Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences)

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Applied Mathematical Sciences I Volume 56

Applied Mathematical Sciences 1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Percus: Combinatorial Methods.

5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-Linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions.

15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I. 19. Marsdeni McCracken: The Hopf Bifurcation and its Applications. 20. Driver: Ordinary rnd Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems.

27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29, de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory.

(continued on inside back cover)

K.W. Chang F.A. Howes

Nonlinear Singular Perturbation Phenomena: Theory and Applications

Springer-Verlag New York Berlin Heidelberg Tokyo

K.W. Chang

F.A. Howes

Department of Mathematics University of Calgary Calgary, Alberta Canada T2N 1N4

Department of Mathematics University of California Davis, California 95616 U.S.A.

AMS Classification: 34D15, 34D20, 34EXX, 35B20, 35B25, 35F99, 35G99

Library of Congress Cataloging in Publication Data Chang, K.W. Nonlinear singular perturbation phenomena. (Applied mathematical sciences ; v. 56) Bibliography: p. Includes indexes. 1. Boundary value problems-Numerical solutions. 2. Singular perturbations (Mathematics) I. Howes, II. Title. III. Series: Frederick A. Applied mathematical sciences (Springer-Verlag New York Inc.); v. 56. QAI.A647 vol. 56 [QA379] 510 s (515.3'5] 84-14014

With 12 Illustrations

© 1984 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York, 10010, U.S.A.

Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.

987654321 ISBN 0-387-96066-X Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-96066-X Springer-Verlag Berlin Heidelberg New York Tokyo

Preface

Our purpose in writing this monograph is twofold.

On the one hand,

we want to collect in one place many of the recent results on the existence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems.

On the other, we hope to

raise along the way a number of questions for further study, mostly questions we ourselves are unable to answer.

The presentation involves a

study of both scalar and vector boundary value problems for ordinary differential equations, by means of the consistent use of differential inequality techniques.

Our results for scalar boundary value problems

obeying some type of maximum principle are fairly complete; however, we have been unable to treat, under any circumstances, problems involving "resonant" behavior.

The linear theory for such problems is incredibly

complicated already, and at the present time there appears to be little hope for any kind of general nonlinear theory.

Our results for vector

boundary value problems, even those admitting higher dimensional maximum principles in the form of invariant regions, are also far from complete. We offer them with some trepidation, in the hope that they may stimulate further work in this challenging and important area of differential equations.

The research summarized here has been made possible by the support over the years of the National Science Foundation and the National Science and Engineering Research Council.

We offer each agency our sincerest

thanks for their generosity and consideration.

We also wish to thank our

colleagues and students who have shared their knowledge of and curiosity about singular perturbation theory with us, especially Bob O'Malley,

v

vi

Adelaida Vasil'eva and Wolfgang Wasow.

This monograph is but a small

token of our appreciation of their friendship and support.

K. W. Chang Calgary

F. A. Howes Davis

Contents Page

PREFACE CHAPTER I.

CHAPTER II.

v INTRODUCTION

1

Notes and Remarks

4

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

6

Scalar Boundary Value Problems Vector Boundary Value Problems Notes and Remarks

13 17

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

18

The Dirichlet Problem: Boundary Layer Phenomena 3.2. Robin Problems: Boundary Layer Phenomena Interior Layer Phenomena 3.3. Notes and Remarks

18 27 32 34

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

37

The Dirichlet Problem: Boundary Layer Phenomena 4.2. Robin Problems: Boundary Layer Phenomena 4.3. Interior Layer Phenomena Notes and Remarks

37 49 55 59

QUADRATIC SINGULAR PERTURBATION PROBLEMS

61

Introduction The Dirichlet Problem: Boundary Layer Phenomena Robin Problems: Boundary Layer Phenomena 5.3. 5.4. Interior Layer Phenomena Notes and Remarks

61

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

91

Introduction A Dirichlet Problem Robin Problems: Boundary Layer Phenomena Interior Layer Phenomena A General Dirichlet Problem A General Robin Problem: Boundary and Interior Layer Phenomena 6.7. A Comment Notes and Remarks

91 93 95 96 98 101 104 105

SINGULARLY PERTURBED SYSTEMS

106

Introduction The Semilinear Dirichlet Problem The Semilinear Robin Problem The Quasilinear Dirichlet Problem Notes and Remarks

106 106 111 114 121

2.1. 2.2.

CHAPTER III.

6

3.1.

CHAPTER IV.

4.1.

CHAPTER V.

5.1. 5.2.

CHAPTER VI.

6.1. 6.2. 6.3. 6.4. 6.5. 6.6.

CHAPTER VII.

7.1. 7.2. 7.3. 7.4.

vii

61 76 83 90

viii

Page CHAPTER VIII.

EXAMPLES AND APPLICATIONS

123

Part I - SCALAR PROBLEMS

123

8.1. 8.2. 8.3.

8.4.

Examples of Semilinear Problems and Applications Examples of Quasilinear Problems and Applications Examples of Quadratic Problems and Applications Examples of Superquadratic Problems and An Application

Part II - VECTOR PROBLEMS B.S.

8.6.

Examples of Semilinear Systems and An Application Examples of Quasilinear Systems and An Application

123 132 141

155 161

161

165

REFERENCES

171

AUTHOR INDEX

177

SUBJECT INDEX

179

Chapter I

Introduction

We are mainly interested in quasilinear and nonlinear boundary value problems, to which some well-known methods, such as the methods of matched asymptotic expansions and two-variable expansions are not immediately applicable.

For example, let us consider the following boundary value

problem (cf. O'Malley [75], Chapter 5)

EYof = Y12,

0 < t < 1,

(A)

y(0,0 = l,y(l,c) = 0.

(B)

In general, it is not obvious that such a nonlinear boundary value problem will have a solution in of

e.

[0,1]

for all sufficiently small values

However, in this case, we can obtain by quadratures the following

exact solution in

[0,1]

y(t,c) = -c ln[t + e-1/c(1-t)]

which is defined for all positive values of An important feature of this solution tion of

(t,e), it behaves nonuniformly as

e.

y(t,e) t

and

is that, as a funcc

approach

0, that

is,

lim+ y(t,e) = 0

for each fixed

t > 0

(1.1)

lim+ y(t,c) = 1 t_0+

for each fixed

c > 0.

(1.2)

but

For decreasing values of

c, the solutions

Figure 1.1.

1

y(t,c)

are as shown in

2

I.

INTRODUCTION

y

0

1

Figure 1.1

From the graph or from the relations (1.1), (1.2), it is clear that, as

a -

to

t

val

0+,

the solution

y(t,e)

approaches

on each closed subinterval of Note that

[0,1].

y(t,e)

u(t) - 0

[6,1], for fixed

on

(0,13

0

uniformly with respect

but not on the whole inter-

(which is the limit of the solution

6 > 0) turns out to be the solution of the

corresponding reduced equation U'2 = 0

of the original equation (A) satisfying the right-hand boundary condition

(B) In order to illustrate the difficulties associated with the application of the methods of matched and two-variable asymptotic expansions, we assume, for the moment, that the exact solution of problem (A), (B) is not known, and so we proceed formally. methods is that the solution of (A), series in

The fundamental hypothesis of these (B) can be described by two power

c, known as the inner and outer expansions.

The outer expan-

sion represents the solution away from regions of nonuniform behavior and is simply a power series in

whose coefficients are functions of

a

t.

On the other hand, the coefficients of the inner expansion are functions

not only of

t

but also of a "stretched" variable

T = e

which can be

arbitrarily large as

a - 0+

(cf. [55], Chapter 2;

[75], Chapter 1) can be regarded as a rescaling

for a certain range of

t.

The variable

parameter which has the effect of enlarging the region of nonuniform behavior.

To fix these ideas, let us examine first the outer expansion

y0(t,e), that is, we substitute

T

Introduction

3

ao

enun(t)

YO(t,c) = u0(t) + eu1(t) + 0

into the differential equation like powers of

e.

ey" = y'2

and equate coefficients of

The first two terms of y0

are easily shown to sat-

isfy the equations u02

= 0

(1.3)

and

u"=u'2 1 1

(1.4)

The solutions of (1.3) are

u0(t) = constant, while the problem of

solving (1.4) is essentially equivalent to solving the original problem We note that

(A), (B). (1.4).

ul(t) = constant

If we require that

y0(t,e)

is one family of solutions of

satisfy one of the original boundary

conditions (B), then the obvious choices for the functions

u0

and

u1

are

u0(t) a 0,

ul(t)

0

(if

y0(l,e) = 0)

u0(t) E 1,

ul(t) = 0

(if

YO(01c) = 1).

y0(l,e) = 0

The first choice here (cf.

If the outer expansion

lows.

turns out to be the proper requirement

[75], Chapter 5) and it can be motivated geometrically as foly0(t,E)

were to satisfy the second choice

y0(0,e) = 1, then we would anticipate that 1im+ y(t,c) = 1, e-*0

In a neighborhood of

for

t

in

t = 1, y(t,e)

[0,1-6], where

d > 0.

must decrease rapidly from one to

zero in order to satisfy the other boundary condition the equation (A) requires that y" < 0.

y'

y(l,e) = 0.

Since

never change sign, we must have

This however is impossible since

ey" = y'2 > 0.

Consequently,

we must select the outer expansion which satisfies the boundary condition y0(l,c) = 0

at

t = 1.

In this case, the outer expansion

zero (up to terms of order 5)

y0(t,e) = -E In t

is also an outer expansion and Y0(I,E) = 0.

y0(t,e)

should be identically

e2); however, the function (cf. [75], Chapter

4

INTRODUCTION

I.

This function has a singularity at

t = 0

for

c > 0

and so one might

be tempted to reject it out of hand as an approximation to The surprising fact is that this function follows from the exact solution. t = 0

y0

y

on

(0,1].

is the outer expansion, as

Indeed, the singularity of

y0

at

is precisely what is needed to cancel the singularity of the inner

expansion there.

The construction of the inner expansion of the solution

y

is

equally fraught with difficulties, as it is not obvious at all what the correct stretched variable

t

should be.

standard change of variable (cf. t =

0

To see this we could make the

[55], Chapter 2;

and

4'(c)

-

0+

[75], Chapter 5)

as

and attempt to determine the asymptotic character of the terms of the transformed differential equation.

c - 0+, *(e)

by comparing

Clearly

ey" = y'2

is equivalent to C

d2y

1

*2(c)

`dt)

and so the change of variable accomplishes nothing.

Using various de-

vices, O'Malley ([75], Chapter 5) is able to construct an inner expansion which in fact has a singularity at larity of the function

y

there.

t - 0

that just cancels the singu-

His methods are nevertheless not ob-

vious a priori, and it is quite conceivable that more complicated problems of the form (A), (B) involving

y'2-nonlinearities could not be solved to

such a degree.

We can, however, solve this particular problem using our method, but rather than discussing this method now, we will defer it until Chapter V, when we deal with a much larger class of related boundary value problems. Notes and Remarks. 1.1.

The methods of matched asymptotic expansions and two-variable expansions have been very successful in solving a variety of difficult problems in engineering and applied science, and they continue to be two of the most powerful weapons in the arsenal of applied mathematicians.

Our discussion of the nonlinear problem (A),

(B) (and

indeed, the point of view taken in this monograph), is not meant to denigrate, in any way, the utility and importance of these stalwarts of asymptotic analysis.

Rather we wish to study certain types

of boundary value problems, for which precise results on the exist-

introduction

5

ence and asymptotic behavior of solutions can be derived from mathematical analysis. 1.2.

Alternative approaches to many of the problems discussed in this book can be found in the monographs of Wasow ([93], Chapter 10), Vasil'eva and Butuzov [88], O'Malley [7S], Habets [28], Habets and Laloy [31], Eckhaus [21] and Kevorkian and Cole [55], and in the survey articles of Vasil'eva [87], Erdelyi [23], O'Malley [73], Carrier [8] and Wasow [94].

In addition, these monographs and

papers contain a wealth of references.

Chapter II

A'priori Bounds and Existence Theorems

§2.1.

Scalar Boundary Value Problems Before discussing in detail the various classes of singularly per-

turbed boundary value problems, let us give an outline of the principal method of proof that we will use throughout.

This method employs the

theory of differential inequalities which was developed by M. Nagumo [66] It enables one to prove the existence

and later refined by Jackson [49).

of a solution, and at the same time, to estimate this solution in terms of the solutions of appropriate inequalities.

Such an approach has been

found to be very useful in a number of different applications (see, for It will be seen that for the general classes of

example, [5] and [83]).

problems which we will study in later chapters, this inequality technique leads elegantly (and easily) to some fairly general results about existence of solutions and their asymptotic behavior.

Many results which have

been obtained over the years by a variety of methods can now be obtained by this method, which we hope will also very clearly reveal the fundamental asymptotic processes at work.

Consider first the general Dirichlet problem x" = f(t,x,x'),

a < t < b, (DP)

x(a) = A,

in which

f

x(b) = B,

is a continuous function on

[a,b] xlR2.

The differential

inequality approach of Nagumo is based on the observation that if there exist smooth (say twice continuously differentiable or a(t)

and s(t)

possessing the following properties:

6

C(2)-) functions

2.1.

Scalar Boundary Value Problems

7

a(t) < B(t)

a(a) < A < B(a),

a(b) f(t,a,a'),

then the problem (DP) has a solution such that

a(t) < x(t) < S(t)

for

in

6(t)

x'.

C(2) ([a,b])

[a,b], provided that

does

f

More precisely, it is suffici-

satisfies what is known as a generalized Nagumo

f

condition with respect to every solution

of class

x = x(t) t

not grow "too fast" as a function of ent to require that

(P)

a

and

of

x = x(t)

there exists a constant

a(t) < x(t) <

satisfying

x" = f(t,x,x')

J c [a,b]

on a subinterval

This simply means that

(cf. [36]).

$

has a bounded derivative, that is,

N = N(a,6)

such that

on

1x'(t)j < N

The

J.

most common type of Nagumo condition is the following: f(t,x,z) = 0(jz12)

jzj - - for all

as

[a,b] x [a,8].

in

(t,x)

Clearly, if

This was originally given by Nagumo [66] himself. x" = f(t,x,x') = O(lx'12)

and if a(t) < x < S(t), then Theorem 2.1.

In summary then, we have

1x'j < N(a,B).

Assume that there exist bounding functions

with the properties (P), and assume that the function generalized Nagumo condition with respect to chlet problem (DP) has a solution

satisfying a(t) < x(t) < 8(t)

a

t

in

a(t)

satisfies a

f p.

of class

x = x(t)

for

and

and

a(t)

Then the Diri-

C(2)([a,b])

[a,b].

Thus the task of estimating solutions of (DP) is reduced to the task of constructing sufficiently sharp bounding functions

and

a(t)

8(t).

In this regard, we note (cf. [31]) that it is possible to obtain the same result as in Theorem 2.1 if the bounding functions piecewise

a

and

a

[a,b], that is, if there is a partition

on

are only {ti}

of

a = t0 < t1 < t2 < ... < to = b, such that on each subinter-

with

[a,b]

val

-C(2)

[ti-1,ti], a

partition points

and ti-1

0

and

are twice continuously differentiable.

ti, the derivatives are the right-hand and

left-hand derivatives, respectively. each subinterval

more that for each

At the

We must of course assume that on

(ti-1,ti), all > f(t,a,a'), 8" < f(t,B,8'), and furthert

in

(a,b), a'(t) < a'(t+ )

and

g'(t) > $'(t+).

Then there is the following companion result. Theorem 2.2.

a

and

0

Assume that there exist piecewise

-C(2)

bounding functions

with the stated properties, and assume that the function

f

II.

8

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

satisfies a generalized Nagumo condition with respect to

a

and

Then

$.

the conclusion of Theorem 2.1 follows.

This result allows the bounding functions to have certain types of It follows from the observation that if

"corners".

are lower [upper] functions, then so is

...,SM}] ..., B}].

{al,...,am}[{si.

max{al,...$am}[min{sl,

We will need this fact in our discussion of interior layer

behavior associated with some singular perturbation problems. Let us now consider a boundary value problem with more general boundary conditions of the form a < t < b,

x" = f(t,x,x'),

(RP)

plx(a) - p2x'(a) = A,

where the constants qi + q2 > 0.

glx(b) + g2x'(b) = B,

pi, qi

Note that if

satisfy

p2, q2 > 0, pi + p2 > 0

p2 = q2 = 0

and

and

pl = qi = 1, then the prob-

lem (RP) reduces to (DP), and so we are really interested in the case p2 + q2 > 0.

when

It turns out fortunately that the Nagumo theory for

(DP) can be extended with the obvious modifications to the problem (RP), as was observed by Heidel [36].

bounding functions

a

and

That is, if there exist piecewise

-C(2)

(a < s) which satisfy the above differen-

D

tial inequalities and the boundary inequalities

pla(a) - p2a'(a) < A <

p10(a) - P20' (a), gla(b) + g2a'(b) < 8 < g16(b) + g20'(b), then the problem (RP) has a solution in

with respect to Theorem 2.3.

a

x = x(t)

[a,b], provided that

and

B

a

and

such that

a(t) < x(t) < $(t)

f

satisfies a generalized Nagumo condition

Q.

For later reference, we call this result

Assume that there exist piecewise

generalized Nagumo condition with respect to problem (RP) has a solution for

t

x = x(t) in

a

of class

and

t

bounding functions

-C(2)

with the stated properties, and assume that

a(t) < x(t) < g(t)

for

f s.

satisfies a

Then the Robin

C(2)([a,b])

with

[a,b].

In studying singularly perturbed boundary value problems one is frequently interested in obtaining theorems which guarantee a priori the existence of solutions and give tions.

an estimation of the location of the solu-

The most common results of this kind for both perturbed and un-

perturbed problems are obtained using some maximum principle argument in which the solution is estimated throughout its interval of existence in terms of its values on the boundary of the interval.

The remainder of

this section is concerned basically with existence and estimation results which follow either directly or indirectly (that is, after a change of

Scalar Boundary Value Problems

2.1.

9

variable) from the one-dimensional maximum principle, and its generalizations as embodied in Theorems 2.1 - 2.3.

We consider first the Dirichlet problem a < t < b,

cy" = f(t,y,y'), y(a,c) = A,

in which

y(b,e) = B,

f = O(Iy'I2)

subsets of

(2.1)

as

(2.2)

that is, for

ly'l

[a,b] x lR, f(t,y,y') = O(ly'12)

as

in compact

(t,y)

The next re-

ly'l - -.

sult is a direct application of the maximum principle (cf. Lemma 2.1. t,y,y'

Assume that the function

and of class 2

[a,b] x]R .

f

is continuous with respect to

with respect to

C(1)

[81]).

y

for

(t,y,y')

m

Assume also that there is a positive constant

y(t,y,0) > m > 0

for

(t,y)

in

[a,b] x1R.

problem (2.1), (2.2) has a unique solution

Then for each

y = y(t,s)

in

in

such that

c > 0, the [a,b]

satis-

fying ly(t,a)t < M/m,

where M = max{ max

lf(t,0,0)1, mIA1, mJBI).

[a,b]

Proof:

Define for

a(t) _ -M/m

t

in

and

[a,b]

Q(t) = M/m.

Then a < 0, a(a) < A < 8(a) ferential inequalities

and a(b) f(t,a,a'), 8" <

by Taylor's Theorem f(t,a,0) = f(t,0,0) + fy(t,&,0)a

where , a <

< 0, is an intermediate point, and so

f(t,a,0) < 1f(t,0,0)1 + ma < M + m(-M/m) < 0 = ca".

