τ )g(τ )dτ

(2.4.7)

belongs to C([0, T ], B). Indeed, we infer from the difference w(t + h) − w(t) = = (S(h) − I)w(t) +

Z t+h

Z t

0 t+h

S(t + h − τ )g(τ )dτ −

S(t + h − τ )g(τ )dτ

that as h → 0, kw(t + h) − w(t)k ≤ k(S(h) − I)w(t)k +

Z t+h t

Z t 0

S(t − τ )g(τ )dτ (2.4.8)

kg(τ )kdτ → 0 (2.4.9)

44

NONLINEAR EVOLUTION EQUATIONS

where we have used the strong continuity of S(t) and the absolute continuity of integral for kgk ∈ L1 [0, T ]. Now it can be seen from the last line of (2.4.4) that for almost every t ∈ [0, T ], dw dt

exists, and it equals S(t)f (0) + = S(t)f (0) +

Z t 0

Z t 0

S(z)f 0 (t − z)dz S(t − τ )f 0 (τ )dτ ∈ C([0, T ], B).

(2.4.10)

Thus, for almost every t, dw = −Aw + f. (2.4.11) dt Since w and f both belong to C([0, T ], B), it follows from (2.4.11) that for almost every t, Aw equals a function belonging to C([0, T ], B). Since A is a closed operator, we conclude that w ∈ C([0, T ], D(A)) ∩ C 1 ([0, T ], B) and (2.4.11) holds for every t. Thus the proof is complete.

¤

It can be seen that when f (t) ∈ C([0, T ], B) or even f ∈ L1 ([0, T ], B), w(t) still belongs to C([0, T ], B). In this case u(t) defined by (2.4.13) no longer is necessarily a classical solution, and we call it a mild solution. Now we give an example to show that only assumption f (t) ∈ C([0, +∞), B) is not enough to guarantee that problem (2.4.1) admits a classical solution. Ex. 2.4.1. We still consider the Banach space B and the operator d A= dx as shown in Ex. 2.3.1. For any g that belongs to B, but does not belong to D(A), we take f (t) = S(t)g where S(t) is given by (2.3.17). Therefore, f ∈ C([0, +∞), B),

Semigroup Method

45

but for all t ≥ 0, f (t) does not belong to D(A). For u0 = 0, it is easy to see that the corresponding problem (2.4.1) has a unique mild solution u(t) = tS(t)g,

(2.4.12)

i.e., u satisfies the integral equation u(t) = S(t)u0 +

Z t 0

S(t − τ )f (τ )dτ

But this mild solution is not a classical solution.

(2.4.13) ¤

When B is a reflexive Banach space, we introduce the following result which was established by Y. Komura and refer the proof to [85], [86]. LEMMA 2.4.1 Suppose that B is a reflexive Banach space and f is an abstract Lipschitz continuous function defined in [0, T ] and valued in B . Then f 0 ∈ L1 ([0, T ], B) and the following holds. Z t f (t) = f (0) + f 0 (τ )dτ. (2.4.14) 0

We infer from Lemma 2.4.1 and Corollary 2.4.2 the following result. COROLLARY 2.4.3 If B is a reflexive Banach space and u0 ∈ D(A), and if f is a Lipschitz continuous function from [0, T ] into B , then the function u given by (2.4.2) is a classical solution to problem (2.4.1).

In the same way as the proof of Corollary 2.4.2, we have the following result. COROLLARY 2.4.4 Suppose that B is a Banach space, f ∈ C([0, T ], B), and for almost every t, f (t) ∈ D(A). Moreover, Af (t) ∈ L1 ([0, T ], B). Then the function u given by (2.4.2) is a classical solution to problem (2.4.1).

Proof. It can be seen from (2.4.4) that for almost every t, the terms in the third line of (2.4.4) have limits, and the following holds: dw = S(0)f (t) + dt

Z t 0

S(t − τ )Af (τ )dτ ∈ C([0, T ], B).

(2.4.15)

46

NONLINEAR EVOLUTION EQUATIONS

From the third line of (2.4.4) we infer that w ∈ D(A) and the following holds for every t: dw = f (t) − Aw(t). dt Thus the proof is complete.

2.5

(2.4.16) ¤

Semilinear Equations

In this section we are concerned with the initial value problem for the semilinear evolution equation:  du   + Au = F (u),    

dt

(2.5.1)

u(0) = u0

where A is a maximal accretive operator from a dense subset D(A) in a Banach space B into B, and F is a nonlinear operator from B into B. As indicated in the previous section, even if F is a continuous operator from B to B, problem (2.5.1) does not necessarily have a classical solution. On the other hand, as indicated in Chapter 1, in general, problem (2.5.1) may not have a global solution, i.e., problem (2.5.1) is not solvable for all t > 0. Therefore, we first consider in this section when problem (2.5.1) admits a local classical (or even mild) solution. Then we further discuss when problem (2.5.1) will admit a global solution or blows up in a finite time. THEOREM 2.5.1 Suppose that F satisfies the global Lipschitz condition, i.e., there is a positive constant L such that for all u, v ∈ B ,

kF (u) − F (v)k ≤ Lku − vk.

(2.5.2)

Furthermore, suppose that u0 ∈ B . Then problem (2.5.1) admits a global mild solution u such that u belongs to C([0, +∞), B) and satisfies the following integral equation: Z t u(t) = S(t)u0 + S(t − τ )F (u(τ ))dτ. (2.5.3) 0

Moreover, let u(t), u ˆ(t) be the global mild solutions corresponding to u0 and

Semigroup Method

47

u ˆ0 . Then for all t ≥ 0, the following estimates hold: ku(t) − u ˆ(t)k ≤ eLt ku0 − u ˆ0 k.

(2.5.4)

Proof. We use the contraction mapping theorem to prove the present theorem. Two key steps for using the contraction mapping theorem are to figure out a closed set of the considered Banach space, and an auxiliary problem so that the nonlinear operator defined by the auxiliary problem maps from this closed set into itself, and turns out to be a contraction. In the following we proceed along this line. Let φ(v) = S(t)u0 +

Z t 0

S(t − τ )F (v(τ ))dτ

(2.5.5)

and E = {v ∈ C([0, +∞), B) | supkv(t)ke−kt < ∞ } t≥0

(2.5.6)

where k is a positive constant such that k > L. In E, we introduce the following norm: kukE = sup e−kt ku(t)k. t≥0

(2.5.7)

Clearly, E is a Banach space. We now show that the nonlinear operator φ defined by (2.5.5) maps E into itself, and the mapping is a contraction. Indeed, for v ∈ E, by (2.5.2), we have kφ(v)k ≤ kS(t)u0 k + ≤ ku0 k +

Z t 0

Z t 0

kS(t − τ )kkF (v)kdτ

kF (v)kdτ ≤ ku0 k +

Z t

≤ ku0 k + C0 t + L sup e−kt kv(t)k · t≥0

0

(Lkv(τ )k + kF (0)k)dτ

Z t 0

ekτ dτ

L ≤ ku0 k + C0 t + ekt kvkE k where C0 = kF (0)k. Thus,

(2.5.8)

kφ(v)kE ≤ sup((ku0 k + C0 t)e−kt ) + t≥0

i.e., φ(v) ∈ E. For v1 , v2 ∈ E, we have kφ(v1 ) − φ(v2 )kE = sup e t≥0

−kt

k

Z t 0

L kvkE < ∞, k

(2.5.9)

S(t − τ )(F (v1 (τ )) − F (v2 (τ )))dτ k

48

NONLINEAR EVOLUTION EQUATIONS

≤ sup e−kt L t≥0

Z t 0

L · (ekt − 1))kv1 − v2 kE k

kv1 − v2 kdτ ≤ sup (e−kt · t≥0

L (2.5.10) ≤ kv1 − v2 kE . k Therefore, by the contraction mapping theorem, (2.5.3) has a unique solution in E. To show that the uniqueness also holds in C([0, +∞), B), let u1 , u2 ∈ C([0, +∞), B) be two solutions of (2.5.3) and let u = u1 − u2 . Then u(t) =

Z t 0

S(t − τ )(F (u1 ) − F (u2 ))dτ.

ku(t)k ≤ L

Z t 0

ku(τ )kdτ.

(2.5.11)

(2.5.12)

By the Gronwall inequality stated in Chapter 1, we immediately conclude that ku(t)k = 0, i.e., the uniqueness in C([0, +∞), B) follows. It remains to prove the stability result (2.5.4). In the same way as above, we have u(t) − u ˆ(t) = S(t)(u0 − u ˆ0 ) +

Z t 0

S(t − τ )(F (u) − F (ˆ u))dτ.

Therefore, ku(t) − u ˆ(t)k ≤ ku0 − u ˆ0 k + L

Z t 0

ku − u ˆkdτ.

(2.5.13)

(2.5.14)

By the Gronwall inequality stated in Chapter 1, (2.5.4) follows. Thus the proof is complete. ¤

COROLLARY 2.5.1 Suppose that in Theorem 2.5.1, F satisfies the global Lipschitz condition and u0 ∈ D(A). Then the mild solution u must be Lipschitz continuous in t.

Proof. For any h > 0, let u ˆ(t) = u(t + h). Then it is easy to see that it is a mild solution to the equation (2.5.1) with initial data u(h). We infer from (2.5.4) that ku(t + h) − u(t)k = kˆ u(t) − u(t)k ≤ eLt ku(h) − u0 k. On the other hand, ku(h) − u0 k = kS(h)u0 − u0 +

Z h 0

S(h − τ )F (u)dτ k

(2.5.15)

Semigroup Method ≤k

Z h 0

S(τ )Au0 dτ k + kF (u0 )kh + L

≤ (kF (u0 )k + kAu0 k)h + L

Z h 0

49 Z h 0

ku − u0 kdτ

ku − u0 kdτ.

(2.5.16)

By the Gronwall inequality, we get ku(h) − u0 k ≤ (kF (u0 )k + kAu0 k)heLh .

(2.5.17)

Thus, for any t1 , t2 ∈ [0, T ], we infer from (2.5.15) and (2.5.17) that ku(t1 ) − u(t2 )k ≤ e2LT (kF (u0 )k + kAu0 k)|t1 − t2 |,

(2.5.18)

and the proof is complete.

¤

COROLLARY 2.5.2 Suppose that as in Theorem 2.5.1, F satisfies the global Lipschitz condition, and suppose that B is a reflexive Banach space, and u0 ∈ D(A). Then the mild solution u is a classical one.

Proof. By Corollary 2.5.1, the mild solution u is Lipschitz continuous in t. Since F satisfies the global Lipschitz condition, it follows that F (u(t)) is Lipschitz continuous in t. Then we immediately infer from Corollary 2.4.3 that the mild solution is a classical one. Thus the proof is complete. ¤ THEOREM 2.5.2 Suppose that F is a nonlinear operator from D(A) into D(A), and satisfies the global Lipschitz condition, i.e., there is a positive constant L such that for all u1 , u2 ∈ D(A),

kF (u1 ) − F (u2 )kD(A) ≤ Lku1 − u2 kD(A) .

(2.5.19)

Then for any u0 ∈ D(A), problem (2.5.1) admits a unique classical solution.

Proof. Let B1 = D(A), A1 = A : D(A1 ) = D(A2 ) 7→ B1 .

(2.5.20)

Then B1 is a Banach space, and A1 , i.e., A is a densely defined operator from D(A2 ) into B1 . In what follows we prove that A1 is m-accretive in B1 = D(A). Indeed, for any x, y ∈ D(A2 ), since A is accretive in B, we have kx − y + λ(Ax − Ay)kD(A) ³

= kx − y + λ(Ax − Ay)k2 + kAx − Ay + λ(A2 x − A2 y)k2

´1 2

50

NONLINEAR EVOLUTION EQUATIONS ³

≥ kx − yk2 + kAx − Ayk2

´1

2

= kx − ykD(A) ,

(2.5.21)

i.e., A1 is accretive in B1 . Furthermore, since A is m-accretive in B, for any y ∈ B, there is a unique x ∈ D(A) such that x + Ax = y.

(2.5.22)

Now for any y ∈ B1 = D(A), equation (2.5.22) admits a unique solution x ∈ D(A). It turns out that Ax = y − x ∈ D(A)

(2.5.23)

Thus x ∈ D(A2 ), i.e., A1 is m-accretive in B1 . Let S1 (t) be the semigroup generated by A1 . If u0 ∈ D(A2 ) = D(A1 ), then u(t) = S1 (t)u0 ∈ C([0, +∞), D(A2 )) ∩ C 1 ([0, +∞), D(A)) is unique classical solution of the problem:

 du   + Au = 0,    

dt

(2.5.24)

u(0) = u0

On the other hand, u(t) = S(t)u0 is also a classical solution in C([0, +∞), D(A)) ∩ C 1 ([0, +∞), B). This implies that S1 (t) is a restriction of S(t) on B1 . By Theorem 2.5.1, problem (2.5.1) admits a unique mild solution u ∈ C([0, +∞), B1 ) satisfying u(t) = S1 (t)u0 +

Z t 0

S1 (t − τ )F (u(τ ))dτ.

(2.5.25)

Since S1 (t) is a restriction of S(t) on D(A), then (2.5.25) can be rewritten as u(t) = S(t)u0 +

Z t 0

S(t − τ )F (u(τ ))dτ.

(2.5.26)

Moreover, we infer from F (u) being an operator from D(A) to D(A) and Corollary 2.4.1 that u is a classical solution to problem (2.5.1). Thus the proof is complete. ¤ THEOREM 2.5.3 Suppose that F ∈ C 1 (B, B), i.e., F (u) is a nonlinear operator from B into B, it is Frechet differentiable at any u ∈ B, and F 0 (u) is continuous at u. Suppose that F (u) satisfies the global Lipschitz condition (2.5.2). Then for any u0 ∈ D(A), problem (2.5.1) admits a unique global classical solution.

Semigroup Method

51

Proof. It is easy to see from the assumptions that for any u ∈ B, the Frechet derivative F 0 (u) of F is a bounded linear operator from B into B satisfying kF 0 (u)k ≤ L. By Theorem 2.5.1, problem (2.5.1) admits a unique global mild solution u ∈ C([0, +∞), B). By Theorem 2.4.1, it suffices to verify that u ∈ C 1 ([0, +∞), B). In doing so, let us consider the following prolonged problem:  dv   + Av = F 0 (u)v,  dt (2.5.27)   

v(0) = F (u0 ) − Au0 ∈ B

where u is the global mild solution to problem (2.5.1). Since F 0 (u) is a linear bounded operator from B to B, applying the same approach as in the proof of Theorem 2.5.1 yields that problem (2.5.27) admits a unique global mild solution v(t) ∈ C([0, +∞), B) satisfying v(t) = S(t)(F (u0 ) − Au0 ) +

Z t 0

S(t − τ )F 0 (u(τ ))v(τ )dτ.

(2.5.28)

Thus, we have

° ° ° ° u(t + h) − u(t) ° − v(t)° ° ° h ° Ã Z t+h °1 ° =° S(t)(S(h) − I)u0 + S(t + h − τ )F (u(τ ))dτ °h 0 ¶ Z t Z t

−

0

S(t − τ )F (u)dτ

− S(t)(F (u0 ) − Au0 ) −

0

° °

S(t − τ )F 0 (u)vdτ ° °

° Z t° ° ° ° ° ° F (u(τ + h)) − F (u(τ )) ° S(h)u0 − u0 0 ° ° dτ ° + Au − F (u)v + ≤° 0° ° ° ° h h 0 ° Z ° °1 h ° ° ° +° S(t + h − τ )F (u(τ ))dτ − S(t)F (u0 )° = I1 + I2 + I3 . (2.5.29) °h 0 °

We infer from u0 ∈ D(A) that as h → 0, I1 → 0.

(2.5.30)

It follows from continuity of u and F that as h → 0, I3 → 0. By the definition of Frechet derivative, we have

(2.5.31)

° ° ° F (u(τ + h)) − F (u(τ )) u(τ + h) − u(τ ) ° 0 ° ° − F (u) ° ° h h ° ° ° u(τ + h) − u(τ ) ° ° · o(ku(τ + h) − u(τ )k) (2.5.32) ≤° ° ° h

52

NONLINEAR EVOLUTION EQUATIONS

where o(ku(τ + h) − u(τ )k) in the above denotes that when ku(τ + h) − u(τ )k → 0, o(ku(τ + h) − u(τ )k) → 0. Since u0 ∈ D(A), we infer from Corollary 2.5.1 that u is Lipschitz continuous in t in any finite interval. Thus it follows that the right hand side term in (2.5.32) converges to zero as h → 0. The integrand of I2 can be rewritten as: ° ° ° F (u(τ + h)) − F (u(τ )) ° 0 ° − F (u)v ° ° ° h ° ° ° F (u(τ + h)) − F (u(τ )) u(τ + h) − u(τ ) ° 0 ° − F (u) ≤° ° ° h h ° µ ¶° ° 0 ° u(τ + h) − u(τ ) +° −v ° (2.5.33) °F (u) °. h

Combining (2.5.29)–(2.5.33) yields

° ° ° u(t + h) − u(t) ° ° − v(t)° ° ° h ° Z t° ° ° u(τ + h) − u(τ ) ° − v(τ )° ≤ ε1 + ε2 + ε3 + L ° dτ, ° h 0

(2.5.34)

where εi , i = 1, 2, 3 converges to zero as h → 0. Applying the Gronwall inequality yields ° ° ° u(t + h) − u(t) ° ° ° ≤ (ε1 + ε2 + ε3 )eLt − v(t) ° ° h

∀t ∈ [0, T ].

(2.5.35)

Thus, in any finite interval [0, T ], du = v(t), dt and the proof is complete.

(2.5.36) ¤

So far all results in this section have been obtained under the assumption that F satisfies the global Lipschitz condition. However, we should notice that this is a restrictive condition. If we let v = 0 in (2.5.2), then kF (u)k ≤ Lkuk + kF (0)k

(2.5.37)

for all u. In other words, the global Lipschitz condition implies that kF (u)k is allowed to have at most a linear growth as kuk goes to infinity. This certainly is a restrictive assumption and it rules out a lot of interesting applications. In what follows, we would like to significantly weaken this assumption such that F merely satisfies the following local Lipschitz condition:

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53

DEFINITION 2.5.1 Suppose F is a nonlinear operator from a Banach space B into B . F is said to satisfy the local Lipschitz condition if for any positive constant M > 0, there is a positive constant LM depending on M such that when u, v ∈ B , kuk ≤ M and kvk ≤ M ,

kF (u) − F (v)k ≤ LM ku − vk.

(2.5.38)

If F merely satisfies the local Lipschitz condition, then as seen from Chapter 1, in general only a local existence result can be obtained. THEOREM 2.5.4 Suppose that A is m-accretive, and F is a nonlinear operator from a Banach space B into B satisfying the local Lipschitz condition. Then for any u0 ∈ B , there is a positive constant T > 0 depending on ku0 k such that problem (2.5.1) in [0, T ] admits a unique local mild solution u ∈ C([0, T ], B) such that Z t u(t) = S(t)u0 + S(t − τ )F (u(τ ))dτ, ∀t ∈ [0, T ]. (2.5.39) 0

Furthermore, if u0 ∈ D(A), then u is Lipschitz continuous in t ∈ [0, T ]. If B is a reflexive Banach space, then u is a classical solution.

Proof. We still use the contraction mapping theorem to prove the present theorem. However, since F no longer satisfies the global Lipschitz condition, we have to use the smallness of the time interval to get contraction of the nonlinear mapping. Let L be the Lipschitz constant corresponding to M = ku0 k + 1. Let T be a positive constant such that T < L1 . Let E = {u ∈ C([0, T ], B) | ku(t)k ≤ M, ∀t ∈ [0, T ]}

(2.5.40)

equipped with the norm kukE = sup ku(t)k. 0≤t≤T

(2.5.41)

Then it is easy to see that E is a closed convex subset of Banach space C([0, T ], B). Consider the nonlinear mapping φ(u) = S(t)u0 +

Z t 0

S(t − τ )F (u(τ ))dτ

(2.5.42)

defined on E. For any u ∈ E, it is easy to obtain the following estimate: kφ(u)kE ≤ ku0 k + T (kF (0)k + LM ).

(2.5.43)

54

NONLINEAR EVOLUTION EQUATIONS

If T is chosen in such a way that µ ¶ 1 1 T < min , , L kF (0)k + LM

(2.5.44)

then φ(u) maps E into itself. Moreover, for any u1 , u2 ∈ E, kφ(u1 ) − φ(u2 )kE = sup k 0≤t≤T

Z t 0

S(t − τ )(F (u1 ) − F (u2 ))dτ k

≤ LT ku1 − u2 kE ,

(2.5.45)

i.e., φ(u) is a contraction in E. Thus φ has a fixed point that is the mild solution to problem (2.5.1). In the same manner as in the proof of Theorem 2.5.1, using the local Lipschitz condition of F and the Gronwall inequality yields the uniqueness. When u0 ∈ D(A), it follows from Corollary 2.5.1 that u is Lipschitz continuous in t ∈ [0, T ]. Furthermore, when B is a reflexive Banach space, we infer from Corollary 2.5.2 that this mild solution is also a classical one. Thus the proof is complete. ¤ The following result due to I. Segal [130] indicates the relationship between global nonexistence and blow-up of a certain norm of solution. THEOREM 2.5.5 Under the same assumptions as Theorem 2.5.4, the solution u can be extended to a maximal mild solution in [0, Tmax ) such that either (i) Tmax = +∞, i.e., problem admits a global mild solution, or (ii) Tmax < +∞, and

lim

ku(t)k = +∞,

t→Tmax −0

i.e., the solution blows up in a finite time Tmax .

Proof. Suppose that there are two solutions u1 , u2 defined on [0, T1 ], and [0, T2 ], respectively. If T1 < T2 , then by uniqueness, u1 (t) = u2 (t), for all t ∈ [0, T1 ]. Thus we can view u2 as an extension of u1 that defines a partial order relationship. By the well-known Zorn Lemma in real analysis, there is a number Tmax , and [0, Tmax ) is the maximal time interval in which the mild solution u exists. In order to prove the present theorem, it suffices to show that when Tmax < +∞, lim

ku(t)k = +∞.

t→Tmax −0

In what follows we use the contradiction argument to show that if lim

ku(t)k = +∞

t→Tmax −0

Semigroup Method

55

does not hold, then we can extend the solution to a larger time interval that contradicts the assumption that [0, Tmax ) is the maximal time interval. Indeed, if lim

ku(t)k = +∞

t→Tmax −0

does not hold, then there is a finite number M > 0, and a sequence tn , tn → Tmax − 0 such that ku(tn )k ≤ M. Consider the following problem:    dv + Av = F (v),  dt   

(2.5.46)

v(0) = u(tn ).

By Theorem 2.5.4, this problem admits a unique local mild solution in [0, δ] with δ depending only on M . We can choose n large enough so that tn + δ > Tmax . Let   u(t),

u(t) =



0 ≤ t ≤ tn ,

v(t − tn ),

tn ≤ t ≤ tn + δ.

(2.5.47)

We now want to prove that u(t) is a mild solution of problem (2.5.1) in [0, tn + δ], i.e., u(t) satisfies that integral equation u(t) = S(t)u0 +

Z t

S(t − τ )F (u(τ ))dτ

0

From u(t) = S(t)u0 +

Z t 0

and v(t) = S(t)u(tn ) +

0 ≤ t ≤ tn + δ.

(2.5.48)

0 ≤ t ≤ tn ,

(2.5.49)

S(t − τ )F (u(τ ))dτ

Z t 0

S(t − τ )F (v(τ ))dτ

0 ≤ t ≤ δ,

(2.5.50)

we infer that for t ∈ [0, tn ], u(t) satisfies (2.5.48). For t ∈ [0, δ], u(t + tn ) = v(t) = S(t)u(tn ) + µ

= S(t) S(tn )u0 + = S(t + tn )u0 +

Z tn 0

Z tn 0

Z t 0

S(t − τ )F (v(τ ))dτ ¶

S(tn − τ )F (u(τ ))dτ

S(t + tn − τ )F (u(τ ))dτ

+

Z t 0

S(t − τ )F (v(τ ))dτ

56

NONLINEAR EVOLUTION EQUATIONS +

Z t+tn tn

S(t + tn − τ )F (v(τ − tn ))dτ

= S(t + tn )u0 + +

Z t+tn tn

Z tn 0

S(t + tn − τ )F (u(τ ))dτ

S(t + tn − τ )F (u(τ ))dτ

= S(t + tn )u0 +

Z t+tn 0

S(t + tn − τ )F (u(τ ))dτ.

(2.5.51)

Combining (2.5.51) with (2.5.48) in [0, tn ] yields that u is a mild solution of problem (2.5.1) in [0, tn + δ]. Thus the proof is complete.

¤

THEOREM 2.5.6 Suppose that F is a nonlinear operator from D(A) to D(A), and satisfies the local Lipschitz condition. Then for any u0 ∈ D(A), problem (2.5.1) admits a unique local classical solution u such that

u ∈ C 1 ([0, Tmax ), B) ∩ C([0, Tmax ), D(A)). Moreover, there is an alternative: either (i) Tmax = +∞, i.e., there is a unique global classical solution, or (ii) Tmax < +∞, and

lim

ku(t)kD(A) = +∞.

t→Tmax −0

Proof. We denote B1 = D(A). As proved in Theorem 2.5.2, A1 = A is m-accretive from D(A2 ) ⊂ B1 into B1 . Combining the results in Theorems 2.5.2, 2.5.4, and 2.5.5 immediately yields the conclusion of the present theorem. ¤ The following two corollaries indicate that under slightly weakened assumptions than the global Lipschitz condition, we still have global existence of solution. COROLLARY 2.5.3 Suppose that F satisfies the local Lipschitz condition, and for all v ∈ B ,

kF (v)k ≤ C1 kvk + C2

(2.5.52)

where C1 , C2 are two positive constants. Then for any u0 ∈ B , problem (2.5.1) admits a unique global mild solution.

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57

Proof. By Theorem 2.5.5, the problem admits a unique mild solution u(t) in [0, Tmax ) such that u(t) = S(t)u0 + Thus, ku(t)k ≤ ku0 k +

Z t 0

Z t 0

S(t − τ )F (u(τ ))dτ

∀t ∈ [0, Tmax ).

(C1 kuk + C2 )dτ ≤ (ku0 k + C2 t) + C1

Z t 0

(2.5.53)

ku(τ )kdτ.

If Tmax < +∞, then by the Gronwall inequality, ku(t)k ≤ (ku0 k + C2 Tmax )eC1 t

∀t ∈ [0, Tmax ),

(2.5.54) (2.5.55)

which contradicts lim

ku(t)k = +∞.

t→Tmax −0

Thus, the proof is complete.

(2.5.56) ¤

COROLLARY 2.5.4 Suppose that B is a Hilbert space, and F satisfies the local Lipschitz condition, and for all v ∈ B ,

Re(F (v), v) ≤ C1 kvk2 + C2 ,

(2.5.57)

where C1 , C2 are two positive constants. Then for any u0 ∈ B , problem (2.5.1) admits a unique global mild solution. Furthermore, if u0 ∈ D(A), then this mild solution is a classical one.

Proof. We first consider the case u0 ∈ D(A). By Theorem 2.5.5 and Corollary 2.5.2, problem (2.5.1) admits a unique classical solution u in [0, Tmax ). Taking the inner product of (2.5.1) with u, then taking its real part yields 1d 1d kuk2 ≤ kuk2 + Re(Au, u) = Re(F (u), u) 2 dt 2 dt ≤ C1 kuk2 + C2 .

(2.5.58)

Applying the Gronwall inequality and Theorem 2.5.5, we can conclude that Tmax = ∞. If u0 ∈ B, then there exists a sequence u0n ∈ D(A) such that u0n → u0 . Let the global classical solution of (2.5.1) with initial data u0n be un (t).

58

NONLINEAR EVOLUTION EQUATIONS

It follows from (2.5.58) corresponding to un that d kun k2 ≤ 2C1 kun k2 + 2C2 . dt Therefore, solving this differential inequality yields kun (t)k2 ≤ ku0n k2 e2C1 t +

C2 2C1 t (e − 1). C1

(2.5.59)

(2.5.60)

This implies that for any T > 0, kun (t)k is uniformly bounded, say, by a positive constant MT depending on T . In what follows we show that un converges in C([0, T ], B), and the limit function u is a mild solution of problem (2.5.1). Indeed, let v = un − um , and v0 = u0n − u0m . Then we have

   dv + Av = F (un ) − F (um ),    

dt

(2.5.61)

v(0) = v0

Then taking the inner product with v, and taking its real part as before, we get 1d kvk2 ≤ Re(F (un ) − F (um ), v) ≤ Lkvk2 , (2.5.62) 2 dt where L is the Lipschitz constant corresponding to MT in Definition 2.5.1. Therefore, for all t ∈ [0, T ], kv(t)k2 = kun (t) − um (t)k2 ≤ e2LT ku0n − u0m k2 → 0.

(2.5.63)

Since un (t) is a classical solution, un also satisfies the integral equation un (t) = S(t)u0n +

Z t 0

S(t − τ )F (un (τ ))dτ.

(2.5.64)

Passing to the limit in (2.5.64) yields that u is a mild solution in any time interval [0, T ], i.e., u is a global mild solution. To prove the uniqueness of the mild solution, we use the integral equation, the local Lipschitz condition and the Gronwall inequality. Since it is the same as before, we leave the details to the reader. Thus, the proof is complete. ¤

Semigroup Method

2.6

59

Analytic Semigroups

In the first four sections we work with C0 -semigroup of contractions in a Banach space. Linear and nonlinear heat equations in which −A is the Laplacian operator pretty well fit into the framework established in previous sections. However, if we consider general second-order parabolic equations in the following form: ∂u + Au = f, ∂t where A is the general second-order elliptic operator Au = −

n X

aij (x)

i,j=1

n X ∂u ∂2u + + c(x)u, bi (x) ∂xi ∂xj i=1 ∂xi

(2.6.1)

(2.6.2)

then A in general is no longer an infinitesimal generator of C0 semigroup of contractions. This is due to the fact that by the Poincar´e inequality, it is easy to see that Aλ u ≡ Au + λu = f is invertible from H 2 ∩ H01 to L2 only when λ is sufficiently large, say, λ ≥ λ0 with λ0 being a given positive constant. Equation (2.6.1) can be converted into ∂v + Aλ0 v = f e−λ0 t (2.6.3) ∂t by changing the dependent variable u = eλ0 t v.

(2.6.4)

Since Aλ0 now is an accretive operator, by the results established in the previous sections, Aλ0 generates a C0 -semigroup S(t) of contractions defined by the mapping from u0 to v(·, t): v(·, t) = S(t)u0 . It turns out that the mapping T (t) from u0 to u(·, t) is given by u(·, t) = T (t)u0 = S(t)eλ0 t u0 . It is easy to see that the mapping T (t) satisfies the properties (i), (ii), (iv) in Section 2.1, and instead of (iii), (iii)’ kT (t)k ≤ eλ0 t .

60

NONLINEAR EVOLUTION EQUATIONS

This shows that we need to consider the general C0 -semigroup S(t) such that kS(t)k ≤ M eωt

(2.6.5)

with M ≥ 1, ω > 0 being two constants. In the same way as for the proof of the Hille-Yosida theorem for the C0 semigroup of contractions, we have the following theorem. THEOREM 2.6.1 A linear operator A is the infinitesimal generator of a C0 -semigroup S(t) defined on a Banach space B satisfying kS(t)k ≤ M eωt with M ≤ 1, ω ≥ 0 being given constants, if and only if (i) A is closed and D(A) is dense in B . (ii) The resolvent set ρ(−A) of −A contains the ray (ω, +∞) and def

kR(λI; −A)n k = k(λI + A)−n k ≤

M ∀λ > ω, n = 1, 2, · · · . (λ − ω)n

Since the proof is quite similar to that for Theorem 2.2.1, we can omit the detail here (see also [118] by A. Pazy). On the other hand, in Section 2.3, we discussed the C ∞ regularity of solution for t > 0 when B is a Hilbert space and the infinitesimal generator A of C0 -semigroup of contractions is maximal monotone and self-adjoint. Clearly, B = L2 , and A = −∆ satisfies this setting. In order to extend such a result to the more general cases, including that A is a general second-order elliptic operator and B = Lp (Ω), 1 < p < ∞, in the following, we will introduce the concepts and results on analytic semigroups. Up to now we dealt with semigroups whose domain of the parameter t was the real nonnegative axis. We will now consider the semigroup whose domain of the parameter is a region in the complex plane that includes the nonnegative real axis. It is clear that the semigroup should be an additive semigroup of complex numbers in the region. DEFINITION 2.6.1

Let

4 = {z : φ1 < arg z < φ2 } with φ1 < 0 < φ2 and for z ∈ 4, let T (z) be a bounded linear operator. The family T (z), z ∈ 4 is an analytic semigroup in 4 if (i) z 7→ T (z) is analytic in 4.

Semigroup Method

61

(ii) T (0) = I and

lim T (z)x = x

z→0,z∈4

for every x ∈ B . (iii) T (z1 + z2 ) = T (z1 )T (z2 ) for z1 , z2 ∈ 4.

A semigroup T (t) is called analytic if its extension T (z) is analytic in 4 containing the nonnegative real axis. In what follows we discuss the characterization of analytic semigroups. First, we consider the case in which kT (t)k ≤ M , i.e., T (t) is a uniformly bounded C0 -semigroup. Then the results for the general C0 semigroup satisfying (2.6.5) will easily follow by the transform T (t) = S(t)e−ωt . THEOREM 2.6.2 Let T (t) be a uniformly bounded C0 -semigroup. Let A be the infinitesimal generator of T (t) and assume 0 ∈ ρ(−A). Then the following statements are equivalent: (a) T (t) can be extended to an analytic semigroup in a sector

4δ = {z : |arg z| < δ} ¯ δ0 , δ 0 < δ, of and kT (z)k is uniformly bounded in every closed subsector 4 4δ . (b) There exists a constant C such that for every σ > 0, τ 6= 0, kR(σ + iτ ; −A)k = k((σ + iτ )I + A)−1 k ≤ (c) There exists 0 < δ <

π 2

C . |τ |

(2.6.6)

and M > 0 such that

ρ(−A) ⊃ Σ = {λ : |arg λ| <

π + δ} ∪ {0} 2

(2.6.7)

and

kR(λ; −A)k ≤

M |λ|

(2.6.8)

for λ ∈ Σ, λ 6= 0. (d) T (t) is differentiable for t > 0 and there is a positive constant C such that

kAT (t)k ≤

C t

∀t > 0.

(2.6.9)

62

NONLINEAR EVOLUTION EQUATIONS

Proof. (a) ⇒ (b). Let 0 < δ 0 < δ be such that ¯ δ0 . ∀z ∈ 4

kT (z)k ≤ C1 ,

For x ∈ B and σ > 0, in the same manner as for the proof of (2.2.4), we have R(σ + iτ ; −A)x = ((σ + iτ )I + A)−1 x =

Z ∞ 0

e−(σ+iτ )t T (t)xdt. (2.6.10)

¯ δ0 , it From the analyticity and the uniform boundedness of T (z) in 4 follows that we can shift the path of integration in (2.6.10) from the positive real axis to any ray ρeiθ , 0 < ρ < ∞, and |θ| ≤ δ 0 . For τ > 0, 0 shifting the path of integration to the ray ρeiδ and estimating the resulting integral, we get kR(σ + iτ ; −A)xk ≤

Z ∞ 0

e−ρ(σ cos δ +τ sin δ ) C1 kxkdρ 0

0

C1 kxk Ckxk . ≤ ≤ 0 0 σ cos δ + τ sin δ τ

(2.6.11)

Similarly, for τ < 0 we shift the path of integration to the ray ρe−iδ and obtain Ckxk kR(σ + iτ ; −A)xk ≤ − τ and thus (2.6.6) follows.

0

(b) ⇒ (c). Since A is by assumption the infinitesimal generator of a C0 -semigroup, it follows from (2.6.10) that C Re λ for Re λ > 0. On the other hand, from (b) it follows that for Re λ > 0, kR(λ; −A)k ≤

kR(λ; −A)k ≤

C . |Im λ|

Hence, kR(λ; −A)k ≤

C1 , |λ|

for Re λ > 0. Let σ > 0 and write the Taylor expansion for R(λ; −A) around λ = σ + iτ : R(λ; −A) =

∞ X

R(σ + iτ ; −A)n+1 (σ + iτ − λ)n .

n=0

(2.6.12)

Semigroup Method

63

This series converges in L(B) for kR(σ + iτ ; −A)k|σ + iτ − λ| ≤ k < 1. Choosing λ = Re λ + iτ in (2.6.12) and using (2.6.6) we can easily deduce that the series converges uniformly in L(B) for k|τ | . C Since both σ > 0 and k < 1 are arbitrary, it follows that ρ(−A) contains the set of all λ with Re λ ≤ 0 satisfying |σ − Re λ| ≤

|Re λ| 1 < . |Im λ| C

(2.6.13)

In particular, ρ(−A) ⊃ {λ : |arg λ| ≤

π + δ} 2

where δ = k arctan C1 , 0 < k < 1. Moreover, in this region, kR(λ; −A)k ≤

C 1 · , 1 − k |τ |

which yields (2.6.8) by combining with (2.6.13). Since by assumption 0 ∈ ρ(−A), (c) follows. (c) ⇒ (d). Set U (t) =

1 2πi

Z Γ

eµt R(µ; −A)dµ

(2.6.14)

where Γ is the path composed from the two rays ρeiθ and ρe−iθ , 0 < ρ < ∞ and π2 < θ < π2 + δ. Γ is oriented so that Im λ increases along Γ. In what follows we prove that U (t) is uniformly bounded for all t > 0. It easily follows from (2.6.7), (2.6.8) that for t > 0 the integral in (2.6.14) converges in the uniform topology. Moreover, since R(λ; −A) is analytic in Σ, we can shift the path of integration in (2.6.14) to Γt where Γt = Γ1 ∪ Γ2 ∪ Γ3 with Γ1 = {re−iθ : t−1 ≤ r < ∞}, Γ2 = {t−1 eiφ : −θ ≤ φ ≤ θ} and Γ3 = {reiθ : t−1 ≤ r < ∞}.

64

NONLINEAR EVOLUTION EQUATIONS

But,

Z

1 1 k eµt R(µ; −A)dµk ≤ 2πi Γ3 2π Z M ∞ ds ≤ e−s ≤ C1 . 2π sin (θ− π2 ) s

Z ∞ t

−1

π

e−rt sin (θ− 2 ) M r−1 dr (2.6.15)

The integral on Γ1 is estimated in the same way and on Γ2 we have k

1 2πi

Z

Γ2

eµt R(µ; −A)dµk ≤

M 2π

Z θ

−θ

ecos φ dφ ≤ C2 .

(2.6.16)

Therefore, the uniform boundedness of U (t) follows. Next we show that for λ > 0, R(λ; −A) =

Z ∞ 0

e−λt U (t)dt.

(2.6.17)

To this end, we multiply (2.6.14) by e−λt and integrate from 0 to T . Using Fubini’s theorem and the residue theorem for analytic function we find that Z T Z 1 1 (e(µ−λ)T − 1)R(µ; −A)dµ e−λt U (t)dt = 2πi Γ µ − λ 0 Z 1 R(µ; −A) = R(λ; −A) + dµ. (2.6.18) e(µ−λ)T 2πi Γ µ−λ But, as T → ∞, k

Z

Γ

e

T (µ−λ) R(µ; −A)

µ−λ

−T λ

dµk ≤ M e

Z Γ

|dµ| → 0. |µ||λ − µ|

(2.6.19)

Therefore, passing to the limit as T → ∞ in (2.6.18) yields (2.6.17). Since U (t) is uniformly bounded, we can differentiate (2.6.17) n − 1 times under the integral sign to get Z

∞ dn−1 n−1 R(λ; −A) = (−1) tn−1 e−λt U (t)dt. dλn−1 0 On the other hand, a straightforward calculation shows that

dn−1 R(λ; −A) = (n − 1)!R(λ; −A)n . dλn−1 Combining (2.6.20) with (2.6.21) yields

(2.6.21)

Z

∞ 1 kR(λ; −A) k = k tn−1 e−λt U (t)dtk (n − 1)! 0 Z ∞ C C ≤ tn−1 e−λt dt = n . (n − 1)! 0 λ n

(2.6.20)

(2.6.22)

Semigroup Method

65

Therefore, by Theorem 2.6.1, A is the infinitesimal generator of a C0 semigroup T (t) satisfying kT (t)k ≤ C. In what follows we prove that 1 T (t) = 2πi

Z

Γ

eλt R(λ; −A)dλ.

(2.6.23)

To prove (2.6.23), we first show that for any bounded linear operator B, if γ > kBk, then e−tB =

1 2πi

Z γ+i∞ γ−i∞

eλt R(λ; −B)dλ.

(2.6.24)

Moreover, the convergence in (2.6.24) is in the uniform operator topology and uniformly in t on bounded intervals. Indeed, for |λ| ≥ r > kBk, we have the expansion R(λ; −B) =

∞ X (−1)k B k k=0

(2.6.25)

λk+1

where the convergence is in the uniform operator topology uniformly for |λ| ≥ r > kBk. Multiplying (2.6.25) by 1 λt e 2πi and integrating over Cr , which is the circle of radius r centered at the origin, term by term yields e

−tB

1 = 2πi

Z

Cr

eλt R(λ; −B)dλ.

(2.6.26)

Here we used the identities 1 2πi

Z

Cr

λ−k−1 eλt dλ =

tk , k!

k = 0, 1, 2, · · · .

(2.6.27)

Since outside Cr the integrand of (2.6.26) is analytic and kR(λ; −B)k ≤ C |λ| , we can shift the path of integration from Cr to the line Re z = γ, using Cauchy’s theorem. Thus, (2.6.24) is proved. Secondly, we proceed to show that for x ∈ D(A), γ > 0, Z t 0

T (s)xds =

1 2πi

Z γ+i∞ γ−i∞

eλt R(λ; −A)x

dλ , λ

(2.6.28)

and the integral on the right converges uniformly in t in bounded intervals. To prove (2.6.28), we use the Yosida approximation Aµ , µ > 0 defined

66

NONLINEAR EVOLUTION EQUATIONS

in Section 2.2. Let µ > 0 be fixed and let δ > kAµ k. Set ρk (s) =

1 2πi

Z δ+ik δ−ik

eλs R(λ; −Aµ )dλ.

(2.6.29)

Integrating both sides of (2.6.29) from 0 to t and interchanging the order of integration we find that Z t 0

ρk (s)ds =

1 2πi

Z δ+ik δ−ik

eλt R(λ; −Aµ )x

dλ 1 − λ 2πi

Z δ+ik δ−ik

Letting k → ∞, it follows from (2.6.24) that

R(λ; −Aµ )x

dλ . λ

(2.6.30)

ρk (s) → e−tAµ x uniformly in t in bounded intervals. The second term on the right-hand side of (2.6.30) converges to zero, using the Cauchy theorem and the estimate Cµ kR(λ; −Aµ )k ≤ |λ| for |λ| ≥ δ. Therefore, passing to the limit as k → ∞ in (2.6.30) yields Z t 0

e

−sAµ

1 xds = 2πi

Z δ+i∞ δ−i∞

eλt R(λ; −Aµ )x

dλ . λ

(2.6.31)

It follows from the property of the Yosida approximation in Section 2.2 that for γ > 0, there is µ0 > 0 such that for 0 < µ ≤ µ0 , {λ : Re λ ≥ γ} is in ρ(−Aµ ) and for x ∈ D(A), kR(λ; −Aµ )k ≤

C (kxk + kAxk) |λ|

(2.6.32)

with C being a positive constant depending only on M and γ. Therefore, for µ ≤ µ0 we can shift the path of integration in (2.6.31) from Re λ = δ to Re λ = γ and obtain Z t 0

e−sAµ xds =

1 2πi

Z γ+i∞ γ−i∞

eλt R(λ; −Aµ )x

dλ . λ

(2.6.33)

It easily follows from (2.6.32) that for γ > 0, the integral on the righthand side of (2.6.33) converges uniformly for µ ≤ µ0 . When µ → 0, the left-hand side and the right-hand side of (2.6.33) converges to the left-hand side and the right-hand side of (2.6.28), respectively. Thus, (2.6.28) is proved. Thirdly, for x ∈ D(A2 ), using (2.6.28) for Ax we get T (t)x − x = −

Z t 0

T (s)Axds = −

1 2πi

Z γ+i∞ γ−i∞

eλt R(λ; −A)Ax

dλ λ

Semigroup Method

67

µ ¶ Z γ+i∞ 1 x λt = e R(λ; −A) − dλ.

(2.6.34)

2πi

λ

γ−i∞

Since 1 2πi

Z γ+i∞ γ−i∞

eλt x

dλ = x, λ

(2.6.35)

combining (2.6.34) with (2.6.35) yields (2.6.23). Once (2.6.23) is proved, differentiating (2.6.23) with respect to t (first just formally) yields Z

1 T (t) = 2πi 0

Γ

λeλt R(λ; −A)dλ.

(2.6.36)

But, the integral (2.6.36) converges in L(H) for every t > 0 since kT 0 (t)k ≤

1 π

Z ∞ 0

M . πt cos θ

M e−ρ cos θt dρ =

(2.6.37)

Therefore, the formal differentiation of T (t) is justified, T (t) is differentiable for t > 0 and (2.6.9) follows. (d) ⇒ (a). First we show that µ

T (n) (t) = −AT

µ ¶¶n

t n

µ

= T0

µ ¶¶n

t n

,

n = 1, 2, · · · .

(2.6.38)

This can be proved by induction on n. For n = 1, it has been proved previously in Section 2.2. If (2.6.38) holds for n and t ≥ s, then T

(n)

µ

(t) = −AT

µ ¶¶n

t n

µ

= T (t − s) −AT

µ ¶¶n

s n

.

(2.6.39)

Differentiating (2.6.39) with respect to t we get µ

T (n+1) (t) = −AT (t − s) −AT

µ ¶¶n

s n

.

(2.6.40)

nt Substituting s = n+1 in (2.6.40) yields the result for n + 1. Once (2.6.38) is proved, we deduce that

kT

(n)

(t)k = kT

0

µ ¶n

t n

k ≤ kT

0

µ ¶

t kn . n

(2.6.41)

Using this fact together with (2.6.9) and n!en ≥ nn , we have 1 kT (n) (t)k ≤ n!

µ

Ce t

¶n

.

(2.6.42)

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NONLINEAR EVOLUTION EQUATIONS

Now we consider the power series T (z) = T (t) +

∞ X T (n) (t) n=1

n!

(z − t)n .

This series converges uniformly in L(H) for |z − t| ≤ Therefore, T (z) is analytic in

(2.6.43) kt Ce

with k < 1.

1 }. Ce It is clear that T (z) extends T (t) to the sector 4. By the analyticity of T (z) it follows that T (z) satisfies the semigroup property and from (2.6.43) it follows that 4 = {z : |arg z| < arctan

T (z)x → x as z → 0 in 4. Moreover, we can deduce from (2.6.43) that in every closed subsector ¯ ε = {z : |arg z| ≤ aretan( 1 ) − ε}, 4 Ce ¯ ε . Thus the whole proof is complete. T (z) is uniformly bounded in 4 ¤

REMARK 2.6.1

For C0 -semigroup S(t) satisfying

kS(t)k ≤ M eωt , by the transform T (t) = S(t)e−ωt , we can immediately obtain the corresponding statements to Theorem 2.6.2 for S(t) as follows: (a) remains unchanged; (b) should be changed into (b)0 for every σ > ω , τ 6= 0

kR(σ + iτ ; −A)k ≤

C . |τ |

(2.6.44)

(c) should be changed into (c)0 ρ(−A) ⊃ Σ = {λ : |arg (λ − ω)| <

π + δ} ∪ {λ = ω} 2

(2.6.45)

and

kR(λ; −A)k ≤

M . |λ − ω|

(2.6.46)

Semigroup Method

69

(d) should be changed into (d)0 kAS(t)k ≤

C1 ω 1 t e t

(2.6.47)

with C1 , ω1 being two positive constants. REMARK 2.6.2

The statement (c)0 is equivalent to

ρ(A) ⊃ Sa,φ = {λ : φ ≤ |arg (λ − a)| ≤ π}

(2.6.48)

and

kR(λ; A)k ≤

M . |λ − a|

(2.6.49)

with φ = π2 − δ ∈ (0, π2 ), and a = −ω. A linear closed densely defined operator satisfying condition (c)0 is usually called the sectorial operator. Theorem 2.6.2 implies that for a linear closed densely defined operator A, the necessary and sufficient condition for A being the infinitesimal generator of an analytic semigroup is that it is a sectorial operator.

The following result shows that a lower-order perturbation to a sectorial operator is still a sectorial operator. THEOREM 2.6.3 Suppose that A is a sectorial operator and B is a linear operator with D(B) ⊃ D(A) such that for any x ∈ D(A),

kBxk ≤ εkAxk + Kε kxk

(2.6.50)

where ε > 0 is an arbitrary small constant and Kε is a positive constant depending on ε. Then A + B is sectorial.

Proof. Since A is a sectorial operator, (2.6.48) and (2.6.49) are satisfied. Without loss of generality, we may assume that for A, a = 0. It easily follows from (2.6.50) that A + B is also a closed operator densely defined on D(A). We can deduce from (2.6.49) that there is φ ∈ (0, π2 ), when λ ∈ S0,φ : kA(λI − A)−1 k ≤ C. It turns out that for λ ∈ S0,φ , kB(λI − A)−1 k ≤ εkA(λI − A)−1 k + Kε k(λI − A)−1 k

(2.6.51)

70

NONLINEAR EVOLUTION EQUATIONS ≤ εC +

Kε (1 + C) . |λ|

(2.6.52)

1 Choosing ε = 2C , for |λ| large enough, i.e., |λ| ≥ R0 > 0 with R0 large enough, we have

k(λI − (A + B))−1 k = k(λI − A)−1 (I − B(λI − A)−1 )−1 k ≤

1+C |λ|

µ

1 K(1 + C) − 2 |λ|

¶−1

≤

C1 |λ|

(2.6.53)

which implies that ρ(A + B) ⊃ Sa,φ with a=−

R0 . cos φ

Moreover, for λ ∈ Sa,φ ⊂ S0,φ , 1 1 |λ − a| M = × ≤ , |λ| |λ − a| |λ| |λ − a| i.e., (2.6.49) is satisfied. Thus, A + B is also a sectorial operator, and it generates the analytic semigroup S(t) = e−t(A+B) . ¤

2.7

Applications

In this section we concisely show how the theory established in previous sections can be applied to initial boundary value problems for concrete evolution equations. 1. We first consider the following initial boundary value problem for the linear heat equation:  ut − ∆u = 0,          

u|Γ = 0,

u|t=0 = u0 (x)

where Ω is a bounded domain in Rn with smooth boundary Γ. (i) We first assume that u0 ∈ L2 (Ω).

(2.7.1)

Semigroup Method

71

Then we choose B = L2 (Ω), A = −∆, and D(A) = H 2 (Ω) ∩ H01 (Ω). Problem (2.7.1) can be converted into the initial value problem for the first-order abstract evolution equation:  du   + Au = 0,  dt (2.7.2)   

u(0) = u0 (x).

Notice that the boundary condition in (2.7.1) now has been realized by H01 in the definition of D(A). In order to apply Theorem 2.3.2, we have to verify that A is m-accretive. For any f ∈ B = L2 (Ω), consider the following boundary value problem:   u − ∆u = f, 

(2.7.3)

u|Γ = 0.

This problem can be converted into the weak formulation: Finding u ∈ H01 (Ω) such that for all v ∈ H01 (Ω), the following holds: a(u, v) =

Z

Ω

(∇u · ∇v + uv)dx =

Z

Ω

f vdx.

(2.7.4)

It is clear that the bilinear form defined on H01 × H01 is coercive, and bounded. Then by the Lax-Milgram theorem stated in Chapter 1, problem (2.7.4) admits a unique weak solution u ∈ H01 . By the regularity result stated in Chapter 1, we can conclude that problem (2.7.3) T admits a unique solution u ∈ H 2 (Ω) H01 (Ω). Multiplying (2.7.3) by u, then integrating with x yields 2

2

kuk + k∇uk =

Z

Ω

f udx ≤ kf kkuk.

(2.7.5)

Here kuk denotes the norm of u in L2 . Thus, kuk ≤ kf k,

(2.7.6)

k(I + A)−1 k ≤ 1.

(2.7.7)

i.e., This proves that A is m-accretive. Notice that B = L2 is a Hilbert space, and A clearly is self-adjoint. Thus by Theorem 2.3.2, we have the following theorem.

72

NONLINEAR EVOLUTION EQUATIONS

THEOREM 2.7.1 For any u0 ∈ L2 , problem (2.7.1) admits a unique classical solution u such that

u ∈ C([0, +∞), L2 ) ∩ C 1 ((0, +∞), L2 ) ∩ C((0, +∞), H 2 ∩ H01 ). Furthermore,

u(x, t) ∈ C ∞ ((0, +∞), C ∞ (Ω)). Proof. By Theorem 2.3.2, there is a unique classical solution u such that u ∈ C k ((0, +∞), D(Aj )), for all k, j = 1, 2, · · ·. By the regularity results for the elliptic boundary value problem stated in Chapter 1, we have D(Aj ) ⊂ H 2j (Ω)

(2.7.8)

as long as we assume that the boundary Γ is C ∞ . Thus the proof is complete. ¤ REMARK 2.7.1 If u0 ∈ D(Ak ), then the classical solution u of problem (2.7.1) has the following regularity up to t = 0:

u∈

k \

C k−j ([0, +∞), D(Aj )).

(2.7.9)

j=0

2. Consider the non-homogeneous problem:

 ut − ∆u = f (x, t),          

u|Γ = 0,

(2.7.10)

u|t=0 = u0 (x).