Similarly, for some intermediate point

n, 0 < n < B,

f(t,4,0) = f(t,0,0) + fy(t,n,0) > -M + m(M/m) > 0 = c!!".

It follows from Theorem 2.1 that for each has a solution

y(t,e)

on

[a,b]

c > 0

the problem (2.1), (2.2)

satisfying

-M/m < y(t,c) < M/m.

The uniqueness of the solution follows from the maximum principle.

10

II.

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

If we assume that

A > 0, B > 0

-M < f(t,0,0) < 0

in

and

[a,b],

then by the proof of Lemma 2.1, we will obtain the following more precise estimate of the solution: 0 < y(t,e) < M/m.

Similarly, if we assume that

A < 0, B < 0

and M > f(t,0,0) > 0

[a,b], then the solution of (2.1), (2.2) satisfies

in

M/m < y(t,e) < 0.

These results suggest the following modifications of Lemma 2.1. Lemma 2.2. t,y,y'

Assume that the function

and of class

[a,b] x]R2.

C(n) (n > 2)

Assume also that

is a positive constant and

m

3nf(t,y,0) > m > 0

with respect to

y

for

(t,y,y')

8yf(t,0,0) > 0

for

1 < j .S n-i

in

[a,b] x]R.

Then for each

the problem (2.1), (2.2) has a solution

y = y(t,e)

in

0 < y(t,e) < (n!m

(t,y)

[a,b]

e > 0,

satisfying

IM)I/n,

where M = max{ max

If(t,0,0)I, (mlAI/n!)n, (mIBI/n!)n}.

[a, b]

Remark. Proof:

We use the notation Define for

a(t) B 0

t

and

in

a

n

2nf/Byn.

for

f

[a,b]

S(t) _ (n!m

lM)1/n

Clearly, a < g, a(a) < A < $(a), a(b) f(t,a,a')

Finally, we see that

since n-l

f(t,0,0) = f(t,0,0) +

E

j=1

in

A > 0, B > 0, f(t,0,0) < 0, and that there

such that for

is continuous with respect to

f

81f(t,0,0)131/j! y

+ 9nf(t,n,0)8n/n!

> -M + (m/n!)n! M/m > 0.

The conclusion of Lemma 2.2 follows by virtue of Theorem 2.1. If we make the change of dependent variable

y

-y

and apply

Lemma 2.2 to the transformed problem, we can obtain an analogous result

Scalar Boundaxy Value Problems

2.1.

for the case

A < 0, B < 0

and

11

f(t,0,0) > 0.

This is given in the next

lemma.

Assume that the function

Lemma 2.3.

and of class

t,y,y'

C(n) (n > 2) with respect to

Assume also that

[a,b] x1R2.

and

1 < j0(je) < n-1

(even), for

(t,y)

Then for each

integer.)

in

y = y(t,e)

such that

2

j0(je)

8y f(t,y,O) > m > 0

[a,b] x1R.

in

for

y

in

(t,y,y')

A < 0, B < 0, f(t,0,0) > 0, and that

there is a positive constant m for

is continuous with respect to

f

(< 0)

(< -m < 0) if n

is odd

denotes an odd (even)

j0(je)

(Here

f(t,0,0) > 0

c > 0, the problem (2.1), (2.2) has a solution

satisfying

[a,b]

-(n:m 1M)1/n < Y(t,t) < 0,

where M

is as defined in the conclusion of Lemma 2.2.

In the previous lemmas, we imposed strong conditions on the partial

f with respect to

derivatives of

In the lemma below we will relax

y.

our conditions on the partial derivatives of

f

with respect to

we will impose a condition on the partial derivative

y, but

and thereby

fy

obtain virtually the same result as in Lemma 2.1. Lemma 2.4. t,y,y'

Assume that the function

and of class

k > 0

for

is continuous with respect to

f

with respect to

for

y,y'

(t,y,y')

Assume also that there are positive constants

[a,b] x1R2.

such that

C(1)

If (t,y,0)I < R in

(t,y,y')

for

[a,b] x1R, and

in

(t,y)

Then for

[a,b] x1R2.

lem (2.1), (2.2) has a solution y = y(t,c)

in

and

k

If ,(t,y,y')I >

0 < e < k2/4t, the prob[a,b]

1)

(if

fy, < 0)

ly(t,c)I < YR 1(2eA(a-t) - 1)

(if

fy, > 0).

ly(t,e)I < YR-1(2ea(t-b) -

R

in

satisfying

or

Here for

+ 0(e)

A = -ft-

is a negative root of the polynomial

0 < c < £0 < k2/4R, Y = max{ max

cX2 + kA + R

If(t,0,0)I, 1IAI(2eX(a-b)-1),

[a, b]

RIBI}

if

fy, < 0, and

y = max{ max

If(t,0,0)I, LIAI,

[a,b]

RIBI (2eX(a-b) Proof:

- 1)

if y, > 0.

Suppose for definiteness that y , < -k < 0.

0 < £ < £0

and

t

in

[a,b]

Define for

12

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

II.

a(t,c) =

-YR-1(2e (t-b)

- 1)

and YR-1(2e(t-b)

0(t,c) =

- 1).

Then, clearly a < 0, a(a) < A < $(a)

and

We will only verify that

choice of y.

.Q" < f(t,0,01)

the verification that

a(b) f(t,a,a') in

by our (a,b), as

in

proceeds analogously.

(a,b)

Differentiating and substituting, we have ca" - f(t,a,a') = ca" - f(t,0,0) - {f(t,a,0) - f(t,0,0)} - {f(t,a,a') - f(t,a,0)} = call - f(t,0,0) - fy(t,g,0)a - y,(t,a,n)a' > -ea2YR-12e;k(t-b) - Y - kYR-12e)L(t-b) + Y kXYk-12ea(t-b) -

= 0,

The conclusion of the lemma in the case that

sat + kA + R = 0.

since

now follows from Theorem 2.1.

fy, < -k < 0

and

0 < s < s0

define for

a(t,c) =

-YR-1(2eA(a-t)

t

in

- 1),

If

fy, > k > 0

then we

[a,b]

s(t,e) = -a(t,c),

and proceed as above.

We consider finally a sufficient condition for the existence of a solution of the problem (2.1), (2.2) which includes the assumptions fy > 0

and

If

l

> 0

of Lemmas 2.1 and 2.3, respectively.

The next

lemma is due essentially to van Harten [33].

Assume that the function

Lemma 2.5. t,y,y'

and of class

[a,b] xlR2.

constant in

m

C(1)

f

is continuous with respect to

with respect to

y,y'

Assume also that there are a constant

[a,b] x1R2

and

v

(t,y,y')

0 < c < El.

y = y(t,c)

Then for in

[a,b]

in

and a positive

fy(t,y,0) + Vf y,(t,y,y') > m + ev2

such that

(2.2) has a solution

for

for

(t,y,y')

0 < c < ell the problem (2.1), satisfying

jy(t,c)I < m-1Net, where N = max{ max

If(t,0,0)e-vtl,

mIAje-av,

mlBle-bv}.

[a,b]

Proof:

The lemma follows by making the change of variable

y = zevt

and

Vector Boundary Value Problems

2.2.

13

applying Lemma 2.1 to the resulting problem for z(a,c) = Ae-av, z(b,c) = Be-bv, where vfy,(t,zevt,n)

-

z, namely

ez" = F(t,z,z',e),

F(t,z,z',e) = f(t,O,O)e-vt +

ev2}z + {fy,(t,zevt,n) - 2ev}z'.

We note that if

while if and

f > m > 0, then we may take v = 0 in Lemma 2.5, y Ify,I > k > 0, then we may take v such that sgn v = sgn fy, for

IvkI > R

t = supIfy(t,y,0)I, in order to derive the results of

Lemmas 2.1 and 2.4, respectively.

Lastly we wish to point out that analogous results hold for the Robin

problem a < t < b,

ey" = f(t,y,y'),

p1Y(a,e) - p2Y'(a,e) = A,

In particular, if

(2.3) (2.4)

g1Y(b,c) + g2Y'(b,e) = B.

f(t,y,y') = O(Iy'12)

as

(y'I

-i- -

for

(t,y)

in

[a,b] x1R, and if pl = ql = 1, then Lemmas 2.1 - 2.4

compact subsets of

hold verbatim for (2.3), (2.4), as the reader can easily verify.

Vector Boundary Value Problems

§2.2.

Analogous results also hold for vector boundary value problems.

The

existence and comparison theorems for vector problems can be regarded as higher dimensional forms of Nagumo's scalar theory.

Unfortunately, how-

ever, the assumptions which are imposed for vector problems are more difficult to verify in practice.

This is due, on the one hand, to our

limited experience in treating boundary value problems for systems of On the other hand, systems of differential equa-

differential equations.

tions are inherently more complicated than scalar equations, and so at best, we can only hope to mimic the scalar theory.

The results which

follow are taken mostly from the papers of Kelley [52], [53], although much of the early work was done by Hartman, [34], [35; Chapter 12] and others (see [5] and [83] for further references).

Consider then the boundary value problem x" = F(t,x,x'),

a < t < b, (DP)

x(a) = A,

where

x(b) = B,

x, A and

B

are vectors in IRN

vector function which is continuous on

and

F = (F1,...,FN)T is an

[a,b] x]R 2N

.

N-

It turns out that

the scalar Nagumo theory can be extended to (DP), provided that the vector function

F

satisfies a growth condition (Nagumo condition) with respect

14

to

II.

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

In this vector setting, we say that

x'.

F

satisfies a Nagumo con-

dition if it satisfies one of the following two conditions, for in compact subsets of

and for all

z

in ]RN

(t,x)

(cf. [53]):

There exist positive, nondecreasing, continuous functions

(1)

on

[a,b] x 1I

such that each component

(0,co)

of

Pi, i = 1,...,N

F

$i

satisfies

IFi(t,x z)I <_ $i(IziI) and

s/$i(s)ds =

J

There exists a positive, nondecreasing, continuous function

(2)

on

such that

(0,co)

IIF(t,x,z)II < $(II=II) and s2/$(s) + 00

s + .

as

Here and below,

is the usual Euclidean norm, that is, IIxII

for any

(xTx)1/2

x', then

independent of

hile, if that

x - (x1.... ,xN)T in IRN. F

depends linearly on

F

satisfies condition (2).

F

etch component 0(Iz. 2)

of

Pi

F

F

is

x', then in general, we can only say Clearly condition (1) is satisfied if

depends only on

Izil + 0, for

as

It is clear that if

satisfies both types of Nagumo condition;

(t,x)

x', and if

Fi(t,x,zi) _

in compact subsets of

[a,b] xIRN

We now extend the scalar notion of a bounding function by supposing that there exist

M

scalar functions

C(2)([a,b] xIRN)

class

whenever

p'' > 0

pt = pR(t,x), t = 1,...,M, of

satisfying pR = 0

and

pR = 0.

(2.5)

Here, the derivatives

P, s atz + [grad pR]TXT and

2

2R + [2 grad apR/at]Tx' + xTHx' + (grad pt]TF(t,x,x')

p'R'

at are taken with respect to the solution of (DP), and matrix of

pk

with respect to

x, (that is, H

that represents the derivative of ing region

grad p1).

H

is the Hessian

is the Jacobian matrix

Next, we define the follow-

2.2.

Vector Boundary Value Problems

{(t,x)

I

in

15

[a,b] x]Rn: p£(t,x) < 0

for

R = 1,...,M),

which plays an important role in the following result, a vector analog of Theorem 2.1, which is taken from Kelley [53]. Theorem 2.4.

M

Assume that there exist

functions

p£ = p£(t,x),

£ = 1,...,M, satisfying (2.5) and that the function

F = F(t,x,x')

isfies a Nagumo condition in the domain initial and terminal pairs

by a smooth path in x = x(t)

for

of class

I xIRN.

(a,A), (b,B)

belong to

and can be joined

I

Then the Dirichiet problem (DP) has a solution

I.

such that

C(2)([a,b])

p£(t,x(t)) < 0

on

[a,b],

R = 1,...,M.

The conclusion of the theorem is that (DP) has a solution (t,x(t))

I; hence, we call

in the region

x(t)

with

an invariant region for

I

Thus the description of the nature of the solution of (DP) is de-

(DP).

pendent on the appropriate comparison functions

p£.

Suppose first we use a single comparison function IIxII - y(t), where

IIxII < y(a)

(2.5), then we obtain

and

grad p = x/IIxII

p

satisfies

To ensure that (2.5) is satisfied,

we proceed to determine the scalar function that

If

[a,b], which gives the magni-

on

IIxII.

p = p(t,x) _

IIxII < y(b).

and

IIxII < y(t)

tude of the bound on the norm

y(t).

A calculation shows

grad(Bp/et) 5 0; furthermore, as the norm

is a convex function, it follows that the Hessian matrix H of

IIxII

(and hence, of

IIxII

all

sat-

Assume also that the

in IRN.

x

-Y" +

p"

p) is nonnegative definite, that is, xTHx > 0

for

The inequality (2.5) then reduces to

(XT/I

IXI I)F(t,x,x') > 0

or

Y"(t) < (xT/I Ixl I)F(t,x,x'), whenever

y(t) = IIxII

tions are imposed on

and

y1(t) _ (xT/IIxII)x'.

F, then

y(t)

If appropriate assump-

can be~determined.

Suppose, on the other hand, we wish to obtain bounds on the individual components of the solution of (DP); then we obviously need more than one comparison function. P£(t,x) = x£ - s£(t)

where

a£ < S£, for

To this end, we define the and

2N

functions

P£+N(t,x) = -x£ + a£(t),

£ = 1,...,N.

well as the boundary inequalities

If these functions satisfy (2.5), as

16

A'PRIORI BOUNDS AND EXISTENCE THEOREMS

II.

at (a) < AR < 0t (a) ,

at (b) < B1 < at (b) ,

then we obtain two-sided bounds of the form

at(t) < xt(t) < St (t) t = 1,...,N, on

for

(cf. Theorem 2.1).

[a,b]

The reader can verify

that the functions above will satisfy (2.5), if at

and

satisfy,

at

respectively, the inequalities aR > FI(t,x,x')

when

xt = at(t),

xt'

gR < Ft(t,x,x')

when

x1 = st(t),

x't = 01(t),

= aR(t)

and

for all

xi (i 0 t) in

[ai(t),si(t)]

and all

x! (i 0 t)

in

IR.

By means of different types of invariant regions in Theorem 2.4, we can prove a priori existence and comparison results for the general singularly perturbed vector problem

Ey" = F(t,y,'), y(a,E) = A,

a < t < b,

y(b,E) = B,

and these can be regarded as analogous to Lemmas 2.1 - 2.5.

The analysis

is rather tedious, although straightforward, and we leave the precise formulation of these results to the reader.

The last existence and comparison theorem of this chapter deals with the following vector Robin problem a < t < b,

x" = H(t,x),

(RP)

Px(a) - x1(a) = A,

where

and H

x, A and

B

are

Qx(b) + x'(b) = B,

N-vectors, P, Q

are constant

(N x N)-matrices,

N-vector function defined and continuous on

is an

[a,b] x1RN

(We have limited ourselves to discussing only systems of singularly perturbed Robin problems whose right-hand sides do not depend on any derivatives, but of course we could have discussed more general systems.) If the matrices

P

and Q

are positive semidefinite in the sense that

there exist nonnegative scalars and

xTQx > gIIxII2, for any

x

p

and

q

such that

a result analogous to Theorem 2.3.

Since we will only seek bounds on

the norm of a solution of (RP), we call a set p(t,x) < 0}

xTPx > p1Ixi12

in IRN, then it is possible to prove

I = {(t,x)

in

[a,b] xe:

an invariant region for (RP) if the scalar function

the following three properties:

p

has

2.2.

Vector Boundary Value Problems

is of class

(1)

p

(2)

pp(a,x(a)) - pp(a,x(a)) < 0,

17

C(2) (1);

qp(b,x(b)) + pp(b,x(b)) < 0; (3)

p" > 0

F.)

Theorem 2.5.

whenever

I

p', p"

(The functions

in place of

in

p = 0

and

are as defined in (2.5), with the function

Then we have the following result (cf.

p(t,x(t)) < 0

in

x = x(t)

H(t,x)

[52], [57]).

Assume that there exists an invariant region

Then the Robin problem (RP) has a solution such that

p' = 0.

for (RP).

I

of class

C(2)([a,b])

[a,b].

This theorem will be used in Chapter VII to estimate the norms of solutions of singularly perturbed Robin problems.

There we will show,

under appropriate assumptions, how to construct functions of the form p(t,x,c) = IIx11 - y(t,e)

which satisfy the conditions (1) - (3).

Notes and Remarks 2.1.

Most of the differential inequality and invariant region results quoted in this chapter can be found in the monographs of Bernfeld and Lakshmikantham [5] and Schr6der [83].

These works also contain

many additional references to the relevant literature, as well as instructive illustrations of other applications of inequality techniques. 2.2.

M. Nagumo in a paper [67] published in 1939 was the first mathematician to apply differential inequalities in the study of a singular perturbation problem, namely the initial value problem f(t,y,y',e), 0 < t < T < co, y(0,e), y'(0,e)

ey" =

prescribed.

This paper

was overlooked until the Soviet mathematician N. I. Bris in [7] used Nagumo's results to study the singularly perturbed boundary problem g2y'(b,e)

ey" = f(t,y,y',E), Ply(a,c) - P2y'(a,e), gly(b,E) + prescribed, by means of a shooting technique.

Most, if

not all, of the present monograph is based on these two seminal papers; the reader would do well to consult them or the brief survey [45].

2.3.

It is possible to extend Theorem 2.5 to the Robin problem for the differential equation

x" = F(t,x,x')

and to more general invari-

ant regions of the type considered in Theorem 2.4; cf.

[52] or [57].

Chapter III

Semilinear Singular Perturbation Problems

The Dirichlet Problem:

§3.1.

Boundary Layer Phenomena

We consider first the semilinear Dirichlet problem ey" = h(t,y),

a < t < b,

y(a,c) = A,

y(b,c) = B,

(DP 1)

where

c

is a small positive parameter and prime denotes differentiation

with respect to are:

Some natural questions to ask regarding this problem

t.

Does the problem have a solution for all small values of

c?

Once

the existence of a solution has been established, how does the solution

behave as

c -+ 0+?

The answers to these questions depend greatly on the function (and also on the boundary values

A, B, if h

h

is a nonlinear function),

as we shall see by examining two simple linear equations in

(0,1)

cyil = y

(El)

cy1t = -Y

(E2)

and

subject to the boundary conditions y(O,e) = 1,

Y(l,e) = 2.

The general solution of (E 1) is

y = Y1(t,e) = cl

c2

and the general solution of (E2) is

y = y2(t,c) = cl cos(t/v) + c2 sin(t//),

18

The Dirichlet Problem:

3.1.

where

and

c1

c2

Boundary Layer Phenomena

are arbitrary constants.

19

Using the boundary conditions

to determine these constants, we find that the solutions are, respectively,

- el/re )-1{(e-1/v - 2)et/re Y1(t,e) _ (e-1/Y'

(2-el/re )e-t1r.-

+

2e-(l-t)/,re- +

and

y2(t,e) = cos(t/re) + {2 - cos(l/v)} Let us examine these solutions more closely. is defined for all

e > 0

y1(t,e) = 0

lira

For (E1), the solution

yl

and, moreover,

for

S < t <

(3.1)

e+O

where

is a fixed constant in (0,1).