Then we infer from Theorem 2.3.2 and Corollary 2.4.3 the following results. THEOREM 2.7.2 If f (x, t) is a Lipschitz continuous function on [0, +∞) valued in L2 , and u0 ∈ L2 , then problem (2.7.10) admits a unique classical solution u such that

u ∈ C([0, +∞), L2 ) ∩ C 1 ((0, +∞), L2 ) ∩ C((0, +∞), H 2 ∩ H01 ).

Semigroup Method

73

3. Now we use a different setting to investigate problem (2.7.1). Let B be the following Banach space: B = C0 (Ω) = {u | u ∈ C(Ω), u|Γ = 0},

(2.7.11)

equipped with the norm kukB = max |u(x)|. Ω

(2.7.12)

We still take A = −∆, but take D(A) = {u | u ∈ C0 (Ω), ∆u ∈ C0 (Ω)}.

(2.7.13)

It is easy to see that D(A) is dense in B. To prove that A is m-accretive, we investigate the solvability of the following problem:  ¯  (I + A)u = u − ∆u = f (x) ∈ C0 (Ω), 

(2.7.14)

u|Γ = 0.

Since for any 2 ≤ p < ∞, f ∈ Lp (Ω), we can easily infer from the uniqueness of solution to problem (2.7.14) in W 2,p ∩ W01,p and the Fredholm alternative results stated in Chapter 1 that problem (2.7.14) admits a unique solution u such that u ∈ W 2,p ∩ W01,p . For p > n, it follows from the Sobolev imbedding theorem that u ∈ C 1,α (Ω) ∩ C0 (Ω). We infer from the equation of (2.7.14) that −∆u = f − u ∈ C0 (Ω), i.e, u ∈ D(A). To prove that k(I + A)−1 k ≤ 1, we multiply the equation of (2.7.14) by |u|2p−2 u, then integrate over Ω to obtain Z 2p − 1 kuk2p + |∇(|u|p−1 u)|2 dx 2p L p2 Ω =

Z

Ω

f |u|2p−2 udx ≤ kuk2p−1 L2p kf kL2p .

(2.7.15)

Thus, 1

kukL2p ≤ kf kL2p ≤ |Ω| 2p kf kC0

(2.7.16)

where |Ω| denotes the measure of the domain Ω. Since (2.7.16) holds for any 2 ≤ p < ∞, letting p → +∞ yields kukC0 ≤ kf kC0 .

(2.7.17)

74

NONLINEAR EVOLUTION EQUATIONS

Thus, A is m-accretive. By Theorem 2.2.2, we immediately have the following result. THEOREM 2.7.3 If u0 ∈ D(A) = {u | u ∈ C0 (Ω), ∆u ∈ C0 (Ω)}, then problem (2.7.1) admits a unique classical solution u such that

u ∈ C 1 ([0, +∞), C0 ) ∩ C([0, +∞), D(A)). Let S0 (t), S2 (t) be the semigroups corresponding to B = C0 setting and B = L2 setting, respectively. Since C0 ⊂ L2 , S0 is the restriction of S2 in C0 . Thus, we have the following. REMARK 2.7.2 If u0 ∈ C0 (Ω), then problem (2.7.1) admits a unique solution u ∈ C([0, +∞), C0 ). Moreover, u ∈ C k ((0, +∞), D(Aj )) for any k, j = 1, 2, · · ·, i.e., for t > 0, u is C ∞ if Γ is C ∞ smooth.

In what follows we apply the results in Section 2.6 to show that the general second-order elliptic operator Lu ≡ −

n X ∂

i,j=1

∂xi

n X ∂u ∂u )+ + c(x)u bi (x) ∂xj ∂xi i=1

(aij (x)

(2.7.18)

with smooth coefficients aij (x), bi (x), c(x) and subject to the Dirichlet boundary condition is a sectorial operator that generates an analytic semigroup in Lp (Ω), 1 < p < ∞. For simplicity of exposition we give only the detailed proof for the case p = 2. For the general cases p 6= 2, we refer the reader to the book [118] by A. Pazy. When p = 2, consider first the operator Au = −

n X ∂

i,j=1

∂xi

(aij (x)

∂u ) ∂xj

with H = L2 (Ω), D(A) = H 2 ∩ H01 (Ω). In the same manner as for the case A = −∆, we can prove that A is maximal monotone and for λ > 0, 1 . λ This implies that A is the infinitesimal generator of a C0 semigroup of contraction. Moreover, Theorem 2.3.2 still holds. The assertion k(λI + A)−1 k ≤

Semigroup Method

75

(2.3.21) in Theorem 2.3.2 says that for any u0 ∈ H, 1 ku0 (t)k = kS 0 (t)u0 k = k − AS(t)u0 k ≤ ku0 k, t i.e., 1 kAS(t)k ≤ , t which implies that condition (b) in Theorem 2.6.2 is satisfied. Therefore, by Theorem 2.6.2, A is a sectorial operator. By the regularity results (1.3.26) for the elliptic operator kukH 2 ≤ CkAukL2

(2.7.19)

and the Gagliardo–Nirenberg inequality 1 1 C2 ε kukL2 ∀u ∈ H 2 ∩ H01 , kukH 1 ≤ CkD2 ukL2 2 kukL2 2 ≤ kD2 ukL2 + 2 2ε

we can easily see that the first-order operator Bu =

n X

bi (x)

i=1

(2.7.20)

∂u + c(x)u ∂xi

is a lower-order operator satisfying the assumption in Theorem 2.6.3. It turns out from Theorem 2.6.3 that Lu = Au + Bu is also a sectorial operator, and it generates an analytic semigroup S(t) = e−tL . 4. Consider the initial boundary value problem for the semilinear heat equation:  u − ∆u = f (u),   t        

u|Γ = 0,

(2.7.21)

u|t=0 = u0 (x)

where Ω is assumed as before to be a bounded domain in Rn with smooth boundary Γ. We can convert this problem into the initial value problem for a semilinear evolution equation:  du   + Au = F (u),  dt (2.7.22)   

u(0) = u0

with A = −∆,

F (u) = f (u).

76

NONLINEAR EVOLUTION EQUATIONS

THEOREM 2.7.4 Suppose that f ∈ C 1 (R), and f 0 (u) is uniformly bounded. Then for any u0 ∈ L2 (Ω), problem (2.7.22) admits a unique mild solution u such that

u ∈ C([0, +∞), L2 ). Furthermore, if u0 ∈ H 2 ∩ H01 , then problem (2.7.22) has a unique classical solution u such that

u ∈ C 1 ([0, +∞), L2 ) ∩ C([0, +∞), H 2 ∩ H01 ). Proof. Let B = L2 (Ω), which clearly is a Hilbert space. As proved before, D(A) = H 2 (Ω) ∩ H01 (Ω), and A is m-accretive. By the assumptions, F is a nonlinear operator from B into B, F ∈ C 1 (B, B), and F satisfies the global Lipschitz condition. Thus, by Theorem 2.5.3, when u0 ∈ D(A) = H 2 ∩ H01 , problem (2.7.22) admits a unique classical solution u such that u ∈ C 1 ([0, +∞), L2 ) ∩ C([0, +∞), H 2 ∩ H01 ). When u0 ∈ L2 , by Theorem 2.5.1, problem (2.7.22) admits a unique global mild solution u such that u ∈ C([0, +∞), B). Thus, the proof is complete.

¤

THEOREM 2.7.5 Suppose

f ∈ C 3 (R),

f (0) = 0,

and n ≤ 3. Then for any

u0 ∈ H 2 ∩ H01 , problem (2.7.22) admits a unique maximal classical solution u such that

u ∈ C 1 ([0, Tmax ), L2 ) ∩ C([0, Tmax ), H 2 ∩ H01 ). Furthermore, there is an alternative: (i) either Tmax = +∞, i.e., there is a unique global classical solution,

Semigroup Method

77

(ii) or Tmax < +∞, and

lim

t→Tmax −0

ku(t)kH 2 = +∞.

(2.7.23)

Proof. We still adopt the previous setting: B = L2 ,

A = −∆, D(A) = H 2 ∩ H01 ,

and F (u) = f (u), and problem (2.7.21) is converted into the equivalent problem (2.7.22). By Theorem 2.5.6, it suffices to prove that F is a mapping from D(A) into D(A), and satisfies the local Lipschitz condition. By the Sobolev imbedding theorem stated in Chapter 1, when n ≤ 3, H 2 (Ω) is continuously imbedded into C(Ω), and H 1 is continuously imbedded into L6 . Let D denote the derivative with respect to x. Then Df (u) = f 0 (u)Du,

D2 f (u) = f 0 (u)D2 u + f 00 (u)DuDu.

(2.7.24)

We infer from this and f (u) ∈ H01 , due to f (0) = 0, that F is a nonlinear operator from D(A) into D(A) satisfying the local Lipschitz condition, which easily follows from the assumption that f ∈ C 3 . Thus the proof is complete. ¤ THEOREM 2.7.6 Suppose that

f ∈ C 1 (R),

f (0) = 0.

Then for any

u0 ∈ C0 (Ω), problem (2.7.22) admits a unique maximum mild solution u such that

u ∈ C([0, Tmax ), C0 (Ω)). Furthermore, there is an alternative: (i) either Tmax = +∞, i.e., there is a global mild solution, (ii) or Tmax < +∞, and

lim

ku(t)kC0 (Ω) = +∞.

t→Tmax −0

(2.7.25)

Proof. We use the following setting for this problem: B = C0 (Ω), D(A) = {u | u ∈ C0 (Ω), ∆u ∈ C0 (Ω)}.

(2.7.26)

As proved before, A is m-accretive. By the assumptions f ∈ C 1 (R), f (0) = 0, we can easily conclude that F is a nonlinear operator from

78

NONLINEAR EVOLUTION EQUATIONS

B into B, and it satisfies the local Lipschitz condition. Then the conclusion of the present theorem follows from Theorem 2.5.5. ¤ In the following we give an example to show how in some cases a priori estimates can be derived to obtain global existence of solution. THEOREM 2.7.7 Suppose

f (u) = −u3 + u, n ≤ 3, and u0 ∈ H 2 (Ω) ∩ H01 (Ω). Then Tmax = +∞, i.e., problem (2.7.21) admits a unique global classical solution u such that

u ∈ C([0, +∞), H 2 ∩ H01 ) ∩ C 1 ([0, +∞), L2 ). Proof. By Theorem 2.7.5, problem (2.7.21) admits a unique maximal classical solution u such that u ∈ C 1 ([0, Tmax ), L2 ) ∩ C([0, Tmax ), H 2 ∩ H01 ). Thus it suffices to prove that ku(t)kH 2 is uniformly bounded. For 0 ≤ t < Tmax , we multiply the equation of (2.7.21) by ut , then integrate over Ω to obtain d dt

Z µ Ω

¶

1 1 1 k∇uk2 + u4 − u2 dx + kut k2 = 0. 2 4 2

(2.7.27)

Thus, from the Young inequality 1 u2 ≤ u4 + 1 4 and (2.7.27), it follows that kukH ≤ C,

Z t

1

0

kut k2 dτ ≤ C,

∀t ∈ [0, Tmax ).

For h > 0, let v(t) = u(t + h). Then v ∈ C 1 ([0, Tmax − h), L2 ) ∩ C([0, Tmax − h), H 2 ∩ H01 ),

(2.7.28)

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79

and it is a solution to the following problem:

  vt − ∆v = −v 3 + v,           

v|Γ = 0,

(2.7.29)

v|t=0 = u(x, h).

Let w = v(t) − u(t) = u(t + h) − u(t). Then

  wt − ∆w + w(v 2 + uv + u2 ) − w = 0,           

w|Γ = 0,

(2.7.30)

w|t=0 = u(x, h) − u0 (x).

Multiplying the above equation by w, then integrating over Ω yields 1d kw(t)k2 + k∇wk2 + 2 dt

Z

Ω

w2 (v 2 + uv + u2 )dx = kwk2 .

(2.7.31)

Since 1 1 u2 + uv + v 2 = (u2 + v 2 ) + (u + v)2 ≥ 0, (2.7.32) 2 2 the second term and the third term on the left side of (2.7.31) are positive. Hence, by the Gronwall inequality we have kw(t)k2 = ku(t + h) − u(t)k2 ≤ CT ku(h) − u0 k2

(2.7.33)

where CT is a positive constant depending only on Tmax . Dividing the above by h2 , then letting h → 0, we infer from u0 ∈ H 2 ∩ H01 that kut (t)k2 ≤ CT k∆u0 − u30 + u0 k2 < +∞.

(2.7.34)

Thus, it follows from the Sobolev imbedding theorem, (2.7.28) and the equation that for all t ∈ [0, Tmax ), ku(t)kH 2 ≤ Ck∆u(t)k ≤ CT (kut (t)k + ku3 (t)k + ku(t)k) ≤ C1 < ∞,

(2.7.35)

where C is a positive constant depending only on Ω, and C1 is a positive constant depending only on Tmax , u0 and Ω. This a priori estimate of

80

NONLINEAR EVOLUTION EQUATIONS

solution indicates that if Tmax < +∞, then there is no possibility to have lim

t→Tmax −0

ku(t)kH 2 = +∞.

Thus Tmax = +∞, and the proof is complete.

¤

5. Consider the following initial boundary value problem for the linear wave equation with dissipation:  utt − ∆u + αut = f (x, t),          

u|Γ = 0,

(2.7.36)

u|t=0 = u0 (x), ut |t=0 = u1 (x)

where α ≥ 0 is a given constant. This is a second-order evolution equation. In order to apply the semigroup method, we introduce the new unknown function ∂u , (2.7.37) v= ∂t and the equation in (2.7.36) can be converted into a system of equations:   ut − v = 0, 

vt − ∆u + αv = f.

(2.7.38)

Introducing the vector function U = (u, v)T , system (2.7.38) can be written as dU + AU = F, (2.7.39) dt where µ ¶µ ¶ µ ¶ 0 −I u −v AU = = . (2.7.40) −∆ α v −∆u + αv Concerning the boundary condition for U , we still have U |Γ = (u, v)T |Γ = 0.

(2.7.41)

The initial condition for U is U |t=0 = (u0 , u1 )T .

(2.7.42)

In order to convert problems (2.7.39), (2.7.41), and (2.7.42) into the initial value problem for an abstract first-order evolution equation, we

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81

introduce the following framework. First, let H be a Hilbert space such that H = H01 (Ω) × L2 (Ω) equipped with the inner product (U1 , U2 )H =

Z

Ω

(2.7.43) Z

∇u1 · ∇¯ u2 dx +

Ω

v1 v¯2 dx.

(2.7.44)

For the matrix operator A defined by (2.7.40), we denote that D(A) = (H 2 (Ω) ∩ H01 (Ω)) × H01 (Ω). Clearly, D(A) is dense in H. In what follows we prove that A is maccretive. It follows from the following identity: Re(AU, U )H = Re(−

Z

Ω

∇v · ∇¯ udx −

= αkvk2 ≥ 0

Z

Ω

∆u¯ v dx) + α

and Lemma 2.2.3 that A is accretive. For any F = L2 (Ω), consider the solvability of the problem

Z

Ω

v¯ v dx (2.7.45)

(f, g)T

AU + U = F.

∈ H01 (Ω) × (2.7.46)

Problem (2.7.46) is equivalent to the following system:   −v + u = f, 

−∆u + (α + 1)v = g.

(2.7.47)

Substituting v obtained from the first equation into the second equation yields −∆u + (α + 1)u = (α + 1)f + g ∈ L2 (Ω).

(2.7.48)

Since the corresponding weak formulation is a(u, w) =

Z

Ω

((α + 1)f + g)wdx, ∀w ∈ H01 (Ω),

where a(u, w) =

Z Ω

(2.7.49)

∇u · ∇w + (α + 1)uwdx,

by the Lax-Milgram theorem stated in Chapter 1, and the regularity of the elliptic problem we can conclude that problem (2.7.48) admits a unique solution u such that u ∈ H 2 (Ω) ∩ H01 (Ω).

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NONLINEAR EVOLUTION EQUATIONS

Then we deduce from the first equation of (2.7.47) that v = u − f ∈ H01 (Ω). This implies that R(I + A) = H, and by Definition 2.2.2, A is maximal monotone. Thus by Theorem 2.2.2, we immediately have the following results. THEOREM 2.7.8 If

u0 (x) ∈ H 2 ∩ H01 , u1 (x) ∈ H01 , then problem

  Ut + AU = 0, 

U |t=0 = (u0 , u1 )T .

(2.7.50)

admits a unique classical solution U such that

U ∈ C 1 ([0, +∞), H) ∩ C([0, +∞), D(A)). Accordingly, the problem  utt − ∆u + αut = 0,          

u|Γ = 0,

(2.7.51)

u|t=0 = u0 (x), ut |t=0 = u1 (x).

admits a unique solution u such that

u ∈ C([0, +∞), H 2 ∩ H01 ) ∩ C 1 ([0, +∞), H01 ), and

v = ut ∈ C 1 ([0, +∞), L2 ) ∩ C([0, +∞), H01 ), i.e., the unique solution u belongs to

C([0, +∞), H 2 ∩ H01 ) ∩ C 1 ([0, +∞), H01 ) ∩ C 2 ([0, +∞), L2 ). REMARK 2.7.3

If

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , and α = 0, then it follows from

Re(AU, U )H = 0

Semigroup Method

83

that

1d kU (t)k2H = 0, 2 dt

(2.7.52)

i.e.,

kU (t)k2H = ku(t)k2H 1 + kut (t)k2L2 = kU (0)k2H = ku0 k2H 1 + ku1 k2L2

(2.7.53) which indicates that for the homogeneous wave equation, the energy is conserved.

6. Consider the initial boundary value problem (2.7.36). Since this problem can be converted into the following initial value problem for the first-order abstract evolution equation,   Ut + AU = F, 

U |t=0 = (u0 , u1 )T ,

(2.7.54)

with F = (0, f )T , we immediately infer from Theorem 2.4.1 the following result. THEOREM 2.7.9 Suppose that

f ∈ C 1 ([0, +∞), L2 ), u0 ∈ H 2 ∩ H01 , u1 ∈ H01 . Then problem (2.7.54) admits a unique solution U such that

u ∈ C 1 ([0, +∞), H01 ) ∩ C([0, +∞), H 2 ∩ H01 ), v ∈ C 1 ([0, +∞), L2 ) ∩ C([0, +∞), H01 ). Accordingly, problem (2.7.36) admits a unique solution u such that

u ∈ C 2 ([0, +∞), L2 ) ∩ C 1 ([0, +∞), H01 ) ∩ C([0, +∞), H 2 ∩ H01 ). 7. Consider the initial boundary value problem for the semilinear wave equation:  utt − ∆u + αut = f (u),          

(x, t) ∈ Ω × R+ ,

u|Γ = 0,

u|t=0 = u0 (x), ut |t=0 = u1 (x)

(2.7.55)

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NONLINEAR EVOLUTION EQUATIONS

where Ω ⊂ Rn is a bounded domain with the smooth boundary Γ. As before, we can convert this problem into the initial value problem for the abstract semilinear first-order evolution equation:   Ut + AU = F (U ), 

U |t=0 = (u0 , u1 )T

(2.7.56)

with F = (0, f (u))T . Then we have the following result. THEOREM 2.7.10 If

f (u) : R 7→ R satisfies the global Lipschitz condition, then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , problem (2.7.55) admits a unique solution u such that

u ∈ C 2 ([0, ∞), L2 ) ∩ C 1 ([0, ∞), H01 ) ∩ C([0, ∞), H 2 ∩ H01 ). If

u0 ∈ H01 , u1 ∈ L2 , then problem (2.7.55) admits a unique mild solution u such that

u ∈ C([0, ∞), H01 ) ∩ C 1 ([0, ∞), L2 ). Proof. By the assumption on f , f is a nonlinear operator from L2 into L2 , and it satisfies the global Lipschitz condition. It turns out that F (U ) = (0, f (u))T is a nonlinear operator from H = H01 × L2 into H, and it satisfies the global Lipschitz condition. Thus the conclusion of the present theorem immediately follows from Corollary 2.5.2 and Theorem 2.5.1, and the proof is complete. ¤ In what follows we consider the cases when the space dimension n = 1, 2, 3.

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85

THEOREM 2.7.11 Suppose that f ∈ C 1 (R), n = 1. Then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , problem (2.7.55) admits a unique maximal solution u such that

u ∈ C 2 ([0, Tmax ), L2 ) ∩ C 1 ([0, Tmax ), H01 ) ∩ C([0, Tmax ), H 2 ∩ H01 ). Moreover, there is an alternative: (i) either Tmax = +∞, i.e., the solution is a global one, (ii) or Tmax < +∞, and

lim

ku(t)kH 1 + kut (t)kL2 = +∞,

t→Tmax −0

(2.7.57)

i.e., the solution blows up in a finite time.

Proof. Since problem (2.7.55) can be converted into problem (2.7.56), by Theorem 2.5.5, it suffices to verify that F (U ) = (0, f (u))T is a nonlinear operator from H = H01 × L2 into H, and it satisfies the local Lipschitz condition. Since n = 1, by the Sobolev imbedding theorem, H01 (Ω) is continuously imbedded into C(Ω). By the assumption f ∈ C 1 , we easily deduce that F satisfies the desired conditions. Thus the proof is complete. ¤ THEOREM 2.7.12 Suppose that n = 2, f ∈ C 1 (R), and for all u ∈ R,

|f 0 (u)| ≤ C(|u|p + 1) with p being any given positive number, and C a given positive constant. Then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , problem (2.7.55) admits a unique maximal solution u such that

u ∈ C 2 ([0, Tmax ), L2 ) ∩ C 1 ([0, Tmax ), H01 ) ∩ C([0, Tmax ), H 2 ∩ H01 ). Moreover, there is an alternative: (i) either Tmax = +∞, i.e., the solution is a global one, (ii) or Tmax < +∞, and

lim

ku(t)kH 1 + kut (t)kL2 = +∞,

t→Tmax −0

(2.7.58)

86

NONLINEAR EVOLUTION EQUATIONS

i.e., the solution blows up in a finite time.

Proof. It suffices to notice that when n = 2, by the Sobolev imbedding theorem, H01 (Ω) is continuously imbedded into Lp (Ω) for any p < ∞. Thus F (U ) is a nonlinear operator from H into H and it satisfies the local Lipschitz condition. The conclusion of the present theorem follows from Theorem 2.5.5. ¤ THEOREM 2.7.13 Suppose that n = 3, f ∈ C 1 (R), and for all u ∈ R,

|f 0 (u)| ≤ C(|u|2 + 1). Then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , problem (2.7.55) admits a unique maximal solution u such that

u ∈ C 2 ([0, Tmax ), L2 ) ∩ C 1 ([0, Tmax ), H01 ) ∩ C([0, Tmax ), H 2 ∩ H01 ). Moreover, there is an alternative: (i) either Tmax = +∞, i.e., the solution is a global one, (ii) or Tmax < +∞, and

lim

ku(t)kH 1 + kut (t)kL2 = +∞,

t→Tmax −0

(2.7.59)

i.e., the solution blows up in a finite time.

Proof. It suffices to verify that F (U ) = (0, f (u))T is a nonlinear operator from H = H01 × L2 into H and satisfies the local Lipschitz condition. Since n = 3, by the Sobolev imbedding theorem, H01 (Ω) is continuously imbedded into L6 (Ω). Therefore, for any u1 , u2 ∈ H01 with ku1 kH 1 ≤ M , ku2 kH 1 ≤ M , we infer from the Cauchy-Schwartz inequality and the assumption on f 0 that kf (u1 ) − f (u2 )kL2 = k ≤

Z 1 0

Z 1 0

f 0 (λu1 + (1 − λ)u2 )dλ(u1 − u2 )kL2

kf 0 (λu1 + (1 − λ)u2 )kL3 dλku1 − u2 kL6

≤ LM ku1 − u2 kH 1 .

(2.7.60)

Thus F satisfies the desired condition, and the conclusion of the present theorem follows from Theorem 2.5.5. ¤

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87

THEOREM 2.7.14 Suppose that

f ∈ C 2 (R), f (0) = 0 and n ≤ 3. Then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , problem (2.7.55) admits a unique maximal solution u such that

u ∈ C 2 ([0, Tmax ), L2 ) ∩ C 1 ([0, Tmax ), H01 ) ∩ C([0, Tmax ), H 2 ∩ H01 ). Moreover, there is an alternative: (i) either Tmax = +∞, i.e., the solution is a global one, (ii) or Tmax < +∞, and

lim

ku(t)kH 2 + kut (t)kH 1 = +∞,

t→Tmax −0

(2.7.61)

i.e., the solution blows up in a finite time.

Proof. By Theorem 2.5.6, it suffices to verify that F (U ) = (0, f (u))T is a nonlinear operator from D(A) = (H 2 ∩ H01 ) × H01 into D(A), and it satisfies the local Lipschitz condition. Since n ≤ 3, by the Sobolev imbedding theorem, H 2 ∩ H01 is continuously imbedded into C0 (Ω), and T H 1 is continuously imbedded into L6 . Therefore, for any 2 u1 , u2 ∈ H H01 with ku1 kH 2 ≤ M , ku2 kH 2 ≤ M , we have kf (u1 ) − f (u2 )kL2 = k

Z 1 0

f 0 (λu1 + (1 − λ)u2 )(u1 − u2 )dλkL2

≤ Cku1 − u2 kL2 ≤ LM ku1 − u2 kH 2 ,

(2.7.62)

and kf 0 (u1 )Du1 − f 0 (u2 )Du2 kL2 ≤ k(f 0 (u1 ) − f 0 (u2 ))Du1 kL2 + kf 0 (u2 )(Du1 − Du2 )kL2 ≤ C1 ku1 − u2 kC0 kDu1 kL2 + C2 ku1 − u2 kH 2 ≤ LM ku1 − u2 kH 2 . (2.7.63) We also notice that the assumption f (0) = 0 guarantees that f (u) ∈ H01 . These imply that F satisfies the desired property. Thus, the proof is complete. ¤ In what follows we also give an example to show how in some cases a priori estimates of solution can be derived to obtain the global existence of solution.

88

NONLINEAR EVOLUTION EQUATIONS

THEOREM 2.7.15 Suppose that n ≤ 3, and f = −u3 + u. Then for any u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , problem (2.7.55) admits a unique global solution u such that

u ∈ C 2 ([0, ∞), L2 ) ∩ C 1 ([0, ∞), H01 ) ∩ C([0, ∞), H 2 ∩ H01 ). Proof. We first notice that the function f = −u3 + u satisfies the assumptions in Theorems 2.7.11–2.7.13. Therefore, it suffices to verify that ku(t)kH 1 + kut (t)kL2 is uniformly bounded. For t ∈ [0, Tmax ), we multiply the equation of (2.7.55) by ut , then integrate over Ω to obtain d dt

Z µ Ω

¶

1 1 1 1 |ut (t)|2 + |∇u(t)|2 + |u(t)|4 − |u|2 dx + αkut k2 = 0. 2 2 4 2

(2.7.64)

It turns out that 1 1 1 1 E(t) = kut (t)k2L2 + k∇u(t)k2L2 + ku(t)k4L4 − kuk2L2 ≤ E(0) < +∞. 2 2 4 2 (2.7.65)

Thus, the conclusion of the present theorem follows from this a priori estimate and the Young inequality. ¤ 8. In the remaining part of this section we show how the semigroup method can be applied to a more complicated system, namely the one-dimensional thermoelastic system. Consider the following initial boundary value problem for the one-dimensional thermoelastic system:   utt − uxx + γθx = f (x, t), 

θt − kθxx + γuxt = g(x, t),

(2.7.66)

subject to the following boundary conditions and initial conditions: u|x=0,l = 0, θ|x=0,l = 0, u|t=0 = u0 (x), ut |t=0 = u1 (x), θ|t=0 = θ0 (x).

(2.7.67) (2.7.68)

In (2.7.66), γ is a given non-zero constant, and k is a given positive constant; u represents deformation of displacement in a rod of length l, and θ is derivation of temperature. The first equation of (2.7.66) is the momentum equation, and the second equation is the energy equation. To convert this initial boundary value problem into the initial value problem for an abstract first-order evolution equation, we introduce new dependent variable v = ut , and a vector function U = (u, v, θ)T .

(2.7.69)

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89

Then system (2.7.66) is converted into Ut + AU = F, where

 



(2.7.70) 



−v 0 −I 0 u 2 2      v = −Dx u + γDx θ  , AU = −Dx 0 γDx 2 θ γDx v − kDx2 θ 0 γDx −kDx and

(2.7.71)

 

0 F = f . g

(2.7.72)

H = H01 × L2 × L2

(2.7.73)

Let equipped with the inner product (U1 , U2 )H =

Z l

(Dx u1 Dx u2 + v1 v2 + θ1 θ2 )dx,

(2.7.74)

D(A) = (H 2 ∩ H01 ) × H01 × (H 2 ∩ H01 ).

(2.7.75)

0

and Clearly, H is a Hilbert space, and D(A) is dense in H. In what follows we prove that A is m-accretive. For U ∈ D(A), by integration by parts we have (AU, U )H =

Z l 0

(−Dx vDx u − vDx2 u + γvDx θ + γθDx v − kθDx2 θ)dx

= kkDx θk2L2 ≥ 0.

(2.7.76)

Then by Lemma 2.2.3, it remains to show that R(I +A) = H. Consider solvability of the following equations: 







u−v f1 (I + A)U =  −Dx2 u + v + γDx θ  =  f2  , f3 γDx v + θ − kDx2 θ

(2.7.77)

where f1 ∈ H01 , f2 ∈ L2 , f3 ∈ L2 . Solving the first equation of (2.7.77) yields that v = u − f1 .

(2.7.78)

Then we replace v by this formula in the second and third equations

90

NONLINEAR EVOLUTION EQUATIONS

to obtain the system of equations for u and θ:

 2 2  −Dx u + u + γDx θ = f2 + f1 ∈ L , 

θ − kDx2 θ + γDx u = f3 + γDx f1 ∈ L2 .

(2.7.79)

This is an elliptic system of two equations. For (u, θ) ∈ H01 × H01 , (v, φ) ∈ H01 × H01 , we introduce the following bilinear form: a((u, θ), (v, φ)) =

Z l 0

(Dx uDx v + uv + kDx θDx φ + θφ + γvDx θ + γφDx u))dx. (2.7.80)

It is easy to see that a((u, θ), (v, φ)) is a bounded bilinear form, and a((u, θ), (u, θ)) =

Z l 0

(|Dx u|2 + u2 + k|Dx θ|2 + θ2 )dx,

(2.7.81)

i.e., it is coercive. The weak formulation for system (2.7.79) is to find (u, θ) ∈ H01 ×H01 such that for all (v, φ) ∈ H01 ×H01 , the following holds: a((u, θ), (v, φ)) =

Z l 0

((f2 + f1 )v + (f3 + γDx f1 )φ)dx.

(2.7.82)

By the well-known Lax-Milgram theorem stated in Chapter 1, problem (2.7.79) admits a unique weak solution (u, θ) ∈ H01 × H01 . This also implies that u ∈ H01 is a weak solution for the single second elliptic equation with respect to v: −Dx2 v + v = f2 + f1 − γDx θ ∈ L2 .

(2.7.83)

By the regularity result for the elliptic equation, we conclude that u ∈ H 2 ∩ H01 . In the same way, we also conclude that θ ∈ H 2 ∩ H01 . Thus, R(I + A) = H, i.e., A is m-accretive. THEOREM 2.7.16 Suppose that

f, g ∈ C 1 ([0, +∞), L2 ). Then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , θ0 ∈ H 2 ∩ H01 , problem (2.7.66) admits a unique solution (u, θ) such that

u ∈ C 2 ([0, ∞), L2 ) ∩ C 1 ([0, ∞), H01 ) ∩ C([0, ∞), H 2 ∩ H01 ), θ ∈ C 1 ([0, ∞), L2 ) ∩ C([0, ∞), H 2 ∩ H01 ).

Semigroup Method

91

Proof. Problem (2.7.66) can be converted into the initial value problem for an abstract evolution equation:   Ut + AU = F, 

U (0) = (u0 , u1 , θ0

)T .

(2.7.84)

As proved previously, A is m-accretive. By the assumptions, we have (u0 , u1 , θ0 )T ∈ D(A), and F = (0, f, g)T ∈ C 1 ([0, ∞), H). Therefore, by Theorem 2.4.1, problem (2.7.84) admits a unique classical solution U from which the conclusion of the present theorem follows. Thus, the proof is complete. ¤ Similarly we can also treat semilinear equations with f , and g being nonlinearly dependent on u, ut , ux , θ, θx . Before going further, we would like to make the following remarks. As we can see from Section 2.5, Theorem 2.5.4 and Theorem 2.5.6 are available for the discussion on existence and uniqueness of maximal classical solution for nonlinear problem (2.7.84) with F = (0, f, g)T . Notice that in order to apply Theorem 2.5.4, F has to be a nonlinear operator from H = H01 × L2 × L2 into itself, and it should satisfy the local Lipschitz condition. Since now n = 1 which, by the Sobolev imbedding theorem, implies that H01 is continuously imbedded into C[0, l]. It turns out that f and g can nonlinearly depend on u with any order growth. However, f , g can nonlinearly depend on θ with at most linear growth. In order to loose this constraint, we may use Theorem 2.5.6. Thus, F has to be a nonlinear operator from D(A) = (H 2 ∩ H01 ) × H01 × (H 2 ∩ H01 ) into itself. In order to satisfy this condition, f and g have to be nonlinear functions such that when u = 0, θ = 0, f = g = 0. Besides, in order to show whether the solution blows up in a finite time Tmax or has a unique global solution, one has to investigate the behavior of the norm ku(t)kH 2 + kut (t)kH 1 + kθ(t)kH 2

92

NONLINEAR EVOLUTION EQUATIONS

as t → Tmax − 0. This is more restrictive than the estimate of ku(t)kH 1 + kut (t)kL2 + kθ(t)kL2 required in Theorem 2.5.4. Having kept these in our mind, we now give an example to illustrate the above idea. THEOREM 2.7.17 Suppose that f = −u3 , g = sin θ . Then for any

u0 ∈ H 2 ∩ H01 , u1 ∈ H01 , θ0 ∈ H 2 ∩ H01 , problem (2.7.66) admits a unique global solution (u, θ) such that

u ∈ C 2 ([0, ∞), L2 ) ∩ C 1 ([0, ∞), H01 ) ∩ C([0, ∞), H 2 ∩ H01 ), θ ∈ C 1 ([0, ∞), L2 ) ∩ C([0, ∞), H 2 ∩ H01 ). Proof. As discussed previously, F (U ) = (0, −u3 , sin θ)T is a nonlinear operator from U = (u, v, θ)T ∈ H = H01 × L2 × L2 into H, and it satisfies the local Lipschitz condition. Therefore, by Theorem 2.5.4, problem (2.7.84) admits a unique maximal solution U such that U ∈ C 1 ([0, Tmax ), H) ∩ C([0, Tmax ), D(A)). In order to prove the existence of global solution, we have to obtain the uniform estimate of kU (t)kH , i.e., ku(t)kH 1 + kut (t)kL2 + kθ(t)kL2 . Taking the inner product of (2.7.84) with U in H, and using (2.7.76), and v = ut , we get µ

¶

1 1 d ku(t)k2H 1 + kut (t)k2L2 + kθ(t)k2L2 + ku(t)k4L4 2 dt 2 Z l ´ 1³ (2.7.85) = sin θθdx ≤ l + kθ(t)k2L2 . 2 0 By the Gronwall inequality, we obtain that for any 0 ≤ t < Tmax , 1 ku(t)k2H 1 + kut (t)k2L2 + kθ(t)k2L2 + ku(t)k4L4 ≤ C 2

(2.7.86)

Semigroup Method

93

with C being a positive constant depending on Tmax , and for finite Tmax , C < ∞. Thus, by Theorem 2.5.4, Tmax must be infinity, and the global existence follows. The proof is complete. ¤

2.8

Bibliographic Comments

The book [118] by A. Pazy provides a comprehensive study of linear semigroup theory with applications to partial differential equations. We refer to the book [155] by K. Yosida for the corresponding part of semigroup theory. We also refer to a recent book [17] by H. Amann for the comprehensive study of related topics, including interpolation and extrapolation of Banach scales, maximal regularity aiming at applications to parabolic equations. The book [64] by D. Henry provides a very nice treatment for the semilinear parabolic equations and its dynamical system in which A is the infinitesimal generator of a linear analytic semigroup. When A is time-dependent, the corresponding theory was developed by H. Tanabe, P.E. Sobolevskii and others; see the book [148] by H. Tanabe, the papers [144], [145] by P.E. Sobolevskii, and the book [17] by H. Amann in this direction; see also the book [155] by K. Yosida for a detailed survey. To apply to quasilinear partial differential equations, one has to establish the corresponding semigroup theory for the case that A is a nonlinear, even multi-valued operator. When B is a Hilbert space or a Banach space whose dual space is uniformly convex, this theory was mainly established by Y. Komura (for the Hilbert space) and T. Kato (for the Banach space whose dual space is uniformly convex). We refer to the papers [85], [86] by Y. Komura and the papers [72]–[77] by T. Kato for the details. We also refer to the book [29] by H. Brezis and the book [155] by K. Yosida in this direction. The papers by T. Kato contain the interesting applications to quasilinear parabolic equations as well as quasilinear hyperbolic equations. Aiming at applications to the porous media equation mentioned in Chapter 1 in which B = L1 (Ω) is a good choice of the framework, M.G. Crandall and T. Liggett extended the nonlinear semigroup theory to the case that B is a general Banach space. We refer to the paper [37] by M.G. Crandall and T. Liggett for

94

NONLINEAR EVOLUTION EQUATIONS

the details, and the book [155] by K. Yosida for a good survey. For a linear mechanical system with various dissipation such as friction, heat conduction, viscosity, it is important to investigate the exponential stability and analyticity of the corresponding semigroup associated with this mechanical system. We refer to the recent book [98] by Z. Liu and S. Zheng and the references cited there. A systematic approach, which combines the corresponding result in semigroup theory with PDE technique, has been developed in this book and related papers to deal with this issue.

Chapter 3 Compactness Method and Monotone Operator Method

In this chapter we introduce two important methods, namely the compactness method and the monotone operator method. These two methods were mainly developed in the 1960s, but they are still powerful tools today to deal with nonlinear evolution equations as well as nonlinear stationary problems.

3.1

Compactness Method

Given an initial value problem or initial boundary value problem for a nonlinear evolution equation, if “good enough” a priori estimates, which is often called the compactness estimates in literature, can be (formally) obtained, then we can use a systematic approach that is called the compactness method to deal with the issue of local existence or global existence. The same technique also applies to nonlinear stationary problems, which often are boundary value problems for nonlinear elliptic equations. The following are three major steps for this method: (i) Use the Faedo-Galerkin method, i.e, choose certain base functions in an appropriate Sobolev space, and solve the approximate problems in any finite dimensional space spanned by finite base functions. This often turns out to be an initial value problem for nonlinear ordinary differential equations. By the well-known local existence theorem for ordinary differential equations, local existence of solution to the approximate problem follows. (ii) Obtain the compactness estimates for the solution of the approximate problem. It also turns out that the solution to the approximate problem globally exists. (iii). Further use of the obtained compactness estimates allows one to 95

96

NONLINEAR EVOLUTION EQUATIONS

choose a subsequence of solutions of the approximate problem obtained in the second step, converging to a solution of the original problem; uniqueness of solution for the original problem has to be proved separately, but the compactness estimates obtained in the second step are still very useful for this purpose. In this section we will illustrate this method through some examples. In the whole process, we will encounter various convergences, and need various compactness lemmas. Therefore, in the following subsection, we first make some preparations for that.

3.1.1

Some Results on Convergence and Compactness

Let B be a Banach space. Usually we will encounter the following three different concepts of convergence. (i) Strong convergence. Let un ∈ B, u ∈ B such that as n → +∞, kun − ukB → 0.

(3.1.1)

Then un is said to strongly converge to u. (ii) Weak convergence. Let un , u ∈ B such that for any f ∈ B 0 as n → +∞, f (un ) → f (u).

(3.1.2)

Then un is said to weakly converge to u. (iii) Weakly star convergence. Let un , u ∈ B, and let B be the dual space of another Banach space B ∗ , i.e., B = (B ∗ )0 . If for any f ∈ B ∗ , as n → ∞, un (f ) → u(f ).

(3.1.3)

Then un is said to weakly star converge to u. It is well known that strong convergence implies weak convergence, and weak convergence implies weakly star convergence. When B is a reflexive Banach space, weak convergence is equivalent to weakly star convergence. Concerning compactness, the first well-known result from functional analysis is the following. LEMMA 3.1.1 Any bounded set in a reflexive Banach space is weakly compact, i.e., any sequence in a bounded set has a weakly converging subsequence.

Ex. 3.1.1. Since for 1 < p < ∞, Lp (Ω) is a reflexive Banach space, any bounded set in Lp is weakly compact. ¤

Compactness Method and Monotone Operator Method

97

LEMMA 3.1.2 Let B be a Banach space such that B = (B ∗ )0 where B ∗ is a reflexive Banach space. Then any bounded set in B is weakly star compact, i.e., any sequence in a bounded set in B has a weakly star converging subsequence.

Proof. It is also well known in functional analysis, and we only point out the following essence of the the proof: because B ∗ is reflexive, there is a finite ε-net, from which the conclusion easily follows. ¤ Ex. 3.1.2. Since for 1 ≤ p < ∞, Lp (Ω) is a reflexive Banach space and L∞ (Ω) = (L1 (Ω))0 , any bounded set in Lp (Ω) with 1 < p ≤ ∞ is weakly star compact. In particular, any bounded set in L∞ (Ω) is weakly star compact. ¤ In addition to the compactness theorem of the imbedding operator for various Sobolev spaces (see Chapter 1), the following result, which was established by J. P. Aubin (see [19]), plays a very important role in the study of nonlinear evolution equations. THEOREM 3.1.1 Let B0 , B , B1 be three Banach spaces where B0 , B1 are reflexive. Suppose that B0 is continuously imbedded into B , which is also continuously imbedded into B1 , and imbedding from B0 into B is compact. For any given p0 , p1 with 1 < p0 , p1 < ∞, let

W = {v | v ∈ Lp0 ([0, T ], B0 ), vt ∈ Lp1 ([0, T ], B1 )}.

(3.1.4)

Then the imbedding from W into Lp0 ([0, T ], B) is compact.

Before giving the proof, let us first make some explanations of this result. For functions v(x, t) with time variable t varying in [0, T ] and space variables x varying in a bounded domain Ω ⊂ Rn , it is often that v can be viewed as an abstract function valued in Banach spaces, as B0 , B, B1 as assumed in the statement of this theorem. Then this lemma simply claims that the compactness in x variables, as indicated by imbedding from B0 into B being compact, together with the compactness in t variable, which follows from imbedding of {u ∈ Lp0 [0, T ], ut ∈ Lp1 [0, T ]} into {u ∈ Lp0 [0, T ]}, implies the compact imbedding of W into Lp0 ([0, T ], B). In order to prove this theorem, we first prove the following lemma.

98

NONLINEAR EVOLUTION EQUATIONS

LEMMA 3.1.3 Let B0 , B, B1 be three Banach spaces. Suppose that B0 is continuously imbedded into B , which is also continuously imbedded into B1 , and imbedding from B0 into B is compact. Then for any η > 0, there is a positive constant Cη depending only on η such that for any v ∈ B0 , the following holds.

kvkB ≤ ηkvkB0 + Cη kvkB1 .

(3.1.5)

Proof. We use the contradiction argument. Suppose that (3.1.5) is not true. Then there is a positive constant η0 and a sequence vk such that kvk kB ≥ η0 kvk kB0 + kkvk kB1 holds for all k ∈ N . Let uk =

vk . kvk kB0

(3.1.6)

(3.1.7)

Then kuk kB0 = 1, and kuk kB ≥ η0 + kkuk kB1 , ∀k ∈ N.

(3.1.8)

On the other hand, since B0 is continuously imbedded into B, there is a positive constant C independent of uk such that kuk kB ≤ Ckuk kB0 = C.

(3.1.9)

We infer from (3.1.8) and (3.1.9) that C → 0, (3.1.10) k i.e., kuk kB1 → 0. It follows from the imbedding from B0 into B being compact that there is a subsequence of uk , still denoted by uk , and an element u0 ∈ B such that kuk kB1 ≤

kuk − u0 kB → 0,

(3.1.11)

which turns out that kuk − u0 kB1 → 0 because the imbedding from B onto B1 is continuous. Thus u0 = 0, and uk → 0 in B. This contradicts kuk kB ≥ η0 + kkuk kB1 ≥ η0 ,

(3.1.12)

as seen from (3.1.8). Thus the proof is complete.

¤

Proof of Theorem 3.1.1. We make the following reduction for the

Compactness Method and Monotone Operator Method

99

proof. It suffices to verify that if vn ∈ W such that ³

kvn kW = kvn k2Lp0 ([0,T ],B0 ) + kvnt k2Lp1 ([0,T ],B1 )

´1

2

≤ M,

(3.1.13)

then there is a subsequence, still denoted by vn , converging in Lp0 ([0, T ], B) strongly. Since W is a reflexive Banach space, by Lemma 3.1.1, there is a subsequence, still denoted by vn , weakly converging in W to v ∈ W . Instead of vn , we consider the difference vn − v, but still denoted by vn . Thus it suffices to verify that if vn weakly converges to zero in W , then vn strongly converges to zero in Lp0 ([0, T ], B). By Lemma 3.1.3, for any η > 0, there is Cη such that kvn kLp0 ([0,T ],B) ≤ ηkvn kLp0 ([0,T ],B0 ) + Cη kvn kLp0 ([0,T ],B1 ) . For any ε > 0, we choose η =

ε 2M .

(3.1.14)

Thus, (3.1.14) turns out to be

ε + Cε kvn kLp0 ([0,T ],B1 ) . (3.1.15) 2 Therefore, we need only to prove that kvn kLp0 ([0,T ],B1 ) → 0. Let kvn kLp0 ([0,T ],B) ≤

p = min (p0 , p1 ).

(3.1.16)

It follows from vn ∈ Lp ([0, T ], B1 ), and vnt ∈ Lp ([0, T ], B1 ) that vn ∈ C([0, T ], B1 ). Therefore, by the well-known Lebesgue theorem in real analysis, it suffices to prove that for any t ∈ [0, T ], kvn (t)kB1 → 0. In what follows we prove that kvn (0)kB1 → 0, but the method clearly applies to other t ∈ [0, T ]. Let φ(t) ∈ C 1 [0, T ] with φ(0) = −1, φ(T ) = 0. For λ ∈ (0, 1), let wn (t) = vn (λt). Thus, vn (0) = wn (0) = = Let αn =

Z T 0

Z T 0

φ0 wn dt +

0

φ wn dt, βn =

Thus, kβn kB1 ≤ C

Z λT 0

Z T 0

Z T 0

Z T 0

(φwn )0 dt

φwn0 dt.

φwn0 dt

=

kvn0 (τ )kB1 dτ

Z T 0

(3.1.17)

λφvn0 (λt)dt.

(3.1.18)

100

NONLINEAR EVOLUTION EQUATIONS 1− p1

≤ C(λT )

ÃZ λT 0

!1

kvn0 (τ )kpB1 dτ

p

1

≤ CT M λ1− p

(3.1.19)

where C is a positive constant independent of T and vn , and CT is a positive constant depending only on T . For any ε > 0, there is a λ > 0 such that ε Cε kβn kB1 ≤ . (3.1.20) 4 Since vn , and also wn weakly converges to zero in Lp ([0, T ], B0 ), it follows that αn → 0 weakly in B0 . Therefore, we infer from the imbedding from B0 into B1 being compact that as n → ∞, kαn kB1 = k

Z T 0

φ0 wn dtkB1 → 0.

(3.1.21)

Combining (3.1.15) with (3.1.19), (3.1.21) completes the proof of the theorem. ¤ REMARK 3.1.1 It can be seen from the proof that if the assumption of reflexivity of B0 , B1 is replaced by the assumption that B0 , B1 are the dual spaces of reflexive Banach spaces B0∗ , B1∗ , respectively, then the conclusion of Theorem 3.1.1 still holds. Accordingly, in the proof, weak convergence should be replaced by weakly star convergence.

In applications, Lp (Ω), (1 ≤ p ≤ ∞) are the spaces we encounter quite often. For these spaces, in addition to three kinds of convergence, namely, strong convergence, weak convergence, and weakly star convergence, as a real function, there is another important concept of convergence, i.e., almost everywhere point-wise convergence, simply denoted by un → u, a.e. In what follows we collect some useful results that will be needed later. LEMMA 3.1.4 Suppose that Ω is a bounded or unbounded domain in Rn . Let un (x), u(x) be real functions in Lp (Ω), (1 ≤ p ≤ ∞) such that un strongly converges to u in Lp (Ω). Then if 1 ≤ p < ∞, un has a subsequence almost everywhere converging to u; if p = ∞, then un itself almost everywhere converges to u.

Proof. This is the well-known result in real analysis, and we can leave the proof to the reader. ¤

Compactness Method and Monotone Operator Method

101

LEMMA 3.1.5 Suppose that Ω is a bounded domain in Rn . Let un (x) be a bounded sequence in Lp (Ω) (1 ≤ p < +∞) such that un almost everywhere converges to u. Then u also belongs Lp (Ω), and un weakly converges in Lp (Ω) to u.

Proof. It follows from the well-known Fato lemma that u belongs to Lp (Ω). Let N be a positive integer, and EN = {x | x ∈ Ω, |un −u| ≤ 1, ∀n ≥ N } =

\

{x | x ∈ Ω, |un −u| ≤ 1}.

n≥N

(3.1.22)

Then EN is a measurable set. It is easy to see that when N 0 ≤ N , EN 0 ⊂ EN , and as N → ∞, mes(EN ) → mes(Ω).

(3.1.23)

Let ΦN = {φ | φ ∈ Lp (Ω), Supp(φ) ⊂ EN } 0

1 p

p0

where is the conjugate number of p, i.e., denotes the support of φ. Let Φ=

[

+

1 p0

(3.1.24)

= 1, and Supp(φ)

ΦN .

(3.1.25)

N ≥1

Then it is easy to see from (3.1.23) that Φ is dense in Lp (Ω). For any φ ∈ Φ, there is an N0 such that φ ∈ ΦN0 , and when n ≥ N0 , n → ∞, we have 0

|φ(un − u)| ≤ |φ|,

φ(un − u) → 0 a.e.

(3.1.26)

By the Lebesgue theorem, it follows that Z

Ω

φ(un − u)dx → 0.

(3.1.27)

We infer from the facts that Φ is dense in Lp (Ω) and un is bounded in Lp that 0

Z

Ω

φ(un − u)dx → 0, ∀φ ∈ Lp (Ω).

Thus the proof is complete.

0

(3.1.28) ¤

REMARK 3.1.2 When p = ∞, then the conclusion becomes that un weakly star converges to u.

It is well known that if a sequence in a reflexive Banach space is weakly compact, then in general there is a weakly convergent subse-

102

NONLINEAR EVOLUTION EQUATIONS

quence. The remarkable point of the above lemma is that with the additional condition of almost everywhere convergence, the sequence itself is weakly convergent. Recall the following definition on the uniformly convex Banach space. DEFINITION 3.1.1 A Banach space B is said to be uniformly convex if for any φ, ψ ∈ B such that kφk = kψk = 1, kφ − ψk ≥ ε > 0, then there is a constant δ > 0 depending only on ε such that kφ + ψk ≤ 2(1 − δ) < 2. LEMMA 3.1.6 Let B be a Hilbert space or a uniformly convex Banach space. Suppose that un , u ∈ B , kun k → kuk, and un also weakly converges to u. Then un must strongly converge to u, i.e., kun − uk → 0.

Proof. When B is a Hilbert space, the conclusion of the present theorem easily follows from the following identity: kun − uk2 = kun k2 + kuk2 − (un , u) − (u, un ) → 0.

(3.1.29)

When B is a uniformly convex Banach space, we use the contradiction argument. We first notice that if u = 0, i.e., u is the trivial zero element, then the conclusion is obvious, and the proof is superfluous. Thus we only have to deal with the case that u is different from the trivial zero element. Let un u vn = , v= . (3.1.30) kun k kuk Then kvn k = kvk = 1, and it easily follows from the assumptions that vn weakly converges to v. If un does not strongly converge to u, then vn does not strongly converge to v, too. Therefore, there is a positive constant ε0 and a subsequence of vn , still denoted by vn such that for all n, m ∈ N , kvn − vm k ≥ ε0 .

(3.1.31)

On the other hand, by the assumption that B is uniformly convex, there is a constant δ, 0 < δ < 1 depending on ε0 such that kvn + vm k ≤ 2(1 − δ) < 2.

(3.1.32)

By the Hahn-Banach theorem, there is f ∈ B 0 such that f (v) = kvk = 1, and kf kB 0 = 1. Thus, f (vn + vm ) → 2f (v) = 2,

(3.1.33)

Compactness Method and Monotone Operator Method

103

and |f (vn + vm )| ≤ kf kB 0 kvn + vm k ≤ 2(1 − δ) < 2, a contradiction. Thus the proof is complete.

(3.1.34) ¤

REMARK 3.1.3 Since Lp (Ω) (1 < p < ∞) is a uniformly convex Banach space, the assertion that un strongly converges to u in Lp follows from the fact that un weakly converges to u and kun k converges to kuk. LEMMA 3.1.7 Let B be a Banach space, and B = (B ∗ )0 with B ∗ being another Banach space. Suppose that for 1 < p ≤ ∞,  p  un → u weakly star in L ([0, T ], B), (3.1.35)  0 un → u0 weakly star in Lp ([0, T ], B). Then

un (0) → u(0) weakly star in B.

(3.1.36)

Proof. Without loss of generality, we can assume that u = 0. We can write un (0) = un (t) −

Z t 0

For any φ ∈ B ∗ , we have

u0n (τ )dτ.

(un (0), φ) = (un (t), φ) − Integrating with respect to t yields T (un (0), φ) =

Z T 0

(un (t), φ)dt −

Z t 0

Z TZ t 0

0

(3.1.37)

(u0n , φ)dτ.

(3.1.38)

(u0n (τ ), φ)dτ dt.

(3.1.39)

By assumption (3.1.35), we have Z T 0

and for all t ∈ [0, T ],

Z t 0

Furthermore, |

Z t 0

(un (t), φ)dt → 0,

(3.1.40)

(u0n (τ ), φ)dτ → 0.

(3.1.41)

(u0n , φ)dτ | ≤

Z t 0

ku0n kB kφkB ∗ dτ

104

NONLINEAR EVOLUTION EQUATIONS ≤ kφkB ∗ T

1 p0

ÃZ T 0

!1

ku0n kpB dτ

p

≤C

(3.1.42)

with C being a positive constant. Thus, by the Lebesgue theorem, the terms on the right-hand side of (3.1.39) converge to zero. The proof is complete. ¤ REMARK 3.1.4 Suppose that 1 < p < ∞, and B is a reflexive Banach space. Then the weakly star convergence in the above lemma is equivalent to the weak convergence.

3.1.2

Method and Examples

In this section we plan to illustrate the main steps of the compactness method through some examples. 1. Consider the following initial boundary value problem for a nonlinear parabolic equation:  3   ut − ∆u + u − u = f (x, t),          

(x, t) ∈ Ω × R+ ,

u|Γ = 0,

(3.1.43)

u|t=0 = u0 (x)

where Ω is a bounded domain in Rn with the smooth boundary Γ, f (x, t) ∈ L2 ([0, T ], L2 (Ω)), and u0 (x) ∈ H01 (Ω) ∩ L4 (Ω). Notice that this problem has been treated in Chapter 2 by the semigroup method. However, the assumptions on f and u0 are more restrictive there. In what follows we solve this problem by just following the three major steps described in the first section of this chapter. (i) First we choose the base functions {wj , j ∈ N } in H01 (Ω) with wj being the eigenfunctions of the Laplacian operator subject to the Dirichlet boundary condition:   −∆wj = λj wj , 

wj |Γ = 0.

(3.1.44)

We also normalize wj such that kwj kL2 = 1. By the elliptic operator theory as described in Chapter 1, {wj , j ∈ N } forms base functions in ¯ j ∈ N . Now we use the H01 ∩ L4 (Ω), and if Γ is C ∞ , then wj ∈ C ∞ (Ω), Faedo-Galerkin method to find the approximate solution. Let m be a

Compactness Method and Monotone Operator Method

105

given positive integer and um =

m X

gim (t)wi (x),

(3.1.45)

i=1

which satisfies the following identities:  3  (umt , wi ) − (∆um , wi ) + (um − um , wi ) = (f, wi ), i = 1, · · · , m 

(um (0), wi ) = ξim , i = 1, · · · , m

(3.1.46)

where (·, ·) denotes the inner product in L2 (Ω), and ξim are given constants such that as m → +∞, m X

ξim wi → u0

i=1

strongly in H01 ∩ L4 .

(3.1.47)

Existence of such ξim follows from u0 ∈ H01 ∩ L4 (Ω) and the fact that {wj , j ∈ N } is the base in H01 ∩ L4 . Thus, (3.1.46) is reduced to the initial value problem for a system of first-order differential equations with respect to gim :  dgim   + λi gim + Gi (g) = fi (t), i = 1, · · · , m,    

where

dt

(3.1.48)

gim (0) = ξim , i = 1, · · · , m, 

Gi (g) = (

m X

gjm (t)wj )3 −

j=1

m X



gjm (t)wj , wi 

(3.1.49)

j=1

is a third-order polynomial of gjm , and fi = (f, wi ) ∈ L2 (0, T ). Thus by the usual Picard iteration method used in ODE, we can conclude that there is a t0 > 0 depending only on |ξim | such that in [0, t0 ], problem (3.1.48) admits a unique local solution gim (t) such that gim ∈ C[0, t0 ], 0 (t) ∈ L2 [0, T ]. and gim (ii) We now try to get the a priori estimates for the approximate solution um (x, t) obtained in the previous step. Multiplying both sides of equations in (3.1.46) by gim (t), and summing with respect to i yields 1d kum k2 + k∇um k2 + kum k4L4 − kum k2 = (f, um ). 2 dt

(3.1.50)

106

NONLINEAR EVOLUTION EQUATIONS

Hereafter we use k · k to denote the norm in L2 (Ω). Integrating with respect to t, we obtain Z

Z

t t 1 k∇um k2 dτ + kum k4L4 dτ kum (t)k2 + 2 0 0 Z t Z t m 1X = ξ 2 + (f, um )dτ + kum k2 dτ 2 i=1 im 0 0 m 1X 1 ξ2 + 2 i=1 im 2

≤

Z t 0

kf k2 dτ +

3 2

Z t 0

kum k2 dτ.

(3.1.51)

By the Young inequality 1 u2 ≤ u4 + 1, 4

(3.1.52)

and the fact that m X i=1

2 ξim

is uniformly bounded, due to u0 ∈ H01 , we deduce that 2

kum (t)k +

Z t 0

2

k∇um k dτ +

Z t 0

kum k4L4 dτ ≤ CT

∀t ∈ [0, t0 ] (3.1.53)

where CT is a positive constant depending only on ku0 k, and T . It follows from (3.1.53) that kum (t)k2 =

m X i=1

2 gim (t) ≤ CT

∀t ∈ [0, T ].

RT 0

kf k2 dτ (3.1.54)

This implies that the solution to the initial value problem for the system of ODE (3.1.48) can be extended to [0, T ], and on [0, T ], we have the following uniform a priori estimates:   um uniformly bounded in L∞ ([0, T ], L2 (Ω)),     

um uniformly bounded in L2 ([0, T ], H01 (Ω)),      u 4 4 m uniformly bounded in L ([0, T ], L (Ω)).

(3.1.55)

Now we would like to get more a priori estimates. In doing so, mul0 , retiplying both sides of the equations in (3.1.46) by λi gim , and gim spectively, and then summing over i, we get (u0m , −∆um ) + k∆um k2 − (u3m , ∆um ) = (f + um , −∆um ) ≤ (kf k + kum k)k∆um k,

(3.1.56)

Compactness Method and Monotone Operator Method

107

and ku0m k2 − (∆um , u0m ) + (u3m , u0m ) = (f + um , u0m ) ≤ kf + um kku0m k.

(3.1.57)

Integrating with respect to t yields Z

Z

t 1 1 t kum ∇um k2 dτ k∇um (t)k2 + k∆um k2 dτ + 3 2 2 0 0 Z t 1 2 ≤ k∇um (0)k + (kf k2 + kum k2 )dτ, (3.1.58) 2 0

and

Z

t 1 1 k∇um (t)k2 + ku0m k2 dτ + kum (t)k4L4 2 4 0 Z t 1 1 2 4 ≤ k∇um (0)k + kum (0)kL4 + (kf k2 + kum k2 )dτ. (3.1.59) 2 4 0

Then we get the following further a priori estimates:

  um uniformly bounded in L∞ ([0, T ], H01 ∩ L4 (Ω)),     

um uniformly bounded in L2 ([0, T ], H 2 ∩ H01 (Ω)),       0 2 2

(3.1.60)

um uniformly bounded in L ([0, T ], L (Ω)).

(iii) Thus, by Lemma 3.1.1 and Lemma 3.1.2, there is a subsequence of um , still denoted by um such that   u → u weakly star in L∞ ([0, T ], H01 ∩ L4 (Ω)),   m   

um → u weakly in L2 ([0, T ], H 2 ∩ H01 (Ω)),       u0 → u0 weakly in L2 ([0, T ], L2 (Ω)). m

(3.1.61)

Taking B0 = H 2 ∩ H01 , B = H01 , and B1 = L2 , by Theorem 3.1.1, there is a subsequence of um , still denoted by um such that um → u strongly in L2 ([0, T ], H01 ).

(3.1.62)

By Lemma 3.1.4, there is a subsequence of um , still denoted by um such that um almost everywhere converges to u in QT = Ω × [0, T ]. It turns out that u3m almost everywhere converges to u3 in QT . On the 4 other hand, (3.1.60) implies that u3m is uniformly bounded in L 3 (QT ). Therefore, we infer from Lemma 3.1.5 that 4

4

u3m → u3 weakly in L 3 ([0, T ], L 3 (Ω)).

(3.1.63)

108

NONLINEAR EVOLUTION EQUATIONS

By Lemma 3.1.7, and Remark 3.1.4, we have um (0) =

m X

ξim wi → u(0) weakly in L2 (Ω)

(3.1.64)

i=1

However, the term on the left-hand side of (3.1.64) strongly converges in H01 ; hence, it also weakly converges to u0 in L2 (Ω). By the uniqueness of the limit, we get u(0) = u0 (x).

(3.1.65)

Passing to the limit in (3.1.46), since each term on the left-hand side of 4 (3.1.46) is weakly convergent in L 3 [0, T ], we obtain that the following 4 holds in L 3 [0, T ]: (u0 , wi ) − (∆u, wi ) + (u3 − u, wi ) = (f, wi ), ∀i = 1, 2, · · · .

(3.1.66)

4 3

Since {wi , i ∈ N } is a base in L (Ω), we infer from (3.1.66) that the 4 4 following holds in L 3 ([0, T ], L 3 (Ω)): u0 − ∆u + u3 − u = f.

(3.1.67)

Since all u0 , ∆u, and f belong to L2 ([0, T ], L2 (Ω)), u3 also belongs to L2 ([0, T ], L2 (Ω)), and (3.1.67) also holds in L2 ([0, T ], L2 (Ω)). Thus, we have the following result. THEOREM 3.1.2 Suppose that u0 ∈ H01 ∩ L4 (Ω), f ∈ L2 ([0, T ], L2 (Ω)). Then problem (3.1.43) admits a unique solution u such that

u ∈ L∞ ([0, T ], H01 ∩ L4 (Ω)) ∩ L2 ([0, T ], H 2 ∩ H01 (Ω)), and

u0 ∈ L2 ([0, T ], L2 (Ω)). Proof. Existence of a solution has been proved through three steps described previously. It remains to prove the uniqueness. Suppose that there are two solutions u1 , u2 . Let u = u1 − u2 . Then u satisfies   ut − ∆u + u31 − u32 − u = 0,           

u|Γ = 0,

u|t=0 = 0.

(x, t) ∈ Ω × R+ , (3.1.68)

Compactness Method and Monotone Operator Method

109

Multiplying the equation of (3.1.68) by u, and integrating over Ω yields 1d kuk2 + k∇uk2 + 2 dt

Z

Ω

u2 (u21 + u1 u2 + u22 )dx = kuk2 .

(3.1.69)

Since the last term on the left-hand side of (3.1.69) is also non-negative, applying the Gronwall inequality yields ku(t)k2 ≤ CT ku(0)k2 = 0

∀t ∈ [0, T ].

Thus the uniqueness follows, and the proof is complete.

(3.1.70) ¤

REMARK 3.1.5 When u0 and f are more regular, we can get more regular a priori estimates of approximate solutions, and hence by passing to the limit, we are led to the existence of a more regular solution.

2. Now we consider the following initial boundary value problem for a nonlinear hyperbolic equation:  utt − ∆u + |ut |ρ ut = f (x, t),          

(x, t) ∈ Ω × R+ ,

u|Γ = 0,

(3.1.71)

u|t=0 = u0 (x), ut |t=0 = u1 (x)

where Ω is a bounded domain in Rn with smooth boundary Γ and ρ > 0 is a given constant. THEOREM 3.1.3 Suppose that f ∈ L2 ([0, +∞), H01 (Ω)), f 0 ∈ L2 ([0, +∞), L2 (Ω)). Then for any u0 ∈ H 2 ∩ H01 (Ω), u1 ∈ H01 ∩ L2(ρ+1) (Ω), problem (3.1.71) admits a unique global solution u such that for any T > 0,

u ∈ L∞ ([0, T ], H 2 ∩ H01 (Ω)), u0 ∈ L∞ ([0, T ], H01 ∩ L2(ρ+1) (Ω)) ∩ Lρ+2 (QT ), utt ∈ L∞ ([0, T ], L2 (Ω)). Proof. We still adopt three major steps to prove this theorem. (i) We still choose {wj , j ∈ N } as the eigenfunctions of the Laplacian operator subject to the Dirichlet boundary condition. By the FaedoGalerkin method, as defined by (3.1.44), for any m ∈ N we look for

110

NONLINEAR EVOLUTION EQUATIONS

the approximate solution um (x, t) =

m X

gim (t)wi (x) satisfying

i=1

 ρ  (umtt , wi ) − (∆um , wi ) + (|umt | umt , wi ) = (f, wi ), i = 1, · · · , m, 

(um (0), wi ) = ξim , (umt , wi ) = ηim ,

i = 1, · · · , m,

where ξim and ηim are chosen such that

 m X    ξim wi → u0 strongly in H 2 ∩ H01 (Ω),     i=1  m  X    ηim wi → u1 strongly in H01 ∩ L2(ρ+1) (Ω)).  

(3.1.72)

(3.1.73)

i=1

This can be achieved because {wi , i ∈ N } is dense in the above spaces. Thus, (3.1.73) is deduced to the initial value problem for a system of nonlinear second-order ordinary differential equations:  2 d gim    + λi gim + Gi (g) = fi (t), i = 1, · · · , m, 2   

dt

gim (0) = ξim ,

0 (0) gim

(3.1.74)

= ηim , i = 1, · · · , m,

where Gi (g) = (|u0m |ρ u0m , wi ) is a differentiable function with respect 0 , and f = (f, w ) ∈ H 1 (0, ∞). Again by the Picard iteration to gim i i method, there is t0 > 0 depending on |ξim | and |ηim | such that problem (3.1.74) admits a unique local solution gim ∈ H 3 (0, t0 ). (ii) Now we try to get a priori estimates of the approximate solutions 0 , then summing with respect to i, we um . Multiplying (3.1.72) by gim get ´ 1d ³ 0 2 kum k + k∇um k2 + 2 dt

Z Ω

|u0m |ρ+2 dx = (f, u0m ).

(3.1.75)

Integrating with respect to t, we obtain Z

´ t 1³ 0 kum (t)k2 + k∇um (t)k2 + ku0m kρ+2 Lρ+2 dτ 2 0 Z t ´ 1³ 0 kum (0)k2 + k∇um (0)k2 , ∀t ∈ [0, t0 ]. = (f, u0m )dτ + 2 0

(3.1.76)

Compactness Method and Monotone Operator Method

111

By (3.1.73), in the same manner as before, for any given T > 0, we get ku0m (t)k2

2

+ k∇um (t)k +

Z t 0

ku0m kρ+2 Lρ+2 dτ ≤ CT

∀t ∈ [0, t0 ], (3.1.77)

where CT is a positive constant depending on ku0 kH 1 , ku1 k, and T . Thus,

m X i=1

2 gim and

m X i=1

RT 0

kf k2 dτ ,

0 (gim )2 are uniformly bounded with respect

to t, for t ∈ [0, T ]. This implies that the solution gim of (3.1.74) can be uniquely extended to [0, T ], and for t ∈ [0, T ], the estimate (3.1.76) still holds. Now we want to derive further a priori estimates. Multiplying (3.1.72) 0 , then summing with respect to i, noticing λ w = −∆w , and by λi gim i i i using integration by parts, we get ´ 1 d ³ k∇u0m k2 + k∆um k2 + 2 dt

Z

Ω

∇(|u0m |ρ u0m ) · ∇u0m dx = (∇f, ∇u0m ). (3.1.78)

Since Z

Ω

∇(|u0m |ρ u0m )

·

∇u0m dx

4(ρ + 1) = (ρ + 2)2

Z ¯ ¯2 ρ ¯ ¯ ¯∇(|u0m | 2 u0m )¯ dx, Ω

(3.1.79)

integrating (3.1.78) with respect to t, and applying the Gronwall inequality, we get k∇u0m k2 + k∆um k2 +

Z tZ ¯ ¯2 ρ ¯ ¯ ¯∇(|u0m | 2 u0m )¯ dxdτ ≤ CT 0

Ω

(3.1.80)

where CT is a positive constant depending only on ku1 kH 1 , ku0 kH 2 , RT 2 0 kf kH 1 dτ and T . Estimates (3.1.77) and (3.1.80) imply that  ∞ 2 T H 1 (Ω)),  0  um uniformly bounded in L ([0, T ], H       (ρ+2) (Q ),   um uniformly bounded in L T    u0m uniformly bounded in L∞ ([0, T ], H01 (Ω)),       ρ 

(3.1.81)

|u0m | 2 u0m uniformly bounded in L2 ([0, T ], H01 (Ω)).

In order to obtain the estimate on u00m , differentiating (3.1.72), then 00 , and summing with respect to i, we multiplying the resultant by gim get ´ 1 ³ 00 kum (t)k2 + k∇u0m k2 + (ρ + 1) 2

Z

Ω

|u0m |ρ |u00m |2 dx = (f 0 , u00m ). (3.1.82)

112

NONLINEAR EVOLUTION EQUATIONS

A straightforward calculation yields Z

Ω

|u0m |ρ |u00m |2 dx =

4 (ρ + 2)2

Z ¯ ³ ´¯¯2 ¯∂ 0 ρ2 0 ¯ ¯ |u | u m m ¯ dx. ¯ ∂t Ω

(3.1.83)

To obtain ku00m (0)k2 , which will be needed when we integrate (3.1.82) with respect to t, we first deduce from (3.1.72) that (u00m (0), wi ) − (∆um (0), wi ) + (|u0m (0)|ρ u0m (0), wi ) = (f (0), wi ). (3.1.84) 00 (0), and summing with respect to i Then multiplying the above by gim yields

ku00m (0)k2 = (∆um (0), u00m (0)) − (|u0m (0)|ρ u0m (0), u00m (0)) + (f (0), u00m (0)). H2

H01 ,

H1

L2ρ+2 ,

u1 ∈ ∩ and f ∈ Since u0 ∈ ∩ follows from the assumptions on f , we have

(3.1.85)

C([0, T ], L2 ),

k∆um (0)k ≤ C, k|u0m (0)|ρ u0m (0)k ≤ C, kf (0)k ≤ C

which

(3.1.86)

where C is a positive constant depending only on ku0 kH 2 , ku1 kL2ρ+2 , and kf (0)k. Thus, by the H¨older inequality, we deduce from (3.1.85) that ku00m (0)k ≤ C.

(3.1.87)

Integrating (3.1.82) with respect to t, we infer from the H¨older inequality and (3.1.87) that for t ∈ [0, T ], ku00m (t)k2

+

k∇u0m (t)k

¯2 Z t¯ ¯ ¯∂ 0 ρ2 0 ¯ ¯ + ¯ ∂t (|um | um )¯ dxdτ ≤ CT . 0

(3.1.88)

Thus we have further uniform a priori estimates:

 00 uniformly bounded in L∞ ([0, T ], L2 (Ω)), u    m  ∂ ³ 0 ρ 0 ´   uniformly bounded in L2 ([0, T ], L2 (Ω)). |um | 2 um

(3.1.89)

∂t (iii) In what follows we use the uniform a priori estimates obtained in the previous step, and take a subsequence of um , if necessary, and still denoted by um for various convergence. It follows from (3.1.81) and (3.1.89) that  T  um → u weakly star in L∞ ([0, T ], H 2 H01 (Ω)),     

u0m → u0 weakly star in L∞ ([0, T ], H01 (Ω)),       u00 → u00 weakly star in L∞ ([0, T ], L2 (Ω)). m

(3.1.90)

Compactness Method and Monotone Operator Method

113

Furthermore, if we take B0 = H01 , B = B1 = L2 , then by Theorem 3.1.1, and (3.1.81), (3.1.89), we get u0m → u0 strongly in L2 ([0, T ], L2 (Ω)).

(3.1.91)

On the other hand, by Lemma 3.1.4, we have u0m → u0

almost everywhere in QT ,

(3.1.92)

which implies that ρ

ρ

|u0m | 2 u0m → |u0 | 2 u0 almost everywhere in QT ,

(3.1.93)

|u0m |ρ u0m → |u0 |ρ u0 almost everywhere in QT .

(3.1.94)

and Again from (3.1.81) and (3.1.89) it follows that

 1 0 ρ 0   |um | 2 um → Φ weakly in H (QT ),  

|u0m |ρ u0m

→ Ψ weakly in L

ρ+2 ρ+1

(3.1.95)

(QT ).

By the Sobolev compact imbedding theorem, we have ρ

|u0m | 2 u0m → Φ strongly in L2 (QT ).

(3.1.96)

Thus, by Lemma 3.1.4 again, we have ρ

|u0m | 2 u0m → Φ almost everywhere in QT .

(3.1.97)

Combining (3.1.97) with (3.1.93) yields ρ

Φ = |u0 | 2 u0 .

(3.1.98)

By Lemma 3.1.5, we infer from (3.1.94) and (3.1.95) that Ψ = |u0 |ρ u0 .

(3.1.99)

ρ+2 ρ+1

< 2 and each term converges

Passing to the limit in (3.1.72), since in L

ρ+2 ρ+1

(0, T ), we get

(u00 , wi ) − (∆u, wi ) + (|u0 |ρ u0 , wi ) = (f, wi ),

i = 1, 2, · · · .

(3.1.100)

Since {wi , i ∈ N } is a base in the corresponding space, we deduce that u satisfies the equation of (3.1.71). From (3.1.90) and Lemma 3.1.7 with B = H01 and B = L2 , respectively, we infer that  1  um (0) → u(0) weakly in H0 (Ω),  0 um (0) → u0 (0) weakly in L2 (Ω).

(3.1.101)

114

NONLINEAR EVOLUTION EQUATIONS

Thus u(0) = u0 , u0 (0) = u1 .

(3.1.102)

Now it remains to prove uniqueness. Let u1 , u2 be two solutions in the class described in the statement of this theorem, and u = u1 − u2 . Then u satisfies u00 − ∆u + |u01 |ρ u01 − |u02 |ρ u02 = 0,

(3.1.103)

u(0) = u0 (0) = 0.

(3.1.104)

and Multiplying (3.1.103) by u0 , then integrating with respect to x, we get ´ 1d ³ 0 2 ku k + k∇uk2 + 2 dt

Z

Ω

(u01 −u02 )(|u01 |ρ u01 −|u02 |ρ u02 )dx = 0. (3.1.105)

Since |λ|ρ λ is a monotone increasing function in λ, the last term on the left-hand side of (3.1.105) is nonnegative. It follows from (3.1.105) that d (ku0 k2 + k∇uk2 ) ≤ 0, (3.1.106) dt which implies that for all t ∈ [0, T ], ku0 (t)k2 + k∇u(t)k2 ≤ 0

(3.1.107)

u(t) = 0.

(3.1.108)

i.e., Thus the proof is complete.

3.2

¤

Monotone Operator Method

In this section we introduce another useful method, namely the monotone operator method to deal with nonlinear evolution equations. Consider the following initial value problem for abstract nonlinear evolution equations:  du   + A(u) = f (t),    

dt

u(0) = u0

(3.2.1)

Compactness Method and Monotone Operator Method

115

where A is a nonlinear operator. Recall that in Chapter 2 we introduced the concept of (linear) monotone operator in a Hilbert space. This definition can be easily extended to a nonlinear operator defined in a real Hilbert space. If A satisfies (A(u) − A(v), u − v) ≥ 0, ∀u, v ∈ D(A),

(3.2.2)

then A is called a monotone operator in the Minty sense. In this section we try to extend this definition so that more applications can be treated.

3.2.1

General Framework

Let V be a real reflexive Banach space, and H is a real Hilbert space. The norm in V is denoted by k · k, and the inner product and norm in H is denoted by ( , ), and | · |, respectively. In this section we always assume that V is dense, and continuously imbedded in H. Thus, the dual space H 0 is also continuously imbedded in V 0 , the dual space of V . The above framework is simply denoted by ∼ H 0 ,→ V 0 . V ,→ H = (3.2.3) We denote by k · k∗ the norm in V 0 . The dual product between V and V 0 is still denoted by ( , ). DEFINITION 3.2.1 V 0 satisfying

Suppose that A is a nonlinear operator from V into

(A(u) − A(v), u − v) ≥ 0, ∀u, v ∈ V.

(3.2.4)

Then A is said to be a monotone operator in the Minty sense.

Consider problem (3.2.1) where A is a nonlinear monotone operator from V into V 0 . THEOREM 3.2.1 Let V be a reflexive, separable Banach space, and H be a Hilbert space such that V ,→ H ∼ = H 0 ,→ V 0 . Suppose that A is a nonlinear operator from V 0 into V satisfying (i) for all v ∈ V ,

kA(v)k∗ ≤ Ckvkp−1

(3.2.5)

where 1 < p < ∞ and C is a positive constant independent of v ; (ii) (hemicontinuous) for all u, v, w ∈ V and λ ∈ R, (A(u + λv), w) is a

116

NONLINEAR EVOLUTION EQUATIONS

continuous function of λ; (iii) (coercive condition) for all v ∈ V ,

(A(v), v) ≥ αkvkp

(3.2.6)

where p is the same as in (i), and α is a positive constant independent of v . 0 Then for any u0 ∈ H , f ∈ Lp ([0, T ], V 0 ) with p1 + p10 = 1, problem (3.2.1) T admits a unique solution u such that u ∈ C([0, T ], H) Lp ([0, T ], V ), u0 ∈ 0 Lp ([0, T ], V 0 ), u(0) = u0 , and the equation in (3.2.1) is satisfied in the 0 sense of Lp ([0, T ], V 0 ).

In order to prove this theorem, we first prove some lemmas for the nonlinear monotone operator. LEMMA 3.2.1 Let A be a nonlinear monotone operator from V into V 0 satisfying the conditions (i), (ii). If un ∈ V strongly converges to u ∈ V , then A(un ) weakly converges to A(u) in V 0 .

Proof. Since un is strongly convergent in V , un must be uniformly bounded in V . It turns out from condition (i) that kA(un )k∗ is uniformly bounded. Thus, if the conclusion of this lemma is not true, then there is a subsequence uµ such that A(uµ ) → f weakly in V 0 , and f is different from A(u). From (A(uµ ), uµ − u) → 0,

(3.2.7)

(A(uµ ) − A(v), uµ − v) ≥ 0,

(3.2.8)

and the monotonicity it follows that for all v ∈ V , (A(uµ ), uµ − v) → (f, u − v) ≥ (A(v), u − v) = lim (A(v), uµ − v), µ→∞

(3.2.9)

i.e., (f − A(v), u − v) ≥ 0.

(3.2.10)

For any w ∈ V and any real number λ, let v = u − λw. Thus, λ(f − A(u − λw), w) ≥ 0.

(3.2.11)

For λ > 0, λ → 0, we infer from condition (ii) and (3.2.11) that (f − A(u), w) ≥ 0.

(3.2.12)

Compactness Method and Monotone Operator Method

117

For λ < 0, λ → 0, the same argument yields (f − A(u), w) ≤ 0.

(3.2.13)

(f − A(u), w) = 0,

(3.2.14)

Thus, for all w ∈ V , which implies that f = A(u), a contradiction. The proof is complete. ¤

REMARK 3.2.1 Under the assumptions of the above lemma, if u(t) is a strongly measurable abstract function from [0, T ] to V , i.e., u(t) is a strong limit of a sequence of abstract step functions, then A(u(t)) is a weakly measurable abstract function from [0, T ] to V 0 , i.e., for v ∈ V , (A(u(t)), v) is an ordinary measurable function. In particular, when V is a separable space, by the Pettis theorem in functional analysis, a strongly measurable function coincides with a weakly measurable function. Thus, A(u(t)) is a strongly measurable abstract function.

The following monotonicity result plays a very important role in the proof of Theorem 3.2.1. LEMMA 3.2.2 Suppose that A is a nonlinear monotone operator from V into V 0 satisfying conditions (i), (ii). Then A possesses the following property M : if  un → u weakly in V,       A(un ) → Ψ weakly in V 0 , (3.2.15)       lim (A(un ), un ) ≤ (Ψ, u), n→∞

then,

Ψ = A(u). Proof. By the monotonicity (A(un ) − A(v), un − v) ≥ 0

∀v ∈ V,

(3.2.16)

we have (A(un ), un ) − (A(un ), v) − (A(v), un − v) ≥ 0.

(3.2.17)

118

NONLINEAR EVOLUTION EQUATIONS

Thus, (Ψ, u) ≥ lim (A(un ), un ) ≥ (Ψ, v) + (A(v), u − v),

(3.2.18)

n→∞

i.e., (Ψ − A(v), u − v) ≥ 0.

(3.2.19)

Let v = u + λw. Using the similar argument as in the proof of Lemma 3.2.1, we get Ψ = A(u). Thus, the proof is complete. ¤ REMARK 3.2.2 Let A be a nonlinear monotone operator from a separable Banach space V to V 0 satisfying conditions (i), (ii). If un ∈ Lp ([0, T ], V ) with 1 < p < ∞ such that

un → u weakly in Lp ([0, T ], V ), A(un ) → Ψ weakly in Lp ([0, T ], V 0 ) 0

with p1 + p10 = 1, and

lim

Z T

n→∞ 0

(A(un ), un )dt ≤

Z T 0

(Ψ, u)dt,

then

Ψ = A(u).

Proof. By Remark 3.2.1, A(un ) is a sequence of strongly measurable abstract functions. It follows from condition (i) that kA(un )kp∗ ≤ C p kun kp (p−1) = C p kun kp . 0

0

0

0

(3.2.20)

Thus, A(un ) ∈ Lp ([0, T ], V 0 ). The remaining part of the proof is just the same as in Lemma 3.2.2. Thus the proof is complete. ¤ After these preparations we can turn to the proof of Theorem 3.2.1. 0

Proof of Theorem 3.2.1. The method of proof is still to use the Faedo-Galerkin method, then pass to the limit. (i) Since V is separable, and V is dense in H, we have a base {wi , i ∈ N } in V , and also in H such that (wi , wj ) = δij , i, j = 1, 2, · · · . For any m ∈ N, we look for the approximate solution um =

m X

gim (t)wi

i=1

Compactness Method and Monotone Operator Method

119

satisfying the following identities:

  (umt , wj ) + (A(um ), wj ) = (f, wj ), j = 1, · · · , m, 

um (0) → u0 strongly in H.

(3.2.21)

Since V is dense in H, there exists {ξim , i = 1, · · · , m} such that um (0) =

m X

ξim wi → u0

i=1

strongly in H. Then (3.2.21) is deduced to the following initial value problem for a system of nonlinear ordinary differential equations on gjm :  dgjm   + Gj (g) = fj (t),    

j = 1, · · · , m,

dt

(3.2.22)

gjm (0) = ξjm , j = 1, · · · , m,

where m X

Gj (g) = (A(

gim wi ), wj ),

i=1

and fj = (f, wj ) ∈ Lp (0, T ). 0

By condition (ii), when gim (t) is a continuous function in t, Gj (g) is also a continuous function in t. (3.2.22) can be converted into equivalent integral equations: gjm (t) = ξjm −

Z t 0

Gj (g)dτ +

Z t 0

fj (τ )dτ,

j = 1, · · · , m.

(3.2.23)

In what follows we use the Leray-Schauder fixed-point theorem to prove the local existence of the solution to (3.2.23). Clearly, the terms on the right-hand side in (3.2.23) define a nonlinear operator T (g) from g = (g1m , · · · , gmm )T ∈ (C[0, t0 ])m into (C[0, t0 ])m . Let Sm = {g | g = (g1m , · · · , gmm )T ∈ (C[0, t0 ])m , kgkC ≤ M }

(3.2.24)

where the norm kgkC is defined as follows: kgkC = max max |gim (t)|, i

0≤t≤t0

(3.2.25)

and M is a given positive constant such that sup|ξim | < M. i

(3.2.26)

120

NONLINEAR EVOLUTION EQUATIONS

Thus, by the H¨older inequality, we have µ

kT (g)kC ≤ sup |ξjm | + j

Z t0

≤ sup|ξjm | + K( j

Z t0 0

1 p

1

0

Z t0

1

0 kf kp∗ dτ ) p0

0

j

0

|fj |dτ +

kf kp∗ dτ ) p0 · t0 + K

0

≤ sup|ξjm | + K(

Z t0

1 p

¶

|Gj (g)|dτ

Z t0

· t0 + K1

0

kA(um )k∗ dτ

Z t0 0

kum kp−1 dτ (3.2.27)

where K and K1 denote positive constants depending on

X

kwi k. Since

i

kum k

p−1

≤K

m X

|gim |p−1 kwi kp−1 ,

(3.2.28)

i=1

it follows from (3.2.27) and (3.2.28) that kT (g)kC ≤

X

Z t0

|ξjm | + K(

0

j

1

1

p−1 · t0 . (3.2.29) kf kp∗ dτ ) p0 · t0p + K2 kgkC 0

When t0 is appropriately small, we can infer from g ∈ Sm that T (g) ∈ 0 Sm . Moreover, T (g) maps Sm into a bounded set in W 1,p (0, t0 ), which is a compact subset in C due to the Sobolev imbedding theorem. Thus, by the Leray-Schauder fixed-point theorem, (3.2.23) admits a solution 0 ∈ Lp0 (0, t ) (i = 1, · · · , m). g ∈ Sm . Moreover, gim 0 To prove the uniqueness, we will use the monotonicity of A. Let (1)

(1) T ) , g (1) = (g1m , · · · , gmm (2)

(2) T g (2) = (g1m , · · · , gmm )

be two solutions, and u(1) m =

m X (1) i=1

gim wi ,

u(2) m =

m X (2) i=1

gim wi ,

(2) um = u(1) m − um .

Then we have

 (1) (2)   (umt , wj ) + (A(um ) − A(um ), wj ) = 0,   u (0) = 0. m (1)

(2)

(3.2.30)

Multiplying (3.2.30) by gjm − gjm , and summing with j from 1 to m,

Compactness Method and Monotone Operator Method

121

we get

 1d  (2) (1) (2)  |um (t)|2 + (A(u(1)  m ) − A(um ), um − um ) = 0,   

2 dt

(3.2.31)

um (0) = 0.

By the monotonicity of A, we get |um (t)|2 =

m X i=1

2 gim (t) ≤ 0,

(3.2.32)

and the uniqueness follows. (ii) In the same manner as above, we get 1d |um (t)|2 + (A(um ), um ) = (f, um ). (3.2.33) 2 dt Using the coercive condition (iii) and the Young inequality, we get εp 1 1d 0 |um (t)|2 + αkum kp ≤ kf k∗ kum k ≤ kum kp + p0 0 kf kp∗ . (3.2.34) 2 dt p ε p Choosing ε small enough such that with respect to t, we obtain |um (t)|2 +

Z t 0

εp p

≤

α 2,

kum kp dτ ≤ K(|um (0)|2 +

and integrating (3.2.34)

Z T 0

kf kp∗ dτ ). 0

(3.2.35)

Thus, gim (t) is uniformly bounded by a positive constant depending R 0 only on |u0 |2 , 0T kf kp∗ dτ, and T . It turns out that the approximate solution um can be extended into [0, T ], and the estimate given by (3.2.35) still holds. So far we have obtained a unique approximate solution um (t) defined on [0, T ]. Moreover,  ∞  um is uniformly bounded in L ([0, T ], H), 

um is uniformly bounded in

Lp ([0, T ], V

).

(3.2.36)

(iii) Therefore, there is a subsequence of um , still denoted by um such that  ∞  um → u weakly star in L ([0, T ], H), 

(3.2.37)

um → u weakly in Lp ([0, T ], V ).

By condition (i), A(um ) is uniformly bounded in Lp ([0, T ], V 0 ). It turns out that there is a subsequence, still denoted by um , such that 0

A(um ) → Ψ weakly in Lp ([0, T ], V 0 ). 0

(3.2.38)

122

NONLINEAR EVOLUTION EQUATIONS

For any fixed j, letting m → ∞ in (3.2.21), we obtain (u0m , wj ) = (f, wj ) − (A(um ), wj ) → (f − Ψ, wj ) weakly in Lp (0, T ). 0

(3.2.39)

On the other hand, it follows from (3.2.37) that (um (t), wj ) → (u(t), wj ) weakly star in L∞ (0, T ).

(3.2.40)

Thus, in the distribution sense, (u0m (t), wj ) → (u0 (t), wj ) in D0 (0, T ).

(3.2.41)

Since {wj , j ∈ N } is a base in V , we have u0 = f (t) − Ψ ∈ Lp ([0, T ], V 0 ). 0

(3.2.42)

Let η(t) be a smooth function in t such that η(0) = 1, η(T ) = 0. Multiplying (3.2.39) by η, and integrating with respect to t yields Z T 0

=

Z T 0

(u0m , wj )ηdt

→

Z T 0

(f − Ψ, wj )ηdt

0

(u , wj )ηdt = −(u(0), wj ) −

Z T 0

(u, wj )η 0 dt.

(3.2.43)

For the term on the left-hand side of (3.2.43), we have Z T 0

Since

(u0m , wj )ηdt = −(um (0), wj ) − Z T 0

0

(um , wj )η dt →

Z T

Z T 0

0

(um , wj )η 0 dt.

(u, wj )η 0 dt,

(3.2.44)

(3.2.45)

we infer from (3.2.44) and (3.2.43) that (um (0), wj ) → (u(0), wj ).

(3.2.46)

On the other hand, it follows from (3.2.21) that (um (0), wj ) → (u0 , wj ).

(3.2.47)

Since {wj , j ∈ N } is dense in H, we get u(0) = u0 .

(3.2.48)

In the same manner, we can prove that um (T ) → u(T ) weakly in H.

(3.2.49)

In what follows we use the monotonicity lemma, Lemma 3.2.2, to prove Ψ = A(u). Once it has been done, we can deduce from (3.2.42) that u 0 satisfies the equation in the sense of Lp ([0, T ], V 0 ), and the existence

Compactness Method and Monotone Operator Method

123

of the solution to problem (3.2.1) follows. Multiplying (3.2.21) by gjm , then summing with respect to j, and integrating over [0, T ] yields 1 1 |um (T )|2 − |um (0)|2 + 2 2 Thus, Z T 0

(A(um ), um )dt =

Z T 0

Z T 0

(A(um ), um )dt =

Z T 0

(f, um )dt. (3.2.50)

1 1 (f, um )dt + |um (0)|2 − |um (T )|2 . (3.2.51) 2 2

Since um (0) → u0 = u(0) strongly in H, Z T 0

(f, um )dt →

Z T 0

(f, u)dt,

(3.2.52) (3.2.53)

and um (T ) → u(T ) weakly in H,

(3.2.54)

we infer from (3.2.51) and the following lower semi-continuity of norms: lim inf |um (T )| ≥ |u(T )| m→∞

that lim

Z T

m→∞ 0 Since u0 =

(A(um ), um )dt ≤

Z T 0

1 1 (f, u)dt + |u(0)|2 − |u(T )|2 . (3.2.56) 2 2

f − Ψ, the above implies that lim

Z T

m→∞ 0

(3.2.55)

(A(um ), um )dt ≤

Z T 0

(Ψ, u)dt.

(3.2.57)

Thus, applying Lemma 3.2.2, and Remark 3.2.2, we get Ψ = A(u).

(3.2.58)

Finally we prove the uniqueness. Let u1 , u2 be two solutions, and let u = u1 − u2 . Then  du   + A(u1 ) − A(u2 ) = 0,    

dt

(3.2.59)

u(0) = 0.

Taking the dual product on the both sides of the equation, we get 1d 2 |u| ≤ 0, 2 dt

(3.2.60)

124

NONLINEAR EVOLUTION EQUATIONS

which implies that |u(t)| = 0, ∀t ∈ [0, T ].

(3.2.61)

Thus, the proof is complete.

¤

REMARK 3.2.3 If the coercive condition (iii) in Theorem 3.2.1 is replaced by the following weaker condition (iii)’, then the conclusion in Theorem 3.2.1 still holds. (iii)’ For all v ∈ V ,

(A(v), v) ≥ α[v]p

(3.2.62)

where [ , ] is the seminorm on V and [v] + |v| is an equivalent norm on V .

Proof. It suffices to notice that now (3.2.34) turns out to be 1d |um |2 + α[um ]p ≤ (f, um ). 2 dt Integrating with respect to t yields 1 |um (t)|2 + α 2

Z t

1 ≤ |um (0)|2 + C 2

0

1 [um ]p dτ ≤ |um (0)|2 + 2

Z t 0

0

kf k∗ kum kdτ

kf k∗ ([um ] + |um |)dτ

Z t

1 ≤ |um (0)|2 + C1 ( 2

Z t

(3.2.63)

0

1

kf kp∗ dτ ) p0 0

µ Z t

(

0

1

[um ]p dτ ) p + (

Z t 0

1

¶

|um |p dτ ) p . (3.2.64)

By the Young inequality, we have Z t

(

0

1

0 kf kp∗ dτ ) p0 (

Z t 0

1 [um ] dτ ) ≤ p0 0 ε p 1 p

p

and Z t

(

0

1

0 kf kp∗ dτ ) p0 (

Z t 0

1 |um | dτ ) ≤ 2 p

1 p

µZ t 0

Z t 0

0 kf kp∗ dτ

0 kf kp∗ dτ

¶ 20 p

εp + p

1 + 2

Z t 0

µZ t 0

[um ]p dτ, (3.2.65) p

|um | dτ

¶2

p

.

(3.2.66)

Choosing ε sufficiently small, we infer from (3.2.64)–(3.2.66) that 1 α |um (t)|2 + 2 2

Z t 0

p

[um ] dτ ≤ C2 + C3

µZ t 0

p

|um | dτ

¶2

p

.

(3.2.67)

Compactness Method and Monotone Operator Method It turns out that p

|um (t)| ≤ C4 + C5

Z t 0

|um |p dτ.

125

(3.2.68)

By the Gronwall inequality, we get |um (t)| ≤ CT , ∀t ∈ [0, T ].

(3.2.69)

Combining (3.2.69) with (3.2.67) yields Z t 0

kum kp dτ ≤ CT , ∀t ∈ [0, T ].

(3.2.70)

The remaining part of the proof is just the same as that in the proof of Theorem 3.2.1, and we can leave it to the reader. Thus, the proof is complete. ¤

3.2.2

Applications

As applications of Theorem 3.2.1, we now give some examples. 1. Consider the following initial boundary value problem for a nonlinear parabolic equation with the Dirichlet boundary condition: Ã¯ !  ¯ n X ¯ ∂u ¯p−2 ∂u ∂ ∂u   ¯ ¯  = f (x, t), −   ∂t i=1 ∂xi ¯ ∂xi ¯ ∂xi   

(x, t) ∈ Ω × R+ ,

 u|Γ = 0,       

u|t=0 = u0 (x)

(3.2.71)

where Ω ⊂ Rn is a bounded domain with a smooth boundary Γ, p (2 ≤ p < +∞) is a given constant. When p = 2, the equation in (3.2.71) turns out to be a non-homogeneous heat equation. THEOREM 3.2.2 0 0 Suppose that u0 ∈ L2 (Ω), f ∈ Lp ([0, T ], W −1,p (Ω)), with p0 being the conjugate number of p, p1 + p10 = 1. Then problem (3.2.71) admits a unique generalized solution u such that

u ∈ C([0, T ], L2 (Ω)) ∩ Lp ([0, T ], W01,p (Ω)), u0 ∈ Lp ([0, T ], W −1,p (Ω)). 0

0

126

NONLINEAR EVOLUTION EQUATIONS

Proof. We take H = L2 (Ω), V = W01,p (Ω) equipped with the equivalent norm: ÃZ n ¯ !1 p X ¯ ∂v ¯¯p ¯ ¯ kvk = . ¯ ∂x ¯ dx Ω i=1

i

(3.2.72)

Then V 0 = W −1,p (Ω). Clearly, V is a separable Banach space satisfying the general framework described in Section 3.2.1. For any v ∈ V , we define 0

Ã¯ ! ¯ ¯ ∂v ¯p−2 ∂v ¯ ¯ A(v) = − . ∂xi ¯ ∂xi ¯ ∂xi i=1 n X ∂

(3.2.73)

Then A is a nonlinear operator from V to V 0 . Indeed, for u, v ∈ V , we have ¯ Z X n ¯ ¯ ∂u ¯p−2 ∂u ∂v ¯ ¯ (A(u), v) = dx. ¯ ¯ ∂xi ∂xi Ω i=1 ∂xi

(3.2.74)

By the H¨older inequality, we get kA(u)k∗ ≤ kukp−1 ,

(3.2.75)

i.e., condition (i) is verified. A straightforward calculation shows that (A(u) − A(v), u − v) = kukp + kvkp −

¯ ¯ Z X Z X n ¯ n ¯ ¯ ∂u ¯p−2 ∂u ∂v ¯ ∂v ¯p−2 ∂v ∂u ¯ ¯ ¯ ¯ dx− dx ¯ ¯ ¯ ¯ ∂xi ∂xi ∂xi ∂xi Ω i=1 ∂xi Ω i=1 ∂xi

≥ kukp + kvkp − kukp−1 kvk − kvkp−1 kuk = (kuk − kvk)(kukp−1 − kvkp−1 ) ≥ 0.

(3.2.76)

Thus, A is a monotone operator. The hemicontinuous condition (ii) is easy to verify, and we leave it to the reader. We infer from (3.2.74) that (A(u), u) = kukp .

(3.2.77)

Therefore, the coercive condition (iii) is verified with α = 1. By Theorem 3.2.1, the conclusion of the present theorem immediately follows. ¤ 2. Consider the following initial boundary value problem for a nonlin-

Compactness Method and Monotone Operator Method

127

ear parabolic equation with the Neumann boundary condition: Ã¯ !  ¯ n ∂ ¯¯ ∂u ¯¯p−2 ∂u ∂u X    − = f (x, t),   ∂t i=1 ∂xi ¯ ∂xi ¯ ∂xi       ¯p−2 n ¯¯ X ¯ ∂u ∂u ¯ ¯  cos (n, xi )|Γ = 0,  ¯ ¯   ∂xi ∂xi  i=1      

(x, t) ∈ Ω × R+ ,

u|t=0 = u0 (x)

(3.2.78)

where Ω again is a bounded domain in Rn with the smooth boundary Γ, and 2 ≤ p < ∞. For this problem, we take V = W 1,p (Ω), H = ¡ ¢0 L2 (Ω), and V 0 = W 1,p (Ω) . Thus, these spaces satisfy the framework described previously. Define the operator A from V into V 0 by ¯ Z X n ¯ ¯ ∂u ¯p−2 ∂u ∂v ¯ ¯ (A(u), v) = dx, ¯ ¯ ∂xi ∂xi Ω i=1 ∂xi

∀u, v ∈ V.

(3.2.79)

It can be verified in the same way as in the above problem that A is a monotone operator satisfying condition (i) with C = 1, and the hemicontinuous condition (ii). For u ∈ V , we have (A(u), u) = where [u] =

¯p Z X n ¯¯ ¯ ¯ ∂u ¯ dx = [u]p ¯ ∂x ¯ Ω i=1

ÃZ

i

!1 ¯ n ¯ p X ¯ ∂u ¯p ¯ dx ¯ ¯ ∂x ¯

Ω i=1

i

(3.2.80)

(3.2.81)

is clearly a semi-norm in W 1,p (Ω). Since 2 ≤ p < ∞, by the GarliadoNirenberg inequality stated in Section 1.4.2, [u] + kukL2 (Ω) is an equivalent norm in W 1,p (Ω). Thus, applying Remark 3.2.3, we immediately have the following result. THEOREM 3.2.3 Suppose that Ω is a bounded domain in Rn with smooth boundary Γ, and p is¢a 0 ¡ given constant with 2 ≤ p < ∞. Then for any f ∈ Lp [0, T ], (W 1,p (Ω))0 , u0 ∈ L2 (Ω), problem (3.2.78) admits a unique generalized solution u such that \

u ∈ C([0, T ], L2 (Ω)) u0 ∈ Lp

0

³

Lp ([0, T ], W 1,p (Ω)), ´

[0, T ], (W 1,p (Ω))0 .

128

3.2.3

NONLINEAR EVOLUTION EQUATIONS

Generalizations and Supplements

In this section we describe some important generalizations and supplements. 1. The case that A is a sum of some monotone operators. Let H be a Hilbert space and Vi (i = 1, · · · , q) be reflexive Banach spaces such that Vi ,→ H and Vi is dense in H. We denote by k · ki the norm in Vi , and k · k∗,i the norm in Vi0 . Let q \

V =

Vi ,

(3.2.82)

i=1

and the norm in V still be denoted by k · k. We define kvk =

q X

kvki .

(3.2.83)

i=1

We assume that V is separable. Thus, we have V ,→ H ∼ = H 0 ,→ V 0 , Vi ,→ H ∼ = H 0 ,→ Vi0 .

(3.2.84)

Suppose that Ai is a nonlinear monotone operator from Vi to Vi0 satisfying the following conditions: (i) For all v ∈ Vi , kAi (v)ki ≤ Ci kvkipi −1

(3.2.85)

where Ci is a positive constant independent of v, and pi (1 < pi < ∞) is a given constant. (ii) Ai (i = 1, · · · , q) is hemi-continuous. (iii) For any v ∈ Vi , (Ai (v), v) ≥ αi kvkpi i

(3.2.86)

where αi is a positive constant independent of v. Let A=

q X

Ai .

i=1

Then we have the following result. THEOREM 3.2.4 Under the above conditions (i), (ii), (iii), for any u0 ∈ H , and

f∈

q X i=1

Lpi ([0, T ], Vi0 ) 0

(3.2.87)

Compactness Method and Monotone Operator Method

129

with p0i being the conjugate number of pi , problem (3.2.1) admits a unique generalized solution u such that

u∈

q \

Lpi ([0, T ], Vi ) ∩ C([0, T ], H),

i=1

u0 ∈

q X i=1

Lpi ([0, T ], Vi0 ). 0

Proof. Since the proof is just the same as that for Theorem 3.2.1, we leave it to the reader. ¤ REMARK 3.2.4 As stated in Remark 3.2.3, if (3.2.86) is replaced by a similar one to (3.2.62):

(Ai (v), v) ≥ αi [v]pi i

(3.2.88)

where [ , ]i is the semi-norm on Vi and [v]i + |v| is an equivalent norm on Vi , then the conclusion of the above theorem still holds.

Now we give an example to show the application of Theorem 3.2.4. 2. Consider the following initial boundary value problem for a nonlinear heat equation:  ∂u   − ∆u + u3 = f (x, t),    ∂t  

(x, t) ∈ Ω × R+ ,

u|Γ = 0,       

(3.2.89)

u|t=0 = u0 (x)

where Ω ⊂ Rn is a bounded domain with the smooth boundary Γ. To treat this problem by the method described in this section, we introduce H = L2 (Ω), V1 = H01 (Ω), V2 = L4 (Ω).

(3.2.90)

The corresponding nonlinear monotone operators A1 , A2 are defined as follows: (A1 (u), v) =

Z

Ω

and (A2 (u), v) =

∇u · ∇vdx, ∀u, v ∈ V1 ,

Z Ω

u3 vdx, ∀u, v ∈ V2 .

(3.2.91)

(3.2.92)

130

NONLINEAR EVOLUTION EQUATIONS

It is easy to verify that A1 , A2 satisfies the conditions (i)–(iii) with p1 = 2, p2 = 4, C1 = C2 = α1 = α2 = 1. Then problem (3.2.89) can be put into the framework just described, with V = H01 ∩ L4 (Ω), 4 A = A1 + A2 , and V 0 = H −1 + L 3 (Ω). Thus, applying Theorem 3.2.4 to problem (3.2.89), we have the following result. THEOREM 3.2.5 Suppose that

u0 ∈ L2 (Ω),

4

4

f ∈ L2 ([0, T ], H −1 ) + L 3 ([0, T ], L 3 (Ω)).

Then problem (3.2.89) admits a unique generalized solution u such that

u ∈ C([0, T ], L2 (Ω)) ∩ L2 ([0, T ], H01 (Ω)) ∩ L4 ([0, T ], L4 (Ω)), 4

4

u0 ∈ L2 ([0, T ].H −1 (Ω)) + L 3 ([0, T ], L 3 (Ω)). REMARK 3.2.5 It has been seen that the same problem (3.2.89) can be treated by different methods: the semigroup method, the compactness method and the monotone operator method. We would like to call the reader’s attention to the difference of assumptions on u0 , f and Ω, and also to the difference of conclusions.

3.3

Bibliographic Comments

We refer to the book [94] by J.L. Lions and the extensive references cited there for the comprehensive study of the compactness method and the monotone operator method. The compactness method stated in this chapter requires that enough “good” a priori estimates can be derived. To treat problems in nonlinear hyperbolic conservation laws and other problems in variational calculus, etc., where enough good a priori estimates are not available, other compactness methods such as the compensated compactness method and the concentrated compactness method have been developed. See the paper [149] by T. Tartar and the paper [113] by F. Murat, and the recent monographs [103] by Y. Lu for the compensated compactness method. We refer to the papers [96], [97] by P.L. Lions for the concentrated compactness method. We also refer to the monographs [49] by L.C. Evans for the comprehensive survey of weak convergence methods with the extensive references cited

Compactness Method and Monotone Operator Method

131

there. We refer to the book [29] by H. Brezis and the more recent book [139] by R.E. Showalter for the monotone operator method with many applications. The Aubin lemma (Theorem 3.1.1) is very useful in many applications, and it was first established by J.P. Aubin in [19]. We also refer to the paper [140] by J. Simon and a more recent paper [18] by H. Amann with the extensive references cited there in this direction.