6

limiting value

The function

nonuniformly in the neighborhoods of

0

attains the

y1

t = 0

and

t = 1

in the following sense: lim+ lim+ y1(t,e) = 1 # 0 = lim+ lim+ y1(t,e) t-+0 t->0 e'0

E-0 and

lim+ lira- y1(t,e) = 2 # 0 = lira

lim+ y1(t,e)

t->l- c0

t-rl

We note that by setting

(cf. Figure 3.1).

0 - h(t,u)

reduced equation

c = 0

whose solution is

in (E1) we obtain the

u - 0.

In the case of (E2), first of all we see that the solution is only defined if 2,...

Thus, for these particular values of

.

Suppose then

solution. Y2

re # (nx)-1, for

sin(1/ f) # 0, that is, if

a

is very small and

y2(t,e)

n = 1,

e, the problem (E2) has no re # (nw)-1.

The function

is a linear combination of two oscillatory functions of arbitrarily

large arguments, and therefore it is densely oscillatory with period 1 and with bounded amplitude (cf. Figure 3.2). for

u a 0

y2

Clearly, it is impossible

to satisfy a limiting relation such as (3.1) above, even though is again the solution of the reduced equation obtained from (E2)

by setting

e = 0.

What sets these two problems apart is of course the

difference in the sign of the coefficient of

y.

Let us now consider the third example in which function of

y

ey"" = Y2,

in

h

is a nonlinear

(0,1),

Y(O,e) = A,

y(l,c) = B.

(E3)

20

III.

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

Figure 3.1

Figure 3.2

This example illustrates how the boundary values the existence and the behavior of solutions as

A and

B

can affect

From our con-

c + 0+.

sideration of the example (El), we expect that for sufficiently small e > 0, the solution of (E3) must remain close to zero, the unique solution of the reduced equation t = 0

and

t = 1.

y2 = 0, except possibly near the endpoints

However, in the neighborhood

of

t = 0

and

t = 1,

the solution must be convex (i.e., y" > 0), as dictated by the differc-ly2

ential equation or

y" =

> 0.

is negative, we would have

B

spectively; see Figure 3.3.

are nonnegative, we have

3.4 shows. tion

near

t = 0

t = 1, re-

On the other hand, if both

e.

y" > 0

For such values of A

y = y3(t,c)

y" < 0

or

In these cases, the problem has no solution

for sufficiently small values of B

A

Thus, if either the boundary value

near and

t = 0

and

A

and

t = 1, as Figure

B, the problem (E3) has a solu-

for all sufficiently small values of

e, and

N

y"<0

y11<0

Figure 3.3

The Dirichiet Problem:

3.1.

Boundary Layer Phenomena

y">0 0

21

I

0

11

Figure 3.4

1i+

y3(t,c)

= 0

d < t <

for

-6,

C-0

where

is a fixed constant in

6

(0,1).

It is interesting to note that the exact solution

y3(t,E)

(for

A,B > 0) can be obtained in terms of elliptic integrals [76], from which we see that y(t,s) - A[l + t(A/6E) 1/2 1in

2

+ B[l + (1-t)(B/6e)1/21-2

[0,1].

However, by simply observing that the solution must satisfy

y" > 0,

we are able to deduce the sign restrictions on the allowable boundary values

A,B.

If we apply this observation and reasoning to the same problem for the differential equation

ey" = y3, we conclude that the problem has a

solution for any value of A

and

B.

Rather than dealing with specific cases, we study now the existence of solutions

y = y(t,c)

of the general problem (DP1) (and related ones)

which behave like the solutions of (E1) and (E3) in the sense that lim+ y(t,E) = u(t) Ey

where

u = u(t)

h(t,u) = 0,

in each closed subinterval of

(a,b),

(3.2)

is a certain solution of the reduced equation a < t < b.

(R1)

If the relation (3.2) holds, we can say that the reduced solution u(t)

is "stable" with respect to the original solution

y(t,E).

To be

III.

22

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

precise, we wish to give explicit definitions of stability for the solution

u(t).

Let

be a solution of (R1) which is continuous in

u = u(t)

[a,b], and let us define the following three domains D0(u) = {(t,Y): a < t < b,

where

d(t)

DO(u)

U1(u), V2(u):

(Y - u(t)l < d(t)),

is a positive continuous function such that

d(t) s IA - u(a)l + 6,

for

d(t) =-d,

for a+6
a < t < a + 6/2

and + d,

d(t) = lB - u(b)

Here

6 > 0

for

is a small constant.

If A > u(a)

B > u(b), we define

and

0 < y - u(t) < d(t)},

D1(u) _ {(t,y): a < t < b,

and if A < u(a)

B < u(b), we define

and

D2(u) = {(t,y): a < t < b,

where

d(t)

b - 6/2 < t < b.

-d(t) < y - u(t) < 0},

is as above.

In the following definitions of stability for the solution we assume that the function

h(t,y)

has the stated number of continuous

partial derivatives with respect to that

q > 0

and

Definition 3.1. [a,b]

y

in

Di(u), i = 0,1

or

2, and

are integers.

n > 2

The function

u = u(t)

is said to be

if there exists a positive constant

ayh(t,u(t)) = 0

u(t),

a < t m > 0 Definition 3.2. [a,b]

in

The function

u = u(t)

is said to be

if there exists a positive constant m ayh(t,u(t)) = 0

for

a2q+lh(t,u(t)) > m > 0 (A-u(a))ayq+2h(t,y) > 0 and

V0(u).

a < t 0

23

V0(u) f1 ([b-d,b] xlR).

The above definitions are motivated by boundary value problems for the differential equation Definition 3.3. if

[a,b]

ey" = y2q+1

The function

(cf.

u = u(t)

u(a) < A, u(b) 0

and

a < t m > 0 Definition 3.4. [a,b]

m

if

in

VI(u).

The function

u = u(t)

is said to be (11)-stable in

and if there exists a positive constant

u(a) < A, u(b) 0

and

a < t m > 0

for

3Y+lh(t,y) > 0

V1(u) fl [([a,a+d) U (b-6,b]) xIR].

a < t < b,

and in

The above definitions are motivated by boundary value problems for the differential equation Definition 3.5.

if

[a,b]

m

ey" = y2n

The function

u(a) > A, u(b) > B

(cf.

u = u(t)

(E3) above).

is said to be (111)-stable in

and if there exists a positive constant

such that

j0(je) h(t,u(t)) > 0

8y

where

j0(je)

if

[a,b]

m

for

a < t m > 0)

The function

u(a) > A, u(b) > B

in

u = u(t)

V2(u),

if n

is even (odd).

is said to be (III n)-stable in

and if there exists a positive constant

such that

2y

0e) h(t,u(t)) > 0

(< 0)

for

a < t m > 0)

3 h(t,y) > 0

in

a < t < b

for

if n

is

even (odd),

and (<

0)

V2(u) fl [([a,a+d) fl (b-d,b]) x 1R],

is

is even (odd).

n

The last two definitions are motivated by boundary value problems for the differential equation

ey" _ -y2n.

With these definitions of stability we proceed now to discuss the We remark that the constant

Dirichlet problem (DP1).

c

in each theorem of

of this chapter is a known positive constant depending on the reduced path under consideration.

Assume that the reduced equation (R1) has an

Theorem 3.1.

stable solution u = u(t) co > 0

such that for

y - y(t,e)

for

in

t

of class

0 < e < co [a,b]

(Iq)- or (Iq)-

Then there exists an

C(2)([a,b]).

the problem (DP1) has a solution

which satisfies cel/(2q+1)

Iy(t,e)-u(t)I < wL(t,c) + wR(t,e) +

where wL(t,c) = IA-u(a)I eXp[-(me-1)1/2(t-a)]

if

q = 0,

wL(t,e) = IA-u(a)I (1+aIA-u(a)Ige-1/2(t-a))-1/q wR(t,e) = IB-u(b)I exp(-(me-1)1/2(b-t)]

if

if

q > 1,

q = 0,

and

wR(t,c) = IB-u(b)I (1+0jB-u(b)Ige-1/2(b-t))-1/q

if q > 1.

Here ml/2q[(q+l)(2q+1)!]-1/2

Q =

and

c

Proof:

is some positive constant.

The theorem follows from Theorem 2.1 of Chapter II, if we can

exhibit, by construction, the existence of the lower and the upper bounding functions

a(t,c)

and

0(t,e)

with the required properties.

Since, by assumption, 2yq+lh(t,y) > m > 0, we must have

h(t,y) -

my2q+l/(2q+1)!, and we are led to consider the differential equation ew" =

m

w2q+1

(2q+1)!

Indeed, the function

wL(t,c)

(3.3)

is nonnegative and is the solution of (3.3)

The Dirichlet Problem:

3.1.

such that

Boundary Layer Phenomena

25

(q+1)(2q+l).11/2IA-u(a)Iq+l.

wL(a,e) - IA-u(a)I, wL'(a,e) _ -

The solution decreases to the right.

is the solution of (3.3) such that [e(q+l)(2q+1)!]1/2I6-u(b)Iq+l.

We now define, for

Similarly, the function

wR(t,e) > 0

wR(b,e) = IB-u(b)I, wR(b,e) _

It decreases to the left.

in

t

[e

e > 0, the functions

and

[a,b]

a(t,e) = u(t) - wL(t,e) - wR(t,e) - r(e), 0(t,e) = u(t) + wL(t,e) + wR(t,e) + r(e).

Here

r(e) _ (ey/m)1/(2q+1), where

is a positive constant which will

y

be specified later. It is obvious that the functions a < A. a(a,e) < A < 0(a,e)

ties:

as easy to prove that

and

call > h(t,a)

some suitable choice of

and consider a(t,e).

have the following proper-

a,$

a(b,e) < B < p(b,e).

and

We treat the case that

y.

in

ce" < h(t,s)

(The verification for

From Taylor's Theorem and the hypothesis that

u(t)

0(t,e) u(t)

It is just (a,b)

for

is (Iq)-stable

follows by symmetry.) (Iq)-stable, we

is

have h(t,a(t,e)) = h(t,a(t,e)) - h(t,u(t))

2q

n

ay h(t,u(t)) [a(t,s) - u(t)]n

E n=l 1

1

u(t)]2q+1

(2q+1)! a2q+lh(t,t(t))(wL+wR+r)2q+1 (2g11,1

(t,(t))

where

is some intermediate "point" between

(t,u(t)), which lies in 0 < c < cO.

T

Since

DO(u)

wL, wR

and

for sufficiently small r (w2q+l

-h(t,a(t,e)) > (2gmm1

(t,a(t,e))

and

e, say

are all positive functions, we have + w22q+1 + r2q+l

and so call - h(t,a(t,e)) > au" - ewL - ew" + (2gm1).(w2q+1+w22q+l+r2q+1)

> -eIu"I + cy

(2q+1)!

by the definitions of wL, wR, and we obtain

call > h(t,a).

r.

Thus, by choosing y > lu"I(2q+1)!,

26

III.

The case that

u(t)

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

(Iq)-stable can be treated analogously, by

is

carrying out the Taylor expansion to

terms; the details are

(2q+2)

left to the reader.

If the reduced equation (R1) has a (IIn)- or (IIn)-stable solution, we have the following result. Theorem 3.2.

Assume that the reduced equation (R1) has a (,In)- or (fn)-

stable solution u(b) 0

in

of class (a,b).

u(a) < A,

such that

C(2) ([a,b])

Then there exists an

co > 0

y = y(t,c)

the problem (DP1) has a solution

such that in

[a,b]

which satisfies 0 < y(t,e) - u(t) < WL(t,c) + wR(t,e) +

ccl/nP

where (A-u(a))(l+v(A-u(a))1/2(n-1)E-1/2(t-a))-2/(n-1)

wL(t,e) _ and (B-u(b))(1+a(B-u(b))1/2(n-1)e-1/2(b-t))-2/(n-1)

wR(t,e) _ Here

a = (n-1)(m/2(m+1)!)1/2 and

c

Proof:

is some positive constant.

The proof of Theorem 3.2 follows in much the same manner the proof

of the previous theorem, once we note that of the differential equation

ew" = n, wn

wL > 0

is now the solution

which satisfies

wL(a,c) = A -

-(2m/e(n+1)!)1/2(A-u(a))1/2(n+l),

u(a)

and

and that

wL(a,c) _

is the solution satisfying

w (b,c) = B - u(b)

(2m/c(n+1)!)1/2(B-u(b))1/2(n+l).

wR > 0

wR(b,e)

and

We then define

a(t,c) = u(t),

S(t,c) = u(t) + wL(t,e) + wR(t,e) + (eym 1)1/n We leave details to the reader

for y > Iu"In!, and proceed as above. except to note that the convexity of

u

implies that

ca" - h(t,a) _

cull - h(t,u) = cu" > 0.

The next theorem is the analog of Theorem 3.2 when the solution of the reduced equation is (IIIn)- or (IIIn)-stable.

by making the change of variable 3.2.

y + -y

It can be proved easily

and immediately applying Theorem

Boundary Layer Phenomena

Robin Problems;

3.2.

Assume that the reduced equation (R1) has a (IIIn)- or

Theorem 3.3.

(III n)-stable solution

u(a) > A, u(b) > B such that for in

[a,b]

27

u = u(t)

u" < 0

and

0 < e < e0

of class in

C(2)

(a,b).

([a,b])

such that

Then there exists an

the problem (DP1) has a solution

e0 > 0

y = y(t,c)

which satisfies

_wL(t,e) - wR(t,e) - cc 1/n < Y(t,e) - u(t) < 0,

where

wL, wR

53.2.

Robin Problems:

and

are the same as in Theorem 3.2.

c

Boundary Layer Phenomena

We turn now to a consideration of the Robin problems

ey"=h(t,Y),

a
Y(a,e) - P1Y'(a,e) = A,

y(b,e) = B,

and

sy"=h(t,y),

a
y(a,e) - P1Y'(a,e) = A, where

pl

and

p2

boundary conditions

Y(b,e) + P2Y'(b,E) = B,

are positive constants. y(a,e)

and

(The Robin problem with the

y(b,e) + p2y'(b,e)

handled by making the change of variable

prescribed, can be

t + a + b - t

in (RP1).)

In

order to motivate our results, we examine briefly the behavior of solutions of the linear equation boundary conditions.

ey" = y

satisfying the above types of

Consider first the problem

0
ey"=y,

(E4)

y(O,e) - y'(O,e) = 1,

y(l,e) = 1.

The exact solution is easily found to be

Y(t,E) =

A-1{(1-e-1/2-ee-1/2) exp[-to-1/2] + (e_ C_ 1/2_1-e-1/2)exp[te 1/2]1,

where A = e-e

-1/2

-1/2

(1-a-1/2) -

ee

(1+e-1/2)

Clearly Y(t,e) - e1/2 exp[-te-1/2] + exp[-(l-t)e-1/2],

28

III.

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

and it follows that lim+ y(t,e) = 0 e+0

where

for

0 < t < 1 - S.

(3.4)

Thus in contrast to the limiting

S, 0 <6 <1, is a fixed constant.

relation (3.1), the solution of (E4) is uniformly close to its limiting

value of

near

0

Similarly the solution

t = 0.

of the

y = y(t,e)

problem

eyu=Y,

0 < t < 1,

Y(O,e) - Y'(O,e) = 1,

(E5)

Y(l,e) + Y'(l,e) = 1,

can be obtained explicitly and it satisfies lim+ y(t,c) = 0 e+0

0 < t < 1.

for

(3.S)

The difference between the relations (3.1) and (3.4), (3.5) stems from the fact that the boundary conditions in (E4) and (E5) prescribe or

y'(l,e)

to be uniformly bounded in

Robin problems cannot

y'(0,e)

e, and so the solutions of these

behave like the solution of (E1) near

and/or

t = 0

t = 1.

In terms of the Robin problem (RP1), a solution reduced equation (R1) is said to be

(I

q)-,

u = u(t)

of the

(Iq)-, (IIn)-, (IIn)-, (IIIn)-

or (III n)-stable if it is so stable in the sense of Definitions 3.1 -

3.6, respectively, with

d(t)

d(t) _ IB-u(b)I + S

[b-6/2, b].

in

satisfying

d(t) _ S

in

[a, b - S]

and

The following theorems are analogs of Theorems 3.1 - 3.3. Assume that the reduced equation (R1) has an (Iq)- or (Iq)-

Theorem 3.4.

stable solution e0 > 0

u = u(t)

such that for

y = y(t,e)

in

[a,b]

of class

0 < e < e0

C(2) ([a,b]).

Then there exists an

the problem (RP1) has a solution

which satisfies cel/(2q+1)

IY(t,e)-u(t)I < vL(t,e) + wR(t,e) + Here vL(t,e) = (e/mp ) vL(t,e) = 0[l +

where

IA-u(a)

+ plu'(a)I exp[-(m/e) q(e(2q+2)!/2m)-1/2Qq(t-a)]-1/q

(t-a]

if

q = 0,

if

q > 1,

3.2.

Robin Problems:

Boundary Layer Phenomena

29

aq+l = {e(2q+2)!/2mp12}1/2IA-u(a) + plu'(a)!,

wR

is as given in Theorem 3.1, and

Proof:

is some positive constant.

c

The proof of this result is not that much different from the

proof of Theorem 3.1 which deals with the Dirichlet problem (DP1).

In-

deed, the function equation

vL > 0 is the decaying solution of the differential ez" _ (2gm1)! z2q+l which satisfies vL(a,e) = -IA-u(a) +

plu'(a)l/pl

and

vL(a,e) = a.

Thus, for

in

t

e > 0

and

[a,b]

we

define

a(t,e) = u(t) - vL(t,e) - wR(t,e) - r(e),

Vt,e) = u(t) + vL(t,e) + wR(t,e) + r(e), where

wR(t,e)

is given in Theorem 3.1 and

Y > lu"1(2q+1)!. Clearly we have (i(a,e) - p1 '(a,e), and

a(b,e) < B < Q(b,e).

u

tial inequalities, let us suppose that sider only

(The verification for a

S.

r(e) = (eym-1)1/(2q+1), for

a < $, a(a,e) - pla'(a,e) < A < As regards the differen-

is (Iq)-stable and let us confollows by symmetry.)

Expand-

ing by Taylor's Theorem we see that 2q

h(t,e) - ce" = h(t,u) +

ayh(t,u(Q-u)J

lil r 1

32q+lh(t,f)(O-u)2q+1

+ (2q+1)!-r

y

- cull - ev" - ew" L R

M

f 2q+l 2q+ll + wR vL

'

(2q+1).

.

e

'

+ (2q+1).

ev" - ew"R L

> 0

by virtue of our assumptions.

mediate point, which lies in say

0 < e < a0.

Here DO(u)

(t,n)

is the appropriate inter-

provided

a

is sufficiently small,

Thus the conclusion of Theorem 3.4 follows from Theorem

2.3.

If the solution

u of (R1) is (,In)- or (11)-stable, then as was

the case with the Dirichlet problem, u must satisfy the additional requirements:

u" > 0

Similarly, if u u" < 0

in

in

(a,b), u(a) - plu'(a) < A

and

u(b) < B.

is (III n)- or (III n)-stable, then we must require that

(a,b), u(a) - plu'(a) > A

and

u(b) > B.

The precise results

30

III.

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

are contained in the following theorems.