Chapter 4 Monotone Iterative Method and Invariant Regions

4.1

Introduction

In this chapter we introduce two other methods, namely the monotone iterative method and invariant regions. These two methods were mainly developed in the 1970s, but they are still important and useful nowadays. The basic strategy of the monotone iterative method for the second-order elliptic equations or the second-order parabolic equations relies on the comparison principle, an invariant of the well-known maximum principle. By the comparison principle one can define the upper solution and lower solution or supersolution and subsolution for nonlinear second-order parabolic equations or nonlinear second-order elliptic equations, and construct the corresponding monotone iterative sequence. If a pair of upper solution and lower solution u, u satisfying u ≤ u can be found, then by the comparison principle, it can be proved that the corresponding sequence un obtained from u or u by the iterative method is monotone, and it stays between u and u, i.e., u ≤ un ≤ u. From the monotonicity and uniform bound of un , the pointwise convergence of un immediately follows. Then one could prove that un has a subsequence that converges in a more regular space, and it turns out that the limit function u is indeed a classical solution of the original problem. The general idea of the monotone iterative method, including the interval iteration method, probably goes back to an early work [34] by German mathematician L. Collatz in 1964. Applications to semilinear second-order elliptic equations were proposed by H. Amann in 1971 [3] where the topological method was also used to conclude existence of multiple solutions. In 1972, D. Sattinger [129] used the monotone iterative method to study the existence of solution to the initial boundary value problem for the semilinear second-order parabolic equation. Since then, this method has been successively extended to semilinear 133

134

NONLINEAR EVOLUTION EQUATIONS

second-order parabolic system, in particular, the reaction-diffusion system. We refer the reader to the book [117] by C.V. Pao for the details. A closely related method, namely the method of invariant regions or invariant sets, was developed in the later 1970s. This method allows one to obtain uniform L∞ norm bounds of solution to some nonlinear evolution equations. Therefore, it also provides a powerful tool to deal with the issue of existence of solutions to these equations. In this aspect, H. Amann [7] and K. Chueh, C. Conley and J. Smoller in [33] made important contributions. In this chapter, we first introduce the monotone iterative method. In the final section of this chapter, the method of invariant regions is also introduced.

4.2 4.2.1

Monotone Iterative Method Monotone Iterative Method for Nonlinear Parabolic Equations

Consider the following initial boundary value problem for the linear second-order parabolic equation:  n n X X  ∂2u ∂u ∂u   − + + c(x, t)u = f (x, t), a (x, t) bi (x, t) Lu ≡  ij   ∂t i,j=1 ∂xi ∂xj i=1 ∂xi      u|Γ = g,       

u|t=0 = φ(x)

(4.2.1)

where Γ is the smooth boundary of a bounded domain Ω ⊂ Rn , and aij (x, t), bi (x, t), c(x, t), f (x, t) are the H¨older continuous functions de¯ T = Ω×[0, ¯ fined on Q T ]. We assume that g, φ are continuous functions, and satisfy the compatibility condition at the corner x ∈ Γ, t = 0. It is always assumed throughout this chapter that there exists a positive constant α > 0 such that for all (x, t) ∈ QT and (ξ1 , · · · , ξn ) ∈ Rn , the following holds: n X

aij (x, t)ξi ξj ≥ α

i,j=1

n X i=1

ξi2 .

Then we have the following comparison principle.

(4.2.2)

Monotone Iterative Method and Invariant Regions

135

LEMMA 4.2.1 If f ≥ 0, g ≥ 0, φ ≥ 0, then for any classical solution u to problem (4.2.1), the following holds:

u(x, t) ≥ 0, ∀(x, t) ∈ QT .

(4.2.3)

u = eλt v(x, t)

(4.2.4)

Proof. Let

where λ is a sufficiently large constant such that λ + c > 0. Thus, v satisfies n n X X ∂v ∂2v ∂v − + + (λ + c(x, t))v aij (x, t) bi (x, t) ∂t i,j=1 ∂xi ∂xj i=1 ∂xi

= f (x, t)e−λt ,

(4.2.5)

v|Γ = ge−λt ,

(4.2.6)

v|t=0 = φ(x).

(4.2.7)

By the maximum principle it is easy to conclude that for all (x, t) ∈ QT , v ≥ 0, which yields that for all (x, t) ∈ QT , u ≥ 0. Indeed, if it is not true, then there is a point (x0 , t0 ), x0 ∈ Ω, 0 < t0 ≤ T such that v(x0 , t0 ) < 0 and v achieves its minimum there. Thus, at this point, we must have vt ≤ 0,

n X

aij

i,j=1

n X ∂v ∂2v ≥ 0, = 0, bi ∂xi ∂xj ∂xi i=1

which contradicts (4.2.5). Thus the proof is complete.

(4.2.8) ¤

REMARK 4.2.1 The above comparison theorem also holds for the Neumann or the third boundary condition, using a strong maximum principle for the linear second-order parabolic equation obtained by L. Nirenberg in 1953.

This comparison principle can also be extended to the case where

136

NONLINEAR EVOLUTION EQUATIONS

Ω = Rn . Consider the following Cauchy problem:

 n n X X ∂2u ∂u ∂u    − + + c(x, t)u = f (x, t), a (x, t) bi (x, t) Lu ≡ ij  ∂t i,j=1 ∂xi ∂xj i=1 ∂xi    

u|t=0 = φ(x).

Then we have the following result.

(4.2.9)

COROLLARY 4.2.1 Suppose that aij , bi , c, f are bounded H¨ older continuous functions defined on Rn × R+ . If f ≥ 0, φ ≥ 0, then for any bounded classical solution u to problem (4.2.9), the following holds:

u(x, t) ≥ 0, ∀(x, t) ∈ Rn × R+ .

(4.2.10)

Proof. We still use a contradiction argument to prove this corollary. Let B > 0 be the bound of u in Rn × R+ . If (4.2.10) is not true, then there exists x0 ∈ Rn , t0 > 0 such that u(x0 , t0 ) < 0.

(4.2.11)

Consider the following domain: QM = {(x, t) | |x−x0 | < M, 0 < t < t0 } where M is a sufficiently large constant. As shown in the proof of Lemma 4.2.1, without loss of generality, we can assume that c ≥ 1 in Rn × R+ . Let B v(x, t) = − 2 (|x − x0 |2 + at) (4.2.12) M where a > 0 is a constant to be chosen later. Then we have Lv = −

n n X X B (a−2 a + 2 bi (xi − x0i ) + c|x − x0 |2 + cat). (4.2.13) ii M2 i=1 i=1

Since aii , bi are bounded in Rn × R+ , it easily follows from (4.2.13) and the Young inequality B2 1 |bi (xi − x0i )| ≤ |x − x0 |2 + 2 2 that if we choose a large enough, then Lv ≤ 0,

(x, t) ∈ QM .

(4.2.14)

Furthermore, the choice of a is independent of M . Now let w = u − v.

(4.2.15)

Monotone Iterative Method and Invariant Regions

137

By this assumption and (4.2.14), we have Lw ≥ 0,

(x, t) ∈ QM .

(4.2.16)

On the lateral boundary |x − x0 | = M , we have w =u+

B (M 2 + at) ≥ 0. M2

(4.2.17)

B|x − x0 |2 ≥ 0. M2

(4.2.18)

On t = 0, we have w =φ+

Applying Lemma 4.2.1, we get that w = u − v ≥ 0 for all (x, t) ∈ QM . In particular, we have u(x0 , t0 ) ≥ v(x0 , t0 ) = −

Bat0 . M2

(4.2.19)

Let M → ∞ in (4.2.19). Then we have u(x0 , t0 ) ≥ 0, a contradiction to (4.2.11). Thus, the proof is complete.

(4.2.20) ¤

We now investigate the following initial boundary value problem for the semilinear parabolic equation:  Lu = f (x, t, u),          

(x, t) ∈ Ω × R+ ,

u|Γ = g(x, t),

(4.2.21)

u|t=0 = φ(x).

DEFINITION 4.2.1 If a smooth function u ∈ C 2,1 , i.e., continuously twice differentiable in x and once differentiable in t, satisfies  Lu ≥ f (x, t, u),          

u|Γ ≥ g(x, t),

(x, t) ∈ Ω × R+ , (4.2.22)

u|t=0 ≥ φ(x),

then u is called an upper solution (or supersolution) to problem (4.2.21). Ac-

138

NONLINEAR EVOLUTION EQUATIONS

cordingly, if a smooth function u ∈ C 2,1 satisfies  Lu ≤ f (x, t, u), (x, t) ∈ Ω × R+ ,          

u|Γ ≤ g(x, t),

(4.2.23)

u|t=0 ≤ φ(x),

then u is called a lower solution (or subsolution) to problem (4.2.21).

Then we have the following result. THEOREM 4.2.1 Suppose that f (x, t, u) is H¨ older continuous in (x, t) ∈ QT , and is differentiable with respect to u ∈ R. Suppose g, φ are continuous functions and satisfy the compatibility conditions at the corner Γ × {t = 0}. If problem (4.2.21) has a pair of upper solution u and lower solution u such that for all (x, t) ∈ QT , u ≤ u, then problem (4.2.21) admits a unique classical solution u(x, t) ∈ C 2,1 (QT ).

Proof. Let M=

sup

(x,t)∈QT ,min u≤u≤max u

¯ ¯ ¯ ∂f (x, t, u) ¯ ¯ ¯ ¯ ¯ ∂u

(4.2.24)

and rewrite the equation in (4.2.21) as LM u ≡ Lu + M u = f (x, t, u) + M u ≡ F (x, t, u).

(4.2.25)

Let u0 (x, t) = u, and u1 (x, t) be the solution to the following linear problem: u1 ∈ C 2,1 satisfies  LM u1 = F (x, t, u),     

u1 |Γ = g(x, t),     

(4.2.26)

u1 |t=0 = φ(x),

and we successively define un (x, t) (n ≥ 2) to be the solution to the following linear problem:  LM un = F (x, t, un−1 ),     

un |Γ = g(x, t),     

un |t=0 = φ(x),

(4.2.27)

Monotone Iterative Method and Invariant Regions

139

By the results on existence and uniqueness of solution to the initial boundary value problem for linear second-order parabolic equation (see, e.g., the book [52], p.69, Theorem 9 by A. Friedman; see also the book [90] by O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uralceva), problem (4.2.26) as well as problem (4.2.27) admits a unique classical solution. In what follows we prove that for all (x, t) ∈ QT , and for n = 1, · · · , u(x, t) ≤ · · · ≤ un (x, t) ≤ un−1 (x, t) ≤ · · · ≤ u1 (x, t) ≤ u(x, t). Indeed, we can infer from (4.2.22) and (4.2.26) that  LM (u − u1 ) ≥ 0,          

(u − u1 )|Γ ≥ 0,

(4.2.28)

(4.2.29)

(u − u1 )|t=0 ≥ 0.

Then by Lemma 4.2.1, for all (x, t) ∈ QT , u1 (x, t) ≤ u(x, t) = u0 (x, t).

(4.2.30)

Similarly, it follows from (4.2.26), (4.2.23) and the definition of M that  LM (u1 − u) ≥ F (x, t, u) − F (x, t, u) ≥ 0,          

(u1 − u)|Γ ≥ 0,

(4.2.31)

(u1 − u)|t=0 ≥ 0.

Thus, applying Lemma 4.2.1 again yields u(x, t) ≤ u1 (x, t).

(4.2.32)

u ≤ un−1 ≤ un−2 ≤ · · · ≤ u.

(4.2.33)

By induction, if for n ≥ 2, Then

 LM (un−1 − un ) = F (x, t, un−2 ) − F (x, t, un−1 ) ≥ 0,          

(un−1 − un )|Γ = 0,

(4.2.34)

(un−1 − un )|t=0 = 0.

Applying Lemma 4.2.1 again yields un ≤ un−1 .

(4.2.35)

140

NONLINEAR EVOLUTION EQUATIONS

In the same manner, it can be proved that u ≤ un .

(4.2.36)

Thus, (4.2.28) holds for all integer n, i.e., {un } is a monotone decreasing sequence. Because u is bounded below, un (x, t) converges to a function u(x, t) point-wise in QT . In what follows we show that u belongs to C 2,1 (QT ) and is a classical solution to the original problem (4.2.21). For simplicity, in the following we will give the proof only for the case that L is the heat operator, i.e., Lu = ut − ∆u, Ω = Rn , and f is solely a function of u. However, for the general case, i.e., for the initial boundary value problem (4.2.21) with smooth bounded domain Ω, the corresponding fundamental solution has similar estimates as for the heat equation. Therefore, the following proof can be carried over to the general case. Under the above assumptions of simplicity, we have ∂v − ∆v + M v. (4.2.37) ∂t Thus, by the fundamental solution to the heat operator, we have LM v ≡

un (x, t) =

Z

2 1 − |x−ξ| −M t 4(t) φ(ξ)dξ n e n 2 R Ct Z tZ 2 1 − |x−ξ| −M (t−τ ) 4(t−τ ) + F (un−1 )dξdτ n e n 2 0 R C(t − τ )

(4.2.38)

where n

C = (4π) 2 .

(4.2.39)

Then, by the Cauchy-Schwartz inequality we have |un (x, t) − um (x, t)|

≤

¯ ¯Z Z ¯ ¯ t 2 1 ¯ ¯ − |x−ξ| −M (t−τ ) 4(t−τ ) e (F (u ) − F (u ))dξdτ =¯ ¯ n n−1 m−1 ¯ ¯ 0 Rn C(t − τ ) 2 ! 10 ÃZ Z t

1 C

1

0

Rn

µZ t Z 0

≤C

0

(t − τ )

np0 2

e

2

−p0 M (t−τ ) − |x−ξ| 4(t−τ )

2

|x−ξ| p − 4(t−τ )

Rn

ÃZ t 0

|F (un−1 ) − F (um−1 )| e 1

(t − τ )

e n(p0 −1) 2

−p0 M (t−τ )

dξdτ

¶1

dξdτ

! 10

dτ

p

p

×

p

×

Monotone Iterative Method and Invariant Regions ÃZ Z T 0

2

|x−ξ| p − 4(t−τ )

Rn

|F (un−1 ) − F (um−1 )| e

141

!1

dξdτ

p

(4.2.40)

where C 0 is a positive constant depending only on n, and p is a positive 0 constant such that p > n+2 2 and p is the conjugate number of p, i.e., 1 1 n p + p0 = 1. In the following we prove that for any bounded set B ⊂ R , un (x, t) uniformly converges to u in B × [0, T ]. As a matter of fact, after changing of variables, we have Z TZ 0

= =

Rn

Z TZ 0

Rn

|F (un−1 ) − F (um−1 )|p e

Rn \BK

Z TZ

+

0

dξdτ

|F (un−1 (x − ξ, τ )) − F (um−1 (x − ξ, τ ))|p e

Z TZ 0

2

− |x−ξ| 4(t−τ )

BK

2

|ξ| − 4(t−τ )

|F (un−1 (x − ξ, τ )) − F (um−1 (x − ξ, τ ))|p e

|F (un−1 (x − ξ, τ )) − F (um−1 (x − ξ, τ ))|p e

dξdτ 2

|ξ| − 4(t−τ )

2

|ξ| − 4(t−τ )

dξdτ

dξdτ (4.2.41)

where BK is a ball centered at the origin with radius K > 0. For any ε > 0, there is a sufficiently large K > 0 such that for all (x, t) ∈ B × [0, T ], we have the following estimate for the first term on the right-hand side of (4.2.41): Z TZ 0

≤ CM

Rn \BK Z TZ 0

|F (un−1 (x − ξ, τ )) − F (um−1 (x − ξ, τ ))|p e

R \BK n

e

2

|ξ| − 4(t−τ )

dξdτ ≤

2

|ξ| − 4(t−τ )

ε 2

dξdτ

(4.2.42)

where CM is a positive constant depending on M . For the second term on the right-hand side of (4.2.41), when n, m are sufficiently large, the following holds uniformly for (x, t) ∈ B × [0, t]: Z TZ

≤

|F (un−1 (x − ξ, τ )) − F (um−1 (x − ξ, τ ))|p e

0

BK

0

ε |F (un−1 (ξ, τ )) − F (um−1 (ξ, τ ))|p dξdτ ≤ , 2 BK ∪B

Z TZ

2

|ξ| − 4(t−τ )

dξdτ (4.2.43)

due to the Lebesgue convergence theorem. Concerning the first term on the right-hand side of (4.2.40), since p > n+2 2 , which implies that n(p0 −1) p n+2 0 p = p−1 < n , and < 1, we conclude that for all 0 ≤ t ≤ T , 2

142

NONLINEAR EVOLUTION EQUATIONS

the first term is uniformly bounded with respect to t, i.e., Z t

1 (t − τ )

0

e−p M (t−τ ) dτ ≤ C1 . 0

n(p0 −1) 2

(4.2.44)

Thus, un uniformly converges in B × [0, T ] to u. Therefore, u is a continuous function in Rn × [0, T ]. Passing to the limit in (4.2.38), as discussed above, u satisfies Z 1 − |x−ξ|2 −M t 4t u(x, t) = φ(ξ)dξ n e Rn Ct 2 Z tZ 2 1 − |x−ξ| −M (t−τ ) 4(t−τ ) + F (u)dξdτ. (4.2.45) n e n 2 0 R C(t − τ ) In what follows we use the bootstrap argument to show that u ∈ C 2,1 , and in B × [0, T ], un converges to u in C 2,1 . By the properties of the fundamental solution to the heat operator, we can conclude that function u given by the right-hand side in (4.2.45) is once continuously differentiable with respect to xi . Then it follows from (4.2.45) and (4.2.38) that ¯ ¯ ¯ ∂(un − u) ¯ ¯ ¯ sup ¯ ¯ ∂xi x∈B,0≤t≤T Z tZ

≤C2

sup

x∈B,0≤t≤T

2 |xi − ξi | − |x−ξ| −M (t−τ ) 4(t−τ ) e |F (un−1 ) − F (u)|dξdτ. n +1 n 2 R (t − τ )

0

(4.2.46)

In the same manner as before, we can obtain similar estimates as (4.2.42), (4.2.43): sup

x∈B,0≤t≤T

≤ C20

ÃZ Z t 0

sup

¯ ¯ ¯ ∂(un − u) ¯ ¯ ¯ ¯ ¯ ∂x i

|ηi |

Rn \BK

ξ∈BK ∪B,0≤τ ≤T

(t − τ )

1 2

e−

|η|2 4

−M (t−τ )

|F (un−1 ) − F (u)|

dηdτ +

Z tZ 0

BK

!

|ηi | 1

(t − τ ) 2

e

−M (t−τ ) − |η| 4

dηdτ (4.2.47)

where C20 is a positive constant depending only on M . For any ε > 0, there is K > 0 large enough such that the first term on the right-hand ε side of (4.2.47) is less than . Since un uniformly converges to u in 2 B × [0, T ] with B being any bounded set in Rn , the second term on the right-hand side of (4.2.47) converges to zero, as n → +∞. Thus,

Monotone Iterative Method and Invariant Regions

143

we can deduce from (4.2.47) that ∂(un − u) (i = 1, · · · , n) ∂xi uniformly converges to zero in (x, t) ∈ B × [0, T ]. It turns out that uxi ∈ C in Rn × R+ . Similarly, by the properties of the fundamental solution to the heat operator, we have ¯ ¯ ¯ ∂ 2 (u − u) ¯ n ¯ ¯ ¯ ¯ ¯ ∂xi ∂xj ¯ ¯ ¯ Z Z ¯ ¯ 2 t ∂(F (u ) − F (u)) xi − ξi − |x−ξ| n−1 ¯ ¯ −M (t−τ ) 4(t−τ ) e dξdτ = ¯C ¯. n +1 ¯ ¯ n ∂ξj 0 R (t − τ ) 2

(4.2.48)

In the same manner as above, we can infer from uniform convergence of ∂(un − u) (i = 1, · · · , n) ∂xi to zero in (x, t) ∈ B × [0, T ] that ¯ ¯ ¯ ∂ 2 (u − u) ¯ n ¯ ¯ ¯ ¯ ¯ ∂xi ∂xj ¯

uniformly converges to zero in B × [0, T ] with B being any bounded set in Rn . From LM un = F (un−1 ), we have ∂(un − um ) ∂t =∆

ÃZ Z t 0

Rn

!

2 1 −M (t−τ ) − |x−ξ| 4(t−τ ) (F (un−1 ) − F (um−1 )) n e 2 C(t − τ )

−M (un − um ) + (F (un−1 ) − F (um−1 )).

(4.2.49)

We can deduce from (4.2.48) that

¯ ¯ ¯ ∂(un − um ) ¯ ¯ ¯ ¯ ¯ ∂t

also uniformly converges to zero in B × [0, T ] as n, m → ∞. Therefore, u ∈ C 2,1 and ∂u ∂un → ∂t ∂t

144

NONLINEAR EVOLUTION EQUATIONS

uniformly in B × [0, T ]. Passing to the limit in (4.2.21), we conclude that u is a classical solution to the Cauchy problem (4.2.21). The uniqueness in the class of bounded functions is easy to prove. Indeed, if there are two solutions u1 , u2 , then u = u2 − u1 satisfies   ut − ∆u = f (u2 ) − f (u1 ) = c(x, t)u, 

(x, t) ∈ Rn × R+ ,

u|t=0 = 0,

where c(x, t) =

Z 1 ∂f 0

∂s

(4.2.50)

(u1 + s(u2 − u1 ))ds.

By Corollary 4.2.1, we immediately get u ≡ 0. Thus, the proof is complete. ¤ Ex. 4.2.1 Consider the following Cauchy problem:  2  Lu ≡ ut − ∆u = u(1 − u ), 

u|t=0 = φ(x)

(4.2.51)

where φ(x) is a bounded continuous function, m ≤ φ(x) ≤ M with m, M being two given constants. For this problem, we can easily find a pair of upper solution u = K > 0 and lower solution u = −K < 0 where K is a sufficiently large constant such that 1 − K 2 < 0, and −K ≤ m ≤ M ≤ K. Then it is easy to see that Lu = 0 > f (u), and Lu = 0 < f (u). Therefore, problem (4.2.51) has a pair of upper solution u and lower solution u such that u < u. Thus, by Theorem 4.2.1, problem (4.2.51) admits a unique bounded classical solution u(x, t). ¤ REMARK 4.2.2 Notice that in the above the following properties of function f (u) = u(1 − u2 ) have been used to find the required pair of upper solution and lower solution: as u → +∞,

f (u) → −∞,

Monotone Iterative Method and Invariant Regions

145

and as u → −∞,

f (u) → +∞. Therefore, the same technique can be applied to the general problems with f (u) having the above properties. We should also notice that for the other class of problems with f (u) having different behavior at infinity: as u → +∞,

f (u) → +∞, and as u → −∞,

f (u) → −∞, we can still find a pair of upper solution u and lower solution u, but they satisfy u < u, not as required as in Theorem 4.2.1. In this case, the monotone iterative method does not work. For instance, for f (u) = −u(1 − u2 ), the solution may even blow up in a finite time.

If for problem (4.2.21) the coefficients in L are independent of t, and f also does not explicitly involve t, then we are led the following initial boundary value problem:  Lu ≡ ut + Au = f (x, u),          

u|Γ = 0,

(4.2.52)

u|t=0 = φ(x)

where Au = −

n X

aij (x)

i,j=1

n X ∂u ∂2u + + c(x)u. bi (x) ∂xi ∂xj i=1 ∂xi

(4.2.53)

If u(x) ∈ C 2 satisfies   Au ≥ f (x, u), 

u|Γ ≥ 0,

(4.2.54)

then u(x) is called an upper solution to the following nonlinear elliptic boundary value problem:   Au = f (x, u), 

u|Γ = 0.

(4.2.55)

146

NONLINEAR EVOLUTION EQUATIONS

Accordingly, if u(x) ∈ C 2 satisfies

  Au ≤ f (x, u), 

u|Γ ≤ 0,

(4.2.56)

then u is called a lower solution to problem (4.2.55). In the same manner as for the proof of Theorem 4.2.1, we can prove the following result. THEOREM 4.2.2 Suppose that u, u ∈ C 2 are the upper solution and lower solution to problem (4.2.55), respectively. Furthermore, u ≤ u holds. Then problem (4.2.55) admits at least a classical solution u(x) such that u ≤ u ≤ u.

Proof. We just briefly outline the proof here. Either from u or from u, we can construct a monotone iterative sequence un (x) staying between u and u. It turns out that un (x) pointwise converges to a function u(x). By the regularity theorem for the elliptic boundary value problem, and the bootstrap argument, we can conclude that un also converges in C 2 to u. Finally, passing to the limit in the iteration equation, we can obtain that u is a classical solution. ¤ We should notice that the nonlinear elliptic boundary value problem may have multiple solutions. This is different from nonlinear parabolic initial boundary value problem. Suppose that the elliptic boundary value problem (4.2.55) has an upper solution u0 (x), and a lower solution v0 (x) such that v0 (x) ≤ u0 (x). Consider the problem (4.2.52) with initial data u0 (x), and v0 (x), respectively. Then it is easy to see that u0 (x) (v0 (x)) is also an upper (lower) solution to problem (4.2.52). By Theorem 4.2.1, there exist classical solutions u(x, t) and v(x, t) to problem (4.2.52) with initial data u0 (x), v0 (x), respectively. The following theorem concerns the long time behavior of u(x, t) and v(x, t) as time goes to infinity. THEOREM 4.2.3 We have

ut ≤ 0,

vt ≥ 0.

Let

u(x) = lim u(x, t), u(x) = lim v(x, t). t→+∞

t→+∞

(4.2.57)

Then u, and u are classical solutions to problem (4.2.55), respectively.

Monotone Iterative Method and Invariant Regions

147

Proof. By the comparison principle Lemma 4.2.1, we can conclude that u(x, t) ≤ u0 (x) for all t ≥ 0. Let h > 0, and wh (x, t) =

u(x, t + h) − u(x, t) . h

(4.2.58)

wh (x, 0) =

u(x, h) − u0 (x) ≤ 0, h

(4.2.59)

Then

and it satisfies

 ∂wh   + Awh = fξ wh ,  Lwh ≡   ∂t  

(4.2.60)

wh |Γ = 0,       

wh |t=0 = wh (x, 0) ≤ 0

where fξ =

Z 1 0

fs (x, su(x, t + h) + (1 − s)u(x, t))ds.

(4.2.61)

By Lemma 4.2.1, for all t > 0, we have wh (x, t) ≤ 0. Let h → 0, then ut ≤ 0 follows. vt ≥ 0 can be proved in the same manner. Thus, we infer from these results that as t → +∞, u(x, t) and v(x, t) converges to a limit function as (4.2.57) described. Moreover, u(x) and u(x) are bounded measurable functions. For all 2 φ(x) ∈ HB = {φ(x) | φ(x) ∈ H 2 , φ|Γ = 0},

we have

Z Ω

=−

ut φdx = −

Z

Ω

Z Ω

uA∗ φdx +

Z

Auφdx +

Ω

Z Ω

f (x, u)φdx

f (x, u)φdx

(4.2.62)

where A∗ is the adjoint operator of A such that ∗

A φ=−

n X ∂ 2 (aij φ)

i,j=1

∂xi ∂xj

−

n X ∂(bi φ) i=1

∂xi

+ cφ.

(4.2.63)

Integrating (4.2.62) with respect to t from 0 to T yields Z

Ω

+

u(x, T ) − u0 (x) φdx = − T

Z Ω

1 T

Z T 0

f (x, u)dt · φdx.

Z

Ω

A∗ φ ·

1 T

Z T 0

udtdx (4.2.64)

148

NONLINEAR EVOLUTION EQUATIONS

It is easy to see that as T → +∞, u(t, T ) − u0 (x) → 0. T Let w(x, T ) = Then 1 u0 (x) ≥ w(x, T ) = T

1 T

Z T 0

Z T 0

(4.2.65)

u(x, t)dt.

(4.2.66)

u(x, t)dt ≥ u(x, T ),

(4.2.67)

and u(x, T ) − w(x, T ) ≤ 0. (4.2.68) T Hence, w(x, T ) is a monotone decreasing function in T . When T → +∞, it has a limit. Now we prove that its limit is u(x), i.e., wT0 (x, T ) =

1 T

Z T

Indeed, w(x, T ) =

u(x, t)dt → u(x).

0

1 T

Z 0

T 2

u(x, t)dt +

1 T 1 = w(x, ) + u(x, T ∗ ) 2 2 2

1 T

(4.2.69)

Z T T 2

u(x, t)dt (4.2.70)

where T2 ≤ T ∗ ≤ T . Let T → +∞, then (4.2.69) follows. In the same manner, we can prove that 1 T

Z T 0

f (x, u)dt → f (x, u(x))

(4.2.71)

as T → +∞. Passing to the limit in (4.2.64), we infer from Lebesgue’s theorem that 0=−

Z

Ω

A∗ φudx +

Z

Ω

f (x, u)φdx

(4.2.72)

i.e., u(x) is a weak solution to the elliptic boundary value problem (4.2.55). By the regularity theorem for the elliptic boundary value problem and the bootstrap argument, we can conclude that u(x) is a classical solution. As a matter of fact, since u is a bounded measurable function, so is the function f (x, u). By the regularity theorem for the elliptic boundary value problem, the weak solution u belongs to W 2,p for any p > 1. Then by the Sobolev imbedding theorem, u belongs to

Monotone Iterative Method and Invariant Regions

149

C 1,α . It turns out that f (x, u) belongs to C α . By the regularity results in the H¨older space, we can conclude that u belongs to C 2,α , i.e., u is a classical solution. The same proof applies to v(x, t). Thus, the proof is complete. ¤

4.2.2

Monotone Iterative Method for Nonlinear Hyperbolic Equations

In this section we illustrate how the monotone iterative method can be applied to some nonlinear hyperbolic problem though the maximum principle in general does not hold for hyperbolic equations. Consider the following Cauchy problem for a semilinear wave equation:  2  utt − a ∆u = f (x, t, u), 

u|t=0 = φ(x),

x ∈ Rn , n ≤ 3, t > 0,

ut |t=0 = ψ(x)

(4.2.73)

where φ(x) ∈ C 2 ,ψ(x) ∈ C 1 , and f ∈ C 1 . For simplicity of exposition, in what follows we mainly discuss problem (4.2.73) in one space dimension, n = 1. But the situations for n = 2 and n = 3 can be discussed in the same manner, and the conclusions remain unchanged. First we consider the following Cauchy problem for the nonhomogeneous wave equation:  2  Lu ≡ utt − a uxx = F (x, t), 

u|t=0 = φ(x),

x ∈ R, t > 0

ut |t=0 = ψ(x).

(4.2.74)

Then we have the following comparison principle. LEMMA 4.2.2 Suppose that F , φ, ψ are nonnegative continuous functions. Then for the classical solution u(x, t), it holds that for all x ∈ R, t ≥ 0, u(x, t) ≥ 0.

Proof. The conclusion follows from the following D’Alembert formula for the classical solution: Z 1 x+at φ(x + at) + φ(x − at) + ψ(ξ)dξ u(x, t) = 2 2a x−at +

1 2a

Z t Z x+a(t−τ ) 0

x−a(t−τ )

Thus the proof is complete.

F (ξ, τ )dξdτ.

(4.2.75) ¤

150

NONLINEAR EVOLUTION EQUATIONS

REMARK 4.2.3 As J.B. Keller [81] pointed out, the comparison principle still holds for the Cauchy problem for the n-dimensional wave equation with n ≤ 3.

As for the parabolic equation, by this comparison principle we can define the upper solution and lower solution as follows. DEFINITION 4.2.2 Suppose that u belongs to C 2 and satisfies  2  Lu ≡ utt − a uxx ≥ f (x, t, u), x ∈ R, t > 0, 

u|t=0 ≥ φ(x), ut |t=0 ≥ ψ(x).

(4.2.76)

Then u is called an upper solution to problem (4.2.73). Lower solution u can be defined similarly by changing the ≥ sign in the above into the ≤ sign.

Then we have the following result. THEOREM 4.2.4 Suppose that φ ∈ C 2 , ψ ∈ C 1 , f ∈ C 1 , fu (x, t, u) ≥ 0, and f (x, t, u) is bounded in R × R+ × R. Then problem (4.2.73) admits a unique classical solution.

Proof. Let T > 0, x0 ∈ R be any given constants. Consider the following characteristic triangle: 4 = {0 ≤ t ≤ T, −at ≤ x − x0 ≤ at}. It is well known for the wave equation that due to finite speed of propagation, we can restrict ourselves to the characteristic triangles. It suffices to prove that the problem is solvable for any triangles 4. Accordingly, we can define the upper solution and lower solution in the characteristic triangle 4. Let M > 0 be the bounds of φ and ψ in this triangle, and f (x, t, u) in R × R+ × R, i.e., |φ|, |ψ|, |f (x, t, u)| ≤ M,

∀(x, t) ∈ 4, u ∈ R.

(4.2.77)

Then the functions u and u defined below, t2 t2 ), u = −M (1 + t + ), (4.2.78) 2 2 are a pair of upper solution and lower solution in 4 such that u ≤ u. Indeed, by (4.2.77), we have u = M (1 + t +

 2  Lu ≡ utt − a uxx = M ≥ f (x, t, u), 

(x, t) ∈ 4,

u|t=0 = M ≥ φ(x), ut |t=0 = M ≥ ψ(x),

(4.2.79)

Monotone Iterative Method and Invariant Regions

151

i.e., u is an upper solution in 4. It can be proved in the same manner that u is a lower solution. As in the case for the parabolic equation, we can define a sequence of functions by the monotone iterative method. Let u(0) = u, and u(n) (n = 1, · · ·) be the solution to the following Cauchy problem:  (n) ≡ u(n) − a2 u(n) = f (x, t, u(n−1) ),  xx  Lu tt   (n) u |t=0 = φ(x),

(n) ut |t=0

(x, t) ∈ 4,

(4.2.80)

= ψ(x).

We can easily deduce from the comparison principle (Lemma 4.2.2) and the assumption fu ≥ 0 that for all (x, t) ∈ 4, u ≤ · · · ≤ u(n) ≤ u(n−1) ≤ · · · ≤ u ≡ u(0) .

(4.2.81)

Thus, u(n) point-wise converges to a bounded function u(x, t) in 4. By the D’Alembert formula (4.2.75), we can also get the expressions of the first-order derivatives of u(n) : u(n) x (x, t) = +

1 2a

Z t³ 0

φ0 (x + at) + φ0 (x − at) ψ(x + at) − ψ(x − at) + 2 2a

f (x + a(t − τ ), τ, u(n−1) (x + a(t − τ ), τ )) ´

−f (x − a(t − τ ), τ, u(n−1) (x − a(t − τ ), τ )) dτ, (n)

ut (x, t) = +

1 2

Z t³ 0

(4.2.82)

a(φ0 (x + at) − φ0 (x − at)) ψ(x + at) + ψ(x − at) + 2 2

f (x + a(t − τ ), τ, u(n−1) (x + a(t − τ ), τ )) ´

+ f (x − a(t − τ ), τ, u(n−1) (x + a(t − τ ), τ )) dτ. (n)

(n)

(4.2.83)

Then we infer from the above that ux , ut are uniformly bounded in the characteristic triangle 4. Thus, it follows from the Arzela-Ascoli compactness theorem and the monotonicity of u(n) that u(n) (x, t) uniformly converges to a continuous function in 4. From the above expressions (4.2.82), (4.2.83) the first-order derivatives of u(n) also uniformly converge in 4. Hence, u ∈ C 1 , and u(n) converges to u in C 1 (4). Passing to the limit in the D’Alembert formula for u(n) , we deduce that u satisfies the D’Alembert formula (4.2.75) in 4. Since u ∈ C 1 in 4, it follows from (4.2.75) that u ∈ C 2 in 4 and u is a classical solution in 4. In what follows we prove the uniqueness of classical solution in 4, using the energy method.

152

NONLINEAR EVOLUTION EQUATIONS

Suppose that there are two classical solutions u1 , u2 in 4. Let u = u1 − u2 .

(4.2.84)

Then u satisfies

 2  Lu ≡ utt − a uxx = f (x, t, u1 ) − f (x, t, u2 ), 

(x, t) ∈ 4,

u|t=0 = 0, ut |t=0 = 0.

(4.2.85)

For 0 ≤ t ≤ T , let E0 (t) = Then dE0 = dt

=

Z x0 +a(T −t) 1 x0 −a(T −t)

Z x0 +a(T −t) x0 −a(T −t)

2

(u2t (x, t) + a2 u2x (x, t))dx.

(4.2.86)

(utt ut + a2 uxt ux )dx

a − (u2t (x0 + a(T − t), t) + a2 u2x (x0 + a(T − t), t)) 2 a 2 − (ut (x0 − a(T − t), t) + a2 u2x (x0 − a(T − t), t)) 2 Z x0 +a(T −t) x0 −a(T −t)

ut (utt − a2 uxx )dx + a2 ux ut (x0 + a(T − t), t)

a −a2 ux ut (x0 − a(T − t), t)− (u2t (x0 + a(T − t), t) 2 +a2 ux (x0 + a(T − t), t)) a − (u2t (x0 − a(T − t), t) + a2 u2x (x0 − a(T − t), t)) 2 =

Z x0 +a(T −t) x0 −a(T −t)

ut (f (x, t, u1 ) − f (x, t, u2 ))dx

a − ((ut (x0 + a(T − t), t) − aux (x0 + a(T − t), t))2 2 +(ut (x0 − a(T − t), t) + aux (x0 − a(T − t), t))2 ) ≤

Z x0 +a(T −t) x0 −a(T −t)

ut (f (x, t, u1 ) − f (x, t, u2 ))dx.

(4.2.87)

Therefore, integrating with respect to t yields E0 (t) ≤ ≤

Z t Z x0 +a(T −τ )

ut (f (x, t, u1 ) − f (x, t, u2 ))dxdτ x0 −a(T −τ ) Z t Z x0 +a(T −τ ) Z Z K t x0 +a(T −τ ) 2 u2t dxdτ + u dxdτ 2 0 x0 −a(T −τ ) 0 x0 −a(T −τ )

0

1 2

(4.2.88)

Monotone Iterative Method and Invariant Regions where K is defined as follows: sup

(x,t)∈4, min(u1 ,u2 )≤u≤max(u1 ,u2 )

153

¯ ¯ ¯ ∂f (x, t, u) ¯ ¯ ¯. ¯ ¯ ∂u

By the Young inequality, we have Z t

u2 (x, t) = 2 ≤ Thus, we have

Z

0 t

0

Z x0 +a(T −t)

≤ ≤

uut (x, τ )dτ

u2 (x, τ )dτ +

Z

0

0

u2t (x, τ )dτ.

(4.2.89)

u2 (x, t)dx

x0 −a(T −t) Z t Z x0 +a(T −t) 0

Z t

(u2 (x, τ ) + u2t (x, τ ))dxdτ

x0 −a(T −t) t Z x0 +a(T −τ ) x0 −a(T −τ )

(u2 + a2 u2x + u2t )dxdτ

(4.2.90)

Let E(t) be the energy function defined as follows: E(t) =

Z x0 +a(T −t) 1 x0 −a(T −t)

2

(u2 + u2t + a2 u2x )dx.

(4.2.91)

Then combining (4.2.90) with (4.2.88) yields E(t) ≤ K1

Z t

E(τ )dτ

(4.2.92)

K1 = max(K + 1, 2).

(4.2.93)

0

where By the initial condition, E(0) = 0. Thus applying the Gronwall inequality to (4.2.92) yields E(t) = 0,

∀0 ≤ t ≤ T,

(4.2.94)

which implies that u ≡ 0 for all (x, t) ∈ 4. Since x0 , T are arbitrary, the existence and uniqueness of the solution to the Cauchy problem (4.2.73) follows. Thus the proof is complete. ¤ REMARK 4.2.4 We should notice that the assumption fu ≥ 0 in Theorem 4.2.4 is rather restrictive, and this assumption is not needed in Theorem 4.2.1 for the parabolic problem because for a second-order parabolic operator

154

NONLINEAR EVOLUTION EQUATIONS

L, the comparison principle still holds for the equation LM u ≡ Lu + M u = f. Nevertheless, we can still find a class of problems that satisfy the assumptions in Theorem 4.2.4. Ex. 4.2.2. Let f (x, t, u) =

g(x, t) 1 + e−u

(4.2.95)

where g ∈ C 1 , g ≥ 0. Then it is easy to verify that f satisfies the assumption fu ≥ 0 in Theorem 4.2.4, and for any φ ∈ C 2 , ψ ∈ C 1 , the functions given by (4.2.78) with M large enough is a pair of upper solution and lower solution satisfying u ≤ u. Thus, for the problem (4.2.73) with f given by (4.2.95) admits a unique classical solution for any given φ ∈ C 2 , ψ ∈ C 1 . ¤

4.3 4.3.1

Invariant Regions Introduction

In this section we introduce the invariant regions, an important method for dealing with nonlinear evolution equations. This invariant regions method can be viewed in some sense as a generalization of the monotone iteration method described in the previous section. For the monotone iterative method to work, a comparison principle has to be established. Moreover, a pair of upper solution u and lower solution u, satisfying u ≤ u has to be found. If it could be done, we would have good information, i.e., a sequence of un could be derived from the monotone iteration and it would stay between u and u, and converge to a solution to the problem considered. However, such a framework sometimes is quite restrictive. Instead, we could seek an alternative, i.e., if initial data of solution stay in a region, could we conclude that the solution for later time would remain in this region? This is certainly a weaker statement than the corresponding one in the monotone iterative method. If such a region does exist, we may call that an invariant region. In other words, existence of an invariant region does not necessarily imply feasibility of the monotone iterative method, but it does imply an a priori estimate of an L∞ norm of solutions to a given

Monotone Iterative Method and Invariant Regions

155

nonlinear evolution equation. Having had an L∞ norm estimate, we could proceed to get higher-order Sobolev norm estimates from which existence of a solution can follow, using the bootstrap argument and the standard approach in functional analysis. As mentioned in the Introduction to this chapter, K. Chueh, C. Conley, and J. Smoller [33] studied the invariant regions in a way that detailed information about the invariant regions was given, while the precise proof on existence of the solution was not shown in detail. On the other hand, the paper [7] by H. Amann was more concentrated on precise detailed proof of the existence and uniqueness of the solution in which the invariant regions yield the L∞ norm estimate. In this section, we follow the approach given by K. Chueh, C. Conley, and J. Smooler with some modifications of proofs.

4.3.2

Methods and Applications

For simplicity of exposition, we restrict ourselves to the following initial boundary value problem for a semilinear parabolic equation in one space dimension:  ∂2u ∂u ∂u   = εD 2 + M + f (u, t),    ∂t ∂x ∂x  

u|Γ = g(t),       

(x, t) ∈ Ω × R+ , (4.3.1)

u|t=0 = u0 (x)

where Ω is a bounded open interval with Γ being its boundary, or Ω = R. ε > 0 is a given positive constant. D(u, x), M (u, x) are matrix functions defined in an open set Rn × Ω, and D ≥ 0, i.e., D is semipositive definite. u = (u1 , · · · , un )T is an unknown vector function, f is a given smooth vector function defined in Rn × R+ , and g is a given vector function. As stated previously, we will follow K. Chueh, C. Conley, and J. Smoller to focus our attention on how a region in Rn can be invariant. Hence, we do not intend to give rigorous proof of the existence and uniqueness of the solution to problem (4.3.1). Instead, we will always assume that problem (4.3.1) admits at least a local classical solution in time in a certain class X of smooth functions, though in principle it can be proved by a contraction mapping theorem, as shown in Chapter 2 in this book. DEFINITION 4.3.1

Suppose Σ is a closed set in Rn . If u0 , g ∈ Σ

156

NONLINEAR EVOLUTION EQUATIONS

implies that for all x ∈ Ω, and t ∈ [0, T ], where existence of the solution to problem (4.3.1) is assumed, solution u(x, t) ∈ Σ, then Σ is called a (positive) invariant region for problem (4.3.1).

We will consider an invariant region in the following form: Σ=

m \

{u | u ∈ U, Gi (u) ≤ 0}

(4.3.2)

i=1

where m ∈ N and each Gi (u) is a smooth function mapping from U into R such that the gradient ¶ µ ∂Gi ∂Gi ,···, (4.3.3) dGi = ∂u1 ∂un does not vanish. Notice that when v ∈ ∂Σ, dGi (v) is the outward normal vector on ∂Σ. Throughout this section, we use the following notations d2 G and d2 G(η, η) for η ∈ Rn : d2 G = (

∂2G ), ∂ui ∂uj

d2 G(η, η) = η T d2 Gη.

(4.3.4) (4.3.5)

DEFINITION 4.3.2 A smooth function G : Rn 7→ R is called quasiconvex at v ∈ Rn if for all vectors η orthogonal to dG(v), d2 G(v)(η, η) ≥ 0; it is called strictly convex if d2 G(v)(η, η) > 0.

Before stating the main result in this section, we introduce the following condition on the set Σ defined by (4.3.2). Condition K. For solution u in class X, there is a compact set K ⊂ Ω such that if x 6∈ K, then u(x, t) ∈ Int(Σ) = Σ\∂Σ. Notice that condition K is a relationship between the set Σ and the behavior of solution u at boundary Γ or at infinity. It amounts to assuming that Gi (g(t)) < 0 for the Dirichlet boundary condition. When Ω = R, condition K requires that for |x| large enough, Gi (u(x, t)) < 0. The invariant region that we are looking for will be selected among sets Σ satisfying condition K. Therefore, throughout this section we always assume that the target set Σ defined by (4.3.2) satisfies condition K. The following result indicates what kind of set Σ defined in the form of (4.3.2) can be an invariant region. First we notice that for Σ being an invariant region, if the initial data u0 stay in Σ, then for a later

Monotone Iterative Method and Invariant Regions

157

time, u must also stay in Σ. For the time being, we assume that for all points x0 ∈ Ω such that u0 (x0 ) ∈ Int(Σ). It will be seen from Remark 4.3.2 and the examples later on that such a assumption is not a severe restriction in applications. THEOREM 4.3.1 Let Σ be defined by (4.3.2) and u0 (x) ∈ Int(Σ), x ∈ Ω. Suppose that for every v0 ∈ ∂Σ (so Gi (v0 ) = 0 for some i), the following conditions hold: (i) For all x ∈ Ω, dGi (v0 ) is a left eigenvector of D(v0 , x) and M (v0 , x) with eigenvalues µ and λ, respectively. (ii)If µ 6= 0, then Gi is quasi-convex at v0 . (iii)For all t ∈ [0, T ], dGi (v0 ) · f < 0, i.e., f is strictly inward to Σ. Then for every ε > 0, Σ is invariant for problem (4.3.1).

Proof. Under condition K, for each x ∈ Ω\K, Gi (u(x, t)) < 0 (1 ≤ i ≤ m). For x ∈ K, by the assumption on initial data and continuity of u and Gi , for any i, 1 ≤ i ≤ m, there is ti (x) > 0 such that when 0 ≤ t < ti (x), Gi (u(x, t)) < 0; Gi (u(x, ti (x))) = 0. Let mi = inf ti (x). x∈K

Then mi ≥ 0. If mi = +∞ for all i, 1 ≤ i ≤ m, then we are done. Otherwise, there is at least one i such that mi < +∞. In what follows we use a contradiction argument to show that it is impossible. For simplicity of notation, we denote Gi by G, mi by t0 and ti (x) by t(x). By the definition on t0 , there is a minimizing sequence xn ∈ K such that t(xn ) → t0 < +∞. Since K is a compact set, there is a subsequence of xn , still denoted by xn such that xn → x0 ∈ K. By continuity of u and G, we have G(u(xn , t(xn ))) → G(u(x0 , t0 )) = 0.

(4.3.6)

By the assumption on the initial data, we get t0 > 0. Furthermore, from the definition on t(x0 ) and t0 we can deduce that t(x0 ) = t0 .

(4.3.7)

158

NONLINEAR EVOLUTION EQUATIONS

Indeed, it is clear from the definition of t0 that t0 ≤ t(x0 ). On the other hand, if t0 < t(x0 ), (4.3.6) contradicts the definition of t(x0 ). Thus, (4.3.7) follows. Thus there is a point x0 ∈ Ω and t0 , 0 < t0 < +∞ satisfying G(u(x0 , t0 )) = 0, G(u(x, t0 )) ≤ 0

(4.3.8)

∀x ∈ Ω,

(4.3.9)

∀0 ≤ t < t0 .

(4.3.10)

and G(u(x0 , t)) < 0

It is clear from (4.3.8) and (4.3.10) that ¯

∂G(u(x0 , t)) ¯¯ ≥ 0. ¯ ∂t t=t0 In what follows we derive from equation (4.3.1) and (4.3.8)-(4.3.10) that ¯ ∂G(u(x0 , t)) ¯¯ < 0, (4.3.11) ¯ ∂t t=t0 which yields a contradiction. Let v(x) = u(x, t0 ). Then it follows from (4.3.8) and (4.3.9) that v(x0 ) ∈ ∂Σ and G(v(x)) achieves its maximum value at x0 . Therefore, the first-order derivatives of G(v(x)) must vanish at x = x0 . It turns out that dG(v(x0 ))vx (x0 ) = 0.

(4.3.12)

Then from G(v(x)) ≤ 0 and the expansion of G(v(x)) around x0 , G(v(x)) = G(v(x0 )) + dG · vx (x0 )(x − x0 ) 1 + (d2 G(vx (x0 ), vx (x0 )) + dG · vxx (x0 ))(x − x0 )2 2 + o(|x − x0 |2 ) ≤ 0, (4.3.13) it follows that d2 G(vx (x0 ), vx (x0 )) + dG · vxx (x0 ) ≤ 0.

(4.3.14)

On the other hand, we can deduce from the equations in (4.3.1) that ¯

∂G(u(x0 , t)) ¯¯ = dG(v(x0 ))ut |t=t0 ,x=x0 ¯ ∂t t=t0 = εdG(v(x0 ))Duxx (x0 , t0 ) + dG(v(x0 ))M ux (x0 , t0 ) + dG(v(x0 )) · f = εµdG(v(x0 )) · vxx (x0 ) + λdG(v(x0 )) · vx (x0 ) + dG(v(x0 )) · f = εµdG(v(x0 )) · vxx (x0 ) + dG · f. (4.3.15)

Monotone Iterative Method and Invariant Regions

159

Since D ≥ 0, its eigenvalue µ ≥ 0. If µ = 0, then we infer from condition (iii): dG · f < 0 that (4.3.11) holds. If µ > 0, then we can deduce from condition (ii) and (4.3.14) that dG(v(x0 )) · vxx (x0 ) ≤ −d2 G(vx (x0 ), vx (x0 )) ≤ 0.

(4.3.16)

Thus, (4.3.11) still follows from (4.3.15). The proof is complete.

¤

REMARK 4.3.1 It is easy to see from the proof of Theorem 4.3.1 that when D is positive definite, assumptions (ii), (iii) in Theorem 4.3.1 can be replaced by the following assumptions: (ii)’ If dGi (v0 )D(v0 , x) = µdGi (v0 ), µ 6= 0, then Gi is strongly convex at v0 . (iii)’ dGi (v0 ) · f ≤ 0.

In the following we give a straightforward useful result. REMARK 4.3.2 Suppose that Σi , i ∈ I are invariant regions. Then Σ, the intersection of Σi , \ (4.3.17) Σ = Σi , i∈I

is also invariant.

Proof. This is immediate from the definition of invariant regions.

¤

COROLLARY 4.3.1 (i) If D and M are diagonal matrices, and fi (u1 , · · · , ui−1 , Ci , ui+1 , · · · , un ) < 0, then

Σi = {u | ui − Ci ≤ 0},

(i = 1, · · · , n)

(4.3.18)

are invariant for all ε > 0. (ii) Suppose that D and M are diagonal matrices. If vector f on ∂Σ is strictly inward to Σ, then the region defined by the following,

Σ=

n \

{u | ai ≤ ui ≤ bi },

(4.3.19)

i=1

where ai and bi are given constants, is invariant for all ε > 0. (iii) If D = I with I being the identity matrix, and M = 0, then any convex closed set Σ ⊂ Rn is invariant for all ε > 0 provided f on ∂Σ is strictly inward to Σ.

160

NONLINEAR EVOLUTION EQUATIONS

Proof. The conclusion in (i) simply follows from the facts that dGi is the left eigenvector of D and M , and Gi = ui − Ci is quasi-convex everywhere since d2 Gi ≡ 0. To prove (ii), we rewrite the region Σ defined by (4.3.19) as Σ=

n \

¯ i (u) ≤ 0} {u | Gi (u) ≤ 0, G

(4.3.20)

i=1

with ¯ i (u) = ui − bi . Gi (u) = ai − ui , G

(4.3.21)

When D and M are diagonal matrices, it is easy to see that dGi and ¯ i are left eigenvectors of D and M . Thus the conclusion follows from dG Theorem 4.3.1. To prove (iii), we use the following result on the convex closed set in Rn : for any convex closed set Σ, there is a family of linear functions Φ such that Σ = {v ∈ Rn | G(v) ≤ 0, G ∈ Φ}

(4.3.22)

G(v) = φˆ · v + φ(0)

(4.3.23)

where with φˆ = (φˆ1 , · · · , φˆn )T ∈ Rn , φ(0) ∈ R being given. Since D = I and M = 0, dG must be a left eigenvector of D and M . Moreover, G must be quasi-convex everywhere since d2 G = 0. Thus, the conclusion follows from Theorem 4.3.1. ¤ In many applications the vector field f satisfies the previous weak condition (iii)’. Remark 4.3.1 tells us that if a stronger condition (ii)’ is satisfied, then under the weak condition (iii)’, Theorem 4.3.1 still holds. It is desirable to find other conditions to replace (ii)’. For this reason, we introduce the following definition. DEFINITION 4.3.3 The system (4.3.1) is said to be f -stable with respect to Σ if there is a smooth vector field h(u), which is bounded on ∂Σ, and is strictly inward to Σ on ∂Σ such that the corresponding solutions uδ to fδ = f + δh(u) converges to f in C 0 topology in any compact subset of Rn × R+ , as δ → 0.

The following result tells us that if the system (4.3.1) is f -stable with respect to Σ, then Theorem 4.3.1 still holds even with the weak condition (iii)’.