Assume that the reduced equation (RI) has a (IIn)- or (IIn)-

Theorem 3.5.

stable solution

u = u(t)

plu'(a) < A, u(b) 0

and

in

such that

C(2)([a,b])

u" > 0

0 < c < e0

such that for

y = y(t,e)

of class

in

(a,b).

u(a) -

Then there exists an

the problem (RP1) has a solution

which satisfies

[a,b]

0 < y(t,e) - u(t) < vL(t,e) + wR(t,e) + ccl/n. Here vL(t,e) = v[l +

?(n-l)(e(n+l)!/2m)-1/2Q1/2(n-1)(t-a)]-2/(n-1)

where

6n+1

wR

= e(n+1)!(A-u(a) + plu'(a))2/2mpl ,

is as given in Theorem 3.2, and

Proof:

is some positive constant.

c

The proof of Theorem 3.5 is almost a repetition of the proof of

the previous theorem, if we define for

in

t

[a,b]

and

c > 0

the

functions a(t,e) = u(t) and

0(t,e) = u(t) + vL(t,e) + wR(t,e) + where

y > lu"In!.

Theorem 3.6.

The details are left to the reader.

Assume that the reduced equation (R1) has a (III n)- or

(III n)-stable solution

u = u(t)

u(a) - plu'(a) > A, u(b) > B an

e0 > 0

y = y(t,e)

(eym-1)1/n,

such that for for

t

in

and

of class

u" < 0

0 < e < e0

[a,b]

C(2) ([a,b]) in

(a,b).

such that

Then there exists

the problem (RPI) has a solution

which satisfies

-vL(t,e) - wR(t,e) - cel/n < y(t,e) - u(t) < 0, where

vL

and wR

are as defined in the conclusion of Theorem 3.5.

The proof follows if we simply let

y - -y

and apply Theorem 3.5

to the transformed problem.

It is now an easy matter to discuss the behavior of solutions of the problem (RP2).

For this problem, a solution

u = u(t)

of the re-

duced equation (R1) is said to be (Iq)-, (IIn)- or (III n)- stable if it

3.2.

Robin Problems:

31

Boundary Layer Phenomena

is so stable in the sense of Definitions 3.1, 3.3 or 3.5, respectively, with

in

d(t) = d

The proofs of the next two results can be pat-

[a,b].

terned after those of Theorems 3.4 and 3.5 and are omitted. Theorem 3.7.

solution

Assume that the reduced equation (Rl) has an (I )-stable

u = u(t)

such that for in

of class

0 < c < c0

C(2)([a,b]).

Then there exists an

the problem (RP2) has a solution

which satisfies

[a,b]

cel/(Zq*1)

jy(t,e)-u(t)j < vL(t,e) + vR(t,e) + Here

is as given in the conclusion of Theorem 3.4 and

vL

vR(t,e) = (e/mp2)1/2IB-u(b)-p2u'(b)I exp[-(m/e)1/2(b-t)]

if

q = 0,

q(e(2q+2)!/2m)-1/2aq(b-t)]-1/q

vR(t,e) = a[l + where

e0 > 0

y = y(t,c)

if

aq+1 = {e(2q+2)!/2mp2}1/2IB-u(b)-p2u'(b)l

and

q > 1, c

is some positive

constant.

Assume that the reduced equation (R1) has a (Ill)-stable

Theorem 3.8.

solution

u = u(t)

u(b) + p2u'(b) 0

and

0 < e < e0

in

(a,b).

such that

u(a) - plu'(a) < A,

Then there exists an

the problem (RP2) has a solution

co > 0

y = y(t,e)

which satisfies

[a,b]

0 < y(t,e) - u(t) < VL(t,e) + vR(t,e) + ccl/n Here

vL

is as given in the conclusion of Theorem 3.5, and

vR(t,c) = a[l + 2(n-l)(e(n+l)!/2m)-1/2v1/2(n-1)(b-t)]-2/(n-1) where

an+l

= e(n+l)!(B-u(b)-p2u'(b))2/2mp2

and

c

is some positive

constant.

The corresponding result for the problem (RP2) in the case that is (III n)-stable and satisfies

and

u" < 0

of variable

in

(a,b)

y -

-y.

u

u(a) - plu'(a) > A, u(b) + p2u'(b) > B

follows from Theorem 3.8 after making the change We leave its precise formulation to the reader.

32

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

III.

Interior Layer Phenomena

§3.3.

The above results deal with stable solutions which are twice continuously differentiable in restriction imposed on

validity of these results.

u'

is of class

or even that the function

[a,b]

(a,b), with

except at the point

with

(a,b)

and

f

of h(t,u) = 0

u2

ul(t),

a < t < t0,

u2(t),

t0 < t < b,

will have the property that and

are stable in

u2

bounded

u

where

(a,b)

It is easy to see that such a situation can arise in Namely, if two

intersect at the point

ui(t0) # u2(t0), then the path

u0(t) _

in

t0

the type of problems we have been considering. u1

u"

Let us now suppose that the continuous function

C(2) ([a,b])

u'(t0) # u'(t0).

solutions

h(t,u) = 0

For example, it is enough to assume that the

is bounded in

is differentiable almost everywhere in

wherever it exists.

of

The smoothness

can be slightly weakened without altering the

u

second derivative of u

u = u(t)

[a,b].

defined by

u0(t)

If both functions

u0(t0-) # uo(t0).

[a,b], then the path

u0

C(2) t0 in

ul

is also stable, and it

is reasonable to expect that, under appropriate restrictions on SIN), y = y(t,c)

there is a solution

of the problem (DP1), (RP1) or (RP2)

such that lim

in each closed subinterval of

y(t,c) = u0(t)

(a,b).

This will turn out to be the case if we supplement the bounding functions a,$

with an "interior layer corrector at

we assume that the reduced path single point throughout

in

t0 [a,b].

u

For ease of exposition,

t0".

is not differentiable at only one

(a,b), and that

u

is either

q-

many points of nondifferentiability and to the case when q-stable in

or n-stable

The extension of our results to the case of finitely

(a,t0)

n-stable in

and

(t0,b)

u

is, say,

is rather straightforward

and will be omitted. Theorem 3.9.

Assume that the reduced equation (R1) has an (Iq)-or (Iq)-

stable solution (a,b)

e0 > 0

where

u = u(t)

such that for

a solution

of class

u'(t0) # u'(t0)

y = y(t,e)

0 < e < for

C(2)([a,b]), except at

and e0 t in

m.

t0

in

Then there exists an

the problem (DP1), (RP1) or (RP2) has [a,b]

which satisfies, respectively,

Interior Layer Phenomena

3.3.

33

cel/(2q+1)

Iy(t,e)-u(t)I < wL(t,e) + wR(t,e) + vI(t,e) +

(DP1) cc1/(2q+1)

Iy(t,e)-u(t)I < VL(t,e) + WR(t,e) + vI(t,e) +

(RP1)

and Iy(t,e)-u(t)I <_ vL(t,e) + vR(t,e) + v1(t,e) + cc1/(2q+1)

wL

Here

and

wR

are as given in Theorem 3.1, vL

and

(RP2) are as given

vR

in Theorem 3.7, and vl(t,e) =

2(m le)l/2iu'(to+)-u1(to-)IeXP[-(me-1)1/2It-toil q(e(2q+2)!/2m)-1/2oglt-toI]-1/q

vl(t,e) = 2 v[l +

if

q = 0,

q > 1,

if

erq+1

where

Iu'(t0+)-u'(t0-)I{e(2q+2)!/2m}1/2

=

and

is some positive

c

constant.

Let us first consider the Dirichlet problem (DP1).

Proof:

pose that

u'(t-) < u'(t+).

(If

u'(to) > u'(t+), we let

Then we define for

obtain this case.)

t

in

[a,b]

and

We can supy - -y

and

e > 0

a(t,e) = u(t) - r(e), 0(t,e) = u(t) + wL(t,e) + wR(t,e) + vl(t,e) + r(e), r(e) = (eym-1)1/(2q+1)

where

not differentiable at a'(t-) < a'(to).

Indeed, for

h(t,a), and so a function in

for y > Iu"I(2q+1):. The function

is

t

in

(a,t0) U (t0,b), we have

ea" >

is a lower solution there (cf. Theorem 2.2).

S, we note that

(a,t0) U (t0,b)

a

t - to; however, this presents no problem because

v1

For the is the solution of ev" _ (2gml)! v2q+1

which satisfies

vi(to,e) _ -vi(t+,e) =

ZIu'(tp)-u'(to)I,

vI(t0,e) = vI(tp,e) = v.

With this function

v1, we see that

Q

is differentiable at

t = t0;

indeed,

6'(to,e) = S'(to,e) = ZIu'(to)+u'(tp)I + wL(to,e) + wR(t0,e)

and, as before, we can show that

e9" < h(t,s).

Thus

(a,Q)

is a bound-

ing pair and the result for the problem (DP1) follows from Theorem 2.2.

Similarly we can treat the problems (RP1) and (RP2); we leave the details to the reader.

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

III.

34

If the reduced path

is (IIn)- or (IIn)-stable, then the require-

u

must be interpreted as follows:

ment of convexity (that is, u" > 0) in

u" > 0

u'(t-) < u'(t0).

and

(a,t0) U (t0,b)

The precise result is contained in the next theorem whose proof is similar to that of Theorem 3.9.

Assume that the reduced equation (R1) has a (IIn)-or (IIn)-

Theorem 3.10.

u'(t0) < u'(t+)

where

(a,b)

of class

u = u(t)

stable solution

u(a) - w1u'(a) < A

(for

C(2)([a,b]), except at 1u"(t0)l < -.

and

wl = pl

t0

0), u(b) + a2u'(b) 0 in (a,t0) U (t0,b). Then there exists an A2 = p2 Co > 0 such that for 0 < c < e0 the problem (DP1), (RP1) or (RP2) has

a solution

for

y = y(t,e)

t

in

[a,b]

which satisfies, respectively,

0 < Y(t,e) - u(t) < wL(t,e) + wR(t,e) + vI(t,e) + cc1/n

(DP1),

0 < Y(t,e) - u(t) < vL(t,e) + wR(t,e) + vI(t,e) + ccl/n

(RP1)

0 < Y(t,e) - u(t) < vL(t,e) + vR(t,e) + vI(t,e) + cc1/n

(RP2).

or

wL

Here

wR

and

are as given in Theorem 3.2, vL

and

vR

are as given

in Theorem 3.8, and 2(n-1)(e(n+1)!/2m)-1/2Q1/2(n

vI(t,e) = 4[1 +

1)It-t0I]-2/(n-1),

where an+1

= e(n+l)'.Iu'(t0) - u'(t0)j2/2m and

c

is some positive constant.

Finally, if the reduced path

u

is (III n)- or (IIIn)-stable, then

the result analogous to Theorem 3.10 is valid provided that AIu'(a) > A, u(b) + w2u'(b) 2.B, u" < 0

in

u(a) -

(a,t0) U (t0,b)

and

u'(to) > u'(t0).

Notes and Remarks 3.1.

The theory of this chapter applies with little change to the more general problem

cy" - h(t,y,e), a < t < b, y(a,e) = A(c), y(b,c) _

B(e).

We need only require that

(t,y)

in

Di(u) (i = 0,1,2)

B(c) = B(0) + 0(1)

h(t,y,c) = h(t,y,O) + o(1)

and that

for all sufficiently small values of

for

and

A(c) = A(0) + 0(1) e.

Notes and Remarks

3.2.

35

The definitions of (Iq)-, (IIn)- and (III n)-stability were intro-

duced by Boglaev [6] and used by him to study the Dirichlet problem Earlier Bris proved Theorem 3.1 in the case of (lo)-stabil-

(DP1).

Among the other work done on the problem (DP1) we mention

ity.

only the papers of Tupchiev [86], Vasil'eva [87], Vasil'eva and Tupchiev [89], Carrier [8], Fife [24], O'Malley [76], Dorr, Parter and Shampine [20], Habets [29], Habets and Laloy [31], Flaherty and O'Malley [25] and Howes [39].

The Robin problems (RP1) and (RP2)

have also been considered by Habets and Laloy in the case of (In)stability. 3.3.

The stability requirements on the solution

u

of the reduced equa-

tion can be relaxed as follows (cf. Fife [24], Flaherty and O'Malley Namely it is enough in the case of (Iq)- or

[25] and Howes [39]).

(Iq)-stability that

8yh(t,u(t)) E 0 a2q+lh(t,u(t))

and for

for

0 < j < 2q,

> m > 0

in

[a,b],

u(a) # A,

h(a,s)ds > 0

for

t

u(a)

in

[A,u(a))

or

(u(a),A]

in

[B,u(b))

or

(u(b),B].

u(b) # B,

or, for

n

I (b)

h(b,s)ds > 0

for

rt

Similar relaxations apply to the cases of (IIn)-stability

(u(a) < A, u(b) < B) 3.4.

and (III)-stability (u(a) > A, u(b) > B).

We have not considered the occurrence of shock layer behavior,

that is, the situation in which a solution y = y(t,e)

of (DP1),

(RP1) or (RP 2) satisfies the limiting relation

lim y(t,e) _ e+0+

where

I ul(t),

a < t < t0,

u2(t),

t0 < t < b,

u1(t0) # u2(t0).

The functions

solutions of the reduced equation (R1).

ul

and

u2

are stable

These phenomena are

studied, for instance, by Vasil'eva [88], Fife [24], O'Malley [76] and Howes [39] to which the reader can refer for details.

36

3.5.

III.

SEMILINEAR SINGULAR PERTURBATION PROBLEMS

Oscillatory phenomena of the type exhibited by the solution of the problem (E2) are discussed for more general problems by Volosov [91] and O'Malley [76].

3.6.

We note that in the case of the Robin problem (RP2), (Iq)-, (IIn)and (III n)-stability are essentially equivalent to their "tilded" counterparts because function

3.7.

u(t), and

Ui(u) (i = 0,1,2) 6

is a

"6-tube" around the

can be taken arbitrarily small.

The theory developed in this chapter for the Robin problems (RP1)

and (RP2) applies with minor modification to the Neumann problem ey" = h(t,y), a < t < b, -y'(a,c) = A, y'(b,e) = B, and related problems.

Chapter IV

Quasilinear Singular Perturbation Problems

§4.1.

The Dirichlet Problem:

Boundary Layer Phenomena

We consider now the singularly perturbed quasilinear Dirichlet problem

ey" = f(t,y)y' + g(t,y) E F(t,y,y'),

a < t < b, (DP2)

Y(a.e) = A, If

y(b,e) = B.

f(t,y) t 0, a great variety of interesting phenomena can occur.

If

f(t,y) E 0, the problem (DP2) is identical to the problem (DP1) already discussed in the previous chapter.

Therefore at points

f(t,y) = 0, we require the function

F(t,y,y')

(t,y)

for which

to be stable with res-

pect to the y-variable (in the sense of Definitions 3.1-3.6), in order that the solutions of (DP2) may behave "reasonably".

There are, however,

qualitative differences between the two problems (DP1) and (DP2) which will be illustrated by the following simple examples. Consider first the linear problems ey" = ±y',

Y(0,c) = 0,

0 < t < 1, Y(l,e) = 1.

The solution of (E6) y(t,e) =

is easily found to be

(1-e-[-e-1/e

+

e-(1-t)/e

e-(1-t)/e,

and so lim

where

y(t,e) = 0

for

0 < t < 1 - 6 < 1,

d, 0 < 6 < 1, is a fixed constant.

37

(4.1)

That is, the solution only

38

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

IV.

exhibits nonuniform convergence at the right endpoint ing function

The limit-

is the solution of the corresponding reduced equa-

u = 0

which satisfies

u' = 0

tion

t = 1.

u(0) = 0 = y(O,e).

On the other hand, the problem (E6) has the solution y(t,e) =

(e-1/e_1)-1(-l+e-t/e)

- 1 -

e-t/e

and so lim+ y(t,e) = 1

for

0 < d < t < 1.

(4.2)

That is, the solution exhibits nonuniform convergence at the left endpoint

Here

t = 0.

satisfying

u = 1

is a solution of the reduced equation

u(l) - 1 = y(l,e).

u' = 0

Thus the nature of the solutions of these

two problems depends critically on the sign of the coefficient of

y'.

It is interesting to note also that the relations (4.1) and (4.2) differ from (3.2) in that nonuniform convergence of the solution (that is, boundary layer behavior) occurs at one endpoint only.

Consider next the problem ey" _ -ty',

-1 < t < 1, (E7)

Y(-',E) = -1,

Y(l,e) = 1.

The exact solution is

Y(t,e) _ -1 + 2(J1

e-s2/2eds)-1 ft 1

e-s2/2eds, 1

from which it follows that -1

lim+ y(t,c) _ e+O

where

1

for

-1 < t < -d < 0,

for 0 < 6 < t < 1,

6, 0 < 6 < 1, is a fixed constant.

the solution of the reduced equation y(-l,e), while 1 = y(l,e).

tu' = 0

is the solution of

ul - -1

satisfying

tu' = 0

is

u1(-1) = -1 =

satisfying

u2(1) _

In this case, the solution exhibits uniform convergence at

both endpoints. left endpoint

u2 - 1

The function

This is not at all surprising, if we note that near the t = -1, the coefficients of

y'

in (E7) and in (E6) are

both positive, and therefore we do not anticipate nonuniform convergence of the solution at the left endpoint

t = -1.

endpoint

y'

t = 1, the coefficients of

Similarly, near the right

in (E7) and in (E6) are both

negative, and so we do not anticipate nonuniform convergence of the solu-

The Dirichlet Problem:

4.1.

tion at the right endpoint

Boundary Layer Phenomena

t = 1.

Instead, the solution exhibits non-

uniform convergence at an interior point continuously in the limit as

39

t = 0, where it switches (dis-

a - 0) from

ul == -1

to

u2

1.

As our third example we take the problem

-1
ey"=ty'+Y, y(-l,e) = 1, The solution

(E8)

y(l,e) = 2.

y = y(t,e)

of (E8) can be shown to satisfy

lim+y(t,e) =0 for -1<-1+6
(4.3)

e+0 where

6, 0 < 6 < 1, is a fixed constant.

negative (positive) near

t = -1

The coefficient of

y'

is

(t = 1), and so the nonuniform conver-

gence of the solution near both endpoints is in accordance with that observed in (E 6).

The limiting function

of the reduced equation not continuous at

u =- 0

is the continuous solution

tu' + u = 0, whose general solution

u = c/t

is

t = 0.

Our final example is ey" = yy',

-1 < t < 1, (E9)

y(-l,e) = A.

y(l,s) = B,

which has been discussed in greater detail by Wasow [94] and O'Malley [75; Chapter 1].

The algebraic sign of the coefficient of

y'

in this

equation depends on the solution under consideration, and so the behavior exhibited by solutions of (E9) is more varied than that previously encountered.

For example, if A > 0

y = y(t,c)

of (E9) satisfies

lim y(t,e) = A (cf. (E6)), while if lim y(t,e) = B e+0+ (cf. (E6)).

y = y(t,e)

lim

e+0+ (cf.

and A > -B, then the solution

for

-l
B < 0

and A < -B, then

for

-1 < -1 + 6 < t < 1

On the other hand, if A = -B > 0 of (E9) satisfies A

for

B

for 0< 6< t< 1,

y(t,e) =

-1 < t < -6 < 0,

(E7)), while if A < 0 < B, then

then the solution

40

IV.

lim

e+0+

y(t,e) =0

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

-1<-l+6
for

(cf. (E8)).

As these examples show, the phenomena exhibited by solutions of the problem (DP2) are more diverse and complicated than those encountered in Our purpose here is to isolate several classes of

the last chapter.

problems for which coherent results are possible and to refer the reader to the relevant literature for discussions of current problems where complete answers are as yet unknown.

We begin our study of the problem (DP2) by associating with it three types of reduced problems, namely, a < t < ti < b,

f(t,u)u' + g(t,u) = 0,

(RL)

u(a) = A, f(t,u)u' + g(t,u) = 0,

a < t2 < t < b, (RR)

u(b) = B, and a < t < b,

f(t,u)u' + g(t,u) = 0,

where

t1, t2

are fixed numbers.

(R)

In what follows, solutions of (RL),

(RR) and (R) will be denoted by uL, UR the function

uR

uL

and

u, respectively.

is only required to exist in

is only defined in

(t2,b] c [a,b]

for

Note that

(a,tl) c [a,b], while

a < t1, t2 < b.

As in the previous chapter, we will only study solutions of these reduced problems which are stable in a sense to be defined momentarily. The definitions of stability will be given in terms of the following domains, where

is a small constant.

d > 0

If a solution the domain

D(uL)

u = uL(t)

of (RL) exists in

D(uL) = {(t,y)

:

a < t < b, Iy-uL(t)I < dL(t)},

where the positive continuous function IB-uL(b)I + d

[a,b], then we define

by

for

b - 6/2 < t < b

Similarly, if a solution we define the domain

D(uR)

and

u = uR(t)

dL(t)

satisfies

dL(t)

d

for

dL(t)

a < t < b - d.

of (RR) exists in

[a,b]

then

by

D(uR) = {(t,y): a < t < b, iy-uR(t)I < dR(t)) where the positive continuous function IA-uR(a)I + 6

for

a < t < a + 6/2

and

dR(t)

satisfies

dR(t) E 6

for

dR(t)

a+6 < t < b.

4.1.

The Dirichiet Problem:

Boundary Layer Phenomena

Finally, for a solution

u = u(t)

41

of (R) we define the domain

D(u)

by

D(u) _ {(t,y): a < t < b,

Iy-u(t)I < d(t)},

where the positive continuous function for d(t)

a < t < a + 6/2, d(t) 6

for

satisfies

d(t)

IB-u(b)I + 6

for

d(t)

_-

b - 6/2 < t < b

IA-u(a)I + 6

and

In addition, we will also consider

a + 6 < t < b - 6.

paths of the form

uL(t), F

u(t),

u0(t) =

uR(t),

a
uR(t),

a
uLM

a
UM.

t L < t
u(t), ul(t) - J and

u2(t)

for

a < tL < tR < b, and we define the additional domains: D(uL,u,uR) = {(t,Y): a < t < b, Iy-u0(t)I < 6), D(uL,uR) = D(uL,u,uR)

with

D(u,uR) _ {(t,Y): a < t < b,

tL = tR,

IY-uI(t)I < dl(t)},

where the positive continuous function

dl

satisfies

dl(t) - IA-u(a)I + 6

for a < t < a + 6/2 and dl (t) - 6 for a + 6 < t < b, and D(uL,u) = {(t,Y): a < t < b,

$y-u2(t)I < d2(t)}

where the positive continuous function

for b - 6/2 < t < b and

d2

satisfies

d2(t)

__

IB-u(b)I + 6

d2(t) -6 for a
We can now define the types of stability which we will require of

solutions of the reduced problems (RL), (RR) and (R). Definition 4.1.

A solution

u = uL(t)

said to be strongly (weakly) stable in constant

k

of the reduced problem (RL) is [a,b]

such that

f(t,y) > k > 0

(f(t,y) 10) in

D(uL).

if there exists a positive

42

IV.

A solution

Definition 4.2.

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

u = uR(t)

of the reduced problem (RR) is

said to be strongly (weakly) stable in constant

if there exists a positive

[a,b]

such that

k

f(t,y) < -k < 0

(f(t,y) < 0) in

A solution

Definition 4.3.

u = u(t)

V(uR).

of the reduced equation (R) is

said to be locally strongly (weakly) stable if there exists a positive constant

and a small positive constant

k

6

such that, if

u(a) # A,

f(t,y) < -k < 0 (f(t,y) < 0) in V(u) A [a, a + 2]

and, if u(b) # B, f(t,y) > k > 0

A solution

Definition 4.4. said to be (I [a,b]

(f(t,y) > 0)

q)_'

in

u = u(t)

V(u) fl [b -

2b].

of the reduced equation (R) is

(Iq)-, (IIn)-, (IIn)-, (III n)- or (III n)-stable in

if the function

h(t,y) = f(t,y)u'(t) + g(t,y)

requirements in the Definitions 3.1 - 3.6 for

satisfies all the

(Iq)-, ..., (III n)-stability,

respectively. Definition 4.5. in

[a,b]

The reduced path

u = u0(t)

is said to be weakly stable

if there exists a small positive constant

6

such that

f(t,y) > 0

in

V(uL,u,uR) A ([a,tL] U [tR -

f(t,y) < 0

in

V(uL,u,uR) n ([t L, tL + 2) U [tR,b]).

Z,

tR])

and

If

tL = tR (= t0), then

V(uL,u,uR) = V(uL,uR); and the reduced path

uL(t),

a < t < t0,

uR(t),

t0 < t < b,

u0(t) _

is said to be weakly stable in

[a,b]

f(t,y) > 0

in

V(uL,uR) n [a,t0]

f(t,y) < 0

in

V(uL,uR) n [t0,b].

if

and

Definition 4.6. in

[a,b]

The reduced path

u = u1(t)

is said to be weakly stable

if there exists a small positive constant

6

such that

The Dirichlet Problem:

4.1.

Boundary Layer Phenomena

f(t,y) > 0

in

D(u,uR) fl [tR

f(t,y) < 0

in

D(uL,u) fl [ti, tL + Z].

43

2, tR]

-

and

Definition 4.7. in

The reduced path

u = u2(t)

is said to be weakly stable

if there exists a small positive constant

[a,b]

f(t,y) > 0

in

D(uL,u) fl [a,tL]

f(t,y) < 0

in

D(uL,u) fl [t L, tL + 6]

6

such that

and

With these definitions we can now discuss the of solutions of the Dirichlet problem (DP2). chapter that the functions of class

f

and

and y

with respect to

C(1)

Moreover, in the cases of that

f

q-

or

g

asymptotic behavior

We assume throughout this

are continuous in

t

and

y

and

in the domain under consideration.

n-stable reduced solutions, we assume

have the required number of continuous partial deriva-

g

tives with respect to

y.

Finally, in the conclusion of each theorem below, the constant

c

is a known positive constant depending on the reduced solution under consideration. Theorem 4.1. tion

Let the reduced problem (RR) have a strongly stable soluof class

u = uR(t)

such that for for

t

in

0 < e < e0

[a,b]

C(2)([a,b]).

Then there exists an

the problem (DP2) has a solution

e0 > 0 y = y(t,c)

which satisfies

(y(t,e) - uR(t)I < wL(t,e) + cc,

where wL(t,e) = IA - uR(a)) exp[X(t-a)] and

X = -k/e + i/k + 0(e).

Proof:

As in the proof of Theorem 3.1, we construct a bounding pair

of functions

(a,B)

which satisfy all the conditions of Theorem 2.1.

Let us first linearize (DP2) about the reduced solution setting

z = y - uR.

This leads to

ez" = f(t,y)z' + hy(t,E)z - euR, z(a,e) = A - uR(a),

z(b,e) = 0,

uR

by

44

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

IV.

and

hy(t,g) = fy(t,)uR + gy(t,&)

where

point between

and

(t,u)

Since

(t, u+z).

by

tion of strong stability, and since in

(t,Q

is some intermediate

f(t,y) < -k < 0

is bounded, say

by defini-

IhyI < £ (R > 0),

V(uR), we are further led to the linear non-homogeneous differential

equation ez" + kz' + Rz = -cult

R*

If

ea2 + kA + t = 0

0 < c < k2/4R., the corresponding auxiliary equation

has two negative roots A _ -k/c + 2./k + 0(c) _ -k/c + 0(1)

XI = -R/k + 0(c) = 0(1).

Indeed, the positive function

wL(t,e) = IA - uf(a)Iexp [a(t-a)]

is the

solution of the corresponding homogeneous differential equation

ew"+kw' +Lw= 0 such that wL'(a,c) = AIA - uR(a)I < 0,

wL(a,c) = IA - uR(a)I,

that is, the solution

Let y > IuRI. is such that

wL

decreases to the right.

r(t,c) = eyZ-(exp[-A1(b-t)] - 1)

Then the function

0 < r < cc

and

-cc < r' < 0

for some

c > 0, and it sat-

isfies

cr" + kr' + Rr = -cy,

r(b) = 0.

We now define the bounding pair a(t,c) = uR(t) - WL(t,c) - r(t,c), p(t,c) = uR(t) + wL(t,c) + r(t,c),

and proceed to show that they satisfy all the required inequalities of Theorem 2.1. e(b,c). for

Clearly, a < Q, a(a,c) < A < S(a,c)

and

a(b,e) < B <

As regards the differential inequalities, we will only verify

a, since the verification for

entiation and Taylor's Theorem (with

0

follows by symmetry. (t,t)

By differ-

as the intermediate point),

we obtain ca" - F(t,a,a') = cu,, - ewL - er" - f(t,a)(a'-uR) -

-cIuRI - ewL - Cr" - kwL - kr' - LWL - kr > -cluRI + cy > 0,

4.1.

The Dirichlet Problem:

Boundary Layer Phenomena

45

since y > Iu''RI. is a lower solution, and similarly

Thus, a

and we conclude from Theorem 2.1 that for (DP2) has a solution

in

y = y(t,c)

$ is an upper solution,

0 < e < k2(4t)-1

the problem

satisfying

[a,b]

a(t,c) < y(t,c) < 0(t,e) In the same way we can prove the following result.

Let the reduced problem (RL) have a strongly stable solu-

Corollary 4.1. tion

0 < c < c0

such that for for

of class

u = uL(t)

in

t

Then there exists an

C(2) ([a,b]).

the problem (DP2) has a solution

co > 0 y = y(t,c)

which satisfies

[a,b]

Iy(t,c) - uL(t)I <_ IB - uL(b)Iexp [A(t-b)] + cc,

where

A = k/c - 1./k + 0(c) = k/c + 0(1).

If the functions

uL

and uR are only weakly or locally weakly

[a,b], then we must in addition assume that

stable in

uL

and

uR

are

y-stable in order to prove the existence of solutions of (DP2) behaving The precise results follow.

as in Theorem 4.1.

Let the reduced problem (RL) or (RR) have a weakly or

Theorem 4.2.

locally weakly stable solution C(2)([a,b]) an

c0 > 0

y = y(t,c)

u = uL(t)

or

which is (Iq)- or (Iq)-stable in such that for for

in

t

0 < e < e0

[a,b]

of class

u = uR(t) [a,b].

Then there exists

the problem (DP2) has a solution

which satisfies

IY(t,c) - uL(t)I < wR(t,c) + r(c) or Iy(t,c) - uR(t)I < wL(t,e) + r(c).

Here, wR

and

wL

are as given in Theorem 3.1 with u

uL

replaced by

ccl/(2q+1)

and

uR, respectively, and

for the case of weak

Ir(E)I <

stability and for the case of locally weak stability with q = 0, while Ir(c)I < ccl/{2q(2q+1)} for the case of locally weak stability with

q > 1. Proof:

The proof will be similar to that of Theorem 3.1.

We shall only

consider the case of the reduced problem (RL) having the solution which is locally weakly stable and can be proved in a similar manner.

We define, for

a < t 0

46

IV.

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

a(t,e) = uL(t) - wR(t,e) - r(e) B(t,e) = uL(t) + wR(t,e) + r(e), where

r(e) = (epy/m)1/(2q+1), p = 1

weak stability with

q > 1, and y > 0

for weak stability and for locally

q = 0, p = 1/(2q)

for locally weak stability with

will be made precise below.

We recall that

IB-uL(b)Iexp[-(m/e)1/2(b-t)]

if

q = 0,

if

q > 1,

wR(t,e) = IB-uL(b)Ie1/2q [el/2+a(b-t)]-1/q

where qIB-uL(b)Iq[(q+l)(2q+l)!]-1/2.

a = ml/2 It only remains to show that ferential inequalities.

verification for

a

a

and

a

satisfy the required dif-

Consider then just the function

follows by symmetry.

0

since the

It follows from Taylor's

Theorem and the hypothesis of (Iq)-stability that 2q jl!

h(t,uL)(Q-uL

ay

JI1 a2q+1

h(t,n)(S-uL)2q+1

+ (2q+1)!

+ f(t,$) wR - euL - ewR p

(2q+1)! -

Thus, if

uL

If(t,s)wRI - eIuLI

is weakly stable, i.e., if

f(t,QwR > 0, we have p

F(t,S,a') - C011 > (2q+i)! - eIuLI > 0

if we choose y > IuLI(2q+1)!.

On the other hand, if uL

ally weakly stable, then

f(t,E)wR 10 in

[b-6/2,b],

but in the rest of the interval

[a,b],

is only loc-

The Dirichlet Problem:

4.1.

Boundary Layer Phenomena

47

IB_uL(b)I(m/e)1/2exp[-(m/e)1/2(b-t)J,

I

if

q = 0,

if

q > i

wR IB-uL(b)IC1/2q oq-1[C1/2

=

so that, for sufficiently small

_ (cle, If(t,C)wR'I < `I l

for some

c1 > 0.

1/2q C

+

6(b-t)]-(q+l)/q,

c,

if

q=0

if

q > 1

Thus

F(t,6,6') - e6" > 0

y > (c1 + IuLI)(2q+1)!. The conclusion of the theorem

if we choose

follows from Theorem 2.1.

Assume that the reduced problem (RL) or (RR) has a weakly

Theorem 4.3.

or locally weakly stable solution which is also (II

C(2)([a,b])

uL > 0

that

in

(a,b)

and

n) uL(b) 0

or

uR > 0

in

such that for

y = y(t,e)

for

t

of class

[a,b].

in

Assume also

(a,b)

and

0 < c < CO [a,b]

which

satisfies

0 < Y(t,e) - uL(t) < wR(t,e) + r(e) or

0 < Y(t,E) - uR(t) < wL(t,e) + r(e). Here wR and

and

wL

are as given in Theorem 3.2 with

u replaced by

uL

uR, respectively, and

Ir(e)i

ccI/n 1/{n(n_1)}

for the weakly stable case,

ce

for the locally weakly stable case.

Let us consider only the case of

Proof:

uR.

We can prove this case by

defining a(t,e) = uR(t),

6(t,e) = uR(t) + wL(t,e) + where

p = 1

(or

(Cpym-1)1/n,

p = 1/(n-1)), if uR

is weakly (or locally weakly)

stable and proceeding as in the proof of Theorem 4.2.

48

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

IV.

Assume that the reduced problem (RL) or (RR) has a weakly

Theorem 4.4.

u = uL(t)

or locally weakly stable solution C(2)([a,b])

uL < 0

in

or

u = uR(t)

and

(a,b)

Then there exists an

uL(b) > B

uR < 0

such that for

e0 > 0

y = y(t,e)

has a solution

or

for

in

t

in

and

(a,b)

uR(a) > A.

the problem (DP2)

0 < e < e0

[a,b]

of class

Assume also that

which is also (III n)- or (III n)-stable.

which satisfies

-wR(t,e) - r(e) < y(t,e) - uL(t) < 0 or

-wL(t,e) - r(e) < y(t,e) - uR(t) < 0,

wR and wL

where

are as given in Theorem 3.2 with

uR, respectively, and

and

Simply let

Proof:

y -

-y

r

u

uL

replaced by

is as given in Theorem 4.3.

and apply Theorem 4.3.

We next consider the solution

u = u(t)

of the reduced equation (R)

which does not satisfy either boundary condition but which is locally The proofs of the following two

strongly or locally weakly stable.

theorems are only a slight modification of the proofs of Theorems 4.2 and 4.3.

Theorem 4.5.

Let the reduced equation (R) have a locally strongly or

locally weakly stable solution (Iq)- or (Iq)-stable in

also

that for t

in

0 < e < e0

[a,b]

u = u(t) [a,b].

of class

C(2)([a,b])

Then there exists an

the problem (DP2) has a solution

which is such

e0 > 0

for

y = y(t,e)

which satisfies

Iy(t,e) - u(t)I < wL(t,e) + wR(t,e) + r(e).

Here, r

is as given in Theorem 4.2, wR = IB-u(b)Iexp [A(b-t)],

wL = IA-u(a)Iexp [a(t-a)], and

wR

X'= -k/e + 0(1), if

u(t)

is locally strongly stable, while

are as given in Theorem 3.1, if

Theorem 4.6.

u = u(t)

is also (IIn)- or (IIn)-stable in

(a,b), u(a) < A and

[a,b]

wL,

is locally weakly stable.

Let the reduced equation (R) have a locally strongly or

locally weakly stable solution

for

u(t)

0 < e < e0

u(b) < B.

[a,b].

of class

Assume also that

Then there exists an

the problem (DP2) has a solution

which satisfies

C(2)([a,b])

u" > 0

e0 > 0

y = y(t,e)

which in

such that for

t

in

4.2.

Robin Problems:

Boundary Layer Phenomena

0 < y(t,e) - u(t) < wL(t,e) + wL(t,e) +

WL, wR

where

stable, while

are as given in Theorem 4.5, if u wL, wR

49

el/n0

is locally strongly

are as given in Theorem 3.2, if u

is locally

weakly stable.

If the function u

is locally strongly or locally weakly stable and

(III n)- or (III n)-stable, then the result analogous to Theorem 4.6 holds,

provided that

u" < 0

in

(a,b), u(a) > A

and

u(b) > B.

We leave its

precise formulation to the reader.

Robin Problems:

§4.2.

Boundary Layer Phenomena

We now turn to the occurrence of boundary layer phenomena for solutions of the Robin problems ey" = f(t,y)y' + g(t,y) = F(t,y,y'), y(a,e) - Ply'(a,e) = A,

a < t,< b,

y(b,e) = B.

ey" = f(t,y)y' + g(t,y),

(RP3)

a < t < b, (RP4)

y(a,e) - Ply'(a,e) = A,

y(b,e) + P2y'(b,e) = B.

with their associated reduced problems f(t,u)u' + g(t,u) = 0,

a < t < t1 < b, (RL)

u(a) - plu'(a) = A,

f(t,u)u' + g(t,u) = 0,

a < t2 < t < b, (RR)

u(b) = B,

f(t,u)u' + g(t,u) = 0,

a < t2 < t < b,

(RR)

u(b) + p2u'(b) = B, and

f(t,u)u' + g(t,u) = 0,

a < t < b.

(We note that the related problem

with

y(a,e), y(b,e) + p2y'(b,e)

the change of variable

uL

and

ey" = f(t,y)y' + g(t,y), a < t < b,

prescribed, can be handled by making

t . a + b - t

to the transformed problem.)

(R)

and applying the results for (RP3)

We denote the solutions of (RL) and (R) by

u, respectively, while the solutions of (RR) and (RR) will be

50

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

IV.

denoted, without any distinction, by

functions

dR, d

and u

uL, uR

ity apply to these

and

Earlier definitions of stabil-

uR.

except that for the case (RP3), the

are uniformly small in

d1

[a,b-6]

dR, d, di = 6)) while for the case (RP4) the functions and

d2

are uniformly small in

(that is,

dL, dR, d, di,

[a,b].

To provide some insight into the results for such problems, we conThe first two problems are

sider three simple examples.

Eyn = ±y'

,

0 < t < 1,

y(O,c) - y'(O,e) = 1,

The exact solution of (E10) is, with A-1[e-1/E

-

Y(t,e) =

(E10)

Y(l,c) = 2.