Monotone Iterative Method and Invariant Regions

161

THEOREM 4.3.2 Suppose that the system (4.3.1) is f -stable with respect to Σ, and conditions (i), (ii) in Theorem 4.3.1, and condition (iii)’ stated in Remark 4.3.1 are satisfied. Then the same conclusion of Theorem 4.3.1 still holds.

Proof. Consider problem (4.3.1) with f being replaced by fδ . Then by Theorem 4.3.1, Σ defined by (4.3.2) is invariant with respect to the approximate solutions uδ because now dG · fδ < 0. Since the system is f -stable, it immediately follows that Σ is also invariant with respect to u. Thus the proof is complete. ¤ In what follows we discuss the necessary conditions for Σ being invariant. THEOREM 4.3.3 Suppose that the matrix D in (4.3.1) is positive definite, and Σ defined by (4.3.2) is invariant with respect to system (4.3.1). Then the following conditions hold at each v0 ∈ ∂Σ (say, Gi (v0 ) = 0): (1) dGi is a left eigenvector of D at v0 . (2) Gi is quasi-convex at v0 . (3) dGi (v0 ) · f (v0 , t) ≤ 0.

Proof. For simplicity of notation, we write Gi = G again. We first prove (1) by a contradiction argument. If dG is not a left eigenvector of D at v0 , which implies that dGD 6= µdG, then we can always find a vectors ξ ∈ dG · ξ < 0,

Rn

(4.3.24)

such that at v0 ,

dG · D · ξ > 0.

(4.3.25)

We can also find a vector η ∈ Rn such that dG · η = 0.

(4.3.26)

For any fixed ε > 0, we can always find a sufficiently large number λ ∈ R such that λεdG · D · ξ + dG · M · η + dG · f > 0, λdG · ξ + d2 G(η, η) < 0.

(4.3.27) (4.3.28)

We consider the cases that v0 is not a “corner” point of Σ, and v0 is a “corner” point, respectively. First, let v0 not be a corner point of Σ, i.e., there is no other j different from i such that Gj (v0 ) = 0. Then it turns out that there is

162

NONLINEAR EVOLUTION EQUATIONS

an (n − 1)-dimensional neighborhood of v0 which lies in (Gi = 0) In this neighborhood, we define a function U as follows:

T

Σ.

1 U (x) = v0 + xη + λx2 ξ. 2

(4.3.29)

h(x) = G(U (x)).

(4.3.30)

Let Then, the expansion of h(x) around x = 0 gives 1 h(x) = G(v0 ) + xdG · η + x2 (d2 G(η, η) + λdG · ξ) + o(|x|2 ) 2 1 = x2 (d2 G(η, η) + λdG · ξ) + o(|x|2 ). (4.3.31) 2 Thus, it follows from (4.3.28) that there is a small δ > 0 such that when |x| ≤ δ, h(x) ≤ 0, i.e., U (x) ∈ Σ. Now we choose such a smooth initial datum u0 that in the δ neighborhood of x = 0 (or equivalently, the δ neighborhood of a point x0 ∈ Ω), u0 (x) = U (x). Let v(x, t) be the solution to problem (4.3.1) with such an initial datum u0 . Then, we deduce from the equation of (4.3.1) that dG · vt = εdG · D · vxx + dG · M · vx + dG · f.

(4.3.32)

Thus, at x = 0, t = 0, we infer from (4.3.27) that dG · vt = ελdG · D · ξ + dG · M · η + dG · f > 0,

(4.3.33)

which contradicts the assumption that Σ is an invariant set, since v = v0 and G(v0 ) = 0 at x = 0, t = 0. In the case where v0 is a corner point, i.e., there are j and i such that Gi (v0 ) = 0 and Gj (v0 ) = 0. In this case, we can choose a sequence vk ∈ Rn such that Gi (vk ) = 0, Gj (vk ) 6= 0 for j 6= i, and vk → v0 . By what we have just proved, we have dGi (vk ) · D(vk ) = µ(vk )dGi (vk ). Passing to the limit, we get the desired result (1). In what follows we proceed to prove (2). We still use a contradiction argument. Suppose that G is not quasi-convex at v0 . Then there exists an η ∈ Rn such that dG(v0 ) · η = 0,

d2 G(η, η) < 0.

(4.3.34)

By (1) that we have already proved, there is µ > 0 such that dG(v0 ) · D(v0 ) = µdG(v0 ).

(4.3.35)

We are always able to choose ξ ∈ Rn such that dG · ξ > 0,

dG · ξ + d2 G(η, η) < 0

(4.3.36)

Monotone Iterative Method and Invariant Regions

163

at v0 . Then we choose λ sufficiently large such that ελ2 µdG · ξ + λdG · M · η + dG · f > 0

(4.3.37)

1 U (x) = v0 + λxη + λ2 x2 ξ. 2

(4.3.38)

at v0 . Let

Thus, by expanding G(U ) around x = 0, we obtain 1 G(U ) = G(v0 ) + λxdG(v0 ) · η + λ2 x2 (dG · ξ + d2 G(η, η)) + o(|x|2 ). 2

(4.3.39)

It follows from (4.3.36) that in a small neighborhood |x| ≤ δ of x = 0, G(U ) ≤ 0, i.e., U ∈ Σ. As before, we choose a smooth initial datum u0 (x) such that in a small neighborhood of a point x0 ∈ Ω, which corresponds to |x| ≤ δ, u0 coincides with U . Let v(x, t) be the corresponding solution to problem (4.3.1). Then we deduce from the equation in (4.3.1) that ¯

∂G(v(x, t)) ¯¯ = dG · vt |x=0,t=0 ¯ ∂t x=0,t=0 = (εdG · Dvxx + dG · M vx + dG · f )|x=0,t=0 = εµλ2 dG · ξ + λdG · M · η + dG · f > 0.

(4.3.40)

Since at x = 0, t = 0, G(v(x, t)) = G(U (0)) = G(v0 ) = 0, it follows from (4.3.40) that in a small neighborhood 0 < t < t0 of t = 0, we must have G(u(x, t)) > 0, which contradicts the fact that Σ is an invariant region. Thus, (2) is proved. Finally, we proceed to prove (3). We still use a contradiction argument. If dG · f > 0 at v0 , then we choose U (x) as follows. 1 U (x) = v0 + ληx + λ2 ξx2 2

(4.3.41)

where η ∈ Rn satisfies dG · η = 0

(4.3.42)

dG · ξ + d2 G(η, η) < 0,

(4.3.43)

and ξ satisfies

and λ is a small positive constant specified later. Expanding G(U (x))

164

NONLINEAR EVOLUTION EQUATIONS

around x = 0 yields 1 G(U (x)) = G(v0 ) + λxdG(v0 ) · η + λ2 x2 (d2 G(η, η) + dG · ξ) + o(|x|2 ). 2 (4.3.44)

We infer from (4.3.42) and (4.3.43) that in a small neighborhood |x| ≤ δ, G(U (x)) ≤ 0, i.e., U (x) ∈ Σ. In the same manner as before, we choose the initial datum u0 , which coincides with U in this small neighborhood. Then for the corresponding solution v(x, t), we have ¯

∂G(v(x, t)) ¯¯ = (dG · vt )|x=0,t=0 ¯ ∂t x=0,t=0 = (εdG · Dvxx + dG · M vx + dG · f )|x=0,t=0 = εµλ2 dG · ξ + λdG · M · η + dG · f > 0

(4.3.45)

provided that λ is sufficiently small. This again yields a contradiction, as proved previously. Thus the theorem is proved. ¤ Notice that for the quasilinear hyperbolic system ut = M ux + f,

(4.3.46)

we often use the so-called vanishing viscosity method to approximate the system. In other words, instead of (4.3.46), we considers the following parabolic system: ut = εDuxx + M ux + f.

(4.3.47)

An interesting question is whether for all ε > 0, the above parabolic system has a uniform invariant region Σ. If so, it would suggest that the original hyperbolic system (4.3.41) also has an invariant region Σ. The following result shows that in addition to (1)–(3) in Theorem 4.3.3, another condition has to be satisfied. THEOREM 4.3.4 Suppose that D is positive definite. If for all ε > 0, Σ is an invariant region, then at v0 ∈ ∂Σ, dG must be a left eigenvector of M .

Proof. We still use a contradiction argument. Suppose that dG is not a left eigenvector of M . Then we can choose a vector η such that dG · η = 0, and dG · M · η > 0.

(4.3.48)

Monotone Iterative Method and Invariant Regions

165

We then choose 1 U (x) = v0 + λxη + λ2 x2 ξ 2 where ξ is chosen in such a way that dG · ξ + d2 G(η, η) < 0.

(4.3.49)

(4.3.50)

Choosing λ in the expression (4.3.49) of U large enough so that λdG · M · η + µdG · ξ + dG · f > 0

(4.3.51)

where µ is the eigenvalue of D corresponding to the eigenvector dG. Thus, for ε = λ12 , we have dG · ut |x=0,t=0 = εµλ2 dG · ξ + λdG · M · η + dG · f > 0,

(4.3.52)

which contradicts the fact that Σ is also an invariant region for ε = λ12 . Thus, the proof is complete. ¤ In summary, we have proved the following result. THEOREM 4.3.5 Suppose that problem (4.3.1) is f -stable and D is positive definite. Then for all ε > 0, Σ defined by (4.3.2) is invariant if and only if at each v0 ∈ ∂Σ, (1) dGi is a left eigenvector of D and M . (2) Gi is quasiconvex at v0 . (3) dGi · f ≤ 0. REMARK 4.3.3 space dimensions:

The previous discussions apply to the system in higher m X ∂u ∂u = εD∆u + +f Mi ∂t ∂x i i=1

(4.3.53)

where D and M i , (i = 1, · · · , m) are n × n matrices. We refer the reader to the paper [33] by K. Chuch, C. Conley, and J. Smoller.

In what follows, we give some examples of applications. Ex.4.3.1. We first consider the following Cauchy problem for a nonlinear parabolic equation:  ∂u ∂u   = ∆u + M + f (u),    

∂t

u|t=0 = u0 (x)

∂x

(4.3.54)

166

NONLINEAR EVOLUTION EQUATIONS

where u ∈ R is a scalar unknown function. Notice that in this particular case, the conditions that dG is a left eigenvector of D and M are always satisfied. Then for constants a, b with a < b, the closed convex set Σ = {u | a ≤ u ≤ b}

(4.3.55)

is invariant provided that f (b) < 0, and f (a) > 0. This result coincides with what we have obtained in Section 4.1 where a more specific method, i.e., the monotone iterative method is used. ¤ Ex.4.3.2. Consider the FitzHugh-Nagumo equations arising from the study of propagation of nerve pulse:  ∂v   = vxx + f (v) − u,    ∂t    

∂u

= σv − γu,    ∂t      

(4.3.56)

v|t=0 = v0 (x), u|t=0 = u0 (x)

where f typically is a cubic function in the form f (v) = −v(v − a)(v − b)

(4.3.57)

with 0 < a < b. Let U = (v, u)T ,

F = (f (v) − u, σv − γu)T .

(4.3.58)

Then the system in (4.3.56) can be written in the form Ut = DUxx + F.

(4.3.59)

This is a special reaction diffusion system with D = diag (1, 0). Consider the following closed convex set Σ = {a1 ≤ v ≤ b1 , a2 ≤ u ≤ b2 }

(4.3.60)

i.e., Σ=

4 \

{Gi ≤ 0}

(4.3.61)

i=1

with G1 = a1 − v, G2 = v − b1 , G3 = a2 − u, G4 = u − b2 .

(4.3.62)

Accordingly, we have dG1 = (−1, 0)T , dG2 = (1, 0)T ,

(4.3.63)

Monotone Iterative Method and Invariant Regions

167

and dG3 = (0, −1)T , dG4 = (0, 1)T .

(4.3.64)

Since M = 0 and D is a diagonal matrix, conditions (i), (ii) in Theorem 4.3.1 are clearly satisfied. In order that dG · F < 0, a1 , b1 , a2 , b2 should be chosen to satisfy  f (b1 ) − u < 0, ∀a2 ≤ u ≤ b2 ,           f (a1 ) − u > 0, ∀a2 ≤ u ≤ b2 ,    σv − γb2 < 0,       

(4.3.65)

∀a1 ≤ v ≤ b1 ,

σv − γa2 > 0, ∀a1 ≤ v ≤ b1 .

Notice that the function f defined by (4.3.57) has the following behavior: f (v) → −∞, v → +∞, f (v) → +∞, v → −∞.

(4.3.66) (4.3.67)

Thus, we can choose b1 > 0 large enough, and a1 < 0 with |a1 | being large enough so that σ (4.3.68) f (b1 ) < 0 < a1 , γ and f (a1 ) >

σ b1 > 0. γ

(4.3.69)

It follows from (4.3.65) that b2 should be chosen to satisfy σ b2 > b1 , γ and a2 should be chosen to satisfy σ a1 > a2 . γ

(4.3.70)

(4.3.71)

It turns out that if we choose a1 < 0, b1 > 0 with b1 and |a1 | large enough so that (4.3.68), (4.3.69) are satisfied, which is always possible due to (4.3.66) and (4.3.67), and we choose b2 > 0 and a2 < 0 to satisfy µ

b2 ∈

¶

σ b1 , f (a1 ) , γ

µ

¶

σ a2 ∈ f (b1 ), a1 , γ

(4.3.72)

then (4.3.65) is satisfied. From the above choices of a1 , b1 , a2 , b2 , we can see that the invariant rectangle Σ can be chosen as large as we want. It

168

NONLINEAR EVOLUTION EQUATIONS

implies that for any initial data being smooth bounded functions, the solution (v, u)T of the FitzHugh-Nagumo equations (4.3.56) will stay in a bounded rectangle for all t > 0 from which we can proceed to get more estimates, and finally obtain the global existence of solution. ¤ Ex.4.3.3. Consider the following Hodgkin-Huxley equations arising from the study of physiological phenomenon of signal transmission across axons:  −1   cut = R uxx + g(u, v, w, z),         vt = ε1 vxx + g1 (u)(h1 (u) − v),

   wt = ε2 wxx + g2 (u)(h2 (u) − w),       

(4.3.73)

zt = ε3 zxx + g3 (u)(h3 (u) − z).

The FitzHugh-Nagumo equations previously discussed is just a simplified model of Hodgkin-Huxley equations. In (4.3.73), c, R are given positive constants and εi (i = 1, 2, 3) are given non-negative constants, and the function g is given by the following expression: g(u, v, w, z) = k1 v 3 w(c1 − u) + k2 z 4 (c2 − u) + k3 (c3 − u)

(4.3.74)

with ci (i = 1, 2, 3) being given constants such that c1 > c3 > 0 > c2 .

(4.3.75)

Besides, gi > 0 (i = 1, 2, 3), and 1 > hi > 0 (i = 1, 2, 3) are given functions. In this model, v, w, z represent chemical concentrations, and they should stay in [0, 1], and u represents electric potential. Let U = (u, v, w, z)T , g f (U ) = ( , g1 (h1 − v), g2 (h2 − w), g3 (h3 − z))T , c

(4.3.76) (4.3.77)

and D = diag ((Rc)−1 , ε1 , ε2 , ε3 )T .

(4.3.78)

Then system (4.3.73) can be written in the form Ut = DUxx + f (U ).

(4.3.79)

Now we are looking for an invariant region Σ given by the following: Σ = {(u, v, w, z) | c2 ≤ u ≤ c1 , 0 ≤ v ≤ 1, 0 ≤ w ≤ 1, 0 ≤ z ≤ 1}

(4.3.80)

Monotone Iterative Method and Invariant Regions

169

where c2 < c2 < 0,

c3 < c1 ≤ c1 .

(4.3.81)

In what follows we prove that for c1 being sufficiently positive, and c2 being sufficiently negative, the closed convex set Σ defined by (4.3.80) is an invariant region. For any δ > 0, consider ¯ ¾ ¯ c2 ≤ u ≤ c1 , −δ ≤ v ≤ 1 + δ, ¯ (u, v, w, z) ¯ . −δ ≤ w ≤ 1 + δ, −δ ≤ z ≤ 1 + δ

½

Σδ =

(4.3.82)

Then, Σδ can be expressed as Σδ = {Gi ≤ 0, i = 1, · · · , 8}

(4.3.83)

with G1 = −v − δ, G2 = v − (1 + δ),

G3 = −w − δ,

G4 = w − (1 + δ), G5 = −z − δ, G6 = z − (1 + δ),

(4.3.84) (4.3.85)

and G7 = c2 − u,

G8 = u − c1 .

(4.3.86)

It is clear that d2 Gi = 0 for all i = 1, · · · , 8. Since D is diagonal and M = 0, conditions (i), (ii) in Theorem 4.3.1 are satisfied. For G1 , we have dG1 |v=−δ = (0, −1, 0, 0)T ,

(4.3.87)

dG1 · f |v=−δ = −g1 (h1 + δ)|v=−δ < 0.

(4.3.88)

and In the same way, we can prove that dG3 |w=−δ < 0,

dG5 |z=−δ =< 0.

(4.3.89)

For G2 , we have dG2 |v=1+δ = (0, 1, 0, 0)T

(4.3.90)

dG2 · f |v=1+δ = g1 (h1 − 1 − δ) < 0.

(4.3.91)

and In the same way, we can prove that dG4 · f |w=1+δ = g2 (h2 − 1 − δ) < 0,

dG6 · f |z=1+δ = g3 (h3 − 1 − δ) < 0. (4.3.92)

170

NONLINEAR EVOLUTION EQUATIONS

Finally, we have dG7 = (−1, 0, 0, 0)T , dG8 = (1, 0, 0, 0)T ,

(4.3.93)

and 1 dG7 · f |u=c2 = (−k1 v 3 w(c1 − c2 ) − k2 z 4 (c2 − c2 ) − k3 (c3 − c2 )) < 0, c

(4.3.94)

1 dG8 ·f |u=c1 = (k1 v 3 w(c1 −c1 )+k2 z 4 (c2 −c1 )+k3 (c3 −c1 )) < 0. (4.3.95) c Notice that if u0 (x) ∈ Σ, then u0 (x) ∈ Int(Σδ ). Thus, by Theorem 4.3.1, Σδ is an invariant region for any δ > 0. Since Σ=

\

Σδ ,

δ>0

it follows from Remark 4.3.2 that Σ is invariant. From the existence of an invariant region Σ, we immediately have a uniform L∞ norm bound. Then we can proceed to obtain the global existence of the solution to the Hodgkin-Huxley equations (4.3.73). ¤ Ex.4.3.4. Consider the following equations: ut = D∆u + (1 − |u|2 )u

(4.3.96)

u = (u1 , u2 , u3 )T

(4.3.97)

where is an unknown vector function, |u|2 = u21 + u22 + u23 , and D = diag (α1 , α2 , α3 )

(4.3.98)

is a given diagonal matrix with αi ≥ 0 (i = 1, 2, 3). System (4.3.96) arises from the study of superconductivity of liquids, and u is called the Ginzburg-Landau order parameter. Let f = ((1 − |u|2 )u1 , (1 − |u|2 )u2 , (1 − |u|2 )u3 )T .

(4.3.99)

In the following we show that when α1 = α2 = α3 > 0, the unit sphere Σ given by the following, Σ = {u | |u|2 ≤ 1},

(4.3.100)

is an invariant region. Notice that on ∂Σ, f given by (4.3.99) vanishes. Therefore, in order to apply Theorem 4.3.1, instead of Σ, we consider the following family of closed convex sets Σδ : Σδ = {u | |u|2 ≤ 1 + δ}.

(4.3.101)

Monotone Iterative Method and Invariant Regions

171

In the previous notation, Σδ can be written as Σδ = {u | Gδ (u) ≤ 0} with Gδ (u) = u21 + u22 + u23 − 1 − δ.

(4.3.102)

Since D is an identity matrix and M = 0, dGδ is clearly a left eigenvector of D and M . Moreover, a straightforward calculation shows that d2 Gδ = 2I, i.e., the quasiconvex condition is satisfied. On ∂Σδ , dGδ · f = 2|u|2 (1 − |u|2 ) = −2δ(1 + δ) < 0.

(4.3.103)

Notice that if u0 (x) ∈ Σ, then u0 ∈ Int(Σδ ) Thus, by Theorem 4.3.1, for any δ > 0, Σδ is an invariant region. Taking intersection of Σδ over all δ clearly yields Σ=

\

Σδ .

(4.3.104)

δ>0

Then we deduce from Remark 4.3.2 that Σ is an invariant region for system (4.3.96). ¤ Ex.4.3.5. Consider the following quasilinear hyperbolic system:   ρt − vx = 0, 

vt + p(ρ)x = 0,

(4.3.105)

where p is a given function in ρ with the following properties: when ρ > 0, p > 0, p0 < 0, and p00 > 0. A typical example is that p(ρ) = ρ−γ with γ ≥ 1. The hydrodynamic system in the Lagrangian coordinates can be written as system (4.3.105), and ρ is density, v is velocity, and p is pressure. In order to investigate system (4.3.105) by the socalled vanishing viscosity method, we consider the following nonlinear parabolic system:   ρt − vx = ερxx , 

vt + p(ρ)x = εvxx ,

(4.3.106)

where ε > is a given constant, and the corresponding terms ερxx , and εvxx are called the artificial viscosity terms for the convenience of mathematical study because even if there exists real viscosity, there should not be such a term as ερxx . Let U = (ρ, v)T , and

µ

M=

(4.3.107) ¶

0 1 . 0 −p (ρ) 0

(4.3.108)

172

NONLINEAR EVOLUTION EQUATIONS

Then (4.3.106) can be written in the form Ut = εIUxx + M Ux .

(4.3.109)

Now we introduce new dependent variables r=v−

Z ρq

−p0 (ξ)dξ,

s=v+

Z ρq

−p0 (ξ)dξ,

(4.3.110)

which are called the Riemann invariants. For any r0 , s0 , let G1 = r − r0 ,

G2 = s0 − s.

(4.3.111)

In what follows we prove that Σ defined by Σ = {G1 ≤ 0, G2 ≤ 0}

(4.3.112)

is an invariant region. Indeed, a straightforward calculation shows that q

q

dG1 = (− −p0 (ρ), 1), and

dG2 = (− −p0 (ρ), −1),

p

dG1 · M = − −p0 dG1 ,

dG2 · M =

p

−p0 dG2

(4.3.113)

(4.3.114)

i.e., dG1 , dG2 are the left eigenvectors of M . Since D = εI, dG1 , dG2 are also the left eigenvectors of D. Furthermore, Ã

2

2

d G1 = d G2 =

00 √p 2 −p0

0

0 0

!

,

(4.3.115)

i.e., d2 G1 and d2 G2 are quasi-convex, since p00 > 0. Notice that in the present problem, f ≡ 0. Let h = (1, 0)T . For any δ > 0, consider the corresponding fδ = f + δh = δ(1, 0)T . Thus,

p

dGi · fδ = −δ −p0 < 0. i = 1, 2.

(4.3.116)

(4.3.117)

Clearly, for each ε > 0, the parabolic system (4.3.106) is f -stable. It turns our from Theorem 4.3.1 that Σ is invariant. It follows from v− that

Z ρq

−p0 (ξ)dξ ≤ r0 , Z ρq

v+

Z ρq

−p0 (ξ)dξ ≥

−p0 (ξ)dξ ≥ s0

s0 − r0 . 2

(4.3.118)

(4.3.119)

Monotone Iterative Method and Invariant Regions

173

Therefore, as long as we appropriately choose s0 and r0 , (4.3.119) gives us a lower bound of ρ: ρ≥δ>0

(4.3.120)

with δ being a positive constant depending only on s0 , r0 . In the case that p(ρ) = ρ−γ with γ > 1, (4.3.118) leads to √ 2 γ −γ+1 v+ ρ 2 ≤ r0 , (4.3.121) γ−1 and v−

√ 2 γ −γ+1 ρ 2 ≥ s0 . γ−1

Hence, we deduce from (4.3.121) and (4.3.122) that √ 4 γ −γ+1 ρ 2 ≤ r0 − s0 . γ−1

(4.3.122)

(4.3.123)

Thus, 0<δ≤ρ

(4.3.124)

provided that r0 > s0 . On the other hand, we can easily deduce from (4.3.121) and (4.3.122) that s0 < v < r0 .

(4.3.125)

Estimates (4.3.124) and (4.3.125) on the solutions of the parabolic system (4.3.106) are uniform with respect to ε, which plays a very important role in the study of the nonlinear hyperbolic system (4.3.105). ¤

REMARK 4.3.4 cients:

For the parabolic system with different viscosity coeffi   ρt − vx = ερxx ,   v + p(ρ)x = µvxx ,   t

(4.3.126)

with ε 6= µ, the set Σ defined by (4.3.112) is no longer an invariant region because in the present case, D = diag (ε, µ), and dGi (i = 1, 2) are no longer the left eigenvectors of D. Actually, for this system, there is no invariant region because there are no left eigenvectors for both D and M . This also indicates that a nonlinear parabolic system does not necessarily have an invariant region.

174

4.4

NONLINEAR EVOLUTION EQUATIONS

Bibliographic Comments

The monotone iterative method goes back to the early book [34] by L. Collatz. We refer to the paper [129] by D.H. Sattinger for the monotone iterative method with applications to nonlinear parabolic equations. See also the papers [3], [4], [5], [6] for the general monotone iteration in ordered Banach spaces with applications to nonlinear elliptic boundary value problems. The monotone iterative method, i.e., the upper solution and lower solution method has been extensively extended to deal with some elliptic and parabolic systems. See the book [117] by C.V. Pao for the comprehensive study of this topic. Concerning the comparison principle on the Cauchy problem for the linear wave equation in the case n ≤ 3, we refer to the paper [81] by J.B. Keller. As far as the invariant regions are concerned, we refer to the paper [33] by K. Chuen, C. Conley and J. Smoller and the paper [7] by H. Amann with applications to nonlinear parabolic equations. See also the corresponding part in the book [143] by J. Smoller. We also refer to the paper [152] by H.F. Weinberger, the paper [127] by R. Renheffer and W. Walter and the paper [24] by J.W. Bebernes and K. Schmitt.

Chapter 5 Global Solutions with Small Initial Data

5.1

Introduction

In the previous chapters we have introduced some useful methods to deal with the issue of global existence and uniqueness of the solution to nonlinear evolution equations with arbitrarily given initial data. However, for a given nonlinear evolution equation, there does not always exist a global solution for arbitrarily given initial data. To see this, in addition to the examples in Chapter 1, let us consider the following initial boundary value problem for a simple nonlinear parabolic equation:  ∂u   − ∆u = u3 − u,    ∂t  

(x, t) ∈ Ω × R+ ,

u|Γ = 0,       

(5.1.1)

u(0) = φ(x)

where Ω ⊂ Rn is a bounded domain with a smooth boundary Γ. In what follows we show that when initial data φ are sufficiently large, then the solution to problem (5.1.1) must blow up in a finite time. Indeed, multiplying the equation in (5.1.1) by u, then integrating over Ω, we get 1d ku(t)k2 + 2 dt

Z Ω

(|∇u|2 − |u|4 + |u|2 )dx = 0.

(5.1.2)

Throughout this chapter we always denote by k · k the norm in L2 . Let E(t) =

Z

1 (|∇u|2 − |u|4 + |u|2 )dx, 2 Ω

(5.1.3)

175

176

NONLINEAR EVOLUTION EQUATIONS

which is called the energy function. Then multiplying the equation in (5.1.1) by ut , and integrating over Ω yields 1 dE + kut k2 = 0. 2 dt This implies that E(t) ≤ E(0) =

Z

(5.1.4)

1 (|∇φ|2 − |φ|4 + |φ|2 )dx. 2 Ω

We infer from (5.1.2) that

Z

1 d 1 kuk2 + E(t) − 2 dt 2 If E(0) < 0, then

Ω

Z

1d 1 kuk2 ≥ 2 dt 2

Ω

|u|4 dx = 0.

|u|4 dx.

(5.1.5)

(5.1.6)

(5.1.7)

By the Cauchy-Schwartz inequality, we get Z Ω

|u|2 dx ≤

µZ

¶1

Ω

|u|4 dx

2

1

|Ω| 2

(5.1.8)

where |Ω| denotes the measure of Ω. Let E0 (t) =

Z

Ω

|u|2 dx.

(5.1.9)

Thus, combining (5.1.8) with (5.1.7) yields dE0 (t) ≥ C1 E02 (5.1.10) dt with C1 being a positive constant depending only on Ω. Then we can get E0 (t) ≥

E0 (0) 1 − C1 E0 (0)t

(5.1.11)

by solving differential inequality (5.1.10). This implies that when t → t0 with 1 t0 = , (5.1.12) C1 E0 (0) the solution u must blow up. Notice that the condition E(0) < 0 means that the initial data are sufficiently large. This can be seen from the following investigations. Consider that φ(x) = ku0 (x) with u0 (x) being given and k being positive constant. Then it follows from (5.1.5) that

Global Solutions with Small Initial Data

177

when k is sufficiently large, E(0) =

Z

1 (k 2 |∇u0 |2 − k 4 |u0 |4 + k 2 |u0 |2 )dx 2 Ω

(5.1.13)

must be negative. On the other hand, when initial data are small, as a consequence of the result in Section 3 of this chapter, problem (5.1.1) admits a unique global solution. This simple example indicates the need for considering global existence of solution to nonlinear evolution equations with small initial data. Investigations on the global existence with small initial data are not new topics. For instance, as early as in 1948, R. Bellman [27] considered global existence of the solution to a semilinear parabolic equation with small initial data. However, from an unpublished manuscript by T. Nishida in 1975, a systematic approach, which combines local existence and uniqueness result with uniform a priori estimates, has been developed in a series of papers. Since the middle of the 1960s, some interesting results on the Cauchy problems for some special nonlinear heat equations and nonlinear wave equations have emerged. For instance, in the papers [55], [56] by H. Fujita, the author considered the following initial value problem:  α+1 ,  ut − ∆u = u 

u|t=0 = ϕ(x),

(x, t) ∈ Rn × R+ ,

x ∈ Rn ,

(5.1.14)

2 , then for any smooth initial data ϕ ≥ 0, ϕ 6≡ 0, and proved that if α < n the solution u must blow up in finite time no matter how small ϕ is. 2 and initial On the other hand, he also proved global existence if α > n 2 data are small. Later on, the blow-up results for the case α = n were obtained by K. Kobayashi et al. in [84] and by F.B. Weissler in [153], respectively. It is interesting to ask why global existence or nonexistence is related to the space dimension n. Does this type of relation between n and global existence pursue general fully nonlinear heat equations with small initial data? It has been discovered that for the Cauchy problem, solutions to linear evolution equations including a linear heat equation, a linear wave equation or a linear Schr¨ odinger equation decay at a rate depending on the space dimension n. For instance, for the Cauchy problem of linear heat equation,   ut − ∆u = 0, 

u|t=0 = ϕ(x),

(x, t) ∈ Rn × R+ ,

(5.1.15)

178

NONLINEAR EVOLUTION EQUATIONS

the decay rate of solution as time goes to infinity depends on the space dimension n, nnamely, the L∞ norm and the L2 norm of solution decays −2 − n4 at the rate t , and t , respectively, as can be seen in the next section of this chapter. This explains why global existence or non-existence of solution to the Cauchy problem for nonlinear heat equation with small initial data depends on the space dimension n. Similar results hold for other linear evolution equations such as the linear wave equation, the linear Schr¨odinger equation, etc. For instance, for the Cauchy problem of linear wave equation:   utt − ∆u = 0, 

(x, t) ∈ Rn × R+ ,

u|t=0 = 0, ut |t=0 = ϕ(x),

(5.1.16)

it has been proved that although the L2 norm of solution u is conserved, n−1 the L∞ norm of solution u decays at a rate t− 2 provided that ϕ(x), say, is a smooth function with compact support. Since the end of the 1970s, study of global existence or blow-up in finite time for nonlinear evolution equations with small initial data has become a hot topic. S. Klainerman in the 1980s obtained the sharp result concerning the global existence of a smooth small solution to the Cauchy problem for fully nonlinear wave equations, showing that if the nonlinear term is of second order, then when n > 3, solution globally exists. Early examples by F. John and others show that for a particular nonlinear wave equation in three dimensions the solution blows up in a finite time. This indicates that the result by S. Klainerman is sharp regarding the space dimension. The sharp result for fully nonlinear heat equations was obtained in the paper [166] by S. Zheng and Y. Chen, and [159] by S. Zheng; see also the paper [120] by G. Ponce. We also refer to the monographs [164] by S. Zheng. In this chapter, we will present the results on the Cauchy problem for fully nonlinear heat equations mainly based on [164]. For the initial boundary value problem with the domain Ω being unbounded, i.e., the complement of another bounded domain in Rn , it has been discovered that the situation is very similar to the corresponding Cauchy problem, i.e., global existence or nonexistence of a solution has exactly the same relationship with the space dimension n as for the Cauchy problem. However, if an initial boundary value problem for nonlinear evolution equations in Ω × R+ with Ω being a bounded domain in Rn is concerned, then the situation is very different from the Cauchy problem regarding global existence or nonexistence of a small solution. The

Global Solutions with Small Initial Data

179

main reason is that the decay rate of the solution to the corresponding linearized equation generally does not depend on the space dimension. In contrast, the long time behavior of the solution to the linearized equation heavily depends on the first eigenvalue of the corresponding elliptic operator subject to certain boundary conditions. In this aspect, let us look at two simple examples. First, consider the initial Dirichlet boundary value problem for the heat equation:  ∂u   − ∆u = 0,    ∂t  

(x, t) ∈ Ω × R+ , (5.1.17)

u|Γ = 0,       

u(0) = φ(x)

where Ω is a bounded domain in Rn with the smooth boundary Γ. Let Φi (x) (i = 1, · · ·) be the normalized eigenfunction corresponding to the i-th eigenvalue λi with distribution 0 < λ1 < λ2 ≤ · · · ≤ λn ≤ · · · → +∞ :   −∆Φi = λi Φi , 

x ∈ Ω,

Φi |Γ = 0.

(5.1.18)

Then, it is easy to verify that the solution to problem (5.1.17) can be expressed by u(x, t) =

∞ X

Ai e−λi t Φi (x)

(5.1.19)

i=1

where Ai =

Z Ω

φ(x)Φi (x)dx.

(5.1.20)

Thus, as time goes to infinity, the solution u to problem (5.1.17) exponentially decays to zero, and the decay rate does not depend on the space dimension n. As will be shown in the second section of this chapter, the nonlinear heat equation with a nonlinear term being at least a second-order perturbation, a global small smooth solution exists for any space dimension n. Now let us look at the second example. Consider the initial Neumann

180

NONLINEAR EVOLUTION EQUATIONS

boundary value problem for the heat equation:  ∂u   − ∆u = 0,    ∂t    

(x, t) ∈ Ω × R+ ,

∂u

(5.1.21)

|Γ = 0,      ∂n    

u(0) = φ(x).

Again, by the separate variable method, the solution u to problem (5.1.21) can be expressed by u(x, t) =

∞ X

Ai e−λi t Ψi (x)

(5.1.22)

i=0

where λi and Ψi (x), i = (0, 1, · · ·) are the corresponding eigenvalues and normalized eigenfunctions to the following eigenproblem:  −∆Ψi = λi Ψi ,     ∂Ψi   |Γ = 0.

(5.1.23)

∂n

Notice the first eigenvalue λ0 of problem (5.1.23) is zero, Ψ0 (x) is a constant, 1 Ψ0 = p , |Ω| and

q

A0 |Ω| =

Z Ω

φ(x)dx.

(5.1.24)

Hence, the solution u to problem (5.1.21) does not converge to zero as time goes to infinity provided that Z

Ω

φ(x)dx 6= 0;

(5.1.25)

instead, the solution converges to 1 |Ω|

Z Ω

φ(x)dx

as t → +∞. This explains why for the nonlinear initial boundary

Global Solutions with Small Initial Data

181

problem,    ∂u − ∆u = u2 ,     ∂t   

(x, t) ∈ Ω × R+ ,

∂u

|Γ = 0,    ∂n      

(5.1.26)

u(0) = φ(x),

which has been investigated in Chapter 1, there is no global solution u for any initial datum φ(x) ≥ 0, φ 6≡ 0, no matter how small φ is. In the third section of this chapter we present global existence and uniqueness results on the initial boundary value problem for a fully nonlinear parabolic equation with small initial data. Of course, one of key ingredients is that solution to the corresponding linearized problem decays exponentially to zero as time goes to infinity. In the final section of this chapter we present some results on the nonexistence of global solutions to nonlinear parabolic equations as well as nonlinear hyperbolic equations with small initial data, based on the paper [167] by S. Zheng and Y. Chen.

5.2

IVP for Fully Nonlinear Parabolic Equations

In this section we consider the following initial value problem for fully nonlinear parabolic equations:  2  ut − ∆u = F (u, Du, D u), 

u|t=0 = ϕ(x)

(5.2.1)

where x = (x1 , · · · , xn ) ∈ Rn , t ∈ R+ , Du = (ux1 , · · · , uxn ), D2 u = (uxi xj ), (i, j = 1, · · · , n) and F, ϕ are smooth functions. The basic assumption on F is F (ω) = F (ω0 , ωi , ωij ) = O(|ω|α+1 ), near ω = 0

(5.2.2)

with α ≥ 1 being an integer. We first consider the decay rate of the solution to the Cauchy problem for the linear heat equation.

182

5.2.1

NONLINEAR EVOLUTION EQUATIONS

Decay Rate of the Solution to the Cauchy Problem for the Linear Heat Equation

Consider the following initial value problem for the heat equation   ut − ∆u = 0, 

x ∈ Rn , t > 0,

(5.2.3)

x ∈ Rn .

u|t=0 = ϕ(x),

It is well known that for any ϕ ∈ Lp (Rn ), 1 ≤ p ≤ ∞, the function u(x, t) given by the following Poisson formula: u(x, t) = (4πt)

− n2

Z

Rn

−|x−y|2 ∆ e 4t ϕ(y)dy =

Z

Rn

K(x − y, t)ϕ(y)dy (5.2.4)

is C ∞ in x ∈ Rn , t > 0, and satisfies the heat equation. Moreover, we have the following lemma. LEMMA 5.2.1 Suppose ϕ ∈ Lp (Rn ), 1 ≤ p ≤ ∞. Then we have for t > 0

ku(t)kLp ≤ kϕkLp ,

(5.2.5)

n Ct−( 2r

kDk u(t)kLq ≤

+

k 2 ) kϕkLp

(5.2.6)

1 1 with 1 q = p − r , 1 ≤ r, q ≤ ∞ and C being a positive constant depending only on p, q, r and k . Hereafter D k denotes the k -th order derivatives with respect to x. Proof. We notice that K(x − y, t) = K(y − x, t) and Z

Rn

K(x − y, t)dy =

Z

Rn

K(x − y, t)dx = 1.

(5.2.7)

1 + 1 = 1, By the H¨older inequality, we have for p p0 |u(x, t)| ≤ µZ

≤

R

µZ

= Thus

R

Z Rn

Z

Rn

K(x − y, t)|ϕ(y)|dy ¶ 1 µZ

p

n

n

p

K(x − y, t)|ϕ(y)| dy K(x − y, t)|ϕ|p dy

p

|u(x, t)| dx ≤

Z Rxn

R

¶1

p

Z Ryn

n

K(x − y, t)dy

¶ 10

.

K(x − y, t)|ϕ(y)|p dydx

p

(5.2.8)

Global Solutions with Small Initial Data =

ÃZ

Z Ryn

183

!

Rxn

K(x − y, t)dx |ϕ(y)|p dy = kϕkpLp

(5.2.9)

and (5.2.5) follows. To prove (5.2.6), we use the following well-known Young inequality (see [126]): kf ∗ gkLq = k

Z

Rn

f (x − y)g(y)dykLq ≤ kf kLr0 kgkLp

(5.2.10)

1 + 1 = 1 + 1, 1 ≤ p, q, r0 ≤ ∞. A straightforward calculation with p q r0 easily shows that µZ

kK(t)kLr0 ≤ C − n2 (1− r10 )

≤ Ct

1 Rn

= Ct

t

nr 0 2

e−

|x|2 r 0 4t

¶ 10

dx

r

n − 2r

(5.2.11)

with 1r + 10 = 1. Similarly, we have r kDxi K(t)kLr0 ≤ C n

ÃZ

µ

1

Rn

t

nr 0 2

|xi | t

¶r0

e

2 0

− |x|4tr

1

≤ Ct− 2r − 2

! 10

dx

r

(5.2.12)

and, in general, n

k

kDk K(t)kLr0 ≤ Ct−( 2r + 2 ) .

(5.2.13)

Combining (5.2.13) with the Young inequality yields (5.2.6). Thus the proof is completed. ¤ LEMMA 5.2.2 Let ϕ(x) be a smooth function with bounded norms appearing below and let k be a positive integer. Then u(x, t) given by (5.2.4) satisfies the following estimates: for t > 0, k

kDk ukL1 ≤ Ct− 2 kϕkL1 , − n+2k 4

k

kD uk ≤ Ct

kDk ukL∞ ≤ Ct−

kϕkL1 ,

n+k 2

kϕkL1 ,

(5.2.14) (5.2.15) (5.2.16)

and for t ≥ 0, k

kDk ukL1 ≤ C(1 + t)− 2 (kϕkL1 + kDk ϕkL1 ), kDk uk ≤ C(1 + t)− k

kD uk ≤ C(1 + t)

−

(5.2.17)

n+2k 4

(kϕkL1 + kDk ϕk),

(5.2.18)

n+2k 4

kϕkW n+k,1 ,

(5.2.19)

184

NONLINEAR EVOLUTION EQUATIONS k

kDk uk ≤ C(1 + t)− 2 kϕkH k , − n+k 2

k

kD ukL∞ ≤ C(1 + t)

(5.2.20) k

(kϕkL1 + kD ϕkH [ n2 ]+1 ),

(5.2.21)

kϕkW k+n,1 ,

(5.2.22)

kDk ukL∞ ≤ C(1 + t)−

n+k 2

kDk ukL∞ ≤ C(1 + t)−

n+2k 4

kϕkH k+[ n2 ]+1 .

(5.2.23)

Proof. (5.2.14)–(5.2.16) directly follow from (5.2.6) and (5.2.17)– (5.2.23) follow from (5.2.5), (5.2.6) and the Sobolev imbedding theorem. ¤ The following lemma is concerned with the weighted norm estimate for the solution to problem (5.2.3). LEMMA 5.2.3 Suppose that ϕ ∈ H s (Rn ) with s being a positive integer. Then problem (5.2.3) admits a unique solution u such that u ∈ C([0, +∞), H s ), Dx u ∈ L2 ([0, +∞), H s ). Moreover, for all t > 0, the following estimate holds: s X

k

k

2

(1 + t )kD u(t)k +

k=0

s Z t X k=0 0

(1 + τ k )kDk+1 uk2 dτ ≤ Ckϕk2H s (5.2.24)

where C is a positive constant independent of u and t.

Proof. Differentiating the heat equation by Dk (k = 0, 1, · · · , s), then multiplying the resultant by Dk u, and integrating over Rn yields 1d kDk u(t)k2 + kDk+1 u(t)k2 = 0. 2 dt

(5.2.25)

Multiplying (5.2.25) by δ k tk , we obtain δk d k k 2 δ k ktk−1 (t kD uk ) − kDk uk2 + δ k tk kDk+1 uk2 = 0. 2 dt 2

(5.2.26)

Summing (5.2.26) up from k = 0 to k = s, choosing δ to be a positive 1 constant such that δ < , then integrating with respect to t, we get s (5.2.24). Thus, the proof is complete. ¤

5.2.2

Global Solution to the Cauchy Problem for Fully Nonlinear Parabolic Equations

The main result in this section is the following theorem:

Global Solutions with Small Initial Data

185

THEOREM 5.2.1

2 . Suppose that Let α and s be integers such that s ≥ n + 4, α > n s+1 F ∈C and F satisfies (5.2.2). Then the Cauchy problem (5.2.1) admits a unique global smooth solution u such that u ∈ C([0, ∞); H s ∩ L1 ) ∩ C 1 ([0, ∞); H s−2 ) ∩ C((0, +∞); W 2,1 ), Du ∈ L2 (R+ ; H s ), provided that kϕkH s ∩L1 = kϕkH s + kϕkL1 is sufficiently small. Moreover, for t > 0,

kDk u(t)k ≤ C(1 + t)−

n+2k 4

kD u(t)kL∞ ≤ C(1 + t) k

kD u(t)kL1 ≤ Ct j

, k = 0, 1, 2,

− n+k 2

k

− k2

, k = 0, 1, 2,

, k = 0, 1, 2, − 2j

kD u(t)k ≤ C(1 + t)

, j = 3, · · · , s.

(5.2.27) (5.2.28) (5.2.29) (5.2.30)

The standard method for proving global existence and uniqueness is to combine a local existence and uniqueness result with uniform a priori estimates. To prove a local existence result, we usually choose a suitable class of functions to which the solution belongs, then plugs an element, i.e., a function v from a closed set of this class into the nonlinear term, say, F for problem (5.2.1), and find a solution u to the auxiliary linear problem. Finally, the solution to the original problem is proved to exist locally in time by showing that the nonlinear mapping from v to u is a mapping from the closed set into itself, and it is a contraction if the time interval is sufficiently small. Obtaining uniform a priori estimates is usually a separate step in the proof of global existence since the special structure of nonlinear problem has to be fully exploited. However, it was first discovered in the papers [162], [163] by S. Zheng and later on used by others (see [93] and the references cited there) that for the proof of global existence of a small solution the two steps described above can be simplified into one step, namely, the globally iterative method in which the linearized problem is solved in any finite time interval [0, T ], and the nonlinear mapping is also considered from function v defined on [0, T ] to function u defined on the same time interval. In contrast to the usual procedure, the contraction of nonlinear mapping is achieved not by the smallness of time interval, but by the smallness of functions u and v. Now let us prepare for the proof of Theorem 5.2.1 by introducing some results concerning estimates of composite functions, the so-called “calculus of inequalities” (see [82], [93], and [120]).

186

NONLINEAR EVOLUTION EQUATIONS

LEMMA 5.2.4 Suppose that

1 1 1 = + , 1 ≤ r, p, q ≤ ∞ r p q

(5.2.31)

and suppose that all the norms appearing below are bounded. Then for any given integer s ≥ 0 we have

kDs (f g)kLr ≤ Cs (kf kLp kDs gkLq + kDs f kLp kgkLq ) , (5.2.32) kDs (f g)kLr ≤ Cs (kf kLp kDs gkLq + kDs f kLq kgkLp ) . (5.2.33) For s ≥ 1, we have

kDs (f g) − f (Ds g)kLr

´

³

≤ Cs kDf kLp kDs−1 gkLq + kDs f kLp kgkLq , s

s

kD (f g) − f (D g)kLr ³

(5.2.34)

´

≤ Cs kDf kLp kDs−1 gkLq + kDs f kLq kgkLp . Proof. By the expression Ds (f g) =

X

(5.2.35)

Dl f Dk g

(5.2.36)

l+k=s

and the H¨older inequality, we have kDs (f g)kLr ≤ Cs

X

kDl f kLp kDk gkLq .

(5.2.37)

l+k=s

It follows from the Gagliardo–Nirenberg inequality that l

s−l

k s

s−k s q

kDl f kLp ≤ CkDs f kLs p kf kLsp kDk gkLq ≤ CkDs gkLq kgkL

(5.2.38) s−l s q

l s

= CkDs gkL kgkLq .

(5.2.39)

Thus, we infer from (5.2.39), (5.2.38) and the Young inequality that s−l

l

l

s−l

kDl f kLp kDk gkLq ≤ CkDs f kLs p kgkLs q kDs gkLsq kf kLsp ≤ C (kDs f kLp kgkLq + kDs gkLq kf kLp ) .

(5.2.40)

Combining (5.2.40) with (5.2.37) yields (5.2.32). To prove (5.2.34), it suffices to apply the Gagliardo–Nirenberg inequality for 1 ≤ l < s, 1 < k < s, l + k = s s−l

l−1

s−1 kDl f kLp ≤ CkDs f kLs−1 p kDf kLp , k s−1 q

s−k−1 s−1 q

kDk gkLq ≤ CkDs−1 gkL kgkL

,

(5.2.41) (5.2.42)

Global Solutions with Small Initial Data

187

then we use the Young inequality to obtain the desired conclusion. To prove (5.2.33) and (5.2.35), we first use the H¨older inequality kDl f Dk gkLr ≤ kDl f kLr1 kDk gkLr2

(5.2.43)

with r1 , r2 being determined by the following formula: µ

¶

1 l 1 l = 1− + , r1 s p sq µ ¶ 1 k 1 k = 1− + . r2 s p sq

(5.2.44) (5.2.45)

Then we apply the Gagliardo–Nirenberg inequality: l

k

k s

l s

kDl f kLr1 ≤ CkDs f kLs q kf kLs p ,

(5.2.46)

kDk gkLr2 ≤ CkDs gkLq kgkLp .

(5.2.47)

Finally, we deduce (5.2.33) and (5.2.35) by applying the Young inequality. The proof is complete. ¤ LEMMA 5.2.5 Suppose that F (w) : Rm 7→ R is a smooth function satisfying

F (0) = 0.

(5.2.48)

For any integer s ≥ 0, if a vector function w ∈ W s,p (Rm ), 1 ≤ p ≤ +∞, and

kwkL∞ ≤ ν0 , then F (w) ∈

W s,p (Rm ),

(5.2.49)

and the following estimate holds:

kF (w)kW s,p ≤ C(ν0 )kwkW s,p

(5.2.50)

where C(ν0 ) is a positive constant depending only on ν0 .

Proof. For simplicity of exposition, we give the proof below only for the case that w is a scalar function. By the chain rule for composite functions, we have Ds F (w) =

X ∂ρF

∂wρ 1≤ρ≤s

(Dw)α1 (D2 w)α2 · · · (Ds w)αs

(5.2.51)

where D, as before, denotes derivative with respect to x, and α1 + · · · + αs = ρ,

s X i=1

iαi = s.

(5.2.52) (5.2.53)

188

NONLINEAR EVOLUTION EQUATIONS

Let pi be such that

ps . i = 1, · · · , s iαi

pi =

(5.2.54)

Then by the H¨older inequality, we have kDs F (w)kLp ≤ C(ν0 )

X

k(Dw)α1 kLp1 · · · k(Ds w)αs kLps

1≤ρ≤s

≤ C(ν0 )

X

s Y

kDi wkαisp . L

1≤ρ≤s i=1

(5.2.55)

i

Finally we deduce (5.2.50) by applying the Gagliardo–Nirenberg inequality: s−i

i

kDi wkL spi ≤ CkDs wkLs p kwkLs∞ .

(5.2.56)

Thus, the lemma is proved.

¤

LEMMA 5.2.6 Suppose that F (w) : Rm 7→ R is a smooth function satisfying

F (w) = O(|w|1+α ),

near w = 0

(5.2.57)

where α ≥ 1 is an integer. Suppose that for a given vector function w(x),

kwkL∞ ≤ ν0 ≤ 1

(5.2.58)

and all the norms appearing below are bounded. Then for any given integer s ≥ 0,

kDs F (w)kLp ≤ Cs kwkαL∞ kDs wkLp ,

(5.2.59)

kF (w)kW s,r ≤ Cs kwkW s,q

(5.2.60)

α Y

kwkLpi ,

i=1

where p ≥ 1 and r, pi and q satisfy α 1 1 1 X = + , 1 ≤ pi , q, r ≤ ∞ r p q i=1 i

(5.2.61)

and Cs is a positive constant depending on s and ν0 . In particular, we have

kF (w)kW s,r ≤ Cs kwkW s,q kwkLp kwkα−1 L∞

(5.2.62)

with r, p, q satisfying (5.2.31).

Proof. For simplicity of exposition, we still assume without loss of generality that w is a scalar function. By assumption (5.2.57), F can

Global Solutions with Small Initial Data

189

be written as F (w) = H(w)w1+α

(5.2.63)

with H(w) 6= 0. We infer from (5.2.33) in Lemma 5.2.4 and Lemma 5.2.5 that kF (w)kW s,r ≤ Cs (kH(w)wkLp kwα kW s,q + kH(w)wkW s,q kwα kLp ) ³

´

≤ Cs kwkLp kwα kW s,q + kwkW s,q kwkα−1 L∞ kwkLp .

(5.2.64)

Repeatedly applying (5.2.32) in Lemma 5.2.4 with r = q, p = ∞, we get ³

kwα kW s,q ≤ Cs kwα−1 kW s,q kwkL∞ + kwkW s,q kwα−1 kL∞ ≤ CkwkW s,q kwkα−1 L∞ .

´

(5.2.65)

Inserting (5.2.65) into (5.2.64) yields (5.2.62). Thus the proof is complete. ¤ In the following, we often need to estimate the integral J=

Z t 0

(1 + t − τ )−β (1 + τ )−γ dτ

(5.2.66)

with positive constants β, γ. Notice that if we make a change in variables: σ = t − τ , then J=

Z t 0

(1 + σ)−β (1 + t − σ)−γ dσ.

(5.2.67)

Therefore, without loss of generality, we can assume that β ≥ γ > 0. We now have LEMMA 5.2.7 Suppose that β > 1, β ≥ γ > 0. Then there is a positive constant C such that for all t ≥ 0,

J ≤ C(1 + t)−γ .