A = e-1/E - 1

2(1+c-1) + e-t/e]

-

e_l

2 - ee-t/c

and therefore

lim y(t,e) = 2

in

[0,1].

The limiting function lem

is, with

uR = 2

is the solution of the reduced prob-

On the other hand, the exact solution of (E0)

u' = 0, u(l) = 2.

e-1/c(1-E-1),

0 = 1 A-1[1

Y(t,c) =

-

2e-1/E(1-c-1) + e-(1-t)/E]

i + e-(1-t)/e and so lim

y(t,c) = 1

0 < t < 1-6 < 1.

for

Here the limiting function reduced problem

uL = 1

is of course the solution of the

u' = 0, u(0) - u'(0) = 1.

The third problem is

0
ey"=y',

Y(0,c) - Y'(0,E) = 1,

Y(l,c) + Y'(l,c) = 2. A = c-1 - el/E(1-e-1),

The exact solution is, with p-1[c-1 Y(t,c) =

-

2e-l/E(1-e-1) +

- 1 + Ee-(1-t)/e ,

and consequently

e-(1-t)/c]

4.2.

Robin Problems:

lim

Boundary Layer Phenomena

y(t,e) = 1

0 < t < 1.

for

Here the limiting function

51

u' = 0, u(0) -

is the solution of

uL = 1

u'(0) = 1, and in contrast with (E1*0), is the uniform limit of the solution

y(t,e)

in the whole interval

[0,1].

We return now to the problem (RP3) to show the relation between its solution and the solutions of the reduced problems (RL) and (R). Theorem 4.7.

Assume that the reduced problem (R) has a strongly or locof class

ally strongly stable solution u = u(t) also (Iq)- or (Iq)-stable in that for in

[a,b]

0 < e < e0

[a,b].

C(2)([a,b])

Then there exists an

which is

e0 > 0

y = y(t,e)

the problem (RP3) has a solution

such for

t

which satisfies cel/(2q+1)

Iy(t,e) - uL(t)I < vL(t.c) + wR(t,e) + where wR(t,e) = IB - u(b)Iexp[-ke-1(b-t)] and e(plk)-1IA-u(a)+plu'(a)Iexp[-ke-1(t-a)].

vL(t,e) = Proof:

We consider only the case of (Iq)-stability, and we define for

a < t 0

p1Q'(a,e), if and

Clearly, we have

will be determined later.

Q(t,e), a(b,e) F(t,a,a')

h(t,y) = f(t,y)u' + g(t,y),

we obtain r ea n - F(t,a,a) =

eun

-

evnL

- ewnR

-

2q

j J 1 h j; 8y(t,u)(a-u)

j=l (2q+1):

ayq+lh(t,n)(a-u)2g+1

- f(t,E)(a'-u')

> eIu"I - evL - ewR + (2gm1),(vLq+l+w2Rq+l} + (2q+1)!

+ f(t,)(vL + w'R),

> eIu"I - evL - ewR + (2Q-+L + f(t,E)(vL + wR)

52

IV.

Now, for

in

t

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

[a, a+6/2]

f(t,Q < -k < 0, and hence,

f(t,E)vI > -kvj = evL.

c, the term

Also, for sufficiently small ITI
c0 > 0

and so in

-ew" + fwR = T(e)

is such that

[a,a+6/2],

call - F(t,a,a') > -clu"I + (2q2q+1)'

- c0e,

> 0,

if we choose y = (co + (u"I)(2q+1)!.

Similarly, for

t

in

[b-6/2,b],

we have

f(t,Q > k > 0, and so

f(t,)wR > kwR

= ewR.

For sufficiently small ITI < clc

for some

c, the term

cl > 0.

-cv" + fvL' = T(e)

Thus, in

satisfies

[b-6/2,b],

ca" - F(t,a,a') > -cfu"I + (2q+1)! - c1e > 0,

y > (cl + Iu"I)(2q+1)!. Finally, in

if we choose

sufficiently small

[a+6/2, b-6/2], for

c, we have

call - F(t,a,a') > -CIu"I + (2q+1)' + T(e) + T(e) l

> 0,

by choosing

y > {cl + co + Iu"I}(2q+1)!.

The result of the theorem now follows from Theorem 2.3.

Following the proofs given above and in Theorem 4.3, we can obtain the next result.

Assume that the reduced problem (R) has a strongly or loc-

Theorem 4.8.

ally strongly stable solution also (IIn)- or (IIn)-stable in

of class

u = u(t) [a,b].

C(2)([a,b])

Assume also that

u(b) < B.

u" > 0

Then there exists an

(a,b), u(a) - plu'(a) < A

and

such that for

the problem (RF3) has a solution

for

t

in

0 < c < c0

[a,b]

which satisfies

which is in

co > 0

y = y(t,c)

Robin Problem:

4.2,

Boundary Layer Phenomena

0 < y(t,e) - u(t) < vL(t,c) + wR(t,e) + where

wR

and

vL

53

el/n,

are as given in Theorem 4.7.

On the other hand, if the strongly or locally strongly stable solution

of the reduced problem (R) is (III n)- or (I'IIn)-stable and

u(t)

satisfies

in

u" < 0

there exists an a solution

(a,b), u(a) - plu'(a) > A

such that for

e0 > 0

y(t,e)

in

[a,b]

and

u(b) > B, then

the problem (RP3) has

0 < e :<.z 0

satisfying

-vL(t,e) - wR(t,e) - ccl/n < y(t,e) - u(t) < 0.

The following results for the weakly or locally weakly stable solution

can be proved using analogous arguments.

u(t)

Let the reduced problem (R) have a weakly or locally weakly

Theorem 4.9.

stable solution (IQ)-stable in

0 < e < e0 [a,b]

u = u(t) [a,b].

of class

which is also (Iq)- or

C(2)([a,b])

Then there exists an

e0 > 0

the problem (RP3) has a solution

such that for

y = y(t,e)

for

t

in

which satisfies IY(t,e) - U(t)j.< vL(t,e) + wR(t,e) + r(e),

where

wR

is as given in Theorem 3.1 and

vL

in Theorem 3.4.

Let the reduced problem (R) have a weakly or locally weakly stable solution stable in

u = u(t)

[a,b]

C(2)([a,b])

[a,b], and, moreover, satisfies

plu'(a) < A and 0 < e < e0

of class

u(b) < B.

which is also (IIn)- or (IIn)u" > 0

Then there exists an

the problem (RP3) has a solution

in

(a,b), u(a) -

co > 0

y = y(t,e)

such that for for

t

in

which satisfies 0 < Y(t,e) - u(t) < vL(t,e) + wR(t,e) + r(e),

where

wR

is as given in Theorem 3.2 and

vL

in Theorem 3.5.

We leave it to the reader to formulate the statement of the results for a weakly or locally weakly stable solution

u(t)

which is also (III

or (III n)-stable.

The following results follow from the proofs of Theorems 4.7 - 4.9. Corollary 4.2.

Suppose the reduced problem (RL) has a solution

u = uL(t)

satisfying the hypotheses in each of the Theorems 4.7 - 4.9, then the conclusions of each of the theorems hold with out the term

vL(t,a).

uL

replacing

u, but with-

n)-

54

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

IV.

On the other hand, if the reduced problem (RR) has a solution

u =

satisfying the hypotheses in each of the Theorems 4.7 - 4.9, then

uR(t)

uR

the conclusions of each of the theorems hold with

replacing

u, but

without the term wR(t,e). With these results, it is now an easy matter to deal with the problem (RP4), and so we omit the proofs of the following results. Theorem 4.10.

Assume that the reduced equation (R) has a locally strongly

(or weakly) stable solution u = u(t) also (Iq)-stable in 0 < e < e0

of class

Then there exists an

[a,b].

the problem (RP4) has a solution

which is

C(2)([a,b])

e0 > 0

such that for for

y = y(t,c)

in

t

which satisfies

[a,b]

ly(t,e) - u(t)l < vL(t,e) + vR(t,e) + r(e).

Here vL = e(plk)-1IA-u(a)+plu'(a)Iexp[-ke-1(t-a)], e(p2k)-llB-u(b)-p2u'(b)lexp[-ke-1(b-t)],

VR =

if u

is locally strongly stable, and

3.7 if u

is locally weakly stable, and

Theorem 4.11.

also (IIn)-stable in

t

in

0 < e < e0

[a,b]

u = u(t)

[a,b].

u(a) - plu'(a) < A and such that for

r

are as given in Theorem

is as given in Theorem 4.2.

Assume that the reduced equation (R) has a locally strongly

(or weakly) stable solution

for

vL, vR

of class

Assume also that

u(b) + p2u'(b) < B.

C(2)([a,b])

u" > 0

in

which is [a,b],

Then there exists an

the problem (RP4) has a solution

e0 > 0

y = y(t,e)

which satisfies

0 < y(t,e) - u(t) + vL(t,e) + vR(t,e) + r(e), where

vL

and

vR

are as given in Theorem 4.10 (or Theorem 3.8) if

is locally strongly (or weakly) stable, and

r

u

is as given in Theorem

4.3.

Corollary 4.3.

Suppose the reduced problem (KL) has a solution

u = uL(t)

satisfying the hypotheses in each of the Theorems 4.9, 4.10.

Then the

conclusions of each of the theorems hold with

u, but without

uL

replacing

the term vL(t,e). On the other hand, if the reduced problem (RR) has a solution u = uR(t)

satisfying the hypotheses in each of the Theorems 4.9, 4.10,

then the conclusions of each of the theorems hold with but without the term

vR(t,e).

uR

replacing

u,

Interior Layer Phenomena

4,3.

The corresponding result

55

for (III n)-stable cases we leave to the

reader to state.

Interior Layer Phenomena

§4.3.

The remaining part of this chapter will be devoted to a discussion of interior crossing phenomena for solutions of the problems (DP2), (RP3) What we have in mind is the following.

and (RP4).

that the reduced problem (RL) has a solution problem (RR) has a solution in

(a,b)

and the reduced

u = uL(t)

which intersect at a point

with unequal slopes, that is, uL(t0) = uR(to)

and

to

uL' (t0)

Then we have the reduced path

uR(t0).

a < t < to,

U (t) u0 (t) =

(R0)

L

uR(t),

where

u = uR(t)

Suppose, for example,

t0 < t < b,

We ask under what conditions will the problem

u6(t0-) # u0(t0).

(DP2) possess a solution lim

y = y(t,e)

such that

y(t,c) = u0(t)?

(4.4)

E->0+

The answer to this question is not as straightforward as it was in the case of the problem (DP1) considered in the previous chapter. all, if the pair of reduced solutions

uL

and

uR

stable, then such a pair can never attract a sol ution in the sense of (4.4).

uL, U

R

crosses at

First of

are both strongly y(t,c)

of (DP2)

The reason for this is simply that if the pair

t0, then in the small interval

t0-6 < t < t0+6,

the Cauchy problem f(t,u)u' + g(t,u) = 0, u(t0) = o(= uL(t0) = uR(to))

must have two distinct solutions, but this

smoothness of f

and

g

is impossible in view of the

and the fact that

jfI > 0.

Consequently, we

can only expect the relation (4.4) from the crossing of two weakly stable solutions of the reduced equation (R)

then it necessarily follows that

since if

f(t0,o) = 0.

uL(t0) = uR(t0) = o,

With these remarks, we

can now discuss some general results for this crossing phenomenon, first for the Dirichlet problem (DP2) and then for the Robin problems (RP3) and (RP4).

56

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

IV.

To give our first results, let us recall from Theorem 3.9 the function (e/m)1/2Iu'(t0)-u'(t0)Iexp[-(e/m)1/2It-t0I],

if

q = 0,

if

q > 1,

vl(t,e)

2 v[1+q(e(2q+2)!/2m)

where aq+1 = lu'(t0) - u'(t0)I{e(2q+2)!/2m}1/2.

Theorem 4.12.

that for t

in

Assume that the reduced path (R0) is of class

(Iq)-stable in

stable and

[a,b].

Then there exists an

the problem (DP2) has a solution

0 < e < c0

C2, weakly such

e0 > 0 y = y(t,c)

for

which satisfies

[a,b]

cel/(2q+1)

ly(t,e) - u0(t)l < vl(t,e) +

where

vl(t,e)

vl(t,e)

is

with

replaced by

- u'(t-)I

lu'(t+)

1 u'(t0) - uL(t0)I Proof:

The proof is similar to that of Theorem 3.9.

uL(t0) < uA(t0), and define for

a(t,e) = u0(t) -

(e1'm

a < t lu"1(2q+1)!.

c > 0

1)1/(2q+1)

e(t,e) = u0(t) + vl(t,e) + (eyra

where

and

We can suppose that

t = t0; indeed, al(to) < a'(t0).

is not differentiable at

a

Nevertheless, by our choice of

y, we

can show as in the proof of Theorem 3.9 that call > F(t,a,a')

that is, a of

vi, a

in

(a,b)

is a lower solution. is differentiable at

eel' < F(t,e,e')

in

{t0},

e, we see that by our choice

As regards

t = t0, and again by our choice of y,

(alb) . {t0}.

Thus the results follow from Theorem 2.2. Theorem 4.13.

Assume that the reduced path (R0) is of class

stable and (IIn)-stable. uL(t0) < uR(t0).

Moreover, assume

Then there exists an

e0 > 0

the problem (DP2) has a solution y = y(t,c) satisfies

uL > 0,

u'R' > 0

C2, weakly and

such that for for

t

in

[a,b]

0 < e < e0

which

4.3.

Interior Layer Phenomena

57

0 < y(t,E) - u0(t) < vl(t,c) + ccl/n,

where

vI(t,e)

is

vI(t,e)

with

in place of

luj(t0) - uL'(t0)I

2

in place of

n-1)

q

and with

lu'(t+) - u'(t0)I.

This result follows by arguing as in the proof of the previous

Proof:

theorem if we define for

a < t 0

a(t,e) = u0(t),

0(t,e) = u0(t) + vl(t,e) + (c 1)1/n, where y > IullIn!

and Z(n-1)(e(n+l)!/2m)-l/201/2(n-l)It-t0l]-2/(n-1)

vl(t,a) = 2 0[l +

with 0n+1

- e(n+1)!Iuj(t0) - uL(t0)l/2m.

If the weakly stable reduced path

R0

is (IIIn)-stable, then a

result analogous to Theorem 4.13 is valid provided that and

(a,t0) U (t0,b)

u0 < 0

in

We leave its precise formulation

uL'(t0) > uj(t0).

to the reader.

The above results can be generalized to the three-branch reduced path f uL(t), u(t), u0(t) =

a < t < tl,

1 uR(t),

t2 < t < b,

t1 < t < t2,

uL, uR

where the solutions

intersect the middle

reduced equation (R), such that u(t2) = uR(t2)

duced path

and

u0(t)

Theorem 4.14.

u'(t2) # uR(t2).

for

in

t

0 < e < c0

[a,b].

u0(t)

v1(t,e)

is

respectively, by with

t0

and

the problem (DP2) has a solution

C2,

c0 > 0

y = y(t,c)

which satisfies

vl(t,e) t1

and

with

t0

and

1/(2q+1)

Iu'(t0)

Iu'(t1) - uL(tl)I, while

lu'(t+) - u'(t_)l

IuR(t2) - u' (t2) I

is of class

Then there exists an

IY(t,e) - u0(t)I < v1(t,e) + v2(t,e) + ce Here

of the

tl = t2, this becomes the re-

If

Assume that the reduced path

[a,b]

u

above.

weakly stable and (Iq)-stable in such that for

solution

uL(tl) = u(t1), uL(tl) # u'(tl),

- u'(tp)I

v2(t,c)

replaced, respectively, by

replaced, is

vl(t,e)

t2

and

58

IV.

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

The corresponding result for the (II)-stable solution

we

u0(t)

will leave to the reader to formulate, but we wish to state the result for the following reduced path I

ul(t) _ where

,

a < t < t2,

uR(t),

t2 < t < b,

u(t)

u'(t2) # u1(t2).

Here, since the reduced solution

satisfy the boundary condition at

does not

u(t)

t = a, we expect that the solution of

(DP2) which follows the reduced path

will behave nonuniformly at

ul(t)

Similar results hold for the "reflected path"

t = a.

uL(t),

a < t < tl,

u(t)

tI < t < b.

u2 (t) _ Theorem 4.15.

,

Assume that the function

u(t)

weakly) stable and that the reduced

path

stable and (Iq)- or (iq)-stable in

[a,b].

such that for for

in

t

0 < e < e0

[a,b]

is locally strongly (or

ul(t)

is of class

C2, weakly

Then there exists an

the problem (DP2) has a solution

c0 > 0

y = y(t,e)

which satisfies cel/(2q+1)

ly(t,E) - ul(t)I < wL(t,E) + v2(t,E) +

Here, wL

is as given in Theorem 4.1 or Theorem 4.3, according as

is locally strongly or weakly stable, and

v2

u

is as given in Theorem 4.14.

We conclude this chapter with the following theorems for the Robin problems (RP3) and (RP4) whose proofs are similar to those of Theorems 4.12 - 4.15.

Theorem 4.16.

Let the assumptions of Theorems 4.12 - 4.14 hold with the

solution of (RL) replaced by the solution of (RL) for the Robin problem (RP3) or with the solutions of (RL) and (RR) replaced, respectively, by the solutions of (RL) and (RR) for the Robin problem (RP4).

Then the

conclusions of Theorems 4.12 - 4.14 hold with (DP2) replaced by (RP3)

or (RP4) . Theorem 4.17.

Let the assumptions of Theorem 4.15 hold for the Robin

problem (RP3) or with the solution of (RR) replaced by the solution of (RR) for the Robin problem (RP4).

Then the conclusions of Theorem 4.15

hold with (DP2) replaced by (RP3) or (RP4) and with Here

vL

strongly (or locally weakly) stable.

replaced by

wL

is given in Theorem 4.7 (or Theorem 3.5) if

u

is locally

VL.

Notes and Remarks

59

Notes and Remarks 4.1.

The theory developed above applies with little change to the more general equation A(c)

and

ey" - f(t,y,c)y' + g(t,y,e)

and boundary data

B(c), provided that

{f,g} (t,Y,e) - {f,g} (t,Y,O) + 0(1), for

(t,y)

in the appropriate domain, and

{A(c),B(e)} = {A(0),B(0)} + 0(1) 4.2.

for

0 < c < e0.

The Dirichlet problem (DP2) has been studied by many people including Tschen [85], von Mises [64], Oleinik and Zizina [70], Coddington and Levinson [14], Bris [7], Wasow [92], Vasil'eva [87], Erd6lyi [22], Willett [96], O'Malley [72], Cole [55; Chapter 2], Eckhaus [21; Chapter 5], Cook and Eckhaus [16], Chang [9], [10], Dorr, Parter, Shampine [20], Ackerberg and O'Malley [1], Habets [28], Habets and Laloy [31], Kreiss and Parter [56], Matkowsky [63], Howes [39] and Olver [71].

The majority of these papers deal with

solutions of the various reduced problems which are strongly stable and/or (IO)-stable.

The corresponding Robin problems (RP3) and (RP4) have also been We mention only the papers by Bris [7],

studied extensively.

Vasil'eva [87], O'Malley [75; Chapter 7], Cohen [15], Keller [51], Macki [62], Searl [84], Dorr, Parter, Shampine [20], Habets and Laloy [31] and Howes [43]. 4.3.

The stability conditions given in Definitions 4.1 and 4.2 can be weakened in the following way, as first observed by Coddington and Levinson [14].

Consider first a solution

u = uR(t)

duced problem (RR).

If there is a positive constant

f(t,uR(t)) < -k < 0

in

[a,b]

such that

and

f(a,s)ds > 0

(uR(a)-A) J

of the rek

for all

in

(uR(a),A]

y = y(t,c)

such that

E

uR(a)

or then the problem (DP2) has a solution lim y(t,e) = uR(t) e+0+

for

[A,uR(a)),

a < a+d < t < b,

Similarly, if there is a positive constant f(t,uL(t)) > k > 0

in

[a,b], where

k

u = uL(t)

such that

is a solution of

the reduced problem (RL), then the problem (DP2) has a solution y = y(t,c)

such that

60

IV.

lim

y(t,e) = uL(t)

QUASILINEAR SINGULAR PERTURBATION PROBLEMS

provided that

a < t < b-d < b

for

(n f(b,s)ds < 0

(uL(b)-B) 1

for all

n

(uL(b),B]

in

uL(b)

or 4.4.

[B,uL(b)).

In considering the asymptotic behavior of solutions of (DP2), (RP3)

and (RP4) in the presence of locally strongly or weakly or locally weakly stable solutions of the corresponding reduced problems, we always assumed an additional form of

y-stability.

If such y-

stability is not assumed then the theory becomes incredibly complicated, even in the linear case, that is, f(t,y) = f(t) g0(t)y.

and

g(t,y)

The reader can consult the book of O'Malley [75; Chapter

8] and the article of Olver [71] for discussions of such problems in the linear case.

A corresponding nonlinear theory is nonexist-

ent, in the sense that if our assumptions of y-stability are dropped, then no conclusions regarding the existence and the behavior of solutions of these problems

4.5.

can be drawn.

It is also possible to weaken our assumptions regarding the Robin problems (RP3) and (RP4) in the case that the reduced solution is strongly stable, as was first observed by Bris [7].

Suppose, for

example, that we consider the problem (RP4) and let

u = uL(t)

a strongly stable solution of the reduced problem (RL).

f(a,uL(a)) + Plhy(a,uL(a)) # 0, for

h(t,y) = f(t,y)uL(t) + g(t,y),

the problem (RP4) has a solution y = y(t,e) lim

y(t,e) = uL(t)

Similarly, if

u = uR(t)

is a strongly stable solution of the ref(b,uR(b)) - p2hy(b,uR(b)) # 0,

then the problem (RP4) has a solution

Note that if uL

or

uR

such that

a < t < b.

for

duced problem (RR) such that

lim y(t,e) = uR(t) e*0+

be

Then if

for

y = y(t,e)

such that

a < t 0. P2 y We have not considered shock layer behavior for solutions of (DP2), (RP3) or (RP4) in this chapter.

The reader can consult Howes [39],

[43] for a discussion of these phenomena.

Chapter V

Quadratic Singular Perturbation Problems

45.1.

Introduction

In this chapter we investigate the asymptotic behavior of solutions of boundary value problems for the differential equation ey" = p(t,y) Y'2 + g(t,y),

a < t < b,

(DE)

The novelty here is the presence of the quadratic term in

y'.

The more

general differential equation eY" = P(t,Y)Y'2 + f(t,Y)Y' + g(t,y)

will not be studied, since it can be reduced to the form (DE) in some cases by the familiar device of completing the square.

Our decision to

study the simpler equation (DE) rather than the more general equation stems from a desire to present representative results for this "quadratic" class of problems without having to deal with extra complexities in notation.

45.2.

The Dirichlet Problem:

Boundary Layer Phenomena

We shall first consider the following Dirichlet problem eY" = p(t,Y)Y'2 + g(t,y),

a < t < b, (DP3)

Y(a,e) = A,

y(b,c) = B.

To motivate some of the results to follow, it is useful to pay attention to the results that have been obtained for the model problem (cf. [27])

61

62

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

ey"=1-y, 0
y(l,e) = B.

y(0,e) = A,

The exact solution of (E12), for tures (cf.

IA-BI < 1, as

[94]), is

y(t,e) = C In cosh[{t - 2(A-B + 1)}/e] + Since

can be found by quadra-

cosh z =

2(ez + e-Z)

?(A+B - 1) + 0(E).

z - , it is obvious that the

2 ez, as

asymptotic behavior of the solution depends critically on the relative

A

position of the boundary values

and

In particular, we can dis-

B.

tinguish the following three cases: (i)

If

IA-BI < 1, then Ciiim0+ y(t,c) = lim eln[Z xp({t-

z(A-B+1)}/E] +

2(A-B+l)

It - 2(A-B+1)I + 2 A-B+1)

where (ii)

A-t,

0 < t < t0,

t+B - 1,

to < t < 1,

t0 = 2(A-B+1).

If A-B = -1, then y(t,e) = t+A = t+B - i

(iii)

If A-B = 1, then y(t,e) = A-t

On the other hand, if

is the solution of (E12). IA-BI > 1

can be found again by quadratures. (iv)

If

If

then the exact solution of (E12)

There are two cases:

B-A > 1, then lim+ y(t,c) = t+B - 1

(v)

is the solution of (E12).

for

d < t < 1, where

0 < d < 1.

B-A < -1, then lim+ y(t,c) = A-t

for

0 < t <

E,+O

In case (i) above, the solution

u0(t) = max{A-t, t+B-1}

in

y(t,E)

follows the angular path

[0,1], while in cases (iv) and (v), the

solution displays the familiar boundary layer behavior at

t = 0

and

The Dirichlet Problem:

5.2.

t = 1, respectively.

Boundary Layer Phenomena

The functions

uL(t) = A-t

63

uR(t) = t+B - 1

and

are of course solutions of the reduced equation

1-u'2=0

(R12)

which satisfy, respectively, uL(0) = A and uR(l) = B.

However, the

equation (R12) has another pair of solutions with these properties,

namely uL(t) = t+A functions

and uR(t) = B+l - t.

As we have just seen, the

uL, uR do not participate in the asymptotic description of Therefore, in any asymptotic theory for the general

solutions of (E12).

problem (DP3), we should give criteria to distinguish possible limiting solutions from all solutions of the corresponding reduced problems. As a first step toward developing such a theory, we define the following reduced problems p(t,u)u'2 + g(t,u) = 0,

a < t < tl < b, (RL)

u(a) = A, p(t,u)u'2 + g(t,u) = 0,

a < t2 < t < b, (RR)

u(b) = B, and

p(t,u)u'2 + g(t,u) = 0,

a < t < b.

(R)

Solutions of (RL), (RR) and (R) will be denoted throughout this chapter by

uL, uR, and

in

u', the reduced equation (R) may have, in addition to the general

In view of the quadratic nonlinearity

u, respectively.

solution, a singular solution

us

(cf.

[48; Chapter 3], [38]).

This

phenomenon of a singular solution did not arise in earlier chapters.

A

singular solution is easily visualized as the envelope of a one-parameter family of solutions of (R) and as such

us = 0, that is, us = const.

A

simple example of such a situation is afforded by the Clairaut equation

u'2-u=0,

(E13)

whose singular solution tions

if

us = 0

is the envelope of the family of solu-

u(t) = q(t+c)2; see Figure 5.1.

u(t0) = 0, then

u(t) =

member of the family smoothly.

p(t0,us)

us

at 0

to, then

(cf.

u'(t0) = 0, that is, any

intersects the singular solution

Similarly, if a solution

singular solution

provided

{u(t)}

Note that for the equation (E13), and

4(t-t0)2

u

us = 0

of the equation (R) intersects a

u(t0) = us

[48; Chapter 3]).

and

u'(t0) = us(t0) = 0,

64

V.

QUADRATIC SINGULAR PERTURBATION PROBLEMS

u

Figure 5.1

We now define the domains in which the functions

p

and

g

have

certain properties relative to solutions of the reduced problems (RL), (RR) and (R). Let

u = uL(t)

be a solution of (RL) for

a < t < b.

If

uL(b) < B,

we define the domain R+(uL

where

d+(t)

{(t,y)

:

a < t < b,

-6 < y - uL(t) < d+(t)},

is a positive continuous function satisfying

d+(t) eB-uL(b)+6 for b-6/2
d+(t) =-6 If

for a
uL(b) > B, we define the domain R (uL) _ {(t,y)

where

dL(t)

-dL(t) I Y - uL(t) < 6},

: a < t < b,

is a positive continuous function satisfying

dL(t) - uL(b) - B + 6

for

b - 6/2 < t < b

and

dL(t)6 for a
u = uR(t)

be a solution of (RR) for

uR(a) < A, we define the domain

a < t < b.

The Dirichlet Problem:

5.2.

R+(uR) _ {(t,y)

where

d+R(t)

Boundary Layer Phenomena

-d < y - uR(t) < d+R(t)},

a < t < b,

:

65

is a positive continuous function satisfying

dR(t) - A - uR(a) +6 for

a < t < a + 6/2

and

d+R(t) r6 for a+d
uR(a) > A, we define the domain a < t < b,

R {uR) _ {{t,Y)

where

dR(t)

-d-RR(t) < y - uR(t) < d),

is a positive continuous function satisfying

dR(t) -uR(a) - A+d for a
for a+ 6 < t < b.

Finally, for a solution u = u(t)

of (R), we define the domains

R+(u), R -(u), R±(u), R+(u), depending on the relative values of u(a), A, u(b)

and If

B.

u(a) < A and

R+ (u) = {(t,Y) where

d+(t)

:

u(b) < B, we define the domain

a < t < b, -d < y - u(t) < d+(t)),

is a positive continuous function satisfying

d+(t) = A-u(a)+d

for

a < t < a + 6/2,

d+(t) = B-u(b)+6

for

b- 6/2 < t < b,

d+(t) - 6

for

a+6 < t < b-6.

and

If

u(a) > A

R (u) = {(t,y)

where

and

d -(t)

u(b) > B, we define the domain

and

:

a < t < b,

-d-(t) < y-u(t) < 6),

is a positive continuous function satisfying a < t < a + 8/2,

d -(t) - u(a) - A+6

for

d -(t) = u(b) - B+d

for b - 6/2 < t < b,

66

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

for

d (t) a 6

If u(a) < A

a+6 < t < b-d.

R±(u) = {(t,y)

where

u(b) > B, we define the domain

and

:

a < t < b,

-d-(t) < y-u(t) < d+(t)),

are positive continuous functions satisfying

d+, d

d+(t) S A-u(a) + 6

for

a < t < a + 6/2,

d+(t) S 6

for

a+d < t < b,

d -(t) = u(b) - B+6

for

b - 6/2 < t < b,

d (t) =d

for a
and

Lastly, if

u(a) > A

R+(u) = {(t,y)

where

:

u(b) < B, we define the domain

and

a < t < b,

-d-(t) < y-u(t) < d+(t)),

are positive continuous functions satisfying

d+, d

d+(t) S u(a) - A+6

for

a < t < a +

d+(t) F 6

for

a+d < t < b,

d -(t) = B - u(b)+d

for b - 6/2 < t < b,

d (t)

for a
6/2,

and

If

u = us

d

is a singular solution of (R), then the corresponding domains

are denoted by R(us), R (us), R+(us)

and

R+(us).

As in the last chapter, we will also investigate paths formed by two or more intersecting solutions of the reduced problems. ing four paths:

uo(t) _

ul(t) =

uL(t),

a < t < to,

uR(t),

t0 < t < b,

uL(t),

a < t < tL,

u(t),

tL < t < tR,

uR(t),

tR < t < b,

u(t),

a < t < tR,

uR(t),

tR < t < b,

u2(t)

For the follow-

The Dirichiet Problem:

5.2.

Boundary Layer Phenomena

67

and

u3 (t)

uL(t),

a < t <

u(t),

tL < t < b,

L

we define the corresponding domains: R(u0(t)) _ {(t,y)

:

R(ul(t)) _ {(t,y)

: a < t < b, ly-ul(t)l < 6},

a < t < b, ly-u0(t)! < 61,

R+(u2(t)) _ {(t,y)

:

a < t < b, -6 < y-u2(t) < dZ(t)

R (u2(t)) _ {(t,y)

:

a < t < b, -d2(t) < y - u2(t) < 6),

R+(u3(t)) _ {(t,Y)

: a < t < b, -6 < y-u3(t) <

d3(t)},

and a < t < b, -d3(t) < y-u3(t) < 6),

R (u3(t)) _ {(t,Y)

where

d2, d2, d3

and

d3

are positive continuous functions satisfying,

respectively, d2 B A-u(a)+6

for

a < t < a + 6/2,

and

d2(t) = 6

for

d2(t) = u(a)-A+6

a+6 < t < b, if u(a) < A; for

a < t < a + 6/2,

and d2(t) = 6

for

a+6 < t < b,

d3+

(t) ° B-u(b)+6

for

if

u(a) > A;

b - 6/2 < t < b,

and d3(t) e 6

for

d3(t) B u(b)-B+6

a < t < b-6,

u(b) < B;

if

for b - 6/2 < t < b,

and

d3(t) B 6

for

a < t < b-6,

if u(b) > B.

We now define the types of stability which solutions of the reduced problems (RL), (RR) and (R) are to possess in order to permit the study of the Dirichlet problem (DP3) and other related boundary problems. Definition 5.1.

exists in

[a,b]

A solution u = uL(t)

of the reduced problem (RL) which

is said to be strongly (weakly) stable in

there exists a positive constant

k

such that

[a,b]

if

68

V.

QUADRATIC SINGULAR PERTURBATION PROBLEMS

2p(t,y)uLI (t) > k > 0 (2p(t,y)uL(t) > 0) A solution

Definition 5.2.

exists in

u = uR(t)

there exists a positive constant

or R (uL).

of the reduced problem (RR) which

Definition 5.3.

A solution

if

[a,b]

such that

k

in

(2p(t,y)uR'(t) < 0)

2p(t,y)uR'(t) < -k < 0

[a,b]

R+(uL)

is said to be strongly (weakly) stable in

[a,b]

said to be (I

in

u = u(t)

R+(uR)

or

R (uR).

of the reduced equation (R) is

(Iq)-, (IIn)-, (IIn)-, (III n)2 or (IIIn)-stable in

q)-,

if the function

+ g(t,y)

h(t,y) = p(t,y) (u'(t))

is (Iq)-,...,

(III n)-stable in the sense of Definitions 3.1 - 3.6, respectively. Definition 5.4.

A solution

u = u(t)

of the reduced equation (R) is

said to be locally strongly (weakly) stable if there exists a positive constant

k

and a small positive constant

2p(t,y)u'(t) < -k < 0 if

such that

(2p(t,y)u'(t) < 0)

in

R(u) n [a,a+6]

u(a) # A, and 2p(t,y)u'(t) > k > 0

if

6

u(b) # B

(2p(t,y)u'(t) > 0)

for R(u) = R+(u), R _(u), R±(u)

Definition 5.5.

The reduced path

(weakly) stable in

or

R(u) n [b-6,b] R+(u).

is said to be strongly

if there exists a positive constant

[a,b]

a small positive constant

k

(and

6) such that

2p(t,uL(t))uL(t) > k > 0 (2p(t,y)uL(t) > 0

u = u0(t)

in

in

in

[a,t0]

R(uL,uR) n [t0-6,t0])

and 2p(t,uR(t))uR(t) < -k < 0 (2p(t,y)u.(t)

Definition 5.6.

< 0

in

R(uL,uR) n [t0,t0+6]).

in

The reduced path

(weakly) stable in

6

in

in

in

[a,tL]

R(uL,u,uR) n [tL-6,tL]),

2p(t,u(t))u'(t) < -k < 0 (2p(t,y)u'(t) < 0

is said to be strongly

such that

2p(t,uL(t))uL(t) > k > 0 (2p(t,y)uL'(t) > 0

u = u1(t)

if there exists a positive constant

[a,b]

small positive constant

[t0,b]

in

[tL,tL+6]

R(uL,u,uR) n [tL,tL+6]),

k

and a

5.2.

The Dirichlet Problem:

Boundary Layer Phenomena

2p(t,u(t))u'(t) > k > 0 (2p(t,y)u'(t) > 0

in

in

69

[tR_ 6't RI

R(uL,u,uR) n ftR_ 6't RD,

and 2p(t,uR(t))uR(t) < -k <0 (2P(t,Y)uR'(t) < 0

Definition 5.7.

in

R(uL,u,uR) n [tR,tR+d]).

in

The reduced path

(weakly) stable in

u = u2(t)

is said to be strongly

if there exists a positive constant

[a,b]

small positive constant

[tR,b]

2p(t,y)u'(t) <.-k < 0

in

(2p(t,y)u'(t) < 0

R±(u,uR) n [a,a+d]),

in

R-(u,uR) n [a,a+d]

in

2p(t,u(t))u'(t) > k > 0 (2p(t,y)u'(t) > 0

and a

k

such that

S

[tR_ S,tRJ

R±(u,uR) n [tR-S,tR])

in

and 2p(t,uR(t))uR(t) < -k < 0 (2p(t,y)uR(t) < 0 Definition 5.8.

in

R±(u,uR) n [tR,tR+S]).

in

The reduced path

(weakly) stable in

[a,b]

a small positive constant

in

in

2p(t,y)u'(t) > k > 0 (2p(t,y)u'(t) > 0

is said to be strongly positive constant

k

and

such that

S

in

[a,tL]

R(uL,u) n [tL-S,tL]),

2p(t,u(t))u'(t) < -k < 0 (2p(t,y)u'(t) < 0

u = u3(t)

if there exists a

2p(t,uL(t))uL(t) > k > 0 (2p(t,y)uL'(t) > 0

[tR,b]

in

[tL,tL+S]

R±(uL,u) n [tL,tL+S]), in

in

R-(uL,u) n [b-6,b]

R±(uL,u) n [b-S,b]).

With these definitions we are now in a position to give results on the types of asymptotic behavior displayed by solutions of the Dirichlet problem (DP3). and

g

pect to

In what follows we tacitly assume that the functions p

are continuous in y

t

and sufficiently differentiable with resNote that in the statement of

in the domains defined above.

each of the theorems below, the constant

c

on the reduced solution under consideration.

is positive, depending only

70

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

Theorem 5.1.

Assume that the reduced problem (RL) or (RR) has a strongly

stable solution

u = uL(t)

or u = uR(t)

of class v

also that there exists a positive constant p(t,y) > v > 0

R (uL) if and p(t,y)

+

in

R

(u L)

uL(b) B, or (ii) p(t,y) > v > 0 in < -v < 0 in R (uR) if uR(a) > A.

Then there exists an (DP3) has a solution

e0 > 0

y = y(t,e)

such that for in

Assume

C(2 ([a,b]).

such that either (i)

if

R+(uR)

0 < e < e0

in

uR(a) < A the problem

satisfying

[a,b]

-cc < y(t,c) - uL(t) < wR(t,e) + cc,

if uL(b) < B,

-wR(t,e) - cc < y(t,e) - uL(t) < cc,

if uL(b) > B,

-CE < y(t,e) - uR(t) < wL(t,e) + cc,

if

uR(a) < A,

-wL(t,e) - ce < y(t,e) - uR(t) < cc,

if

uR(a) > A,

or

where -ev-1ln{(b-a)-1[b-t+(t-a)exp(-IB-uL(b)Ive

wR(t,e)

1)]},

and

wL(t,e) = -ev-lln{(b-a)-1[t-a+(b-t)exp(-IA-uR(a)Ive-1)]}. Proof:

As the proofs of all cases are similar, we shall only give the

proof for the case where the reduced equation (RR) has a solution u = uR(t)

such that

uR(a) < A

We first linearize about

and

uR

p(t,y) > v > 0

by setting

in

z = y - uR.