(5.2.68)

Proof. A straightforward calculation gives J=

Z

0

t 2

(1 + t − τ )−β (1 + τ )−γ dτ + −β

≤ C(1 + t)

Z 0

t 2

−γ

(1 + τ )

Z t t 2

(1 + t − τ )−β (1 + τ )−γ dτ −γ

dτ + C(1 + t)

Z t t 2

(1 + t − τ )−β dτ. (5.2.69)

190

NONLINEAR EVOLUTION EQUATIONS

Thus (5.2.68) easily follows from (5.2.69) and the assumption on β and γ. ¤ After these preparations we are now in a position to prove Theorem 5.2.1. Proof of Theorem 5.2.1. In the following we use the globally iterative method to prove Theorem 5.2.1. Let T > 0 be an arbitrary positive number and let s be an integer such that s ≥ n + 4. We introduce:

¯ ¾ ¯ v ∈ C([0, T ]; H s ∩L1 )∩L2 ([0, T ]; H s+1 )∩C((0, T ]; W 2,1 ) ¯ XT = v(x, t) ¯ kvkX < ∞ ½

T

(5.2.70)

equipped with the weighted norm: kvkXT = sup

2 X

0
+ max

0≤t≤T

+

s X

Ã 2 X³

k

t 2 kDk v(t)kL1

(1 + t)

n+2k 4

k=0 k

(1 + t) 2 kDk v(t)k +

kDk v(t)k + (1 + t) ÃZ

t s+1 X

n+k 2

kDk v(t)kL∞

´

! 12  (1 + τ )k−1 kDk v(τ )k2 dτ  .

0 k=1

k=3

(5.2.71)

Then it is easy to see that XT is a Banach space. For 0 < ε < 1, we denote by ST,ε a closed convex subset of XT : ST,ε = {v(x, t) |v(x, t) ∈ XT , v|t=0 = ϕ(x), kvkXT ≤ ε } . R

(5.2.72)

Since the function v given by v = Rn K(x − y, t)ϕ(y)dy belongs to XT and v|t=0 = ϕ, it is easy to see that for any given T and ε, ST,ε is nonempty provided that ϕ is sufficiently small in H s ∩ L1 . For any v ∈ ST,ε , we consider the following auxiliary linear problem:   ut − ∆u = F (Λv), 

u|t=0 = ϕ(x)

(x, t) ∈ Rn × R+ ,

(5.2.73)

where we simply denote Λv = (v, Dv, D2 v). The unique classical solution u of problem (5.2.73) is given by u=

Z

Rn

K(x − y, t)ϕ(y)dy +

Z tZ 0

Rn

K(x − y, t − τ )F (Λv(y, τ ))dydτ (5.2.74)

Global Solutions with Small Initial Data

191

where K(x−y, t) is the fundamental solution to the heat equation given in (5.2.4). We now prove that u also belongs to XT . Indeed, it is easy to see from Lemmas 5.2.1–5.2.3 that the function uI given by uI =

Z

K(x − y, t)ϕ(y)dy

Rn

(5.2.75)

satisfies the homogeneous heat equation and belongs to XT , and the estimate kuI kXT ≤ CkϕkH s ∩L1

(5.2.76)

holds. From now on we denote by C a positive constant independent of T, u and v, which may vary in different places. By the Duhamel principle, the function uII given by Z tZ

uII =

0

Rn

K(x − y, t − τ )F (Λv(y, τ )) dy dτ

(5.2.77)

satisfies the nonhomogeneous heat equation with homogeneous initial data. By v ∈ XT and the assumption on F we can conclude that F (Λv) ∈ C([0, T ]; H s−2 ∩W s−2,1 )∩L2 ([0, T ]; H s−1 )∩L1 ([0, T ]; W s−1,1 ). It turns out that we have the following estimates on uII : (i) Energy norm estimates. For 0 ≤ k ≤ 2 and t > 0, it follows from (5.2.19), (5.2.20) and Lemma 5.2.6 that kDk uII (t)k ≤ C +C

Z t

≤C +C

t 2

Z

Z

t 2

0

n+2k 4

kF kW k+n,1 dτ

k

(1 + t − τ )− 2 kF kH k dτ t 2

0

Z t t 2

(1 + t − τ )−

n+2k 4

kΛvkα−1 L∞ kΛvk kΛvkH k+n dτ

k

(1 + t − τ )− 2 kΛvkαL∞ kΛvkH k dτ.

Noticing that



kΛv(t)kH k+n ≤ C  ³

(1 + t − τ )−

2 X

j=0

n

kDj v(t)k + ´ 3

≤ Cε (1 + t)− 4 + (1 + t)− 2 ,

k+n+2 X

(5.2.78) 

kDj v(t)k

j=3

(5.2.79)

192

NONLINEAR EVOLUTION EQUATIONS

we deduce from (5.2.78) that k

kD uII (t)k ≤ Cε +(1 + τ )−( + Cε ≤ Cε

α+1

Z t t 2

α+1

+ Cε

n(α−1) 2

α+1

+ n4 + 32 )

´

(1 + t − τ )−

n+2k 4

k

Z

t 2

³

³

0

(1 + t)

³

(1 + τ )−

nα 2

dτ

(1 + t − τ )− 2 (1 + τ )−( − n+2k 4

³

t 2

0

(1 + t)

α+1

Z

+ n4 ) −( nα 2

(1 + τ )−

nα 2

nα 2

+ n4 )

+ (1 + τ )−(

+ (1 + τ )−(

−( nα + 23 ) 2

+ (1 + t)

´Z t t 2

n(α−1) 2

nα 2

+ 32 )

´

+ n4 + 32 )

´

dτ dτ

k

(1 + t − τ )− 2 dτ. (5.2.80)

Since 0 ≤ k ≤ 2, nα > 2, we can deduce from (5.2.80) that kDk uII (t)k ≤ Cεα+1 (1 + t)−

n+2k 4

.

(5.2.81)

In the same manner as we derived the energy estimates for uI , we get the following energy estimates for the solution uII to the nonhomogeneous equation: s X

(1 + tk )kDk uII (t)k2 +

k=0

≤C ≤C +

ÃZ

t

0

µZ t 0

s−1 XZ t k=0 0

µ

kuII k kF kdτ +

k=0 0

s−1 XZ t k=0 0

(1 + τ k )kDk+1 uII k2 dτ !

(1 + τ

k+1

!

(1 + τ Z t 0

k+1

k 2 )kΛvk2α L∞ kD Λvk dτ

kuII k(1 + τ )−(

s−2 XZ t k=0 0

nα 2

+ n4 )

dτ n

k

2

)kD F k dτ

kuII k kΛvkαL∞ kΛvkdτ

≤ C εα+1 +ε2(α+1)

s Z t X

(1 + τ k+1 )(1 + τ )−nα−k− 2 dτ

Global Solutions with Small Initial Data Z t

+ε2α

Ã

0

(1 + τ s )(1 + τ )−nα kDs+1 vk2 dτ

193

¶

! n 4

α+1

≤C ε

2(α+1)

sup (1 + τ ) kuII k + ε

0≤τ ≤t

≤ Cε2(α+1) .

(5.2.82)

Again, in the above Lemmas 5.2.5–5.2.7 have been used. (ii) L∞ norm estimates. For 0 ≤ k ≤ 2 and t > 0, it follows from Lemma 5.2.2 and Lemma 5.2.5 that kDk uII kL∞ ≤C

Z

(1 + t − τ )−

0

+C ≤C

t 2

Z t

ÃZ

+ ≤ Cε +

t 2

t 2

n

(1 + t − τ )−

n+k 2

kΛvkα−1 L∞ kΛvk kΛvkH n+k dτ !

n 4

(1 + t − τ )− kΛvkαL∞ kDk ΛvkH n dτ

α+1

ÃZ 0

Z t t 2

kF kW n+k,1 dτ

(1 + t − τ )− 4 kDk F kH n dτ

t 2

0

Z t

n+k 2

t 2

(1 + t − τ )− − n4

(1 + t − τ )

≤ Cεα+1 (1 + t)−

n+k 2

³

n+k 2

³

(1 + τ )−

nα 2

+ (1 + τ )−(

−( nα + n+2k ) 2 4

(1 + τ )

+ n4 + 32 )

n(α−1) 2

−( nα + 23 ) 2

+ (1 + τ )

.

´

´

dτ

!

dτ (5.2.83)

Therefore, sup (1 + t)

0≤t≤T

n+k 2

kDk uII kL∞ ≤ Cεα+1 , 0 ≤ k ≤ 2.

(iii) L1 norm estimates. By Lemma 5.2.5 and Lemma 5.2.2, we have for 0 ≤ k ≤ 2 : kDk uII (t)kL1

(5.2.84)

194 ≤

NONLINEAR EVOLUTION EQUATIONS Z t 0

≤C

k

(1 + t − τ )− 2 kF kW k,1 dτ

Z t 0

k

α−1 (1 + t − τ )− 2 kΛvkH k kΛvkkΛvkL ∞ dτ

≤ Cεα+1

Z t 0

k

³

(1 + t − τ )− 2 (1 + τ )−

nα 2

+ (1 + τ )−(

n(α−1) 2

+ n4 + 32 )

´

dτ.

(5.2.85)

Applying Lemma 5.2.7 to (5.2.85) yields k kDk uII (t)kL1 ≤ C(1 + t)− 2 εα+1 , 0 ≤ k ≤ 2, t ≥ 0.

(5.2.86)

Combining the results obtained in the previous three steps with the estimate (5.2.76) on the solution uI to the homogeneous equation yields ³

´

kukXT ≤ C1 kϕkH s ∩L1 + εα+1 . Thus if we take

µ

(5.2.87)

¶

1 1 ε0 = min 1, ( )α , 2C1

(5.2.88)

then when ε ≤ ε0 and kϕkH s ∩L1 ≤

ε , 2C1

(5.2.89)

we have kukXT ≤ ε.

(5.2.90)

For any given T > 0, the auxiliary linear problem (5.2.73) defines a nonlinear operator N : v ∈ ST,ε 7→ u ∈ ST,ε . In what follows we prove that N is a contraction provided that ε is suitably small. Indeed, for any v1 , v2 ∈ ST,ε , we denote u1 = N v1 , u2 = N v2 , v = v1 − v2 and u = u1 − u2 . Then u satisfies   ut − ∆u = F (Λv1 ) − F (Λv2 ), 

u|t=0 = 0.

We can write F (Λv1 ) − F (Λv2 ) =

Z 1 0

F 0 (σΛv1 + (1 − σ)Λv2 )Λvdσ

(5.2.91)

(5.2.92)

Global Solutions with Small Initial Data

195

with n X

F 0 Λv = Fω v +

Fωi vxi +

i=1

n X

Fωij vxi xj .

(5.2.93)

i,j=1

In the same manner as above, we can obtain the energy norm estimates, L∞ norm estimates and L1 norm estimates again. (i) Energy norm estimates. For 0 ≤ k ≤ 2, t ≥ 0, Z

k

kD u(t)k ≤ C +C

Z t t 2

≤C +C

(1 + t − τ )−

0

t 2

n+2k 4

kF (Λv1 ) − F (Λv2 )kW k+n,1 dτ

k

n+2k 4

¡

¢

kF 0 k kΛvkH k+n + kΛvk kF 0 kH k+n dτ

¡

k

¢

(1 + t − τ )− 2 kF 0 kL∞ kΛvkH k + kΛvkL∞ kF 0 kH k dτ Z

α

t 2

0

(1 + t − τ )−

+(1 + τ )−( +Cε

0

t 2

Z t

α

(1 + t − τ )−

(1 + t − τ )− 2 kF (Λv1 ) − F (Λv2 )kH k dτ

Z

≤ Cε

t 2

Z t t 2

³

n(α−1) 2

+ n4 )

³

n+2k 4

(1 + τ )−(

kΛvk + (1 + τ )−( ³

k

(1 + t − τ )− 2 (1 + τ )−

+ (1 + τ )−(

+ 32 )

n(α−1) 2

n(α−1) 2

nα 2

+ (1 + τ )−(

³

+ n4 )

n(α−1) 2

kΛvkH k+n

+ 32 )

´

kΛvk dτ ´

kΛvk + kD3 vk + kD4 vk

n(α−1) 2

+ n4 )

´

´

kΛvkL∞ dτ



≤ Cεα (1 + t)− ×

ÃZ

+Cε

t 2

0 α

n+2k 4

(1 + τ )−

³

n 4

sup (1 + τ ) kΛvk + (1 + τ )

k+n+2 X

0≤τ ≤t nα 2

dτ +

+ n4 ) −( nα 2

(1 + t)

3 2

Z

t 2

0

(1 + τ )−(

n(α−1) 2

−( nα + 32 ) 2

+ (1 + t)

+ n4 + 32 )

´

!



kDj vk

j=3

dτ

Ã n

sup (1 + τ ) 4 kΛv(τ )k

0≤τ ≤t

196

NONLINEAR EVOLUTION EQUATIONS + sup (1 + τ )

4 X

3 2

0≤τ ≤t

×

Z t t 2



n 2

kDj v(τ )k + sup (1 + τ ) kΛv(τ )kL∞  0≤τ ≤t

j=3 k

(1 + t − τ )− 2 dτ

≤ Cεα (1 + t)−

n+2k 4

kvkXT .

(5.2.94)

Therefore, sup

2 X

(1 + t)

n+2k 4

0≤t≤T k=0

kDk u(t)k ≤ Cεα kvkXT ,

(5.2.95)

In the same manner as before, we get s X

(1 + tk )kDk u(t)k2 +

k=0

≤C

µZ t 0

+ ≤C

k=0 0

+

0

k=0 0

(1 + τ k )kDk+1 uk2 dτ

kuk kF (Λv1 ) − F (Λv2 )k dτ

s−1 XZ t

µZ t

s Z t X

!

(1 + τ

k+1

k

2

)kD (F (Λv1 ) − F (Λv2 ))k dτ

kuk kF 0 kL∞ kΛvk dτ

s−1 XZ t k=0 0

(1 + τ

k+1

³

k

2

) kD Λvk

kF 0 k2L∞

≤ Cε sup (1 + τ ) ku(τ )k sup (1 + τ ) kΛv(τ )k s−1 XZ t k=0 0

0≤τ ≤t

³

kDk Λv1 k2 + kDk Λv2 k2

≤ Cε

kvk2XT

0

kΛvk2L∞

´

(1 + τ )−

!

dτ

n(α+1) 2

dτ

(1 + τ k+1 )(1 + τ )−nα kDk vk2H 2 dτ

+ C sup (1 + τ )n kΛv(τ )k2L∞

2α

Z t

0≤τ ≤t

0≤τ ≤t

+ Cε2α

0 2

+ kD F k

n 4

n 4

α

k

+Cε

2α

ÃZ t 0

´³

s−1 XZ t k=0 0

(1 + τ k+1 )(1 + τ )−n

2(α−1)

kΛv1 kL∞

kvk2H 2 dτ +

2(α−1)

+ kΛv2 kL∞

s−1 XZ t k=1 0

´

dτ !

k−1

(1 + τ )

kD

k

vk2H 2 dτ

Global Solutions with Small Initial Data

197

+ Cε2α sup (1 + τ )n kΛvk2L∞ 0≤τ ≤t

≤ Cε2α kvk2XT .

(5.2.96)

(ii) L∞ norm estimates. For 0 ≤ k ≤ 2, we have k

kD u(t)kL∞ ≤ C +C ≤C

Z t t 2

ÃZ

+

Z t t 2

0

(1 + t − τ )−

≤ C(1 + t) ³

0

+ Cε

ε

n+k 2

³

n 4

ÃZ

¡

kF (Λv1 ) − F (Λv2 )kW n+k,1 dτ

¢

kF 0 k kΛvkH n+k + kΛvk kF 0 kH n+k dτ ´

t 2

0

(1 + τ )−(

−( n(α−1) + n4 ) 2

(1 + t)−(

nα 2

n

nα 2

+ 32 )

0≤τ ≤t

n 2

+(1 + τ ) kΛv(τ )kL∞ ≤ C(1 + t)−

n+k 2

³

³

sup

0≤τ ≤t

´

sup

n(α−1) 2

+ n4 )

+ (1 + τ )

+ n+2k ) 4

+(1 + τ ) 2 kΛv(τ )kL∞ +(1 + t)−(

kΛvkH n+k dτ

−( n(α−1) + 32 ) 2

(1 + τ )

n+2k 4

´

!

kΛvk dτ

kDk v(τ )kH 2−k

3

(1 + τ ) 2 kD3 v(τ )kH n+k−1

´´ Z t t 2

n

(1 + t − τ )− 4 dτ

εα kvkXT .

(5.2.97)

(iii) L1 norm estimates. For 0 ≤ k ≤ 2, t > 0, we have kDk u(t)kL1 ≤ C

Z t 0

!

kF 0 kL∞ kDk ΛvkH n + kΛvkL∞ kDk F 0 kH n dτ

(1 + τ )

Ã α

n+k 2

n

(1 + t − τ )−

t 2

(1 + t − τ )−

(1 + t − τ )− 4 kDk (F (Λv1 ) − F (Λv2 ))kH n dτ

− n+k α 2

+

t 2

t 2

0

Z

Z

k

(1 + t − τ )− 2 kF (Λv1 ) − F (Λv2 )kW k,1 dτ

198

NONLINEAR EVOLUTION EQUATIONS

≤C

Z t

≤ Cε

0 α

k

Z t 0

+ Cεα

³

(1 + t − τ )− 2 (1 + τ )−( k

+(1 + τ )−(

α

¢

¡

(1 + t − τ )− 2 kF 0 kH k kΛvk + kΛvkH k kF 0 k dτ

Z t 0

n(α−1) 2

+ 32 )

´

n(α−1) 2

+ n+2k ) 4

kΛvk dτ

(1 + t − τ )− 2 (1 + τ )−( k

n(α−1) 2

+ n4 )

kΛvkH k dτ

Ã − k2

≤ Cε (1 + t)

! n 4

3 2

3

sup (1 + τ ) kΛv(τ )k + sup (1 + τ ) kD v(τ )kH 1

0≤τ ≤t

0≤τ ≤t

k

≤ Cεα (1 + t)− 2 kvkXT .

(5.2.98)

Combining (5.2.95)–(5.2.98) yields kukXT ≤ Cεα kvkXT .

(5.2.99)

Since the constant C appearing in (5.2.99) is independent of ε, T, u and v, we can take ε small enough, if necessary, so that 1 Cεα ≤ . 2

(5.2.100)

Thus the nonlinear operator N is a contraction from ST,ε to ST,ε . It turns out that by the contraction mapping theorem, it has a unique fixed point u ∈ ST,ε . Since T is arbitrary, it is the unique global solution to the Cauchy problem (5.2.1). The decay rates (5.2.27)–(5.2.30) are deduced from the definition of ST,ε . Thus the proof of Theorem 5.2.1 is complete. ¤ that the usual energy method will produce good terms R t It isk clear 2 dτ with 1 ≤ k ≤ s + 1, and when F depends only on derivakD uk 0 R R R R tives of u, the terms 0t R uF dxdτ and 0t R Dk+1 uDk−1 F dxdτ (1 ≤ Rt k 2

k ≤ s) can be bounded by 0 kD uk dτ, (1 ≤ k ≤ s) provided that u is small. Based on this observation, S. Zheng in [159] further proved that if F does not explicitly involve u, then for any n ≥ 1 and any ϕ small only in H s , the Cauchy problem (5.2.1) admits a unique global smooth solution. More precisely, we have the following theorem.

Global Solutions with Small Initial Data

199

THEOREM 5.2.2 Suppose that (i) F = F (Du, D2 u) does not involve u explicitly; (ii) F satisfies (5.2.2) with α ≥ 1; (iii) F ∈ C s+1 with an integer s > n 2 + 3, ϕ(x) ∈ s n H (R ). Then for any n ≥ 1, the Cauchy problem (5.2.1) admits a unique global smooth solution u such that

u ∈ C([0, ∞); H s ) ∩ C 1 ([0, ∞); H s−2 ),

Du ∈ L2 ([0, ∞); H s )

provided that kϕkH s is sufficiently small. Moreover, the solution of (5.2.1) has the following decay rates: j

kDj u(t)k = O(t− 2 ), (1 ≤ j ≤ s).

(5.2.101)

Proof. Indeed, when F does not explicitly involve u, we can use only the energy estimates. In this case we define s X

kukXe = sup T

and

k 2

k

(1 + t )kD u(t)k +

Ã s Z X T

0≤t≤T k=0

k=0 0

!1

k

(1 + τ )kD

k+1

2

uk dτ

2

(5.2.102)

) ( ¯ ¯ u ∈ C([0, T ]; H s ) ∩ L2 ([0, T ]; H s+1 ), ¯ . SeT,ε = u ¯ ¯ u|t=0 = ϕ(x), kukX eT ≤ ε

(5.2.103)

For v ∈ SeT,ε , let u be the solution to the problem:  e   ut − ∆u = F (Λv),   u|

t=0

(x, t) ∈ Rn × R+ ,

(5.2.104)

= ϕ(x)

e = (Dv, D2 v). where Λv Then in the same manner as before, we obtain Ã

kuk2Xe T

≤C

kϕk2H s +

Ã

≤ C kϕk2H s + +

Z T 0

Z T 0

kukkF kdτ +

k=1 0

! k

(1 + τ )kD

k−1

2

F k dτ

e α ∞ kΛvk e dτ kuk kΛvk L !

s Z T X k=1 0

s Z T X

e 2α∞ kDk−1 Λvk e 2 dτ (1 + τ k )kΛvk L

Ã

e α−1 n ≤ C kϕk2H s + sup ku(τ )k sup kΛvk H [ 2 ]+1 0≤τ ≤T

0≤τ ≤T

Z T 0

e 2 [ n ]+1 dτ kΛvk H 2

200

NONLINEAR EVOLUTION EQUATIONS

+ sup

0≤τ ≤T

³

e 2α[ n ]+1 τ kΛvk H

2

Ã

≤C

kϕk2H s Ã

≤C

kϕk2H s

+ε

α+1

s Z T ´X k=1 0

!

(1 + τ

k−1

)kD

k−1 e

2

Λvk dτ

! 2(α+1)

sup ku(τ )k + ε

0≤τ ≤T

!

ε2(α+1) δ sup ku(τ )k2 + + ε2(α+1) . + 2 0≤τ ≤T 2δ

1 , we deduce from (5.2.105) that Taking δ = C ³

(5.2.105)

´

kuk2Xe ≤ C kϕk2H s + ε2(α+1) .

(5.2.106)

T

Let

µ

ε0 = min 1, (

¶

1 1 ) 2α . 2C

Then it turns out from (5.2.106) that when kϕkH s ≤

(5.2.107) √ε 2C

0 , ≤ √ε2C

the nonlinear operator N defined by (5.2.104) maps SeT,ε into itself. Moreover, in the same manner as before, by taking ε small enough if necessary, we deduce that N is a contraction. The remaining part of the proof is the same as for the proof of Theorem 5.2.1. ¤

5.3

IBVP for Fully Nonlinear Parabolic Equations

In this section we consider the following initial boundary value problem for fully nonlinear parabolic equations:  n  ∂u − X ∂ (a (x) ∂u ) + c(x)u = F (x, u, Du, D2 u),   ij  ∂t ∂xi ∂xj i,j=1   u| = 0,   Γ

(5.3.1)

u|t=0 = u0 (x)

where Γ is the smooth boundary of a bounded domain Ω ⊂ Rn , x = (x1 , · · · , xn ), Du = (ux1 , · · · , uxn ), D2 u = (uxi xj ), (i, j = 1, · · · , n). We make the following assumptions on problem (5.3.1): (i) Suppose that for simplicity of exposition, F ∈ C ∞ , u0 , aij , c ∈ ¯ C ∞ (Ω).

Global Solutions with Small Initial Data

201

(ii) c(x) > 0 and there is a positive constant α such that n X

aij (x)ξi ξj ≥ α

ij=1

n X i=1

¯ ξi2 , ∀ξ ∈ Rn , x ∈ Ω.

(5.3.2)

As can be seen from Section 2.7, the assumption (ii) implies that the operator Au = −

n X

∂u ∂ (aij (x) ) + c(x)u ∂xi ∂xj i,j=1

(5.3.3)

with D(A) = H 2 ∩ H01 is a sectorial operator. Moreover, the first eigenvalue λ1 of A is strictly positive. Let s be an integer with s > n 2 + 1. We can successively derive from (5.3.1) the derivatives of u with respect to t at t = 0: u1 (x) = ut |t=0

¯

∂ s u ¯¯ = F (x, u0 , Du0 , D u0 ) − Au0 , · · · , us (x) = , ∂ts ¯t=0 2

(5.3.4)

which can be explicitly expressed by u0 , aij , c, F and their derivatives with respect to x. By assumption (i), it is easy to see that ¯ (1 ≤ j ≤ s). uj (x) ∈ C ∞ (Ω), (5.3.5) (iii) The following compatibility conditions are satisfied: uj (x) ∈ H01 (Ω), (0 ≤ j ≤ s).

(5.3.6)

O(|ω|2 )

(iv) F (x, ω0 , ωi , ωij ) = near ω = 0 uniformly with respect to 2 ¯ x ∈ Ω where ω = (ω0 , ω1 , ωij ) ∈ R × Rn × Rn . More precisely, |Dk F (x, ω0 , ωi , ωij )| ≤ Const.|ω|2 , as |ω| ≤ 1, k ≤ 2s + 1,

(5.3.7)

k

|D Dω F (x, ω0 , ωi , ωij )| ≤ Const.|ω|, as |ω| ≤ 1, k + 1 ≤ 2s + 1, (5.3.8)

|D

k

Dωl F (x, ω0 , ωi , ωij )|

≤ Const., as |ω| ≤ 1, k + l ≤ 2s + 1, (5.3.9)

¯ uniformly with respect to x ∈ Ω. Then the main result in this section is the following. THEOREM 5.3.1 Suppose that the assumptions (i)–(iv) are satisfied. Then problem (5.3.1) admits a unique global smooth solution u such that

u∈

s \

C k ([0, ∞); H 2(s−k)+1 ),

k=0

202

NONLINEAR EVOLUTION EQUATIONS u(k) =

∂ku ∈ L2 (R+ ; H 2(s−k+1) ), (0 ≤ k ≤ s) ∂tk

provided that ku0 kH 2s+1 is sufficiently small. Moreover, ku(k) kH 2(s−k)+1 exponentially decay to zero as t → ∞. REMARK 5.3.1 It can be seen from the proof of Theorem 5.3.1 given below that the assumption on the positivity of c(x) can be weakened and the assertion of the theorem still holds as long as λ1 > 0. On the other hand, as will be shown in the final section of this chapter, if λ1 ≤ 0, the solution u may blow up in finite time no matter how small the initial data are. REMARK 5.3.2 We can also deal with the following second initial boundary value problem:  n  ∂u − X ∂ (a (x) ∂u ) + c(x)u = F (x, u, Du, D2 u),   ij  ∂t  ∂xi ∂xj  i,j=1  n X (5.3.10) ∂u cos(n, x )| = 0,  Bu|Γ = aij (x) ∂x  i Γ  j   i,j=1  

u|t=0 = u0 (x)

where Γ is the smooth boundary of a bounded domain Ω ⊂ Rn and n is the outward normal direction to Γ. Then under the same assumptions (i), (ii), (iv) and a compatibility condition similar to (iii), similar assertions to Theorem 5.3.1 still hold (see [163]). REMARK 5.3.3 Theorem 5.3.1 shows that in contrast to the Cauchy problem described in the previous section, the result on global existence of a solution to problem (5.3.1) is independent of the space dimension n. This is essentially due to the fact that the solution of the linearized problem has an exponential decay rate as t → ∞, which will be shown in the next section.

5.3.1

Decay Rate of the Solution to IBVP for Linear Parabolic Equations

Consider the following initial boundary value problem: ut −

n X

∂u ∂ (aij (x) ) + c(x)u = 0, (x, t) ∈ Ω × (0, ∞), ∂xi ∂xj i,j=1 (5.3.11)

u|Γ = 0, t > 0,

(5.3.12)

Global Solutions with Small Initial Data u|t=0 = u0 (x), x ∈ Ω

203 (5.3.13)

where Ω ⊂ Rn , for simplicity, is a bounded domain with C ∞ boundary ¯ satisfying Γ and aij (x), c(x) ≥ 0 are C ∞ functions in Ω n X

aij (x)ξi ξj ≥ α

i,j=1

n X i,j=1

¯ ξi2 , α > 0, x ∈ Ω.

(5.3.14)

Let Au = −

n X

∂u ∂ (aij (x) ) + c(x)u ∂x ∂x i j i,j=1

(5.3.15)

with D(A) = H 2 ∩ H01 . Therefore, for u ∈ D(A) ⊂ L2 Z



(5.3.16) 

n X

∂u ∂u  (Au, u) = + c(x)u2  dx ≥ 0. aij ∂x ∂x Ω i,j=1 j i

(5.3.17)

Thus, as proved in Chapter 2, A is a maximal accretive operator in L2 and generates a C0 -semigroup S(t). Moreover, as proved in Chapter 2, A is a sectorial operator and S(t) is an analytic semigroup. For s ∈ R, we can define D(As ). LEMMA 5.3.1 Let s ≥ 0 be an integer and λ1 > 0 be the first eigenvalue of A. Then, for 1 any u0 ∈ D(As+ 2 ) ⊂ H 2s+1 , there is a positive constant C depending only on α, the coefficients and Ω such that for all t ≥ 0, s X k=0

kDtk u(t)kH 2(s−k)+1 ≤ Ce−λ1 t ku0 kH 2s+1 .

(5.3.18)

Proof. It follows from (5.3.11) that 1d kuk2 12 + kAuk2 = 0. D(A ) 2 dt

(5.3.19)

1 d kuk2 12 + λ1 kuk2 12 ≤ 0, D(A ) D(A ) 2 dt

(5.3.20)

≤ e−2λ1 t ku0 k2

(5.3.21)

Hence,

kuk2

1

D(A 2 )

1

D(A 2 )

.

204

NONLINEAR EVOLUTION EQUATIONS

Let uk = Dtk u, 0 ≤ k ≤ s.

(5.3.22)

Then uk satisfies the prolonged system: duk + Auk = 0, dt uk (0) = (−A)k u0 .

(5.3.23) (5.3.24)

In the same manner as before, we obtain for 0 ≤ k ≤ s kDtk u(t)kD(A 21 ) = kuk (t)kD(A 21 ) ≤ e−λ1 t kAk u0 kD(A 12 ) .

(5.3.25)

Since uk = Dtk u satisfies Auk = −Dtk+1 u, (x, t) ∈ Ω × (0, ∞), uk |Γ = 0, t > 0.

(5.3.26) (5.3.27)

For any fixed t, applying the regularity results for the above elliptic boundary value problem stated in Chapter 1, we obtain for k = s − 1, kus−1 (t)kH 3 ≤ CkDts ukH 1 ≤ Ce−λ1 t kAs u0 kH 1 ≤ Ce−λ1 t ku0 kH 2s+1 , and for k = s − 2, . . . , 0 successively we have

(5.3.28)

kuk (t)kH 2(s−k)+1 ≤ CkDtk+1 ukH 2(s−k−1)+1 ≤ Ce−λ1 t ku0 kH 2s+1 . (5.3.29) Thus the proof is complete.

¤

REMARK 5.3.4 For other initial boundary value problems with the Neumann or Robin boundary conditions the same conclusion holds provided that the corresponding first eigenvalue of A is positive.

5.3.2

Global Solution to IBVP for Fully Nonlinear Parabolic Equations

In order to prove Theorem 5.3.1, we first introduce some function spaces, and prove some properties of composite functions. Let λ1 > 0 be the first eigenvalue of the operator A defined in (5.3.3). We now introduce the following Banach space: For any fixed T > 0, ¯ k ) ¯∂ v ¯ ∂tk ∈ C([0, T ]; H 2(s−k)+1 ∩H01 )∩L2 ([0, T ]; H 2(s−k+1) ), XT = v(x, t) ¯ ¯ 0≤k≤s (5.3.30) (

Global Solutions with Small Initial Data

205

equipped with the weighted norm Ã

kvkXT =

sup e

s X ∂kv 2βt

k

0≤t≤T

k=0

(t)k2H 2(s−k)+1 ∂tk

Z T X s ∂kv 2 k + k 0

k=0

∂tk

where β is a positive number satisfying β < λ1 . Let

H

!1 2(s−k+1)

dτ

2

(5.3.31)

¯ ) ( ¯ ¯ v ∈ X , v| ∂kv ¯ = u (x), (1 ≤ k ≤ s), = u , ¯ ¯ t=0 0 ∂tk T k ST,µ = v ¯ t=0 ¯ kvkX ≤ µ < 1 T

(5.3.32)

where uk (x) (1 ≤ k ≤ s) are defined by (5.3.4) and µ is a small positive constant specified later on. It is easy to see that for fixed µ and T , if ku0 kH 2s is sufficiently small, then s X uj (x)tj j=0

j!

∈ ST,µ

and ST,µ is nonempty. Clearly, ST,µ is a closed convex subset of XT . For any v ∈ ST,µ , consider the following linear auxiliary problem:  n X  ∂u ∂ def   (aij (x) ) + c(x)u = F (x, v, Dv, D2 v) = f (x, t), u −  t   ∂xi ∂xj  i,j=1     u|Γ = 0,       

u|t=0 = u0 (x).

(5.3.33)

We need the following lemma. LEMMA 5.3.2 Suppose v ∈ XT . Then f (x, t) = F (x, v, Dv, D2 v) satisfies

∂kf ∈ C([0, T ]; H 2(s−k)−1 ) ∩ L2 ([0, T ]; H 2(s−k) ), 0 ≤ k ≤ s − 1, ∂tk ∂sf ∂ts

(5.3.34)

∈ L2 ([0, T ]; L2 (Ω)).

(5.3.35)

Moreover, for v ∈ ST,µ ,

max

0≤t≤T

s−1 X k=0

e2βt k

∂kf 2 k 2(s−k)−1 ≤ Cµ4 , ∂tk H

(5.3.36)

206

NONLINEAR EVOLUTION EQUATIONS Z T X s 0

e2βt k

k=0

∂kf 2 k 2(s−k) dτ ≤ Cµ4 ∂tk H

(5.3.37)

where C > 0 is a constant independent of v, T, t, and µ.

Proof. We first prove (5.3.34). For k = 0, this follows from v ∈ XT and the well-known result that H l forms a Banach algebra as l > n 2 (see [1]). For 1 ≤ k ≤ s − 1, it can be seen that Dx2(s−k)−1 (i+j)

where Fω to ω and

∂kf ∂tk

=

X



D m F i+j x

i,j,m

ω

i Y σ=1

Dxασ v

j Y l=1



Dxβl Dtνl v 

(5.3.38)

stands for the (i+j)th-order derivatives of F with respect

0 ≤ i ≤ 2(s − k) − 1, 1 ≤ j ≤ k, 1 ≤ i + j ≤ 2(s − k) − 1 + k, (5.3.39)

ν1 + · · · + νj = k, 0 ≤ m ≤ 2(s − k) − 1, 0 ≤ ασ ≤ 2(s − k) + 1, (0 ≤ σ ≤ i), 0 ≤ βl ≤ 2(s − k) + 1 (1 ≤ l ≤ j), 2(s − k) − 1 ≤

i X

ασ +

σ=1

j X

βl ≤ 2(s − k) − 1 + 2(i + j).

(5.3.40) (5.3.41) (5.3.42) (5.3.43)

l=1 Dxασ v

do not appear. We use the GagliardoIn the case i = 0, the terms Nirenberg inequality and the H¨older inequality to estimate the L2 norm of ∂kf Dx2(s−k)−1 k ∂t in (5.3.38). It is clear that for all σ, ασ < 2s + 1, and for all l, βl ≤ 2s − 2νl +1. Furthermore, the equal sign can only be held when i = 0, j = 1. We first choose aσ , a ˜l so that they satisfy βl ασ ≤ aσ ≤ 1, ≤a ˜l ≤ 1 (5.3.44) 2s + 1 2(s − νl ) + 1 and i X

ασ +

σ=1

=

i X σ=1

j X l=1

βl +

(i + j − 1)n 2

aσ (2s + 1) +

j X l=1

a ˜l (2s − 2νl + 1).

(5.3.45)

Global Solutions with Small Initial Data

207

In what follows we prove that such a choice can always be made. Indeed, for aσ = 1, a ˜l = 1 (1 ≤ σ ≤ i, 1 ≤ l ≤ j), it follows from (5.3.39)–(5.3.43) that i X

(2s + 1) +

σ=1

j X

(2s − 2νl + 1) = (2s + 1)(i + j) − 2k

l=1

= (2s + 1)(i + j − 1) + 2s + 1 − 2k n > ( + 2)(i + j − 1) + 2(s − k) + 1 2 j i X X (i + j − 1)n . ≥ βl + ασ + 2 σ=1 l=1

(5.3.46)

βl σ , a On the other hand, for aσ = 2sα+ 1 ˜l = 2(s − νl ) + 1 , (1 ≤ σ ≤ i, 1 ≤ l ≤ j), we have i X

aσ (2s + 1) +

σ=1

<

i X

j X

a ˜l (2s − 2νl + 1) =

l=1

ασ +

σ=1

j X l=1

βl +

i X

ασ +

j X

σ=1

βl

l=1

(i + j − 1)n . 2

(5.3.47)

Thus, (5.3.46), (5.3.47) imply that the hyperplane (5.3.45) must intersect the set ½

¯ ¯ ασ

(aσ , a ˜l ) ¯¯

2s + 1

≤ aσ ≤ 1,

βl ≤a ˜l ≤ 1 2(s − νl ) + 1

¾

(5.3.48)

where 1 ≤ σ ≤ i, 1 ≤ l ≤ j. This proves the claim. Once we have chosen aσ , a ˜l , then pσ , p˜l defined by 1 ασ − aσ (2s + 1) 1 = + , 1 ≤ σ ≤ i, pσ n 2

(5.3.49)

βl − a ˜l (2s − 2νl + 1) 1 1 = + , 1≤l≤j p˜l n 2

(5.3.50)

must satisfy pσ ≥ 2, p˜l ≥ 2,

i X 1

p σ=1 σ

+

j X 1 l=1

p˜l

1 = . 2

(5.3.51)

Having chosen pσ , p˜l , we infer from the H¨older inequality and the

208

NONLINEAR EVOLUTION EQUATIONS

Sobolev imbedding theorem that kDxm Fω(i+j)

i Y σ=1

Dxασ v

≤ max |Dxm Fω(i+j) | ¯ Ω

i Y σ=1

≤ C max |Dxm Fω(i+j) | ¯ Ω

×

j Y l=1 ¯ Ω

×

l=1

Dxβl Dtνl vk

kDxασ vkLpσ

i Y σ=1

j Y l=1

kDxβl Dtνl vkLp˜l

kDx2s+1 vkaσ kvk1−aσ

kDx2(s−νl )+1 Dtνl vka˜l kDtνl vk1−˜al

≤ C max |Dxm Fω(i+j) | j Y

j Y

i Y

(kDx2s+1 vk + kvk)

σ=1

(kDx2(s−νl )+1 Dtνl vk + kDtνl vk).

(5.3.52)

l=1

Then we can easily deduce from the definition of C([0, T ]; H 2(s−k)−1 ), (5.3.38) and (5.3.52) that ∂kf ∈ C([0, T ]; H 2(s−k)−1 ), (1 ≤ k ≤ s − 1) ∂tk and (5.3.36) follows from (5.3.52) and assumption (iv). Similarly, we can prove that ∂kf ∈ L2 ([0, T ]; H 2(s−k) ), (0 ≤ k ≤ s) k ∂t and the estimate (5.3.37) holds. The lemma is proved. ¤ To prove Theorem 5.3.1, for 0 ≤ k ≤ s, we consider the prolonged system:  ∂u(k) + Au(k) = ∂ k f ,     ∂t ∂tk   

u(k) |Γ = 0,        (k) u

(x, t) ∈ Ω × R+ , (5.3.53)

|t=0 = uk (x)

where u(0) = u, A is defined in (5.3.3) and uk (x) are given in (5.3.4). It easily follows from the compactness method in Chapter 3 that there is a unique solution u(k) ∈ C([0, T ]; H01 ) ∩ L2 ([0, T ]; H 2 ) (0 ≤ k ≤ s)

Global Solutions with Small Initial Data

209

satisfying 1 d (k) 2 ∂kf ku kH 1 + kAu(k) k2 = (Au(k) , k ) 2 dt ∂t

(5.3.54)

1

where the equivalent norm kukH 1 = (Au, u) 2 is used. By the Young inequality, we have d (s) 2 ∂sf ku kH 1 + 2kAu(s) k2 ≤ 2kAu(s) kk s k dt ∂t ≤2

∂sf (λ1 − β) kAu(s) k2 + Ck s k2 . λ1 ∂t

(5.3.55)

Since kAuk2 ≥ λ1 kuk2H 1 , it turns out from (5.3.55) that e2βt ku(s) (t)k2H 1 ≤ kus k2H 1 + C and ku(s) k2H 1 +

Z t 0

Z t 0

e2βτ k

µ

ku(s) k2H 2 dτ ≤ C kus k2H 1 +

∂sf 2 k dτ ∂ts

(5.3.56)

¶ Z t ∂sf k s k2 dτ

∂t

0

(5.3.57)

where the fact that kAu(s) k is equivalent to ku(s) kH 2 has been used. k We now prove that for 1 ≤ k ≤ s, u(k) = ∂ ku . Indeed, let ∂t v (s−1) = us−1 +

Z t 0

u(s) dτ.

(5.3.58)

Then ∂v (s−1) = u(s) , ∂t −Av

(s−1)

(5.3.59)

= −Aus−1 − A

Z t 0

u(s) (τ ) dτ.

(5.3.60)

Since A generates a C0 -semigroup, it easily follows from the semigroup theory that Z t

−A

0 (s)

=u

(s)

u

(τ ) dτ = u

(s)

(t) − us −

Z t s ∂ f 0

¯

∂ts

dτ

∂ s−1 f (t) ∂ s−1 f ¯¯ . (t) − us − + ¯ ∂ts−1 ∂ts−1 ¯t=0

(5.3.61)

Since v ∈ ST,µ , by the definition of uk , (1 ≤ k ≤ s), we have ¯

uk+1

∂ k f ¯¯ , (0 ≤ k ≤ s − 1). = −Auk + ¯ ∂tk ¯t=0

(5.3.62)

210

NONLINEAR EVOLUTION EQUATIONS

Therefore, it turns out from (5.3.58)–(5.3.62) that v (s−1) also satisfies (5.3.53) with k = s − 1. By uniqueness, we must have v (s−1) = (s−1) u(s−1) , i.e., ∂u∂t = u(s) . Applying the same argument to u(k) (k = k s − 1, · · · , 0) successively, we obtain u(k) = ∂ ku (1 ≤ k ≤ s). ∂t For t ∈ [0, T ], consider the elliptic problems induced from (5.3.53):  ∂ k f − u(k+1) , (0 ≤ k ≤ s − 1),    Au(k) = ∂tk

(5.3.63)

  (k)  u |Γ = 0.

Then applying the standard elliptic estimates stated in Chapter 1 yields Ã

!

ku(s−1) (t)kH 3 ≤ C ku(s) (t)kH 1 Ã

ku

(s−2)

(s−1)

(t)kH 5 ≤ C ku

∂ s−1 f + k s−1 kH 1 , ∂t

(t)kH 3

∂ s−2 f + k s−2 (t)kH 3 ∂t

(5.3.64) !

!

Ã

∂ s−2 f ∂ s−1 f + k s−1 kH 1 + k s−2 kH 3 , ∂t ∂t

≤ C ku(s) (t)kH 1 ··· ···

Ã

(5.3.65) !

s−1 X

∂kf + k k kH 2(s−k)−1 . (5.3.66) ∂t k=0

ku(t)kH 2s+1 ≤ C ku(s) (t)kH 1

Thus, combining (5.3.64)–(5.3.66) with (5.3.56) yields max

0≤t≤T

s X k=0

e2βt ku(k) (t)k2H 2(s−k)+1

Ã

≤ C kus k2H 1 + max

0≤t≤T

+

Z T 0

e2βτ k

∂sf ∂ts

Similarly, we have Z t 0

ku(s−1) k2H 4 dτ ≤ C

≤C

Ã

kus k2H 1

··· ···

+

Z tÃ 0

e2βt k

k=0

!

k2 dτ

ÃZ

k

s−1 X

t

0

∂sf ∂ts

∂kf 2 k 2(s−k)−1 ∂tk H

.

(5.3.67)

! Z t s−1 f ∂ ku(s) k2H 2 dτ + k s−1 k2H 2 dτ 0

2

k +k

∂ s−1 f

k2 2 ∂ts−1 H

!

∂t

!

dτ

,

(5.3.68)

Global Solutions with Small Initial Data Z t 0

Ã

kuk2H 2s+2 dτ ≤ C kus k2H 1 +

s Z t X ∂kf 2 k k

k=0 0

dτ

H 2(s−k)

∂tk

211

!

. (5.3.69)

Adding together with (5.3.57) yields s Z t X k=0 0

Ã

ku(k) k2H 2(s−k+1)

kus k2H 1

dτ ≤ C

!

s Z t X ∂kf 2 k + k

∂tk

k=0 0

H 2(s−k)

Ã

+

Z T 0

≤ C kus k2H 1 + max

s−1 X

0≤t≤T

e

2βτ

k

∂sf ∂ts

2

k dτ +

e2βt k

k=0 Z s X T ∂kf

k=0 0

k

.

(5.3.70)

Combining (5.3.67) with (5.3.70) yields kuk2XT

dτ

∂kf 2 k 2(s−k)−1 ∂tk H !

k2 2(s−k) ∂tk H

dτ

.

(5.3.71)

For 0 ≤ k ≤ s−1, it follows from equation (5.3.1) that (refer to (5.3.38) and (5.3.52)) Ã

Dx2(s−k)−1 uk+1 (x)

=

Dx2(s−k)−1

¯

∂ k f ¯¯ −Auk (x) + ¯ ∂tk ¯t=0

!

= −Dx2(s−k)−1 Auk (x) +

X

Dxm Fω(i+j) (x, u0 , Dx u0 , Dx2 u0 )

i Y σ=1

Dxασ u0

j Y l=1

Dxβl uνl . (5.3.72)

By assumption (iv) and (5.3.52) we can successively obtain that when ku0 kH 2s+1 ≤ µ < 1, kuk+1 kH 2(s−k)−1 ≤ Cku0 kH 2s+1 , (0 ≤ k ≤ s − 1)

(5.3.73)

where C > 0 is again independent of µ. Thus it follows from (5.3.71), (5.3.73) and Lemma 5.3.2 that ³

kuk2XT ≤ C1 ku0 k2H 2s+1 + µ4

´

(5.3.74)

with C1 > 0 independent of T, µ, v, u. Let w1 , w2 be two solutions of (5.3.33) corresponding to v1 , v2 ∈ ST,µ . Then in the same way as above we obtain kw1 − w2 k2XT ≤ C2 µ2 kv1 − v2 k2XT

(5.3.75)

with C2 > 0 also being a constant independent of T, µ, v1 , v2 . It turns

212

NONLINEAR EVOLUTION EQUATIONS

out from (5.3.74) and (5.3.75) that there exists a positive constant µ0 : µ ¶ 1 1 µ0 = min 1, √ , √ (5.3.76) 2 C1 2 C2 such that for µ ≤ µ0 and ku0 kH 2s+1 ≤ √µ , v ∈ ST,µ , the solution 2 C1 u defined by problem (5.3.33) also belongs to ST,µ . Moreover, the nonlinear operator N : v ∈ ST,µ 7→ u ∈ ST,µ is a contraction. Thus the well-known contraction mapping theorem yields that N has a unique fixed point. Since T is arbitrary and independent of µ, the global existence and uniqueness of a smooth solution follows. The exponential decay of ° ° ° ∂ku ° ° ° ° k° ° ∂t °

H 2(s−k)+1

is directly derived from the weighted norm defined in (5.3.31). Thus the proof of Theorem 5.3.1 is complete. ¤ As an application, we reconsider the previous problem (5.1.1). The corresponding linearized problem is  ∂u   − ∆u + u = 0,     ∂t 

u|Γ = 0,       

(5.3.77)

u(0) = φ(x).

Since the operator −∆ + I subject to the Dirichlet boundary condition is positive definite, the first eigenvalue of this operator λ1 > 0. In this problem, F = u3 clearly satisfies the assumption (iv) made in the beginning of this section. Therefore, as long as the initial datum u0 satisfies the compatibility condition (iii), and ku0 kH 2s+1 is sufficiently small, problem (5.1.1) admits a global smooth small solution. Moreover, as time goes to infinity, the solution exponentially decays to zero.

5.4

Nonexistence of Global Solutions with Small Initial Data

Concerning the initial value problem for nonlinear parabolic equations with small initial data, as shown in Section 2 of this chapter, global existence or nonexistence of a classical solution heavily depends

Global Solutions with Small Initial Data

213

on the space dimension n. In this section we are concerned with the nonexistence of global solution to the initial boundary value problems for nonlinear parabolic equations with small initial data. The results presented in this section show that for small initial data, the question of whether solutions to the initial boundary value problems for nonlinear parabolic equations globally exist or blow up in finite time is very closely related to whether the corresponding elliptic operators subject to the boundary conditions are positive definite. It turns out that this question is also closely related to whether solutions to the linearized problems have exponential decay rates. The same question for the nonlinear hyperbolic equations is also discussed in this section. In the past forty years a large literature has developed concerning the nonexistence of global solution or blow-up in finite time. For instance, to name just a few, see [91], [92] by H. A. Levine; [71] by S. Kaplan; [57], [58] by R.T. Glassay; [21], [22] by J.M. Ball; and the references cited therein. We refer to [21], [22] for the distinction between two different concepts: the nonexistence of global solutions and blow-up in finite time. Most conclusions concerning nonexistence of global solution or blow-up in finite time were drawn under the assumption on initial data that the appropriate “energy function” is negative at t = 0. As was shown in the beginning of this chapter, this assumption amounts to the initial data being “large”. In contrast, for the following problem,  ut − ∆u = u2 ,      ¯  ∂u ¯¯ = 0, ∂n Γ      

(5.4.1)

u|t=0 = u0 , u0 ≥ 0, u0 6≡ 0,

the operator −∆ subject to the Neumann boundary condition is not positive definite and the solution to the corresponding linearized problem does not decay to zero as time goes to infinity. It turns out, as shown in Chapter 1, that the problem (5.4.1) does not have a global classical solution no matter how small and smooth the initial data u0 are. It can be seen from previous discussions that as far as the initial boundary value problem for nonlinear parabolic equations with small initial data is concerned, global existence or nonexistence depends on whether the solution to the linearized problem has an exponential decay rate. We now extend the above results on problem (5.4.1) to more general initial boundary value problems for nonlinear parabolic equations as well as nonlinear hyperbolic equations.

214

NONLINEAR EVOLUTION EQUATIONS

More precisely, we consider the following initial boundary value problem for nonlinear parabolic equations:  n X  ∂u ∂   (aij (x) ) − cu = f (x, t, u), u −  t   ∂xi ∂xj  i,j=1     Bu|Γ = 0,       

(5.4.2)

u|t=0 = u0 (x).

Here Au = −

n X

∂u ∂ (aij (x) ) ∂x ∂x i j i,j=1

(5.4.3)

is a self-adjoint elliptic operator with smooth coefficients aij (x) (i, j = 1, · · · , n), and the boundary operator Bu denotes u or n X

aij (x)

i,j=1

∂u cos(n, xi ) + σ(x)u ∂xj

with smooth σ ≥ 0 and n being the exterior normal to the smooth boundary Γ of a bounded domain Ω ⊆ Rn . Let λB 1 be the first eigenvalue of the following eigenvalue problem:   Aϕ = λϕ, 

Bϕ|Γ=0 = 0.

(5.4.4)

It is well known (see, e.g., [35]) that the corresponding eigenfunctions span the one-dimensional subspace and we have corresponding eigenfunction ϕB 1 such that  B ϕ (x) > 0, x ∈ Ω,    1 Z    ϕB 1 dx = 1.

(5.4.5)

Ω

Now we have the following nonexistence result. THEOREM 5.4.1 p Suppose that c = c(t) ≥ λB 1 , f (x, t, u) ≥ b|u| with given constants b > ¯ 0, p > 1. Suppose u is a classical solution to problem (5.4.2) in Ω×[0, Tmax ). R B Then when Ω u0 ϕ1 dx > 0, no matter how small the initial data u0 are, we must have Tmax < ∞, i.e., nonexistence of a global classical solution.

Global Solutions with Small Initial Data

215

Proof. Multiplying the equation and the initial condition in (5.4.2) by ϕB 1 and integrating with respect to x, we obtain Z  B )v(t) + b  |u|p ϕB v (t) ≥ (c(t) − λ  1 dx, 1   t Ω Z     v(0) = u0 (x)ϕB 1 dx > 0

(5.4.6)

Ω

where v(t) =

Z Ω

uϕB 1 dx.

(5.4.7)

Applying the Jensen inequality (see Chapter 1), we get Z

Ω

|u|p ϕB 1 dx

Z

≥(

Ω

p |u|ϕB 1 dx)

Z

=(

¯Z ¯p ¯ ¯ ¯ = vp. ≥ ¯¯ uϕB dx 1 ¯

Ω

p |uϕB 1 |dx)

(5.4.8)

Ω

Combining (5.4.6) with (5.4.8) yields

 p v ≥ (c(t) − λB  1 )v(t) + b|v| ,   t Z    v(0) = u0 ϕB 1 dx > 0.

(5.4.9)

Ω

Therefore, by (5.4.9) we have v(t) ≥ v(0) > 0, f or t ∈ [0, Tmax )

(5.4.10)

vt ≥ bv p > 0, f or t ∈ [0, Tmax ).

(5.4.11)

and Since p > 1, we deduce from (5.4.11) that Tmax < ∞, i.e., nonexistence of a global classical solution. ¤. It can be easily seen that the assumption c ≥ λB 1 implies n X ∂u ∂ (aij (x) ) − cu subject that the corresponding elliptic operator −

REMARK 5.4.1

i,j=1

∂xi

∂xj

to the boundary condition Bu|Γ = 0 is not positive definite. It turns out that the solution to the linearized problem does not decay to zero. REMARK 5.4.2 The above result is sharp in the sense that if c = , then it follows from Theorem 5.3.1 that problem (5.4.2) with Const. < λB 1

216

NONLINEAR EVOLUTION EQUATIONS

f = f (u) being smooth, f (u) = O(u2 ) near u = 0, must have a global smooth solution provided that the initial data are small.

Similarly, we can discuss the initial boundary value problem for nonlinear hyperbolic equations:  n X  ∂u ∂   (aij (x) ) − c(t)u + αut = f (x, t, u, ut , ∇u), u −  tt   ∂xi ∂xj  i,j=1     Bu|Γ = 0,       

u|t=0 = u0 (x), ut |t=0 = u1 (x).

(5.4.12)

The assumptions on the operators A, B and Ω are the same as before. We also assume that c, f and initial data u0 , u1 are smooth given functions and α is a positive constant. The dissipative term αut usually is a good one to stabilize the system (see, e.g., [106]). However, the following result shows that when the corresponding elliptic operator is not positive definite, the dissipative term αut is not strong enough to prevent the solution from developing a singularity in the finite time even if the initial data are small and smooth. THEOREM 5.4.2 p1 p2 with constants Suppose that c = c(t) ≥ λB 1 , α > 0, f ≥ b1 |u| + b2 |ut | bi > 0, pi > 1 (i = 1, 2). RSuppose that u(x,Rt) is a classical solution in B Ω × [0, Tmax ). Then when Ω u0 ϕB 1 dx > 0, Ω u1 ϕ1 dx > 0, no matter how small and smooth the initial data are, we must have Tmax < +∞.

Proof. In the same manner as before, we get

 p1 p2 v (t) − (c(t) − λB  1 )v(t) + αvt (t) ≥ b1 |v| + b2 |vt (t)| ,   tt Z Z    v(0) = dx > 0, v (0) = u0 ϕB u1 ϕB t 1 1 dx > 0. Ω

(5.4.13)

Ω

We first claim that v(t) ≥ v(0) > 0, vt (t) > 0, t ∈ [0, Tmax ).

(5.4.14)

Indeed, it follows from v(0) > 0, vt (0) > 0 that there is a neighborhood of t = 0 such that when t stays in this small neighborhood, v(t) > 0, vt (t) > 0. If (5.4.14) is not true, then there is a t∗ ∈ [0, Tmax ) such that vt (t) > 0, for t ∈ [0, t∗ ) and vt (t∗ ) = 0. Thus, it follows from

Global Solutions with Small Initial Data

217

(5.4.13) that vtt + αvt (t) > 0, t ∈ [0, t∗ ).

(5.4.15)

Therefore, d αt (e vt ) > 0, t ∈ [0, t∗ ), dt ∗ vt (t) > vt (0)e−αt > 0, t ∈ [0, t∗ ), ∗ vt (t∗ ) ≥ vt (0)e−αt > 0,

(5.4.16) (5.4.17) (5.4.18)

a contradiction. To prove the theorem, we need the following comparison lemma (see the book [25], p. 139, Theorem 5).

LEMMA 5.4.1 Suppose that p(t), q(t) ∈ C, u, v ∈ C 2 and satisfy (i) utt + p(t)ut − q(t)u > 0, t ≥ 0; (ii) vtt + p(t)vt − q(t)v = 0, t ≥ 0; (iii) q(t) ≥ 0, t ≥ 0; (iv) u(0) = v(0), ut (0) = vt (0). Then u(t) > v(t), for t > 0.

It follows from (5.4.13), (5.4.14) and c(t) ≥ λB 1 that vtt + αvt − b1 v p1 −1 (0)v ≥ b2 vtp2 > 0.

(5.4.19)

Let w(t) be the solution to the following problem:  p −1  wtt + αwt − b1 v 1 (0)w = 0, 

w(0) = v(0), wt (0) = vt (0).

(5.4.20)

Thus w(t) = C1 eµ1 t + C2 eµ2 t where

 p 1  2 p −1   µ1 = (−α + α + 4b1 v 1 (0)) > 0,  2   p   µ2 = 1 (−α − α2 + 4b1 v p1 −1 (0)) < 0

2

(5.4.21)

(5.4.22)

218 and

NONLINEAR EVOLUTION EQUATIONS  vt (0) − µ2 v(0)   > 0,  C1 = p 2  α + 4b1 v p1 −1 (0)    µ v(0) − vt (0)   .  C2 = p 21 α + 4b1 v p1 −1 (0)

(5.4.23)

Therefore, we conclude from (5.4.21) that w(t) → +∞, as t → ∞.

(5.4.24)

By Lemma 5.4.1, we have v(t) > w(t), ∀ t > 0.

(5.4.25)

We now use a contradiction argument to prove that Tmax < ∞. If Tmax = ∞, then by (5.4.25) we get v(t) → ∞, as t → ∞.

(5.4.26)

By the Young inequality, we get for any δ > 0, vt ≤

δ p2 p2 1 v + 0 p0 p2 t p2 δ 2

(5.4.27)

where p02 > 1, p12 + 10 = 1. Taking δ small enough, we have p2 b2 p2 (5.4.28) v + Cδ 2 t where Cδ > 0 is a constant depending on δ. Combining (5.4.19) with (5.4.28) yields αvt ≤

b2 p2 v + b1 v p1 −1 (0)v − Cδ . (5.4.29) 2 t On the other hand, it follows from (5.4.26), (5.4.29) that there exists t0 ∈ R+ such that vtt ≥

b2 p2 v , ∀t ≥ t0 . (5.4.30) 2 t Since b2 > 0, p2 > 1, the function v satisfying (5.4.30) must blow up in a finite time. This contradicts Tmax = ∞. Thus the proof is complete. vtt ≥

¤

REMARK 5.4.3 The above result is sharp in the sense that if the corresponding elliptic operator is positive definite (i.e., c = Const < λB 1 ), as shown in [106], problem (5.4.12) admits a unique global solution provided that

Global Solutions with Small Initial Data

219

the initial data are smooth, small and the right-hand side term f is smooth and is at least second order about u and its derivatives near the origin. For more details, see [106].

5.5

Bibliographic Comments

A systematic approach dealing with the global existence of small smooth solutions to nonlinear evolution equations has been well developed since the middle of the 1970s. In this direction, we refer to a series of papers [106], [107], [108] by A. Matsumura; [109], [110], [111], [112] by A. Matsumura and T. Nishida; [79], [80] by S. Kawashima; [142] by M. Slemrod; [156] by S. Zheng; [170], [171] by S. Zheng and W. Shen; [125] by R. Racke; [164] by S. Zheng; and the references cited therein for the equations of compressible viscous and heat-conductive fluids, the equations in nonlinear thermoelasticity and nonlinear thermoviscoelasticity, and other various nonlinear hyperbolic-parabolic coupled systems. We also refer to the papers [82], [83] by S. Kleinerman for the study of the Cauchy problem for fully nonlinear wave equations; see also the books [146] by W. Strauss, and [93] by Li Tatsien and Chen Yunmei. For the study of the Cauchy problem for fully nonlinear parabolic equations, we refer to the papers [166] by S. Zheng and Y. Chen, [159] by S. Zheng and the monographs [164] by S. Zheng; see also the paper [120] by G. Ponce and the monographs [93] by Li Tatsien and Chen Yunmei. Concerning the blow-up results for the Cauchy problem for nonlinear wave equations, we refer to the book [70] by F. John and the references cited therein; see also the book [146] by W. Strauss. On the blow-up results for the solutions to the Cauchy problem for nonlinear parabolic equations, we refer to the papers [55], [56] by H. Fujita; [84] by K. Kobayashi et al.; [153] by F.B. Weissler; and the [63] by K. Hayakawa. There are a great many references in the literature concerning the nonexistence of global solution or blow-up in finite time for nonlinear hyperbolic equations and nonlinear parabolic equations. For instance, to name just a few, see [91], [92] by H. A. Levine; [71] by S. Kaplan; [57], [58] by R.T. Glassay; [21], [22] by J.M. Ball; and the references cited therein. Concerning the blow-up results for the initial boundary

220

NONLINEAR EVOLUTION EQUATIONS

value problems for the nonlinear hyperbolic equations and nonlinear parabolic equations with small initial data, we refer to the paper [167] by S. Zheng and Y. Chen. The results in Section 5.4 are based on that paper. Section 5.2 is mainly based on the monographs [164] by S. Zheng. The results in Section 5.3 are based on the papers [162], [163] by S. Zheng. There are a great many references in the literature concerning the existence of solutions to the initial value problems and initial boundary value problems for quasilinear parabolic equations. We refer to the books [52] by A. Friedman; [90] by O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Uralceva; and the references cited therein for the earlier results, and refer to a series of papers [8]–[17] by H. Amann for recent developments. We refer to the paper [88] by N.V. Krylov for the first initial boundary value problem for other classes of fully nonlinear parabolic equations. What they are all concerned with is the existence and uniqueness of solutions with arbitrary initial data. In order to obtain global existence, the price to pay is to put various restrictions on nonlinear terms such as the sign condition, monotonicity or convexity (concavity) conditions, growth condition, etc. The viscosity solution method is a powerful tool for the study of nonlinear second-order parabolic equations. In this direction we refer to the paper [36] by M.G. Crandall, H. Ishii and P.L. Lions and the references cited therein.

Chapter 6 Asymptotic Behavior of Solutions and Global Attractors

6.1

Introduction

For a given nonlinear evolution equation, once it is known that a solution exists for all time t > 0, a natural and interesting question is to ask about the asymptotic behavior of the solution as t → +∞. Let us first look at two simple examples. Ex. 6.1.1. Consider the following initial boundary value problem for the linear heat equation:  ut − uxx = 0,          

u|x=0,π = 0,

(6.1.1)

u|t=0 = u0 (x).

By the method of separate variables, as shown before, the solution can be explicitly expressed by the following formula: u(x, t) =

∞ X

Ak e−k t sin kx 2

(6.1.2)

k=1

where

Z

2 π u0 (x) sin kxdx. (6.1.3) Ak = π 0 Thus, it is easy to see that as time goes to infinity, the solution decays to zero exponentially. The corresponding stationary problem is   −uxx = 0, 

u|x=0,π = 0

(6.1.4)

and zero is its only solution. This clearly indicates that as time goes 221

222

NONLINEAR EVOLUTION EQUATIONS

to infinity, the solution to problem (6.1.1) converges to a stationary solution, which is also called an equilibrium. The same conclusion holds for the initial Neumann or Robin boundary value problems. Ex. 6.1.2. Consider the following initial boundary value problem for the wave equation:   utt − a2 uxx = 0,           

u|x=0,π = 0,

(6.1.5)

u|t=0 = u0 (x), ut |t=0 = u1 (x).

By the same method of separate variables, the solution can be expressed in the form u(x, t) =

∞ X

(Ak cos akt + Bk sin akt) sin kx

(6.1.6)

k=1

where Ak =

2 π

Z π 0

u0 (x) sin kxdx,

Bk =

2 akπ

Z π 0

u1 (x) sin kxdx.

(6.1.7)

Although problems (6.1.5) and (6.1.1) have the same stationary problem (6.1.4) for which zero is the unique stationary solution, it is clear that the solution to problem (6.1.5) behaves like a periodic function or its composition, and it will not converge to the equilibrium. We can name at least two reasons for such big differences. First, problem (6.1.1) is a dissipative system, i.e., its energy E(t) defined by E(t) =

1 2

Z π 0

u2 dx

(6.1.8)

is decreasing along time. Actually, as shown in Chapter 5, the energy decays exponentially. However, problem (6.1.5) is not a dissipative system, and it is well known that the energy E(t) =

1 2

Z π 0

(u2t + a2 u2x )dx

(6.1.9)

conserves for all time. Secondly, for problem (6.1.1), the heat operator has the “smoothing” property, i.e., the solution will be in C ∞ for t > 0 even for rough initial data. This can also be easily seen from the expression (6.1.2). While for the hyperbolic problem (6.1.5), the solution does not have

Asymptotic Behavior of Solutions and Global Attractors

223

such a property, and singularity of initial data will propagate along the characteristic lines. The situation for nonlinear evolution equations is far more complicated than linear evolution equations. However, the above two reasons still play important roles. This probably explains why for nonlinear evolution equations most concern and interest in the literature so far have been put on the systems defined by nonlinear parabolic equations or other dissipative systems including nonlinear hyperbolic equations with various dissipation, though there are some investigations in the literature on the existence of periodic solutions to nonlinear wave equations, e.g., see [30]. In this chapter we are only concerned with dissipative systems. The study of asymptotic behavior of the solution to nonlinear evolution equation as time goes to infinity can be divided into two categories. The first category is to investigate the asymptotic behavior of the solution for any given initial datum. The second category is to investigate the asymptotic behavior of all solutions when initial data vary in any bounded set. To explain the difference of two categories in detail, let us consider the following initial boundary value problem:  ut − ∆u = f (x, u),          

u|Γ = 0,

∀(x, t) ∈ Ω × (0, +∞) (6.1.10)

u|t=0 = u0 (x)

where Ω ⊂ Rn is a bounded domain with smooth boundary Γ, and f is a given smooth function in its independent variables. Let us consider the first category problem. Suppose that for any given initial datum, this problem admits a unique global solution u in certain function space. Then there are three possibilities as t → +∞: (i). u becomes unbounded in a certain norm; (ii). u stays bounded, but does not converge to a function in x in a certain norm; (iii). u converges to a function in x, which is an equilibrium. In 1978, H. Matano [105] investigated the problem (6.1.10) in one space dimension, i.e., Ω = (a, b), and proved that under a very general assumption on f , the possibility (ii) does not take place. More precisely, if f belongs to C 2 , and if u stays bounded as time goes to infinity, then u must converge to an equilibrium. The same is true for problems with the Neumann boundary condition or Robin boundary condition.

224

NONLINEAR EVOLUTION EQUATIONS

If we define the ω-limit set ω(u0 ) as follows: ω(u0 ) = {ψ | ∃tn , s.t. u(·, tn ) → ψ,

as tn → +∞}

(6.1.11)

then the above results by H. Matano simply amount to saying that if u stays bounded for all time, then the ω-limit set ω(u0 ) consists of a single point. Since then, a great effort has been made to extend the above result to a higher-space dimension case. It has been proved in the literature that for ‘generic’ domain Ω, and ‘generic’ nonlinear term f , the same conclusion is true. However, this kind of result does not lead to a definite conclusion for a particularly given problem. More strikingly, in 1996, P. Polacik and K.P. Rybakowski [119] constructed an example in which f belongs to C m with m arbitrary, but fixed, and Ω is a unit disk in R2 , showing that the corresponding initial Dirichlet boundary value problem (6.1.10) has a bounded solution whose ω-limit set is diffeomorphic to the unit circle S 1 . In the literature many assumptions have been made to assure that all bounded solutions of problem (6.1.10) are convergent. Among these attempts, in 1983 L. Simon [141] proved that convergence takes place if the nonlinear term f is analytic in u. His basic strategy was to extend the Lojasiewicz inequality [99], [100], [101] for the analytic function defined in Rn to the infinite-dimensional space. Since this remarkable contribution, a lot of work in this direction have been done. We refer the reader to the bibliographical comments section of this chapter for more detailed information. In Section 6.3 of this chapter, we will introduce this approach initiated by L. Simon and prove the Lojasiewicz-Simon inequality, following a simplified argument introduced in [67]. We usually view the solution u(x, t) as an orbit in a certain Sobolev space, starting from the initial datum u0 . In the investigation of the first category, we are concerned with the asymptotic behavior of the single orbit starting from an arbitrary, but fixed initial datum. On the other hand, we need to investigate the asymptotic behavior of a family of orbits starting from initial data varying in any bounded set of a Sobolev space. This kind of investigation is important both from the point of view of theoretical study and applications. We refer the reader to the book by R. Temam [151] for more explanation on this aspect. More precisely, for a given nonlinear evolution equation like (6.1.10), once existence and uniqueness of global solution for any initial data in a certain Sobolev space are known, then one may ask a series of further questions. Recall that global attractor is defined in the literature to

Asymptotic Behavior of Solutions and Global Attractors

225

be a compact invariant set for the nonlinear semigroup S(t) defined by the solution to this system such that all orbits starting from any bounded set in a certain Sobolev space will be attracted by this global attractor. Then the first question is: Is there a global attractor? If there does exists such a global attractor, one may further ask about its structure, and its Hausdorff dimension or fractal dimension, etc. This is the study of the asymptotic behavior in the second category. From a mathematical point of view the distinction between the two categories is clear. It turns out that in the second category the asymptotic behavior should be uniform with respect to all orbits. In Section 6.4 of this chapter, we will briefly introduce some results in this aspect especially for the gradient system. In Section 6.2 of this chapter, in connection with the first category problem, we will introduce a lemma in analysis that was first established by W. Shen and S. Zheng [135] in 1993, then extended to a certain extent by P. Krejci and J. Sprekels [87] in 1998. Some nontrivial applications will also be given in that section.

6.2

A Lemma in Analysis and Its Applications

The following lemma was first established by W. Shen and S. Zheng [135] in 1993. LEMMA 6.2.1 Let T be given with 0 < T ≤ +∞. Suppose that y(t), h(t) are nonnegative continuous functions defined on [0, T ] and satisfy the following conditions:

dy ≤ A1 y 2 + A2 + h(t), dt

Z T 0

y(t)dt ≤ A3 ,

Z T 0

h(t)dt ≤ A4

(6.2.1) (6.2.2)

where Ai (i = 1, · · · , 4) are given nonnegative constants. Then for any r > 0 with 0 < r < T , the following estimate holds: µ ¶ A3 y(t + r) ≤ + A2 r + A4 eA1 A3 t ∈ [0, T − r). (6.2.3)

r

Furthermore, if T = +∞, then

lim y(t) = 0.

t→+∞

(6.2.4)

226

NONLINEAR EVOLUTION EQUATIONS

Before giving the proof of this lemma, let us make some explanations first. It is well known from analysis that for a nonnegative continuous R function y(t) defined on [0, +∞), the sole condition 0+∞ y(t)dt < +∞ cannot guarantee that (6.2.4) holds. Usually, in the literature, an additional condition like Z +∞ 0

|y 0 (t)|dt < +∞

(6.2.5)

is introduced to assure that (6.2.4) holds. However, this additional condition is not very convenient to use because in many applications y(t) usually is the square of a norm of the unknown function in a certain Sobolev space, and it is more difficult to verify condition (6.2.5) than a differential inequality like (6.2.1) when the energy method is used. Proof of Lemma 6.2.1. The proof of (6.2.3) is quite similar to that for the uniform Gronwall inequality introduced in Chapter 1. For the variable s staying between [t, t + r], we multiply the differential inequality dy ≤ A1 y 2 (s) + A2 + h(s) (6.2.6) ds R by exp (− ts A1 y(τ )dτ ), then we integrate the resultant with respect to s from s to t + r to obtain y(t + r) ≤ y(s)e

R t+r s

A1 y(τ )dτ

+ (A2 r + A4 )e R t+r

R t+r t

A1 ydτ

≤ (y(s) + A2 r + A4 )e t A1 ydτ ≤ (y(s) + A2 r + A4 )eA1 A3 .

(6.2.7)

Then integrating with respect to s from t to t + r yields (6.2.3). The proof of (6.2.4) is quite different from the above. We use a contradiction argument. First we infer from (6.2.3) that µµ

¶

A3 dy ≤ A1 + A2 r + A4 eA1 A3 dt r = Ar + h(t), ∀t ≥ r where

µµ

Ar = A1

¶

¶2

A3 + A2 r + A4 eA1 A3 r

+ A2 + h(t) (6.2.8) ¶2

+ A2 .

(6.2.9)

If (6.2.4) is not true, then there exists a positive constant a > 0 and a sequence tn → +∞ such that a a , tn+1 ≥ tn + , ∀n ∈ N tn ≥ r + 4Ar 4Ar

Asymptotic Behavior of Solutions and Global Attractors y(tn ) ≥

a > 0, 2

∀n ∈ N.

227 (6.2.10)

Integrating (6.2.8) from t to tn yields y(tn ) − y(t) ≤ Ar (tn − t) +

Z tn t

Thus, we have a a − y(t) ≤ y(tn ) − y(t) ≤ + 2 4

h(τ )dτ,

Z tn tn −

a 4Ar

tn −

a ≤ t < tn . (6.2.11) 4Ar

h(τ )dτ,

t ∈ [tn −

(6.2.12)

It turns out that y(t) +

Z tn a tn − 4A

h(τ )dτ ≥ r

a , 4

a , tn ). 4Ar

tn −

a ≤ t < tn . 4Ar

(6.2.13)

For any T > 0, let nT = max{n | n ∈ N, r +

a ≤ tn ≤ T }. 4Ar

(6.2.14)

Then, lim nT = +∞.

(6.2.15)

T →+∞

Hence, we infer from (6.2.2) and (6.2.13) that aA4 ≥ A3 + 4Ar ≥ ≥

Z T 0

a y(τ )dτ + 4Ar ÃZ tn

X

1≤n≤nT a2

16Ar

a tn − 4A

r

Z T 0

h(τ )dτ

a y(τ )dτ + 4Ar

nT → +∞,

a contradiction. Thus, the lemma is proved.

!

Z tn a tn − 4A

h(τ )dτ r

(6.2.16) ¤

REMARK 6.2.1 Suppose T = +∞. Then the conclusion (6.2.4) still holds even if the condition (6.2.1) is replaced by the following weaker one: Z t Z t y(t) − y(s) ≤ f (y(τ ))dτ + h(τ )dτ, ∀0 ≤ s < t < +∞ (6.2.17) s

s

where f (y) is any given nonnegative, nondecreasing continuous function. This can be proved by the same method as presented here. See P. Krecji and J. Sprekels [87], and S. Zheng [165].

228

NONLINEAR EVOLUTION EQUATIONS

The above lemma has many applications to the study of global existence and long time behavior of solutions to nonlinear evolution equations. When the energy method is used, as indicated previously, y(t) often is the square of a norm of dependent function in a certain Sobolev space. Then the uniform estimate (6.2.3) can be used to combine a local existence and uniqueness result to conclude global existence of the solution. It can also be used to conclude existence of an absorbing set in the study of global attractor, as shown in [136], [137]. The conclusion (6.2.4) and its extension as shown in Remark 6.2.1 can be used to study the asymptotic behavior of solution. In what follows we will give some examples to show the applications of Lemma 6.2.1. Ex. 6.2.1. Let us first consider a simple initial boundary value problem:   u − ∆u + u3 − u = 0,   t         

(x, t) ∈ Ω × R+ ,

u|Γ = 0,

(6.2.18)

u|t=0 = u0 (x)

where Ω is a bounded domain in Rn with the smooth boundary Γ. The global existence and uniqueness of the solution can be proved either by using the semigroup method introduced in Chapter 2, or the compactness method introduced in Chapter 3, or the monotone iterative method introduced in Chapter 4. Furthermore, it can be shown that for t > 0, the solution is in C ∞ , using the bootstrap argument. In what follows we apply Lemma 6.2.1 to show that kut (t)k → 0,

as t → +∞.

For simplicity of exposition, we assume that n ≤ 3 and u0 ∈ H 2 ∩ H01 (Ω). Multiplying the equation in (6.2.18) by ut , then integrating with respect to x yields d dt

Z

1 1 1 ( |∇u|2 + u4 − u2 )dx + kut k2 = 0. 4 2 Ω 2

(6.2.19)

Integrating (6.2.19) with respect to t, we deduce that for t > 0, Z t 0

kut k2 dτ +

Z

1 1 1 ( |∇u|2 + u4 )dx ≤ kuk2 + C 4 2 Ω 2

(6.2.20)

where C is a positive constant depending on u0 . By the Young inequal-

Asymptotic Behavior of Solutions and Global Attractors ity,

Z

1 u dx ≤ 4 Ω we can deduce that for all t > 0, 2

Z t 0

Z Ω

u4 dx + |Ω|,

kut k2 dτ ≤ C1

229

(6.2.21)

(6.2.22)

where C1 is a positive constant independent of t. Let y(t) = kut k2 . Then (6.2.22) tells us that T = +∞, and the first condition in (6.2.2) is satisfied. In order to verify condition (6.2.1), we differentiate the equation in (6.2.18) with respect to t, then we multiply the resultant by ut , and integrate with respect to x to obtain Z 1d 1 1 1 2 kut k + ( |∇ut |2 + 3u2 u2t )dx = kut k2 ≤ kut k4 + . (6.2.23) 2 dt 2 2 Ω 2 This indicates that condition (6.2.1) and the second one in (6.2.2) are satisfied with A1 = 1, A2 = 1, h(t) ≡ 0. Thus it immediately follows from Lemma 6.2.1 that as t → +∞, y(t) = kut (t)k2 → 0.

(6.2.24)

Ex. 6.2.2. Next, consider the following abstract initial value problem for a nonlinear evolution equation:   ut + A(u) = f (t), 

u|t=0 = u0

(6.2.25)

which has already been investigated in Section 3.2 of Chapter 3. We still use the same framework established there. Let V be a reflexive, separate Banach space, and H be a Hilbert space such that V ,→ H ∼ = H 0 ,→ V 0 . A is still assumed to be a nonlinear operator from V into V 0 satisfying conditions (i), (ii), and (iii) in Theorem 3.4, and u0 is assumed to 0 belong to H. We now assume that f ∈ Lp ([0, +∞), V 0 ). Then we have the following result. THEOREM 6.2.1 Let these assumptions on A, u0 and f be satisfied. Then as t → +∞,

ku(t)kH → 0.

(6.2.26)

230

NONLINEAR EVOLUTION EQUATIONS

Proof. First by Theorem 3.4, problem (6.2.25) admits a unique global solution u such that u ∈ C([0, +∞), H) ∩ Lp ([0, +∞), V ),

ut ∈ Lp ([0, +∞), V 0 ). 0

Taking the dual product of the equation in (6.2.25) with u, we get 1d ku(t)k2H + (A(u), u) = (f (t), u) ≤ kf kV 0 kukV . 2 dt By condition (iii) on A, αkukpV ≤ (A(u), u),

(6.2.27)

(6.2.28)

we can deduce from (6.2.27) and the Young inequality that 0 ε 1 1d ku(t)k2H + αkukpV ≤ kukpV + kf kpV 0 . p0 2 dt p p0 ε p

Taking ε =

αp 2

(6.2.29)

in (6.2.29), we obtain

0 d ku(t)k2H + αkukpV ≤ Ckf kpV 0 (6.2.30) dt where C is a positive constant independent of u and t. Integrating (6.2.30) with respect to t yields

ku(t)k2H

≤ C1 ,

Z t 0

kukpV dτ ≤ C1 , ∀t > 0

(6.2.31)

where C1 is a positive constant depending only on Z +∞

ku0 kH ,

0

kf kpV dτ. 0

We now consider the situations p ≥ 2 and 1 < p < 2, respectively. First, by the continuous imbedding from V into H, there is a positive constant C2 such that kukH ≤ C2 kukV .

(6.2.32)

In the case p ≥ 2, we infer from (6.2.31) that Z t 0

ku(t)kpH dτ

≤

C2p

Z t 0

kukpV dτ ≤ C3 .

(6.2.33)

Let y(t) = ku(t)kpH . Then (6.2.33) tells us that

Z t 0

y(τ )dτ ≤ C3 ,

(6.2.34)

(6.2.35)

Asymptotic Behavior of Solutions and Global Attractors

231

p−2 i.e., the first one in (6.2.2) is satisfied. Multiplying (6.2.30) by kukH , and using (6.2.31), we get 0 dy (6.2.36) ≤ C4 kf kpV 0 . dt This implies that condition (6.2.1) is satisfied with A1 = A2 = 0 and 0 h(t) = C4 kf kpV 0 . Thus, by Lemma 6.2.1, we immediately get (6.2.26). When 1 < p < 2, we deduce from (6.2.31) that

Z t 0

2−p kuk2H dτ ≤ sup ku(τ )kH

Z t

0≤τ ≤t

0

kukpH dτ ≤ C5 .

(6.2.37)

Thus, if we define y(t) = ku(t)k2H ,

(6.2.38)

then (6.2.37) implies that the first one of condition (6.2.2) is satisfied. In this case, (6.2.30) and (6.2.31) mean that condition (6.2.1) is satisfied 0 with A1 = A2 = 0, and h(t) = Ckf kpV 0 . Thus, (6.2.26) follows from Lemma 6.2.1. The theorem is proved. ¤ The following example shows the application to the study of asymptotic behavior for strong solutions of the Navier-Stokes equations with external forces (see [165]). Ex. 6.2.3. Consider the following initial value problem for the nonstationary Navier-Stokes equations in the whole space R2 :   ut − µ∆u + (u · ∇)u + ∇P = f,           

(x, t) ∈ R2 × R+ ,

∇ · u = 0, (x, t) ∈ R2 × R+ ,

(6.2.39)

u|t=0 = u0 (x), x ∈ R2

where the vector function u and the scalar function P are unknown, and u0 and f are given vector functions. The pressure P is determined up to a constant by the vector function u. In what follows we apply Lemma 6.2.1 to prove the following result. THEOREM 6.2.2 Let p, 2 < p < +∞ be given. Suppose that

u0 ∈ Lp ∩ L2 (R2 ),

∇ · u0 = 0,

and

f ∈ L1 ([0, +∞), Lp ∩ L2 (R2 )) ∩ L∞ ([0, +∞), Lp (R2 )).

232

NONLINEAR EVOLUTION EQUATIONS

Then problem (6.2.39) admits a unique solution u such that

u ∈ C([0, +∞), Lp ∩ L2 (R2 )) ∩ L2 ([0, +∞), H 1 (R2 )). Moreover, as t → +∞,

ku(t)kLp → 0.

(6.2.40)

Proof. Since our purpose for this example is to illustrate the application of Lemma 6.2.1, we will frequently refer to the existing results for the Navier-Stokes equations in the literature. Define H ≡ {u ∈ L2 , ∇ · u = 0},

V ≡ {u ∈ H 1 , ∇ · u = 0}.

(6.2.41)

Since problem (6.2.39) is the initial value problem for the non-stationary Navier-Stokes equations in two space dimensions, it is well known that for u0 ∈ H, f ∈ L1 ([0, +∞), L2 ), problem (6.2.39) admits a unique solution u such that u ∈ C([0, +∞), H) ∩ L2 ([0, +∞), V ), ut ∈ L1 ([0, +∞), H −1 ) (see, e.g., [150]). Moreover, the following energy estimates hold: for all t ≥ 0, ku(t)k ≤ ku0 k + µ

Z t 0

Z ∞ 0

kf (s)kds,

k∇u(s)k2 ds ≤ (ku0 k+

Z ∞ 0

(6.2.42)

kf (s)kds)

Z ∞ 0

kf (s)kds.(6.2.43)

Hereafter we denote by k · k the L2 norm. Estimates (6.2.42) and (6.2.43) can be easily obtained by multiplying the equation in (6.2.39) by u, then integrating over R2 . On the other hand, under the additional conditions on u0 and f as stated in this theorem, it follows from Theorem 2.2 in [26] on the local existence and uniqueness that the solution u also belongs to C([0, T0 ), Lp ∩ H) for some T0 > 0. Therefore, to prove that u ∈ C([0, +∞), Lp ), it suffices to prove that kukLp is uniformly bounded. First, by the Gagliardo-Nirenberg inequality stated in Chapter 1, we have kukLp ≤ Ck∇uk

p−2 p

2

kuk p

(6.2.44)

with C being a positive constant independent of u and t. Then we infer from (6.2.42)–(6.2.44) that Z ∞ 0

kukqLp dt ≤ C

(6.2.45)

2p . p−2

(6.2.46)

with q=

Asymptotic Behavior of Solutions and Global Attractors

233

Since n = 2, it turns out that 2 n + = 1. q p

(6.2.47)

Hence by Theorem 0.1 in [26], we can deduce that ku(t)kLp is uniformly bounded by a positive constant C depending only on µ and ku0 kLp , kf kL1 ([0,+∞),Lp ) ,

Z ∞ 0

kukqLp dt.

Thus, estimate (6.2.45) implies the global existence. In order to apply the strong version of Lemma 6.2.1 as stated in Remark 6.2.1, in what follows we want to show that y(t) = ku(t)kαLp p satisfies conditions (6.2.17) and (6.2.2). It easily follows with α = p−2 from p > 2 that α > q. Hence, we infer from (6.2.45) and uniform boundedness of ku(t)kLp that 2

Z ∞ 0

y(t)dt ≤

sup ku(t)kα−q Lp 0≤t<+∞

Z ∞ 0

kukqLp dt ≤ C,

(6.2.48)

i.e., the first one of condition (6.2.2) is satisfied. To verify that y(t) satisfies condition (6.2.17), in what follows we first proceed to estimate the approximate smooth solution, then pass to the limit. More precisely, we approximate u0 and f by a sequence of smooth divergence free vector fields uk0 and fk with compact support in x such that uk0 → u0

strongly in Lp ∩ L2 ,

fk → f

strongly in L1 ([0, +∞), Lp ∩ L2 ),

fk → f

weakly star in L∞ ([0, +∞), Lp ).

and Then for each fk , uk0 , since n = 2, the corresponding problem (6.2.39) for uk admits a unique global smooth solution uk . Now we proceed to obtain some estimates on uk . Multiplying the first equation in the corresponding problem (6.2.39) for uk by |uk |p−2 uk and integrating with respect to x, after integration by parts, we get 1 d p−2 kuk kpLp + µNp (uk ) + 4µ 2 Mp (uk ) = I1 + I2 , p dt p where Np (uk ) =

Z R

2

|∇uk |2 |uk |p−2 dx,

(6.2.49)

(6.2.50)

234

NONLINEAR EVOLUTION EQUATIONS Mp (uk ) =

Z

p

R2

|∇|uk | 2 |2 dx,

(6.2.51)

and I1 , I2 are integrals involving pk , and fk , respectively. Following Lemma 1.1 and Lemma 1.2 in [26], we have the estimates: (p − 2)2 kuk kp+2 Lp+2 , µ

|I1 | ≤ C

(6.2.52)

and p−1 |I2 | ≤ kfk kLp kuk kL p .

(6.2.53)

p

Let gk = |uk | 2 . By the Gagliardo-Nirenberg inequality for n = 2, we have kgk k

2

L

2(p+2) p

p

≤ Ck∇gk k p+2 kgk k p+2

(6.2.54)

i.e., 2

p p kuk kp+2 Lp+2 ≤ Ckuk kLp (Mp (uk )) .

(6.2.55)

Notice that for almost x ∈ R2 , p p p |∇|uk | 2 | ≤ |uk | 2 −1 |∇uk |. 2 Combining (6.2.56) with (6.2.55) yields

(6.2.56)

p2

2

p p−2 p kuk kp+2 Lp+2 ≤ Ckuk kLp (Np (uk )) ≤ εNp (uk ) + Cε kuk kLp

(6.2.57)

with ε being any small positive constant. Then we deduce from (6.2.49), (6.2.52), (6.2.53) and (6.2.57) that 2

p p−2 µ 1 d p−1 kuk kpLp + Np (uk ) + 4µ 2 Mp (uk ) ≤ Ckuk kLp−2 p + kfk kLp kuk kLp . p dt 4 p

Therefore, for α =

p2 p−2 ,

(6.2.58)

we have 2

p +2p d α−1 + kfk kLp kuk kL kuk kαLp ≤ C(kuk kLp−2 p p ). dt

(6.2.59)

Let yk (t) = kuk (t)kαLp .

(6.2.60)

Integrating (6.2.59) with respect to t yields that for any 0 ≤ s < t < +∞, yk (t) − yk (s) ≤ C

Z t s

p+2

(yk p + yk + kfk kαLp )dτ.

(6.2.61)

Asymptotic Behavior of Solutions and Global Attractors

235

On the other hand, by the energy estimates it is easy to conclude from the first equation of (6.2.39) that uk → u

in C([0, +∞), L2 ),

(6.2.62)

uk → u

in L2 ([0, +∞), H 1 ).

(6.2.63)

and Indeed, we can easily deduce from the first equation of (6.2.39) and the Gagliardo-Nirenberg inequality that 1d ku − uk k2 + µk∇(uk − u)k2 2 dt ≤ kuk − uk2L4 k∇uk k + kfk − f kkuk − uk ≤ Ckuk − ukk∇(uk − u)kk∇uk k + kfk − f kkuk − uk µ ≤ k∇(uk − u)k2 + C1 kuk − uk2 k∇uk k2 2 +kfk − f kkuk − uk, (6.2.64) which implies that

R

d kuk − uk ≤ C2 k∇uk k2 kuk − uk + kfk − f k. dt

(6.2.65)

Since 0t k∇uk k2 dτ is uniformly bounded, it follows from (6.2.65) that when k → +∞, for all t > 0, kuk − u(t)k ≤ (kuk0 − u0 k +

Z t 0

Z t

kfk − f kdτ ) · exp (

0

k∇uk k2 dτ ) → 0. (6.2.66)

Integrating (6.2.64) with respect to t, then combining the resultant with (6.2.66) yields that as k → +∞, for any t > 0, Z t 0

k∇(uk − u)k2 dτ → 0.

(6.2.67)

From the Gaglirado-Nirenberg inequality (6.2.44) we infer that uk → u

in Lq ([0, +∞), Lp )

(6.2.68)

2p where q = p−2 . Since kuk kLp and kukLp are uniformly bounded, as proved before, it follows from (6.2.68) that for any T > 0 and any 1 < q 0 < ∞,

in Lq (0, T ). 0

kuk kLp → kukLp

(6.2.69)

Since the strong convergence in Lq (0, T ) implies convergence almost 0

236

NONLINEAR EVOLUTION EQUATIONS

everywhere in t, passing to the limit in (6.2.61) yields y(t) − y(s) ≤ C

Z t s

(y

p+2 p

+ y + kf kαLp )dτ.

(6.2.70)

By continuity of y(t), (6.2.70) holds for any 0 ≤ s < t < +∞. This implies that condition (6.2.17) is satisfied. Thus, applying the strong version of Lemma 6.2.1 as stated in Remark 6.2.1, we get (6.2.40). The theorem is proved.

6.3

Convergence to Stationary Solutions

In this section we are concerned with convergence of the solution of a nonlinear evolution equation to a stationary solution as time goes to infinity. For simplicity of exposition, we consider the following problem:   u − ∆u = u − u3 ,   t         

(x, t) ∈ Ω × R+ ,

u|Γ = 0,

(6.3.1)

u|t=0 = u0 (x)

where Ω is a bounded domain in Rn , n ≤ 3 with smooth boundary Γ. Then the main theorem in this section is the following. THEOREM 6.3.1 For any given u0 ∈ H = H 2 ∩H01 , there is an equilibrium ψ ∈ C ∞ satisfying  3  −∆ψ = ψ − ψ , x ∈ Ω, (6.3.2) 

ψ|Γ = 0

such that the solution u to problem (6.3.1) converges to ψ in the following sense:

lim ku(·, t) − ψkH = 0.

t→+∞

(6.3.3)

The method used in this section was initiated by L. Simon and applies to the problem (6.1.10) with general f (x, u), which is analytic in u and differentiable in x (see, e.g., [67]). Before giving the detailed proof, we first introduce some basic concepts on gradient systems and the Lojasiewicz–Simon inequality.

Asymptotic Behavior of Solutions and Global Attractors

6.3.1

237

Some Basic Concepts

First let us consider the general situation. Let H be a Banach space. Suppose that for a given nonlinear evolution equation with or without boundary conditions, for any initial datum u0 ∈ H, there is a unique global solution u(t) ∈ H for all t ≥ 0 that defines a nonlinear C0 semigroup S(t): S(t) : u0 7→ u(t) = S(t)u0 ∈ H. When t → +∞, even if u stays bounded, as indicated in Section 6.1, various possibilities can happen. To study the asymptotic behavior of u(t) as time goes to infinity, it suffices to investigate the structure of ωlimit set ω(u0 ) defined in (6.1.11). We easily infer from the continuity of S(t) that ω(u0 ) is positively invariant, i.e., for all t ≥ 0, S(t)ω(u0 ) ⊂ ω(u0 ). A set X in H is called invariant if for all t ≥ 0, S(t)X = X. Under certain assumptions on the system (see Lemma 6.3.2 and Theorem 6.4.2 later on) X = ω(u0 ) is invariant. In short, under certain assumptions we can describe the long time behavior of the solution to nonlinear evolution equation by an invariant set X such that S(t)X = X,

∀t ≥ 0,

dist(u(t), X) ≡ inf d(u(t), y) → 0 y∈X

as t → +∞

(6.3.4) (6.3.5)

where d(x, y) is the distance of x, y ∈ H in H. Thus, if u converges to a stationary solution us as time goes to infinity, then the set X is reduced to {us }; if as time goes to infinity, u(t) − ϕ(t) → 0

(6.3.6)

where ϕ(t) is a periodic solution of the given nonlinear evolution equation (see [151]), then X is reduced to X = {ϕ(t), t ∈ R}

(6.3.7)

i.e., X is in fact a closed curve; if more generally, ϕ(t) = g(ω1 t, · · · , ωn t)

(6.3.8)

with g being periodic with period 2π in each variable, then the set X is still the orbit of ϕ given as in (6.3.7), but now this orbit lies on an n-dimensional torus. However, a chaotic behavior may also happen as

238

NONLINEAR EVOLUTION EQUATIONS

time goes to infinity where u(t) looks completely random for all time. Then the structure of the set X is very complicated. Typically, it can be a fractal set such as a Cantor set or the product of a Cantor set with an interval. In this situation, the flow shows a chaotic appearance. To make further study on X = ω(u0 ), we now introduce some basic concepts in the dynamical system. For u0 ∈ H, as mentioned before, the set [

[

S(t)u0 =

u(t)

t≥0

t≥0

is called the orbit starting from u0 . The ω-limit set ω(u0 ) is defined as follows. \ [

ω(u0 ) =

S(t)u0 ,

s≥0 t≥s

which is clearly equivalent to the previous definition (6.1.11). In the above the closure is taken in H. Similarly we can define the ω-limit set ω(A) with A being a set in H as follows: ω(A) =

\

E(s)

s≥0

where E(s) =

[

S(t)A.

t≥s

It is easy to see that ϕ ∈ ω(A) is equivalent to the following statement: there is a sequence ϕn ∈ A and a sequence tn → +∞ such that as n → +∞, S(tn )ϕn → ϕ. If u0 is a fixed point of S(t), i.e., S(t)u0 = u0 ,

∀t ≥ 0

then u0 is also called the stationary solution of the corresponding nonlinear evolution equation or its equilibrium. If S(t) is defined for all t ∈ R, then u(t) = S(t)u0 , t ∈ R is called a complete orbit through u0 . As seen in Chapter 2, for the hyperbolic problem, there is a complete orbit. For the ω-limit set ω(A), we have the following result. LEMMA 6.3.1 Suppose that S(t) is a nonlinear C0 -semigroup, and A is a nonempty set in

Asymptotic Behavior of Solutions and Global Attractors

239

H . Then ω(A) is positively invariant, i.e., for all t ≥ 0, S(t)ω(A) ⊂ ω(A).

(6.3.9)

Proof. For any fixed t > 0, if ψ ∈ S(t)ω(A), then there exists φ ∈ ω(A) such that ψ = S(t)φ.

(6.3.10)

By the definition of ω(A), there exists a sequence φn ∈ A and tn → +∞ such that S(tn )φn → φ.

(6.3.11)

Thus, S(t)S(tn )φn = S(tn )S(t)φn = S(tn )φ˜n → S(t)φ.

(6.3.12)

In the above, we denote φ˜n = S(t)φn .

(6.3.13)

ψ = S(t)φ ∈ ω(A)

(6.3.14)

Thus, follows. The lemma is proved.

¤

The following result gives a sufficient condition so that ω(A) is a nonempty, compact invariant set. LEMMA 6.3.2 Suppose that A is a nonempty set in H . If there is t0 > 0 such that [

S(t)A

t≥t0

is relatively compact in H , then ω(A) is a nonempty, compact invariant set. Furthermore, if A is connected, then ω(A) is also connected.

Proof. Since A is nonempty, the set

[

E(s) =

S(t)A

t≥s

is nonempty for each s ≥ 0. It turns out that the closure of E(s) is a nonempty compact set as s ≥ t0 , and it decreases as s increases. Hence, ω(A) =

\

s≥0

E(s)

240

NONLINEAR EVOLUTION EQUATIONS

is a nonempty compact set. In what follows we prove that ω(A) is invariant. Suppose that φ ∈ ω(A). Then there is a sequence φn ∈ A and tn → +∞ such that S(tn )φn → φ.

(6.3.15)

By the assumption, for any fixed t, when tn ≥ t + t0 , S(tn − t)φn is relatively compact in H. Hence, there is a subsequence of tn and φn , still denoted by tn and φn , and an element ψ ∈ H such that S(tn − t)φn → ψ.

(6.3.16)

This implies that ψ ∈ ω(A). However, we have S(tn )φn = S(t)S(tn − t)φn → S(t)ψ = φ,

(6.3.17)

which implies that φ ∈ S(t)ω(A), i.e., ω(A) ⊂ S(t)ω(A). Combining this with the result in Lemma 6.3.1 yields that ω(A) is invariant. It remains to prove that when A is connected, ω(A) =

\

E(s)

(6.3.18)

s≥0

is also connected. Since A is connected, and S(t) is continuous, it follows that E(s) is also connected. We now use a contradiction argument for the proof. If ω(A) is not connected, then there are two open sets U1 , U2 such that U1 ∩ U2 = ∅, ω(A) ⊂ U1 ∪ U2 ,

U1 ∩ ω(A) 6= ∅ U2 ∩ ω(A) 6= ∅.

Hence, there is a small constant ε > 0 and V1ε , the ε-neighborhood of U1 ∩ ω(A), and V2ε , the ε-neighborhood of U2 ∩ ω(A) such that V1ε ∩ V2ε = ∅,

V1ε ∩ ω(A) 6= ∅,

V2ε ∩ ω(A) 6= ∅,

and V1ε ∪ V2ε contains the ε-neighborhood of ω(A). We infer from (6.3.18) that there is s0 > 0 such that when s ≥ s0 , E(s) enters the ε-neighborhood of ω(A) ⊂ V1ε ∪ V2ε . This contradicts the fact that E(s) is connected. The lemma is proved. ¤ When A = {x}, we have the similar result. COROLLARY 6.3.1 Suppose that H is a complete metric space and S(t) is a nonlinear C0 semigroup defined on H . Let x ∈ H . If there is t0 ≥ 0 such that [

S(t)x

t≥t0

Asymptotic Behavior of Solutions and Global Attractors

241

is relatively compact in H , then the ω -limit set ω(x) is a compact, connected invariant set.

6.3.2

Gradient Systems

DEFINITION 6.3.1 Suppose that H is a complete metric space and S(t) is a nonlinear C0 -semigroup defined on H . A continuous function V : H 7→ R is called a Lyapunov function with respect to S(t) if the following two conditions are satisfied: (i) For any x ∈ H , V (S(t)x) is monotone non-increasing in t. (ii) V (x) is bounded from below, i.e., there is a constant C such that for all x ∈ H , V (x) ≥ C .

The following result is an important property of the Lyapunov function. LEMMA 6.3.3 Suppose that H is a complete metric space and S(t) is a nonlinear C0 semigroup defined on H . If there is a Lyapunov function V , then for x ∈ H , V is constant on ω(x).

Proof. Since V (S(t)x) is monotone non-increasing in t, and V is bounded from below, it follows that lim V (S(t)x)

t→+∞

exists, and we denote it by V∞ . For any ψ ∈ ω(x), there is a sequence tn → +∞ such that S(tn )x → ψ. By the continuity of V , we deduce that V∞ =

lim V (S(tn )x) = V (ψ).

tn →+∞

Thus, the lemma is proved.

(6.3.19) ¤

DEFINITION 6.3.2 Suppose that H is a complete metric space, S(t) is a nonlinear C0 -semigroup defined on H and V (x) is a Lyapunov function. Then the nonlinear semigroup S(t), or more precisely system (H, S(t), V ) is called a gradient system if the following conditions are satisfied:

242

NONLINEAR EVOLUTION EQUATIONS

(i) for any x ∈ H , there is t0 > 0 such that [

S(t)x

t≥t0

is relatively compact in H ; (ii) if for t > 0, V (S(t)x) = V (x), then x is a fixed point of the semigroup S(t). Accordingly, the orbit u(t) is called a gradient flow.

Then we have the following result. THEOREM 6.3.2 Suppose that (H, S(t), V ) is a gradient system. Then for any x ∈ H , ω(x) is a connected compact invariant set, and it consists of the fixed points of S(t).

Proof. By Corollary 6.3.1, ω(x) is a connected, compact invariant set. For any ψ ∈ ω(x), S(t)ψ ∈ ω(x). We deduce from Lemma 6.3.3 that V (ψ) = V (S(t)ψ) for all t ≥ 0. Since (H, S(t), V ) is a gradient system, it follows that ψ is a fixed point of S(t). The theorem is proved. ¤ COROLLARY 6.3.2 Suppose that (H, S(t), V ) is a gradient system and the set of fixed points of S(t) is discrete. Then for any x ∈ H , as t → +∞, S(t)x converges to an equilibrium.

In what follows we prove that problem (6.3.1) defines a gradient system. It follows from Chapter 2 that for u0 ∈ H 2 ∩ H01 , problem (6.3.1) admits a unique global solution u such that u ∈ C([0, +∞), H 2 ∩ H01 ) ∩ C 1 ([0, +∞), L2 ). Let H = H 2 ∩ H01 (Ω). Then the solution u of problem (6.3.1) defines a C0 -semigroup on H. If we multiply the equation of (6.3.1) by ut and integrate over Ω, then we get Z

d 1 1 1 ( |∇u|2 + u4 − u2 )dx + kut k2 = 0. dt Ω 2 4 2 It is easy to see that the functional E defined as follows Z 1 1 1 E(u) = ( |∇u|2 + u4 − u2 )dx 4 2 Ω 2

(6.3.20)

(6.3.21)

is a Lyapunov function defined on H. Indeed, condition (i) in Definition 6.3.1 is clearly satisfied. Condition (ii) in Definition 6.3.1 follows from

Asymptotic Behavior of Solutions and Global Attractors

243

the following Young inequality: 1 u2 ≤ u4 + 1. 4 Integrating (6.3.20) with respect to t yields E(u(t)) +

Z t 0

kut k2 dτ = E(u0 ).

(6.3.22)

This indicates that if there is t0 > 0 such that E(u(t0 )) = E(S(t0 )u0 ) = E(u0 ), then for all 0 ≤ t ≤ t0 , ut = 0. Therefore, u0 must be an equilibrium. To prove that (H, S(t), E) is a gradient system, it remains to verify that there is t0 > 0 such that [

S(t)u0

t≥t0

is relatively compact in H. We use a density argument. First we assume that u0 ∈ D(A2 ), i.e., u0 ∈ H, ∆u0 ∈ H. Then by the result in Chapter 2, we have a more regular global solution u such that u ∈ C([0, +∞), D(A2 )) ∩ C 1 ([0, +∞), H) ∩ C 2 ([0, +∞), L2 ). We differentiate the equation in (6.3.1) with respect to t, then multiply the resultant by ut and integrate over Ω to get Z 1d kut k2 + k∇ut k2 + 3 u2 u2t dx = kut k2 . (6.3.23) 2 dt Ω Integrating with respect to t and using the equation in (6.3.1), as shown in Chapter 2 and Ex. 6.2.1, we can get Z t 0

k∇ut k2 dτ ≤ C, ku(t)kH 2 ≤ C ∀t ≥ 0.

(6.3.24)

Hereafter C is a positive constant depending only on ku0 kH 2 . Differentiating the equation in (6.3.1), then multiplying the resultant by −∆ut , and integrating over Ω, we get 1d k∇ut k2 + k∆ut k2 2 dt Z =−

It follows that

Ω

(ut − 3u2 ut )∆ut dx

≤ Ckut kk∆ut k 1 C2 ≤ k∆ut k + kut k2 . 2 2 d k∇ut k2 + k∆ut k2 ≤ C 2 kut k2 . dt

(6.3.25)

(6.3.26)

244

NONLINEAR EVOLUTION EQUATIONS

Multiplying (6.3.26) by t, then integrating with respect to t yields tk∇ut k2 + ≤

Z t 0

Z t 0

τ k∆ut k2 dτ

k∇ut k2 dτ + C 2

≤ C + C 2t

Z t

≤ C + C 3 t.

Z t 0

τ kut k2 dτ

kut k2 dτ

0

∀t > 0

(6.3.27)

Thus, for t ≥ δ > 0, we have C C + C 3 ≤ Cδ = + C 3. (6.3.28) t δ We can deduce from the equation in (6.3.1) and the regularity theorem for the elliptic problem k∇ut k2 ≤

 3  −∆u = u − u − ut , 

x∈Ω

u|Γ = 0,

(6.3.29)

stated in Chapter 1 that ku(t)kH 3 ≤ C(kut kH 1 + ku − u3 kH 1 ) ≤ Cδ0 ,

∀t ≥ δ > 0.

(6.3.30)

In the case that u0 ∈ H, by the density theorem, there is a sequence u0n ∈ D(A2 ) such that u0n converges to u0 in H. It easily follows that (6.3.30) still holds as u0 ∈ H. This proves that [

S(t)u0

t≥δ

is relatively compact in H. Therefore, the system defined by problem (6.3.1) is a gradient system. In the one-dimensional case, the corresponding stationary problem to problem (6.3.1) becomes a two-points boundary value problem for a nonlinear ordinary differential equation:   −uxx = f (x, u), 

u|x=a = ux=b = 0.

x ∈ (a, b)

(6.3.31)

H. Matano in 1978 proved that the set of equilibria is discrete, and any bounded global solution to problem (6.1.10) must converge to an equilibrium as time goes to infinity. The later conclusion is a consequence of Corollary 6.3.2 and the fact that set of equilibria is discrete since

Asymptotic Behavior of Solutions and Global Attractors

245

the system defined by problem (6.1.10) can be verified to be a gradient system as before. However, as mentioned in Section 6.3.1, in a higher space dimension case the ω-limit set may be a continued set. It turns out that Corollary 6.3.2 does not apply to problem (6.1.10) in a higher space dimension case. Since f = u − u3 is an analytic function in u, in what follows we use a method that was initiated by L. Simon [141] and recently well developed, to prove Theorem 6.3.1.

6.3.3

The Lojasiewicz–Simon Inequality

We first introduce the Lojasiewicz inequality for the analytic function defined on Rm : Suppose that F : Rm 7→ R is an analytic function near its critical point a (i.e., ∇F (a) = 0). Then there is a positive constant σ and θ ∈ (0, 12 ) depending on a such that when kx − akRm ≤ σ, |F (x) − F (a)|1−θ ≤ k∇F (x)kRm .

(6.3.32)

For the details of the proof, we refer to [99], [100], and [101]. In the case m = 1, by Taylor’s expansion near x = a: F (x) − F (a) =

F (n) (ξ) (x − a)n , n!

and ∇F (x) = F 0 (x) =

F (n) (η) (x − a)n−1 (n − 1)!

with n ≥ 2, the assertion easily follows with θ = small positive constant chosen to satisfy

n−1 n

− ε where ε is a

¯ ¯ ¯ ¯ ¯ (n) (ξ) ¯ ¯ F (n) (η) ¯ ¯ ¯ ¯ |x − a| ¯ ¯≤¯ ¯ ¯ n! ¯ ¯ (n − 1)! ¯ ε ¯F

in the small neighborhood |x − a| ≤ σ. Notice that if (6.3.32) is relaxed to be the following one |F (x) − F (a)|1−θ ≤ Ck∇F (x)kRm

(6.3.33)

with a positive constant C, then ε can be chosen as zero, and θ is allowed to be 21 . The following, which is now called the Lojasiewicz–Simon inequality, is an extension of the Lojasiewicz inequality to the infinite-dimensional space.

246

NONLINEAR EVOLUTION EQUATIONS

LEMMA 6.3.4 Let E be the set of equilibrium points of problem (6.3.1):

E = {φ | ∆φ + φ − φ3 = 0, φ|Γ = 0} and ψ ∈ E . Then there is a small positive constant σ and θ ∈ (0, pending on ψ such that for all u ∈ H = H 2 ∩ H01 , ku − ψkH ≤ have

k∆u + u − u3 k ≥ |E(u) − E(ψ)|1−θ

1 2)

deσ , we

(6.3.34)

where E(u) is defined by (6.3.21).

Proof. Let us first consider the linearized problem of (6.3.2) near the equilibrium ψ:  2  Lw ≡ −∆w − w + 3ψ w = 0, 

x ∈ Ω,

w|Γ = 0.

(6.3.35)

It is easy to see that the operator L defined on H 2 ∩ H01 ⊂ L2 is a selfadjoint operator. It follows from the theory of linear elliptic boundary problems that there is a positive constant λ > 0 such that λI + L is invertible. Therefore, there is a Fredholm alternative result for the problem:   Lw = h, 

x ∈ Ω,

w|Γ = 0.

(6.3.36)

More precisely, if Ker(L) = ∅, then for any h ∈ L2 , problem (6.3.36) admits a unique solution w ∈ H 2 ∩ H01 ; otherwise, dim(Ker(L)) is finite, and the necessary and sufficient condition for the solvability of problem (6.3.36) is h ∈ (Ker(L))⊥ . Let (φ1 , · · · , φm ) be the normalized orthogonal basis of Ker(L) in L2 , and Π be the projection from L2 onto Ker(L). Define the operator L˜ from H = H 2 ∩ H01 into L2 as follows: ˜ = Πw + Lw. Lw (6.3.37) Then L˜ : H 7→ L2 is a one-to-one and onto operator. Now we define u = v + ψ and M (v) ≡ −∆u − u + u3 : H 7→ L2 .

(6.3.38)

Asymptotic Behavior of Solutions and Global Attractors

247

It is easy to see that DM (0) = L where DM denotes the Frechet derivative of M . Let N v = M (v) + Πv. Then ˜ DN (0) = L. Since L˜ is a one-to-one and onto operator, by the local inversion theorem in nonlinear analysis, there is a neighborhood W1 (0) of the origin in H and a neighborhood W2 (0) of the origin in L2 , and an inverse mapping Ψ : W2 (0) 7→ W1 (0) such that N (Ψ(g)) = g,

∀g ∈ W2 (0),

(6.3.39)

Ψ(N (v)) = v,

∀v ∈ W1 (0).

(6.3.40)

and u3

Since f = u − is analytic in u, the operators M , N and its inverse mapping Ψ are also analytic. Furthermore, there is a positive constant C such that kΨ(g1 ) − Ψ(g2 )kH ≤ Ckg1 − g2 kL2 ,

∀g1 , g2 ∈ W2 (0),

(6.3.41)

kN (v1 ) − N (v2 )kL2 ≤ Ckv1 − v2 kH ,

∀v1 , v2 ∈ W1 (0).

(6.3.42)

and Hereafter we use C to denote a positive constant that may vary in different places. Let ξ = (ξ1 , · · · , ξm ) ∈ Rm ,

Πv =

m X j=1

When ξ is sufficiently small, clearly we have m X

ξj φj ∈ W2 (0).

j=1

Now define Γ : Rm 7→ R as follows. Γ(ξ) = E(Ψ(

m X

j=1

ξj φj ) + ψ).

ξj φj .

248

NONLINEAR EVOLUTION EQUATIONS

Clearly, Γ(ξ) is analytic in a small neighborhood of the origin in Rm . We deduce from (6.3.39) that in the small neighborhood of the origin, we have DN (v) · DΨ(g) = I,

(6.3.43)

i.e., for v ∈ W1 (0), DN (v) ∈ L(H, L2 ), and for g ∈ W2 (0), DΨ(g) ∈ L(L2 , H). A straightforward calculation shows that m X ∂Γ = DE(Ψ( ξj φj ) + ψ) · DΨ · φj . ∂ξj j=1

(6.3.44)

On the other hand, we infer from (6.3.21) that DE(w) · v =

Z

Ω

(∇w · ∇v + (w3 − w)v)dx.

(6.3.45)

In particular, for w ∈ H, by integration by parts, we have DE(w) · v =

Z

Ω

(−∆w + w3 − w)vdx.

(6.3.46)

Since ψ is an equilibrium, it follows from (6.3.44) that ξ = 0 is the critical point of Γ(ξ). Thus, for all j, 1 ≤ j ≤ m, we infer from (6.3.44), (6.3.38), (6.3.46), and kφj k = 1 that ¯ ¯ m m ¯ ∂Γ ¯ X X ¯ ¯ ξj φj ))kkDΨ( ξj φj )kL(L2 ,H) ¯ ¯ ≤ kM (Ψ( ¯ ∂ξj ¯ j=1 m X

≤ CkM (Ψ(

j=1

ξj φj ))k.

(6.3.47)

j=1

Since Πv =

m X

ξj φj ,

j=1

we have |∇Γ(ξ)| ≤ CkM (Ψ(Πv))k = CkM (Ψ(Πv)) − M (v) + M (v)k ≤ C(kM (v)k + kM (Ψ(Πv)) − M (v)k).

(6.3.48)

Asymptotic Behavior of Solutions and Global Attractors

249

It follows from N (v) = M (v) + Πv and (6.3.42) that kM (v1 ) − M (v2 )k ≤ kΠv1 − Πv2 k + kN (v1 ) − N (v2 )k ≤ kv1 − v2 k + Ckv1 − v2 kH ≤ (C + 1)kv1 − v2 kH . (6.3.49) Similarly, we have kΨ(Πv) − vkH = kΨ(Πv) − Ψ(N (v))kH ≤ CkΠv − N (v)k = CkM (v)k. Therefore, we can deduce from (6.3.48)–(6.3.50) that

(6.3.50)

|∇Γ(ξ)| ≤ CkM (v)k + (C + 1)kΨ(Πv) − vkH = CkM (v)k + (C + 1)kΨ(Πv) − Ψ(N (v))kH ≤ CkM (v)k + (C + 1)CkΠv − N (v)k = CkM (v)k + C(C + 1)kM (v)k ≤ CkM (v)k. (6.3.51) On the other hand, for t ∈ [0, 1], when v ∈ W1 (0), v + t(Ψ(Πv) − v) ∈ W1 (0). For u = v + ψ, we infer from (6.3.49) and (6.3.50) that |E(u) − Γ(ξ)| = |E(u) − E(Ψ(Πv) + ψ)| ¯ ¯Z 1 ¯ ¯ d ¯ E(u + (1 − t)(Ψ(Πv) − v))dt¯¯ =¯ 0 dt ¯Z 1 ¯

= ¯¯

0

¯ ¯

DE · (Ψ(Πv) − v)dt¯¯

≤ max kM (v + (1 − t)(Ψ(Πv) − v))kkΨ(Πv) − vk 0≤t≤1

≤ (kM (v)k + (C + 1)kΨ(Πv) − vkH ) · kΨ(Πv) − vkH ≤ (kM (v)k + C(C + 1)kM (v)k) · CkM (v)k ≤ CkM (v)k2 . (6.3.52) By the Lojasiewicz inequality, there is a small constant σ > 0 and θ ∈ (0, 12 ) such that |∇Γ(ξ)| ≥ |Γ(ξ) − Γ(0)|1−θ .

(6.3.53)

Thus, we can choose σ smaller, if necessary, so that when |ξ| ≤ σ, Πv =

m X

ξj φj ∈ W2 (0). We infer from (6.3.53) and the definition of

j=1

Γ(ξ) that

|∇Γ(ξ)| ≥ |Γ(ξ) − E(ψ)|1−θ .

(6.3.54)

250

NONLINEAR EVOLUTION EQUATIONS

Applying the elementary inequality |a + b|1−θ ≤ 2|a|1−θ + |b|1−θ , we can deduce from (6.3.51) and (6.3.52) that CkM (v)k ≥ |∇Γ(ξ)| 1 1 ≥ |E(u) − E(ψ)|1−θ − |Γ(ξ) − E(u)|1−θ 2 2 1 1 1−θ − C 1−θ kM (v)k2(1−θ) . (6.3.55) ≥ |E(u) − E(ψ)| 2 2 1 Since θ < 2 , we have 2(1 − θ) > 1. Then it follows from (6.3.55) that when v ∈ W2 (0), kM (v)k ≥ C|E(u) − E(ψ)|1−θ .

(6.3.56)

Let ε be a given sufficiently small positive constant. Then we can choose σ smaller, if necessary, so that when kvkH ≤ σ, C|E(u) − E(ψ)|−ε ≥ 1.

(6.3.57)

Combining (6.3.57) with (6.3.56) yields kM (v)k ≥ |E(u) − E(ψ)|1−θ

0

(6.3.58)

with

1 0 < θ0 = θ − ε < . 2 Thus, the lemma is proved. REMARK 6.3.1

If (6.3.34) is replaced by

k∆u + u − u3 k ≥ C|E(u) − E(ψ)|1−θ , then the possible range of θ becomes θ ∈ (0, of the Lojasiewicz inequality.

6.3.4

¤

1 2 ],

(6.3.59)

as noticed for the statement

Proof of Theorem 6.3.1

We now turn to prove Theorem 6.3.1. The whole proof consists of several steps. 1. As proved in Section 6.3.2, problem (6.3.1) defines a gradient system. Therefore, it follows from Theorem 6.3.2 that for any given initial datum u0 ∈ H, the ω-limit set ω(u0 ) consists of equilibria. Hence there is at least one point ψ ∈ ω(u0 ) and a sequence tn → +∞ such that u(·, tn ) → ψ

in H.

(6.3.60)

Asymptotic Behavior of Solutions and Global Attractors

251

In what follows we prove that ω(u0 ) consists of only one point ψ and (6.3.3) holds. 2. Multiplying the equation in (6.3.1) by ut , then integrating over Ω yields dE(u) + kut k2 = 0. dt Then integrating with respect to t yields Z +∞ 0

If we can prove that

kut k2 dτ < +∞.

Z +∞ 0

kut kdτ < +∞,

then it follows from ku(t) − u(s)k ≤

Z t s

kut (τ )kdτ

(6.3.61)

(6.3.62)

(6.3.63)

(6.3.64)

that as t → +∞, u(t) converges to ψ in L2 . By the relative compactness of [

u(·, t),

t≥t0

we obtain the desired conclusion (6.3.3). Unfortunately, (6.3.62) does not imply the needed formula (6.3.63). Then the Lajasiewicz–Simon inequality stated in Lemma 6.3.4 plays an important role here. 3. Since E(u(t)) is decreasing in t, for all t ≥ 0, we have E(u(t)) − E(ψ) ≥ 0. If there is t0 ∈ R+ such that E(u(t0 )) = E(ψ), then it follows from (6.3.22) that for all t ≥ t0 , ut = 0 and u = ψ, and we are done. Hence, in the following we need only to consider the case that E(u(t)) > E(ψ) for all t ≥ 0. Owing to the equation in (6.3.1), we can rewrite (6.3.61) in the form: d (E(u) − E(ψ)) + kut kk − ∆u + u3 − uk = 0. (6.3.65) dt For θ > 0 being the number appearing in Lemma 6.3.4, we have d d (E(u) − E(ψ))θ = θ(E(u) − E(ψ))θ−1 (E(u) − E(ψ)). dt dt

(6.3.66)

252

NONLINEAR EVOLUTION EQUATIONS

Thus, it follows from (6.3.65) that d (E(u)−E(ψ))θ +θ(E(u)−E(ψ))θ−1 k−∆u+u3 −ukkut k = 0. (6.3.67) dt This implies that if there is a sufficiently large T > 0 such that for all t > T , ku − ψkH ≤ σ, then by Lemma 6.3.4, the following Lojasiewicz– Simon inequality holds: k − ∆u + u3 − uk ≥ (E(u) − E(ψ))1−θ .

(6.3.68)

Thus, we can infer from (6.3.67) that kut k ∈ L1 (T, +∞), and by the previous discussion in step 2, we are done. 4. We now use a simplified argument proposed by M.A. Jendoubi in [67]. This argument is based on the observation that after a certain time, the orbit will fall into a small neighborhood of ψ and stay there forever. Since u(·, tn ) → ψ in H, E(u(tn )) → E(ψ). Hence, for any ε > 0, ε < σ, there is an integer N such that when n ≥ N , ε 1 ε ku(·, tn ) − ψkH ≤ , (E(u(tn )) − E(ψ))θ ≤ . (6.3.69) 2 θ 2 For n ≥ N , let t¯n = sup{t ≥ tn | ku(·, s) − ψkH < σ, ∀s ∈ [tn , t]}. If there is an integer n0 ≥ N such that t¯n0 = +∞, then for t ≥ tn0 , by the previous discussion in step 3, we have kut k ∈ L1 (tn0 , +∞), and we are done. Otherwise, we have t¯n < +∞ for all n ≥ N . In what follows we prove that it is impossible. By the Lojasiewicz–Simon inequality stated in Lemma 6.3.4, for t ∈ [tn , t¯n ], we have −

d d (E(u) − E(ψ))θ = −θ E(u) · (E(u) − E(ψ))θ−1 dt dt = θkut kk − ∆u + u3 − uk(E(u) − E(ψ))θ−1 ≥ θkut k. (6.3.70)

Integrating with respect to t yields Z t¯n tn

1 kut kdτ ≤ (E(u(tn )) − E(ψ))θ . θ

It turns out from (6.3.69) and (6.3.71) that for all n ≥ N , ku(t¯n ) − ψk ≤

Z t¯n tn

kut kdτ + ku(tn ) − ψk

(6.3.71)

Asymptotic Behavior of Solutions and Global Attractors

253

1 ≤ (E(u(tn )) − E(ψ))θ + ku(tn ) − ψk θ ≤ ε,

(6.3.72)

i.e., as n → +∞, u(t¯n ) → ψ

in L2 .

By the relative compactness of the orbit, there is a subsequence of u(t¯n ), still denoted by itself, such that u(t¯n ) → ψ

in H.

≥ N such that when n ≥ N 0 , σ ku(t¯n ) − ψkH < , 2 ¯ which contradicts the definition of tn . Thus the theorem is proved. ¤

Hence, there is an integer

N0

The following corollary shows the decay rate of the solution u converging to ψ as time goes to infinity. COROLLARY 6.3.3 If θ = 21 in the Lojasiewicz–Simon inequality stated in Lemma 6.3.4, then the solution u to problem (6.3.1) converges to ψ in L2 exponentially as time goes to infinity. More precisely,

ku(·, t) − ψk ≤ C0 e−C1 t

(6.3.73)

with two positive constants C0 , C1 . On the other hand, if θ ∈ (0, 12 ) in the Lojasiewicz–Simon inequality stated in Lemma 6.3.4, then the solution u to problem (6.3.1) converges to ψ in L2 at a polynomial rate. More precisely, as t → +∞, θ

ku(·, t) − ψk ≤ C(1 + t)− 1−2θ

(6.3.74)

where C is a positive constant.

Proof. It follows from Lemma 6.3.4 that when t is large enough, we have k − ∆u + u3 − uk = kut k ≥ C|E(u) − E(ψ)|1−θ .

(6.3.75)

Let y(t) = (E(u) − E(ψ))θ . We infer from (6.3.67) that dy + Ckut k ≤ 0. dt

(6.3.76)

254

NONLINEAR EVOLUTION EQUATIONS

Therefore, integrating with respect to t yields Z +∞

1 y(t). C t On the other hand, combining (6.3.76) with (6.3.75) yields ku(·, t) − ψk ≤

kut (τ )kdτ ≤

(6.3.77)

1−θ dy (6.3.78) + Cy θ ≤ 0. dt When θ = 12 , (6.3.73) immediately follows from (6.3.78) and (6.3.77). For 0 < θ < 12 , the proof of (6.3.74) is more involved, and we use a comparison principle for nonlinear ordinary equations. Consider the following initial value problem for a nonlinear ordinary differential equation:  1−θ   dz + Cz θ = 0,  dt (6.3.79)

  

z(0) = y(0) > 0.

A straightforward calculation shows that θ

z(t) = (C1 + C2 t)− 1−2θ

(6.3.80)

where C1 = y(0)

2θ−1 θ

,

Let

C2 =

C(1 − 2θ) . θ

w(t) = z(t) − y(t). Then w satisfies

 dw   + Cf 0 (ξ)w ≥ 0,    

dt

(6.3.81)

(6.3.82)

(6.3.83)

w(0) = 0

where

1 − θ 1−2θ ξ θ , (6.3.84) θ and ξ stays between y and z. Therefore, it follows from (6.3.83) that f 0 (ξ) = Rt

w(t)e

0

Cf 0 (ξ(τ ))dτ

≥ 0.

(6.3.85)

Thus, w(t) ≥ 0,

∀t ≥ 0.

(6.3.86)

Combining (6.3.86) with (6.3.80) yields (6.3.74). The proof is completed. ¤

Asymptotic Behavior of Solutions and Global Attractors

255

REMARK 6.3.2 Using the equation in (6.3.1), we can proceed to obtain the same decay rate for the higher-order norms of the solution u. We leave the proof of this point to the reader.

6.4

Global Attractors

In this section we are concerned with the asymptotic behavior of solutions to nonlinear evolution equations in the second category. More precisely, we would like to study the existence of a global attractor. DEFINITION 6.4.1 Suppose that H is a complete metric space, and S(t) is a nonlinear C0 -semigroup of operators defined on H . A set A ⊂ H is called an attractor if the following hold: (i) A is an invariant set, i.e.,

S(t)A = A,

∀t ≥ 0.

(ii) A possesses an open neighborhood U such that for any element u0 ∈ U , as t → +∞, S(t)u0 converges to A, i.e.,

dist(S(t)u0 , A) = inf d(S(t)u0 , y) → 0. y∈A

If A is an attractor, then the maximal open set U satisfying (ii) is called the basin of attraction of A. According to the above definition, it can be also said that A attracts points of U. If a subset B ⊂ U satisfies d(S(t)B, A) ≡ sup

inf d(x, y) → 0,

x∈S(t)B y∈A

as t → +∞,

then we say that A uniformly attracts B, or simply A attracts B. DEFINITION 6.4.2 If A is a compact attractor, and it attracts bounded sets of H , then A is called a global or universal attractor.

It is easy to see that a global attractor is maximal among all bounded attractors or bounded invariant sets in the sense of inclusion. Hence, a global attractor is also called a maximal attractor. In order to establish the existence of a global attractor, a crucial step is to show the existence of an absorbing set.

256

NONLINEAR EVOLUTION EQUATIONS

DEFINITION 6.4.3 Suppose that B0 is a subset in H , and U is an open set containing B0 . If for any bounded set B ⊂ U , there exists t1 (B) ≥ 0 such that when t ≥ t1 (B), S(t)B ⊂ B0 , then we say that B0 is absorbing in U . Sometimes we also say that B0 absorbs bounded sets of U .

It can be seen from the definitions for global attractor and absorbing set that for a nonlinear C0 -semigroup S(t), the existence of a global attractor implies the existence of a bounded absorbing set. As a matter of fact, for any ε > 0, let Vε be the ε-neighborhood of A: Vε =

[

B(x, ε)

x∈A

where B(x, ε) denotes a ball with center x and radius ε. Then Vε is a bounded set. Since A is a global attractor, then for any bounded set B, d(S(t)B, A) → 0.

as t → +∞

Therefore, for any ε > 0, there is t(ε) > 0 such that when t ≥ t(ε), ε d(S(t)B, A) ≤ . 2 Thus, S(t)B ⊂ Vε . This implies that Vε is an absorbing set. Notice that an absorbing set is not necessarily required to be an invariant set. In contrast, we have the following result. THEOREM 6.4.1 Suppose that H is a Banach space and S(t) is a nonlinear C0 -semigroup defined on H , satisfying the following conditions: (i) there exists a bounded absorbing set B0 ; (ii) for any bounded set B , there is t0 (B) ≥ 0 depending on B such that [

S(t)B

t≥t0 (B)

is relatively compact in H . Then A = ω(B0 ) is a global attractor.

Proof. If conditions (i), (ii) are satisfied, then by Lemma 6.3.2, A = ω(B0 ) is a nonempty, compact invariant set. To prove that A is a global attractor, we use a contradiction argument. If there is a bounded set B such that when time goes to infinity, dist(S(t)B, A) does not converge

Asymptotic Behavior of Solutions and Global Attractors

257

to zero, then there exists δ > 0 and a sequence tn → +∞ such that dist(S(tn )B, A) ≥ δ > 0. Thus, for every n, there is bn ∈ B such that δ > 0. (6.4.1) 2 Since B0 is an absorbing set, it follows that when n is large enough (i.e., when tn ≥ t1 (B)), dist(S(tn )bn , A) ≥

S(tn )B ⊂ B0 . It turns out that S(tn )bn ∈ B0 . By condition (ii), S(tn )bn is relatively compact. Hence, there is a subsequence of tn , still denoted by itself, and a point β ∈ H such that β=

lim S(tn )bn =

tn →+∞

lim S(tn − t1 (B))S(t1 (B))bn .

tn →+∞

Then it follows from S(t1 (B))bn ∈ B0 that β ∈ A = ω(B0 ) which contradicts (6.4.1). Thus the theorem is proved. REMARK 6.4.1

¤

The global attractor A = ω(B0 ) is connected.

Proof. If B0 is an absorbing set, then a ball B containing B0 is clearly connected, and it is also an absorbing set. Since A = ω(B0 ) is maximal, we have A = ω(B). By Lemma 6.3.2, A is connected. ¤ Condition (ii) in Theorem 6.4.1 can be weakened to a certain extent. More precisely, we have the following result. THEOREM 6.4.2 Suppose that H is a Banach space and S(t) is a nonlinear C0 -semigroup defined on H , satisfying the following conditions: (i) there exists a bounded absorbing set B0 ; (ii)’ for any t ≥ 0, S(t) can be written in the form

S(t) = S1 (t) + S2 (t) where S1 (t) satisfies the condition (ii) in Theorem 6.4.1, and S2 (t) is a continuous mapping from H into H , and satisfies the following condition:

γK (t) = sup kS2 (t)φkH → 0, φ∈K

as t → +∞

258

NONLINEAR EVOLUTION EQUATIONS

where K is any bounded set in H . Then A = ω(B0 ) is a global attractor and it is connected.

Proof. First we prove that A = ω(B0 ) is a nonempty compact invariant set. To this end, we notice that for any φn ∈ B0 and tn → +∞, we have kS2 (tn )φn k → 0. Thus, S1 (tn )φn is convergent if and only if S(tn )φn is convergent. It turns out from the proof of Lemma 6.3.2 that A = ω(B0 ) is a nonempty compact invariant set. The conclusion that ω(B0 ) attracts bounded sets can be proved in the same way as that in Theorem 6.4.1 provided that we use the fact that for bn being bounded, S2 (tn )bn → 0, and, hence, the relative compactness of S1 (tn )bn implies the same assertion for S(tn )bn . Finally, that A is connected follows from Remark 6.4.1. Thus, the theorem is proved. ¤ In what follows we present two examples to show the applications of Theorem 6.4.1 and Theorem 6.4.2. Ex.6.4.1. We still consider the following initial boundary value problem for a nonlinear parabolic equation:   ut − ∆u = u − u3 ,           

(x, t) ∈ Ω × R+

u|Γ = 0,

(6.4.2)

u|t=0 = u0 (x)

where Ω is a bounded domain in Rn , n ≤ 3 with a smooth boundary Γ. Then we have the following result concerning the existence of a global attractor for the semigroup defined by problem (6.4.2). THEOREM 6.4.3 Let H = H 2 ∩ H01 . Then problem (6.4.2) has a global attractor A in H that is compact and connected.

Proof. As proved in the previous section, for any u0 ∈ H, this problem admits a unique global solution, and the corresponding nonlinear semigroup defined by the solution u: u(t) = S(t)u0

Asymptotic Behavior of Solutions and Global Attractors

259

is a C0 -semigroup on H. Moreover, for any bounded set B and for any δ > 0, [

S(t)B

t≥δ

is bounded in H 3 . It turns out that the condition (ii) in Theorem 6.4.1 is satisfied. To prove that problem (6.4.2) admits a compact connected global attractor, it remains to prove that there is an absorbing ball. Notice that by the definition, the radius of an absorbing ball should be independent of the bounded set B, and the time that all orbits starting from B enter this absorbing ball should be uniform. It turns out that the a priori estimates carried out in Chapter 2, which are good for the proof of global existence, are not suitable for the proof of the existence of an absorbing set because positive constants appearing in the proof do depend on the norm of the initial data in H. Therefore, in order to prove the existence of an absorbing ball, one often tries to establish the following type of differential inequality: dE + C1 E(t) ≤ C2 , ∀t ≥ 0 dt where E(t) is a certain energy function of the solution u, and it is equivalent to the norm (or norm square) of the solution in H, and C1 , C2 are two positive constants independent of initial data u0 . If such a differential inequality can be established, then it immediately leads to E(t) ≤ E(0)e−C1 t +

C2 , C1

∀t ≥ 0

by solving this differential inequality. Thus, for initial data in any bounded set B, which yields that E(0) is bounded by a constant depending on B, there is a uniform time t1 (B) depending on B such that when t ≥ t1 (B), E(t) ≤

2C2 , C1

which implies that after certain time t1 (B), the orbits starting from B 2 uniformly enter an absorbing ball of radius 2C C1 . We now use this idea to establish the existence of an absorbing ball for problem (6.4.2). From now on, we use C to denote positive constants in various places that do not depend on initial data. Multiplying the equation in (6.4.2) by u, then integrating over Ω yields 1 d kuk2 + k∇uk2 + 2 dt

Z

Ω

u4 dx = kuk2 ≤

1 2

Z

Ω

u4 dx +

|Ω| . 2

(6.4.3)

260

NONLINEAR EVOLUTION EQUATIONS

Thus,

Z

d kuk2 + 2k∇uk2 + u4 ≤ C. (6.4.4) dt Ω Similarly, multiplying the equation in (6.4.2) by ut , then integrating over Ω yields Z Z d 1 4 1 1 1 2 2 ( |∇u| + u )dx + kut k = uut dx ≤ kut k2 + kuk2 . (6.4.5) dt Ω 2 4 2 2 Ω Hence, Z d 1 (|∇u|2 + u4 )dx + kut k2 ≤ kuk2 . (6.4.6) dt Ω 2 Summing (6.4.4) and (6.4.6) up, we get µ

d 1 kuk2 + k∇uk2 + dt 2

¶

Z

Ω

u4 dx +2k∇uk2 +

Z

Ω

u4 dx+kut k2 ≤ kuk2 +C.

By the Young inequality, 1 u2 ≤ u4 + 1, 4 we can deduce from (6.4.7) that µ

Z

(6.4.7)

¶

d 1 u4 dx kuk2 + k∇uk2 + dt 2 Ω Z 1 +kuk2 + 2k∇uk2 + u4 dx + kut k2 ≤ C. 2 Ω

(6.4.8)

In order to get the H 2 norm estimate of the solution, differentiating the equation in (6.4.2) with respect to t, then multiplying the resultant by ut , and integrating over Ω, we get Z 1d kut k2 + k∇ut k2 + 3u2 u2t dx = kut k2 . (6.4.9) 2 dt Ω Multiplying (6.4.9) by yields

1 2,

then adding the resultant up with (6.4.8) dE + E(t) ≤ C dt

where

E(t) = kuk2 + k∇uk2 +

1 2

(6.4.10)

Z

1 u4 dx + kut k2 . 4 Ω

(6.4.11)

Thus, we have E(t) ≤ E(0)e−t + C1

(6.4.12)

Asymptotic Behavior of Solutions and Global Attractors where

261

Z

1 1 u40 dx + k∆u0 + u0 − u30 k2 . 2 Ω 4 It turns out from the Sobolev imbedding theorem that for initial data varying in a bounded set in H, there is a positive constant M depending on this bounded set such that E(0) = ku0 k2 + k∇u0 k +

E(0) ≤ M. Therefore, there is tM = ln

M C1

such that when t ≥ tM , E(t) ≤ 2C1 .

(6.4.13)

By the regularity result for the elliptic boundary problem, for t ≥ tM we have ku(t)k2H 2 ≤ Ck∆u(t)k2 ≤ C2 (kut k2 + ku3 k2 + kuk2 ) ≤ C3 .

(6.4.14)

We infer from (6.4.14) that when t ≥ tM , the√orbits starting from the bounded set enter the absorbing ball of radius C3 . Therefore, by Theorem 6.4.1, we can conclude that problem (6.4.2) has a global attractor that is compact and connected. The proof is complete. ¤ Ex. 6.4.2. We now consider the following initial boundary value problem for a nonlinear wave equation:   utt + ut − ∆u = u − u3 ,           

(x, t) ∈ Ω × R+

u|Γ = 0,

(6.4.15)

u|t=0 = u0 (x), ut |t=0 = u1 (x)

where Ω is a bounded domain in R3 with a smooth boundary Γ. Let H = H01 (Ω) × L2 (Ω) equipped with the following norm: k(u, v)T k2H = k∇uk2 + kvk2 . For (u0 , u1 )T ∈ H we first prove the following result. LEMMA 6.4.1 Suppose that (u0 , u1 )T ∈ H. Then problem (6.4.15) admits a unique global solution u such that

(u, ut )T ∈ C([0, +∞), H)

262

NONLINEAR EVOLUTION EQUATIONS

and S(t) defined by the solution

U = (u, ut )T = S(t)(u0 , u1 )T is a nonlinear C0 -semigroup on H . Moreover, The following identity holds for all t > 0: µ ¶ Z d 1 1 1 1 k∇uk2 + kut k2 + u4 dx − kuk2 + kut k2 = 0. (6.4.16)

dt

2

2

4

2

Ω

Proof. We use the density argument. For any (u0 , u1 )T ∈ H, there (n) (n) is a sequence (u0 , u1 )T ∈ (H 2 ∩ H01 ) × H01 converging to (u0 , u1 )T (n) (n) in H. Accordingly, for initial data (u0 , u1 )T , by Theorem 2.7.15 in Chapter 2, we have a sequence of unique global solutions u(n) to the corresponding problem (6.4.15) such that u(n) ∈ C([0, +∞), H 2 ∩ H01 ) ∩ C 1 ([0, +∞), H01 ) ∩ C 2 ([0, +∞), L2 ), and the equation in (6.4.15) is satisfied in the sense of C([0, +∞), L2 ). (n) Multiplying the corresponding equation for u(n) by ut , then integrating with respect to x and t yields that for all t > 0, µ

¶

1 1 (n) k(u(n) , ut )T k2H + ku(n) (t)k4L4 − ku(n) (t)k2 + 2 2 µ ¶ 1 1 (n) (n) (n) (n) k(u0 , u1 )T k2H + ku0 k4L4 − ku0 k2 . = 2 2

Z t 0

(n)

kut k2 dτ (6.4.17)

Let v = u(n) − u(m) . Then v satisfies

 vtt + vt − ∆v = v − vF,            

(x, t) ∈ Ω × R+

v|Γ = 0,

(6.4.18) (n)

(m)

(n)

(m)

v|t=0 = u0 (x) − u0 (x), vt |t=0 = u1 (x) − u1 (x)

where F = (u(n) )2 + u(n) u(m) + (u(m) )2 .

(6.4.19)

Multiplying (6.4.18) by vt , then integrating with respect to x, we get for all t ≥ 0, d dt

µ

¶

1 k(v, vt )T k2H + kvt k2 2

Asymptotic Behavior of Solutions and Global Attractors Z

1 1 (vvt − vvt F )dx ≤ kvt k2 + kv − vF k2 2 2 Ω 1 ≤ kvt k2 + C(kvk2 + kvk2L6 kF k2L3 ). 2

=

263

(6.4.20)

It easily follows from (6.4.17) that for all t ≥ 0, u(n) (t) is uniformly bounded in H 1 . Then we deduce from the Sobolev imbedding theorem H 1 ,→ L6 (n = 3) that kF k2L3 ≤ C.

(6.4.21)

Combining (6.4.20) with (6.4.21) yields

´ d ³ kvk2H 1 + kvt k2 + kvt k2 ≤ Ckvk2H 1 . dt

(6.4.22) (n)

Applying the Gronwall inequality yields that for any T > 0, (u(n) , ut )T is a Cauchy sequence in C([0, T ], H). It turns out from the correspond(n) ing equation (6.4.15) that utt is a Cauchy sequence in C([0, T ], H −1 ). Passing to the limit, the global solution u in the desired class follows. Furthermore, the similar identity to (6.4.17) holds for u, which yields (6.4.16). The uniqueness and C0 continuity of S(t) can be proved by similar energy identity and the Gronwall inequality, and we omit the details here. Thus the lemma is proved. ¤ Now we have the following result concerning the existence of a global attractor for problem (6.4.15). THEOREM 6.4.4 For problem (6.4.15), the C0 -semigroup S(t) defined in H = H01 × L2 by

U = (u, ut )T = S(t)(u0 , u1 )T has a global attractor A that is compact and connected.

Proof. Notice that problem (6.4.15) is a hyperbolic problem, and we cannot expect that condition (ii) in Theorem 6.4.1 will be satisfied because it is well known that “singularity” of the solution will propagate along the characteristic lines. However, we will show that for this problem the conditions (i) and (ii)’ in Theorem 6.4.2 are satisfied. Therefore, problem (6.4.15) will still have a global attractor. To this end, we first prove that there exists an absorbing ball. As shown in (6.4.16), the following holds: d dt

µ

1 1 1 kut k2 + k∇uk2 + 2 2 4

Z

Ω

¶

u4 dx + kut k2

264

NONLINEAR EVOLUTION EQUATIONS Z

1 uut dx ≤ (kut k2 + kuk2 ). (6.4.23) 2 Ω Similarly, multiplying the equation in (6.4.15) by u, then integrating over Ω yields ¶ µ Z Z d 1 kuk2 + uut dx + k∇uk2 + u4 dx = kut k2 + kuk2 . (6.4.24) dt 2 Ω Ω 1 Multiplying (6.4.24) by , then adding up with (6.4.23), we get 4 Z d 1 1 1 E(t) + kut k2 + k∇uk2 + u4 dx dt 2 4 4 Ω 1 1 1 (6.4.25) ≤ ( + )kuk2 + kut k2 2 4 4 where Z Z 1 1 1 1 1 E(t) = kut k2 + k∇uk2 + u4 dx + kuk2 + uut dx. (6.4.26) 2 2 4 Ω 8 4 Ω Since Z 1 1 uut dx ≤ kut k2 + kuk2 , 2 2 Ω it is easy to see that Z 3 1 1 E(t) ≥ kut k2 + k∇uk2 + u4 dx ≥ Ck(u, ut )T k2H (6.4.27) 8 2 4 Ω where C is a positive constant. Then we can easily deduce from the Young inequality =

Z

Ω

2

u dx ≤ ε

Z

Ω

u4 dx + C

and (6.4.25) that dE + C1 E(t) ≤ C2 . (6.4.28) dt Thus, the existence of an absorbing ball follows from (6.4.28) and (6.4.27). In the following, we show how to decompose the semigroup S(t) into S1 (t) + S2 (t). First, consider the following initial boundary value problem:   wtt + wt − ∆w = −w3 ,           

(x, t) ∈ Ω × R+

w|Γ = 0,

w|t=0 = u0 (x), wt |t=0 = u1 (x).

(6.4.29)

Asymptotic Behavior of Solutions and Global Attractors

265

We would like to show that w decays to zero exponentially as time goes to infinity. As a matter of fact, the basic strategy is just the same as before. We have similar identities: µ ¶ Z d 1 1 1 2 2 4 kwt k + k∇wk + w dx + kwt k2 = 0, (6.4.30) dt 2 2 4 Ω and µ ¶ Z Z d 1 2 kwk + wwt dx + k∇wk2 − kwt k2 + w4 dx = 0. (6.4.31) dt 2 Ω Ω 1 , then adding up with (6.4.30), we get 4 Z d 3 1 1 2 2 Ew (t) + kwt k + k∇wk + w4 dx = 0 (6.4.32) dt 4 4 4 Ω

Multiplying (6.4.31) by

where 1 1 1 1 Ew (t) = kwt k2 + k∇wk2 + kwk2 + 2 2 8 4

Z Ω

wwt dx+

1 4

Z Ω

w4 dx. (6.4.33)

It is easy to see from the Young inequality 1 |ab| ≤ |a|2 + |b|2 4 that there is a positive constant γ > 0 such that Ew ≥ γ(kwt k2 + k∇wk2 ).

(6.4.34)

On the other hand, by the Poincar´e inequality kwk ≤ Ck∇wk,

(6.4.35)

we can easily deduce from (6.4.32) that d Ew + C2 Ew (t) ≤ 0 (6.4.36) dt where C2 is a positive constant independent of t and w. Therefore, Ew (t) ≤ Ew (0)e−C2 t

(6.4.37)

and S2 (t) defined by (w, wt )T = S2 (t)(u0 , u1 )T satisfies the condition stated in Theorem 6.4.2. Let v =u−w where u is the unique global solution to problem (6.4.15). Then v

266

NONLINEAR EVOLUTION EQUATIONS

satisfies

  v + vt − ∆v = u − (v + w)3 + w3 ,   tt         

(x, t) ∈ Ω × R+

v|Γ = 0,

(6.4.38)

v|t=0 = 0, vt |t=0 = 0.

For the initial data (u0 , u1 )T belonging to a bounded set in H = H 1 × L2 , it follows from uniform estimates obtained by integrating (6.4.16) with respect to t and (6.4.37) that (v, vt )T also belongs to a bounded set in H, i.e., there is a positive constant C depending on the norm of (u0 , u1 ) in H such that k(v, vt )T kH ≤ C.

(6.4.39)

In what follows we would like to show that v is more regular. More precisely, (v, vt )T will belong to a bounded set in C([0, +∞), H 2−ε ) × C([0, +∞), H 1−ε ) for any ε ∈ ( 21 , 1). Recall that the operator −∆ subject to the Dirichlet boundary condition is positive definite in L2 , and as shown in Chapter 1, we cans define (−∆)s for any s ∈ R. In the above we denote H s = D((−∆) 2 ). To prove that (v, vt ) is more regular, we still use the density argument. For the time being, we assume that the initial data (u0 , u1 )T ∈ H 2 ×H 1 . As shown before, we have (v, vt )T , (w, wt )T ∈ C([0, +∞), H 2 ×H 1 ). Let z = vt . Then (z, zt )T ∈ C([0, +∞), H) and z is a unique solution to the following problem:   ztt + zt − ∆z = ut − 3((v + w)2 − w2 )ut − 3w2 z,           

(x, t) ∈ Ω × R+

z|Γ = 0,

z|t=0 = 0, zt |t=0 = u0 .

(6.4.40)

We now proceed to get the uniform estimate of (z, zt )T in H 1−ε × H −ε . Let I1 = kut k2H −ε , I2 = k3((v + w)2 − w2 )ut k2H −ε , I3 = k3w2 vt k2H −ε .

(6.4.41)

Multiplying (6.4.40) by zt in H −ε , we obtain d dt

µ

¶

1 1 kzk2H 1−ε + kzt k2H −ε + kzt k2H −ε ≤ C(I1 + I2 + I3 ). 2 2

(6.4.42)

By the Sobolev imbedding theorem (n = 3), H ε ,→ Lp (Ω) with 0

1 p0

=

Asymptotic Behavior of Solutions and Global Attractors 1 2

− 3ε . Hence, Lp ,→ H −ε with

1 p

=

1 2

267

+ 3ε . Thus, we have

I1 ≤ Ckut k2Lp ≤ C1 kut k2 ,

(6.4.43)

and I2 ≤ Ckv(v + 2w)ut k2Lp ≤ Ckut k2 kv(v + 2w)k2Lpq 1 q

(6.4.44)

p 2.

where q is a positive constant such that = 1 − Hereafter we denote by Ci positive constants depending at most on the norm k(u0 , u1 )T kH . Clearly, we have 1 1 ε 1 = − = . pq p 2 3 Therefore, by the H¨older inequality, we deduce that I2 ≤ Ckut k2 k(|v| + |w|)k2L6 kvk2Lspq where

1 s

=1−

1 2ε .

(6.4.45)

It easily follows that spq =

6 . 2ε − 1

On the other hand, by the Sobolev imbedding theorem, H 2−ε ,→ Lr (Ω) with 1r = 21 − 2−ε 3 , i.e., r = spq. It turns out from (6.4.45) that I2 ≤ Ckut k2 (kvk2H 1 + kwk2H 1 )kvk2H 2−ε .

(6.4.46)

Now we estimate kvkH 2−ε in terms of kvtt kH −ε . It follows from the equation in (6.4.38) and (v, vt )T , (w, wt )T being bounded in H that kvkH 2−ε = k∆vkH −ε ≤ kvtt kH −ε + kvt kH −ε + ku3 − w3 − ukH −ε ≤ kzt kH −ε + C2 .

(6.4.47)

Combining (6.4.46) with (6.4.47) yields I2 ≤ C3 kut k2 kvtt k2H −ε + C3 kut k2 .

(6.4.48)

Now we proceed to estimate I3 . In the same manner as before, we have I3 ≤ C4 kw2 vt k2Lp ≤ C5 kwk4L6 kvt k2Lpq where

(6.4.49)

1 1 1 ε 1 1−ε 1 = − = + = − . pq p 3 6 3 2 3

By the Sobolev imbedding theorem, we have H 1−ε ,→ Lpq . Thus, we infer from (6.4.49) that I3 ≤ C6 kwk4H 1 kvt k2H 1−ε .

(6.4.50)

268

NONLINEAR EVOLUTION EQUATIONS

Since kwkH 1 is uniformly bounded, combining (6.4.42) with (6.4.43), (6.4.46) and (6.4.50) yields ´ d ³ kzk2H 1−ε + kzt k2H −ε + C7 dt ≤ C6 (kzt k2H −ε + kzk2H 1−ε + C7 )(kut k2 + kwk2H 1 ) + C8 kut k2 . (6.4.51)

It follows from (6.4.16) and (6.4.32) that Z +∞ 0

kut k2 dτ ≤ C9 ,

Z +∞ 0

kwk2H 1 dτ ≤ C10 .

(6.4.52)

Thus, we infer from (6.4.51) and (6.4.52) that kzt k2H −ε + kzk2H 1−ε + C7 ≤ C11 ,

∀t ≥ 0.

(6.4.53)

Combining (6.4.53) with (6.4.47) yields kvk2H 2−ε ≤ C12 ,

∀t ≥ 0.

(6.4.54)

For the initial data (u0 , u1 )T in a bounded set of H, we can choose a (n) (n) sequence (u0 , u1 )T ∈ (H 2 ∩ H01 ) × H01 converging to (u0 , u1 )T in H. (n) Accordingly, for the solution (u(n) , ut ), the corresponding (6.4.53), (6.4.54) will be satisfied. It can be easily proved that passing to the limit yields that (6.4.53), (6.4.54) also hold for v. This implies that (v, vt )T = S1 (u0 , u1 )T , ∀t ≥ 0 is bounded in H 2−ε × H 1−ε , hence, is relatively compact in H, i.e., condition (ii)’ is satisfied. Thus, by Theorem 6.4.2, the C0 -semigroup S(t) associated with problem (6.4.15) has a compact global attractor that is also connected. The proof is complete. ¤ Let u0 be an equilibrium. Then we define its stable manifold M− (u0 ) and unstable manifold M+ (u0 ) as follows. DEFINITION 6.4.4 The stable manifold of u0 , M− (u0 ), is the (possibly empty) set of points u∗ ∈ X, that belong to a complete orbit {u(t), t ∈ R}, i.e., u∗ = u(t0 ) for some t0 and such that

u(t) = S(t − t0 )u∗ → u0 as t → +∞. The unstable manifold of u0 , M+ (u0 ), is the (possibly empty) set of points u∗ that belong to a complete orbit {u(t), t ∈ R}, i.e., u∗ = u(t0 ) for some t0 and such that

u(t) = S(t − t0 )u∗ → u0 as t → −∞.

Asymptotic Behavior of Solutions and Global Attractors

269

A stationary point or equilibrium u0 is called stable if M+ (u0 ) = ∅ (empty set), and unstable otherwise. It is easy to follow from the definition that M− (u0 ) and M+ (u0 ) are invariant. From a graphic point of view, if M+ (u0 ) = ∅, then all complete orbits through u0 are inward to u0 . Otherwise, there is at least one complete orbit through u0 going outward from u0 . Of particular interest for the understanding of the dynamics are the heteroclinic orbits that go from the unstable manifold of a stationary point u∗ to the stable manifold of another stationary point u∗∗ , u∗ 6= u∗∗ . When u∗∗ = u∗ , such a curve is called a homoclinic orbit. Similarly for a set B ∈ X, we can define its unstable set M+ (B) and stable set M− (B) as follows. DEFINITION 6.4.5 Let B be a set in X . M+ (B) denotes the unstable set of B , which is the (possibly empty) set of point u∗ belonging to a complete orbit {u(t), t ∈ R} and such that d(u(t), B) → 0 as t → −∞. The stable set M− (B) of B can be defined in a similar way.

In what follows we give the results concerning the structure of a global attractor for a gradient system. First, we prove the following result. LEMMA 6.4.2 Suppose that X is a Banach space, and S(t) is a C0 -semigroup on X that possesses a global attractor A. Let B be a compact invariant set of S(t). Then

M+ (B) ⊂ A.

(6.4.55)

M+ (A) = A.

(6.4.56)

In particular, we have

Proof. Let u∗ ∈ M+ (B) and let U = {u(t), t ∈ R} be the complete orbit containing u∗ with, say, u∗ = u(0). It is clear that S(t)U = U. Since d(u(t), B) → 0,

270

NONLINEAR EVOLUTION EQUATIONS

as t → −∞, and d(u(t), A) → 0, as t → +∞, it follows from boundedness of B and A, and continuity of S(t) that U is bounded. Since A is a global attractor, d(S(t)U, A) → 0, t → +∞.

(6.4.57)

Then it follows from S(t)U = U that U ⊂ A. Thus, (6.4.55) is proved. To prove (6.4.56), it remains to show that A ⊂ M+ (A). Suppose that u∗ ∈ A. Then there is a complete orbit {u(t), t ∈ R} ⊂ A with, say, u∗ = u(0). The construction of the complete orbit contained in A can be done by the following procedure stated in the proof of Proposition 1.3 in Chapter 3 of the book [20] by A.V. Babin and M.I. Vishik: Let u(t) = S(t)u0 , ∀t ≥ 0. Since A is invariant, u(t) ∈ A for all t ≥ 0. For t < 0, we first recurrently define a sequence of points u−k , k ∈ N such that S(1)u−k = u−k+1 . Owing to A being invariant, this can be done, and u−k ∈ A for all k ∈ N . Then we define u(−k + τ ) = S(τ )u−k ∀τ ∈ [0, 1]. In this way, a complete orbit laying on A is constructed. Now d(u(t), A) = 0,

∀t < 0.

By the definition of unstable set, we immediately have that u∗ ∈ M+ (A). The lemma is proved. ¤ THEOREM 6.4.5 Suppose that X is a Banach space, and S(t) is a nonlinear C0 -semigroup defined on X , which possesses a Lyapunov function V and a global attractor A. Let E be the set of equilibria of S(t). Then

A = M+ (E).

(6.4.58)

Furthermore, if E is discrete, then A is the union of E and the heteroclinic curves joining one point of E to another point of E , and [ M+ (z). (6.4.59) A= z∈E

Asymptotic Behavior of Solutions and Global Attractors

271

Proof. First we show that E ⊂ A. Indeed, for any x ∈ E, we have d(S(t)x, A) = d(x, A) → 0

as t → +∞.

Thus, x ∈ A, i.e., E ⊂ A. Since E ⊂ A, it is easy to see that M+ (E) ⊂ M+ (A). By Lemma 6.4.2, M+ (A) = A. Therefore, M+ (E) ⊂ A. We now prove the opposite inclusion. Suppose that u0 is a point of A. Then, as proved in Lemma 6.4.2, u0 belongs to a complete orbit {u(t), t ∈ R} contained in A with u(0) = u0 . Let γ=

\

{u(t), t < s}.

s<0

Since A is compact and the sets {u(t), t < s} are connected and contained in A, we can conclude that γ is compact and connected. As in the proof of Lemma 6.3.2 we can show that γ is invariant: S(t)γ = γ

∀t ≥ 0.

(6.4.60)

Since V is decreasing along the orbit, and V is bounded from above on the compact set A, the limit of V (u(t)) as t → −∞ exists. V |γ = lim V (u(t)) = sup V (u(t)). t→−∞

t∈R

(6.4.61)

It follows from (6.4.60), (6.4.61) and the definition of a Lyapunov function that γ consists only of equilibria, i.e., γ ⊂ E. Since u0 ∈ M+ (γ),it follows that u0 ∈ M+ (E). Thus A ⊂ M+ (E). This proves (6.4.58). When E is discrete, since γ is connected, it follows that the set γ defined previously reduces to one equilibrium z, and the whole family u(t) converges to z as t → −∞. On the other hand, by Corollary 6.3.2, the ω-limit set ω(u0 ) consists only of a single equilibrium z 0 . The whole orbit u(t) ∈ A is the heteroclinic curve joining z to z 0 . The theorem is proved. ¤

6.5

Bibliographic Comments

Lemma 6.2.1 was first established in [135] by W. Shen and S. Zheng. The result stated in Remark 6.2.1 was first given by P. Krecji and J. Sprekels [87], and its simplified proof was given by S. Zheng [165] with the same spirit as in the proof of Lemma 6.2.1. Theorem 6.2.2 as well

272

NONLINEAR EVOLUTION EQUATIONS

as the corresponding results for the three-dimensional case was first established by S. Zheng [165]. The result of Ex. 6.4.2 was previously presented in the book [20] by A.V. Babin and M.I. Vishik, which extends the corresponding results shown in the books [59] by J.K. Hale and [151] by R. Temam. The approach presented in Section 6.3 for studying convergence of solutions to nonlinear parabolic equations in a higher-space dimension with analytic nonlinearity was first proposed in [141] by L. Simon. We refer to [99], [100], [101], and [102] for the Lojasiewicz inequality in the finite dimensional case. The Simon’s approach was much simplified in the paper [67] by M.A. Jendoubi. For the recent development of Simon’s approach, see also [32], [50], [61], [62], [66], [68], [128], [154] and the references cited there. We refer to the books [20] by A.V. Babin and M.I. Vishik, [59] by J.K. Hale, [151] by R. Temam, [131] by G. Sell and Y. You, and the references cited there for the thorough study of infinite-dimensional dynamical systems. In addition to the study of global attractors and inertial manifolds, we refer to [45], [46] for the study of inertial sets.

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Index C0 -semigroup, 21 ω-limit set, 238 absorbing set, 255 analytic semigroups, 61 asymptotic behavior of a solution, 223 Aubin Lemma, 97 compact attractor, 255 compactness method, 95 complete orbit, 238 decay rate of a solution, 181 Faedo-Galerkin method, 95 Gagliardo–Nirenberg interpolation inequalities, 10 global attractor, 7, 224 global Lipschitz condition, 46 globally iterative method, 185 gradient system, 241 Gronwall inequality, 18 Hille–Yosida theorem, 26 infinitesimal generator, 24 interpolation spaces, 17 invariant regions, 133 invariant set, 237 Lax–Milgram theorem, 15 local Lipschitz condition, 52 Lojasiewicz inequality, 224, 245 Lojasiewicz–Simon inequality, 224, 245, 252

Lyapunov function, 241 maximal accretive operator, 32, 42 mild solution, 38 monotone iterative method, 133 monotone operator, 33 monotone operator in the Minty sense, 115 monotone operator method, 95 nonexistence of a global solution, 213 sectorial operator, 69 semigroup method, 21 stable manifold, 268 stationary point, 269 strong convergence, 96 the lower solution, 133 the upper solution, 133 uniformly convex Banach space, 102 universal attractor, 255 unstable manifold, 268 weak convergence, 96 weakly star convergence, 96 Yosida approximation, 28

287