R+(uR).

This leads to

ez" = p(t,y)z'2 + 2p(t,y)u'Rz' + Ey(t,&)z - cull ,

z(a,e) = A - uR(a),

where

(t,t)

Since

p(t,y) > v

and since

z(b,c) = 0,

is some intermediate point between

gy

and

2puj < -k < 0

is bounded, say

(t,uR)

and

(t,uR+z).

by definition of strong stability,

IgyI < k

(k > 0) in

R+(uR), we are

further led to the nonlinear differential equation ez" =

z'2 - kz' - kz - cull R*

The nonlinear (quadratic) part and the linear part will be utilized to construct the bounding pair of functions.

wL(t,e) _ -(e/v)ln

{(b-a)-1[t-a

Indeed the function

+ (b-t)exp(-IA-u R(a)Ive-1)]}

5.2.

The Dirichlet Problem:

Boundary Layer Phenomena

71

is the solution of the nonlinear boundary value problem ew" = Vw'2,

w(a,e) = IA-uR(a)I, If

w(b,c) = 0,

0 < c < k2 /4Z, the characteristic equation

ea2 + kA + R = 0

has two

negative roots A = -t/k + 0(c)

Then, for any

and

XI = -k/e + 0(c).

y > 0, the function

r(t,e,y) = eyt-1(exp[-A(b-t)] - 1)

is the solution of the linear non-homogeneous equation er" = -kr' - tr - ey such that

0 < r < cc

r(b,e,y) = 0, for some

and

-cc < r' < 0,

c > 0.

We now define the bounding pair a(t,e) = uR(t) - r(t,c,y) e(t,e) - uR(t) + wL(t,e) + r(t,e,Y),

where y

and

y

are positive constants to be chosen so that the bound-

ing pair satisfy the required inequalities of Theorem 2.1. It follows from the definition that

a(b,e) < B < p(b,e), since

uR(b) = B.

a < g, a(a,e) < A < $(a,e)

Differentiating and applying

Taylor's Theorem, we have call - p(t,a)a'

2

- g(t,a) = euR - er" P(t,uR r)(uR r')2 - g(t,uR r) = Eu" - er" - 2puRr' - pr,2 + g(t,uR) - g(t,uR r)

-eIuRI - (Sr" + kr' + in

-

E(Y - IuRI - IPIc2E) > 0, if we choose Y = IuRI + 1, for

0 < c
c

sufficiently small, say

pr12

and

72

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

es" - p(t,s)s'2 - g(t,s) = eu" + ew" + er" - pu'2 - pw'2 R

L

R

p1''2

L

- 2puj(wL+r') - 2pwLr' - g(t,s) < eu'R' + swti + et" - vwL2 + kr' + R(wL+P),

< eIu"I + RwL - e'f < 0,

if we choose for some

sufficiently small, say

c

K > 0, and then choose

0 < c < e2, so that

RwL < eK

lu"I + K.

R 0 < e < c0 = min{c1,e2}, the bounding pair satisfies the

Thus, for

required inequalities of Theorem 2.1 and the conclusion follows. A similar result holds if the functions weakly stable in pect to

and

are only

uR

[a,b], provided that some form of stability with res-

is imposed.

y

uL

A precise discussion is contained in the follow-

ing theorems.

Assume that the reduced problem (RL) or (RR) has a weakly

Theorem 5.2.

stable solution

p(t,y)(B-uL(b)) > 0

that in

u = uL(t)

R±(uR) n [a,a+d].

stable in

or in

Assume also that

Then there exists an

[a,b].

of class

u = uR(t)

R±(uL) n [b-6,b]

or

C(2 ((a,b])

such

p(t,y)(A-uR(a)) > 0

or uR is (Iq)- or (I

uL

e0 > 0

0 < e < e0, the problem (DP3) has a solution

q)-

such that for

y = y(t,s)

in

[a,b]

satisfying _ce1/(2q+1)

< y(t,e) - uL(t) < wL(t,e) + cel/(2q+1)

-wL(t,e) - cel/(2q+l) < y(t,e) - uL(t) <

Cel/(2q+l)

if uL(b) < B,

if uL(b) > B,

or -cc l/(2q+1)

< y(t,e) - uR(t) < wL(t,e) + cel/(2q+1)

_wL(t,e) - Cel/(2q+l) < y(t,e) - uR(t) < Cel/(2q+1)

and wR

where the functions

wL

Theorem 3.1 with

replaced by

Proof:

u

if

uR(a) < A,

if uR(a) > A,

are as defined in the conclusion of uR

and

uL, respectively.

Let us give the proof for the case when (RL) has an (I0)-stable

solution

u = uL(t)

b-6 < t < b.

such that

uL(b) > B

and

p < 0

in

The proofs for the other cases are analogous.

As noted in the proof of Theorem 3.1, the function wR(t,e) =

JB-uL(b)lexp[-(me-I)1/2(b-t)]

R (uL)

for

The Dirichlet Problem:

5.2.

Boundary Layer Phenomena

73

is the solution of

ew" = mw, w(b,E) = IB - uL(b)I = uL(b)-B,

(me-1)1/2IB-uL(b)I.

w'(b,e) =

Let us define the bounding pair a(t,e) = uL(t) - wR(t,e) - eym 0(t,e) = uL(t) + EMm 1,

where M = Since

and y

IuLI,

is a positive constant to be chosen later.

clearly satisfies all the required inequalities of Theorem

0

2.1, we turn our attention to

Theorem (with

a.

Differentiating and applying Taylor's

as the intermediate point), we obtain

(t,E)

ea" - p(t,a)a'2 - g(t,a) =

cut, - ewR - p(t,a)uL2 - p(t,a)w'2

+ 2p(t,a)uLwR - g(t,a) > -CM - mwR

-

pwR2 + 2puLwR'

+ [Py(t,E)'L2 + gy(t,E)](wR + Eym > -CM - pw'R2 + cY,

by the weak stability of

puLwR > 0

since

by the (I0)-stability of

uL.

uL, and

pyuL2 + gy > m > 0

Now, in the interval

[b-6/2,b], we have

and so the desired inequality follows by setting y > M.

p < 0

remaining interval

pwR2 < c1c

for

set y > M + c1.

[a,b-6/2), there exist

c1 > 0

and

e0 > 0

In the

such that

0 < e < co; and so the desired inequality follows if we Theorem 5.2 now follows from Theorem 2.1.

A similar argument allows us to prove the following theorem. Theorem 5.3.

Assume that the reduced problem (RL) or (RR) has a weakly

stable solution

u = uL(t)

or

also (IIn)- or (IIn)-stable in uR > 0 [b-6,b]

e0 > 0

in

or

(a,b), uL(b) 0

p(t,y) > 0

in

which is

or R+(uL) f1

Then there exists an

the problem (DP3) has a solution

satisfying

0 < y(t,E) - uL(t) < wR(t,E) + cc1/n or

of class

Assume also that

74

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

0 < Y(t,e) - uR(t) < wL(t,e) + cel/n, wL

where the functions u

Theorem 3.2 with

wR

and

are as defined in the conclusion of

replaced by

uR

uL, respectively.

and

The next result follows from Theorem 5.3 with the change of variable

y + -Y Assume that the reduced problem (RL) or (RR) has a weakly

Theorem 5.4.

u = uL(t)

stable solution

or

in

u'R' < 0

R (uL)

fl

solution

or

uR(a) > A, and

p(t,y) < 0

in

R-(uR) n [a,a+d].

or

such that for

c0 > 0

y = y(t,e)

in

[a,b]

0 < e < e0

which

C(2)([a,b])

Assume also that

[a,b].

(a,b), uL(b) > B

[b-6,b]

exists an

of class

u = uR(t)

or

is also (IIIn)- or (IIIn)-stable in

uL < 0

p(t,y) < 0

in

Then there

the problem (DP3) has a

satisfying

-wR(t,e) - cel/n < y(t,e) - uL(t) < 0 or

-wL(t,e) - ccl/n < y(t,c) - uR(t) < 0,

where the functions

wL

Theorem 3.2 with

replaced by

u

and

wR

are as defined in the conclusion of uR

and

uL, respectively.

In the next two theorems, the reduced equation (R) has a solution u = u(t)

which in general satisfies neither of the boundary conditions

but which is locally strongly or weakly stable.

The proofs are similar

to those of Theorems 5.1 and 5.2 and are omitted. Theorem S.S.

Assume that the reduced equation (R) has a locally strongly

(weakly) stable solution

p(t,y) >_v > 0

of class

u = u(t)

(p(t,y) > 0) in

such that

C(2) (ja,b])

if u(a) < A

R+(u) fl {[a,a+6] U [b-6,b))

u(b) < B, p(t,y) < -v < 0 (p(t,y) < 0) in R _(u) fl {[a,a+6] U [b S,b]} if u(a) > A and u(b) > B, p(t,y) > v > 0 (p(t,y) > 0) in R±(u) fl [a,a+d] and p(t,y) < -v < 0 (p(t,y) < 0) in R±(u) fl [b-6,b] if u(a) < A

and

and

u(b) > B, and

p(t,y) > v > 0 u(b) 0) in

for a positive constant

(Iq)-stable in

0 < e < e0

p(t,y) < -v < 0

[a,b].

(p(t,y) < 0) in

R+(u) v.

fl

[b-6,b]

if

R+(u) fl [a,a+6]

u(a) > A

Assume also that

Then there exists an

the problem (DP3) has a solution

e0 > 0

y = y(t,c)

u

is

and (I

in

[a,b]

satisfying -cc I/(2q+1) < y(t,e) - u(t) < wL(t,e) + wR(t,e) + cel/(2q+l)

if

u(a) < A and u(b) < B;

q)-

such that for

and

5.2.

The Dirichiet Problem:

ccl/(2q+1)

-wL(t,e) -

if

Boundary Layer Phenomena

75

< y(t,e) - u(t) < wR(t,e) + ce1/(2q+1)

u(a) > A and u(b) < B; -wR(t,e) - ccl/(2q+l) < y(t,e) - u(t) < wL(t,e) + cc

if

1/ (2q+ 1)

u(a) < A and u(b) > B; or -wL(t,e) - wR(t,E) - cel/[2q+1) < y(t,e) - u(t) < cel/(2q+1)

if

u(a) > A

and

u(b) > B.

Here

wL(t,e) _ -CV_ lln{(b-a)-1[t-a + (b-t)exp(-jA-u(a)jvc-l)]} and

wR(t,e) = -CV_ 1ln{(b-a)-1[b-t + (t-a)exp(-IB-u(b)Ive-1)]}

if u

wL, wR

is locally strongly stable, and

conclusion of Theorem 3.1 if Theorem 5.6.

Assume that the reduced equation (R) has a locally strongly

(weakly) stable solution (IIn)- or (IIn)-stable in

of class

u = u(t) [a,b].

[b-d,b]} for a positive constant 0 < e < c0

(p(t,y) > 0) v.

which is also

C(2)([a,b])

u" > 0

Assume also that

u(a) < A, u(b) < B, p(t,y) > v > 0

that for

are as defined in the

is locally weakly stable.

u

in

in

(a,b),

R+(u) fi {[a,a+d] U

Then there exists an

the problem (DP3) has a solution

e0 > 0

y = y(t,e)

such in

satisfying

[a,b]

cel/nI

0 < y(t,e) - u(t) < wL(t,c) + wR(t,e) +

where

wL, wR

are as defined in Theorem 5.5 (Theorem 3.2) for the

locally strongly (weakly) stable solution.

An analogous result holds when the locally strongly or weakly stable function

u

and

fl {[a,a+d] U [b-d,b]}.

R _(u)

variable

y - -y

p(t,y) < -v < 0

2p(t,y)us(t) = 0

tainly locally weakly stable. is

in

in

and applying Theorem 5.6 to the transformed problem.

u = us = const., then

us

or

It can be proved by making the change of

We remark that if the solution u

if

u" < 0

p(t,y) < 0

is (III n)- or (III n)-stable, provided that

(a,b), u(a) > A, u(b) > B

of (R) is a singular solution in

[a,b]

and so

us

is cer-

Consequently Theorems 5.S and 5.6 apply,

q- or n-stable and satisfies the additional geometric condi-

tions of these theorems.

76

§5.3.

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

Robin Problems:

Boundary Layer Phenomena

We turn now to a consideration of similar phenomena for the Robin problems a < t < b,

Ey" = P(t,y)y'2 + g(t,y),

(RP5)

y(a,e) - ply'(a,e) = A,

y(b,c) = B,

and a < t < b,

ey" = P(t,y)y'2 + g(t,y),

(RP6)

y(a,e) - Ply'(a,e) = A, where

and

pl

p2

y(b,e) + P2y'(b,e) = B,

are positive constants.

ey"

(The related problem

p(t,y)y'2 + g(t,y), a < t < b, y(a,e), y(b,e) + p2y'(b,e) can be studied by making the change of variable

t - a+b-t

the theory of (RP5) to the transformed problem.)

prescribed,

and applying

The associated reduced

problems are then p(t,u)u'2 + g(t,u) = 0,

< b,

a < t < t 1

(RL)

u(a) - Plu'(a) = A, p(t,u)u'2 + g(t,u) = 0,

a < t2 < t < b, (RR)

u(b) = B, p(t,u)u'2 + g(t,u) = 0,

a < t2 < t < b, (RR)

u(b) + P2u'(b) = B, and p(t,u)u'2 + g(t,u) = 0,

whose solutions are denoted by

a < t < b,

(R)

uL, uR, and

u, respectively.

The

definitions of stability given at the beginning of this chapter are assumed to apply to the functions

uL, uR

and

In the case of (RP5) the domains

fications.

u with the following modiR+(uR)

and

R (uR)

are

replaced by R(uR) = {(t,y): a < t < b,

(y - uR(t)I < 6),

while in the definition of the domains

R+(u), R _(u), R±(u)

and

R+(u)

the error function is assumed to be uniformly small (that is, bounded Similarly, in the case of (RP6) the error

above by

6) in

function

in each of the domains is assumed to be uniformly small in

[a,b-6].

[a,b].

Boundary Layer Phenomena

Robin Problems:

5.3.

77

The theory for the problems (RP5) and (RP6) is not as straightforward as for their counterparts in Chapter IV, in view of the nonlinear (quadratic) dependence on

y'.

We shall discuss the Robin problem (RP5) first.

results relate to the solution Theorem 5.7.

Assume that the reduced problem (RL) has a strongly (weakly)

stable solution

u = uL(t)

(Iq)-stable in

v

of class

p(t,y) < -v < 0

there exists an

(p(t,y) > 0)

(p(t,y) < 0)

in

in

which is also (Iq)- or

R+(uL)

R (uL)

in

such that

e0 > 0

y = y(t,e)

has a solution

C(2)([a,b])

Assume also that there exists a positive constant

[a,b].

p(t,y) > v > 0

such that

and

The following two

of the reduced problem (RL).

uL

for

if

uL(b) B.

Then

0 < e < e0, the problem (RP5) satisfying

[a,b]

-Ce1/(2q+1) < y(t,e) - uL(t) < wR(t,e) + Cel/(2q+1),

if

uL(b) < B,

_WR(t,e) - Cel/(2q+l) < y(t,e) - uL(t) < ccl/(2q+l),

if

uL(b) > B,

and

wL

where

Proof:

are as defined in Theorem 5.1 (Theorem 3.1) with

u

uL, for the strongly (weakly) stable solution.

We only give the proof for the case when (RL) has a strongly stable

solution with

wR

and

replaced by

u = uL(t), which is also (10)-stable and satisfies

p > v > 0

in

uL(b) < B,

R+(uL).

Define the functions

a(t,e) = uL(t) - eMm 0(t,e) = uL(t) + wR(t,e) + eMm 1,

where M > Iu"J.

It is straightforward to show that

pla'(a,e) < A < B(a,e) - p1e'(a,e), and

a,$

a < s, a(a,e) -

a(b,e) < B < e(b,e); moreover,

satisfy the required differential inequalities (cf. the proofs of

Theorem 5.1 and 5.2).

Thus the conclusion of Theorem 5.7 follows from

Theorem 2.3.

Similarly the following result can be proved by applying Theorem 2.3. Theorem 5.8.

Assume that the reduced problem (RL) has a strongly (or

weakly) stable solution u = uL(t) (IIn)- or (ftn)-stable in uL(b) v > 0

e0 > 0

such that for

y = y(t,e)

in

of class

[a,b]

C(2)([a,b])

Assume also that

[a,b]. (or

p(t,y) 10)

in

which

uL > 0 R+(uL).

in

is also (a,b),

Then there

0 < e < co, the problem (RP5) has a satisfying

78

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

0 < y(t,c) - uL(t) < wR(t,e) + cel/n

where wR ing

is as defined in Theorem 5.1 (or Theorem 3.2 with

u) for the strongly (or weakly) stable solution

uL

replac-

uL.

A corresponding result holds for (IIIn)- or (III n)-stable solutions uL < 0

uL, provided that (or

p(t,y) < 0) in

in

(a,b), uL(b) > B

and

p(t,y) < -v < 0

We leave the precise formulation of this

R (uL).

result to the reader.

Let us now study the case when the reduced problem (RR) has a strongly stable solution

u

has a solution

uR(t).

We expect that if uR

satisfies an

t = a, then the boundary value problem (RP5)

appropriate inequality at y - y(t,c)

.satisfying

in [a,b].

lim+ y(t, e) = uR(t) e-'0

More precisely, we have the following result. Theorem 5.9.

stable.

Assume that the reduced problem (RR) has a solution

of class

u = uR(t)

Let

which is strongly stable and

C(2)([a,b])

max

L =

Ip(t,y)I.

(I0)-

Then, if

R(uR)

kI = k - Lp1'JuR(a) - pluR(a) - Al > 0, tr+ere exists an

a solution

e0 > 0

y = y(t,c)

such that for in

[a,b]

(t)

0 < c < cos the problem (RP5) has

satisfying

Iy(t,c) - uR(t)I < VL(t,c) + eMm 1 where VL(t,c) = c(plkl)-IIuR(a) - p1uR(a)-Alexp[-kl(t-a)c-1]. Note:

The inequality in (t) can be motivated by considering, in place

of (RP5), the initial value problem ey" = p(t,y)y'2 + g(t,y),

a < t < a+d, (I)

y(a,e) = uR(a),

y'(a,e) = pII[uR(a) - A].

It follows from the study of initial value problems (cf. (61], [93; Chapter 10], [87; Chapter 1]) that the solution converge to

uR

as

y = y(t,c)

of (I) will

a - 0+, provided the "initial jump" Iy'(a,e) - uj(a)I

is sufficiently small, that is, provided

IuR(a) - pluR(a)-AI < kpl/L.

Robin Problems:

5.3.

Proof:

Boundary Layer Phenomena

We prove only the case for which p(t,y) < 0

for the case

79

p(t,y) > 0, since the proof uR

Linearizing about

is similar.

by setting

z = y - uR, we are led to the Robin problem ez" = p(t,y)(z'+2u'R)z' + [p (t,E)uR2 + gy(t,E)]z - euR z(a) - p1z'(a) = A - uR(a) + p1uR'(a) = y

z(b) = 0, where

(t,)

Since

2puR < -k

is some intermediate point between and

pyuR2 + gy > m > 0

and

(t,uR)

(t,uR+z).

by assumption, we are further

led to the equation ez" = [p(t,y)z' - k]z' + mz - euR.

Indeed, we note that the positive function of

ev" = -k1v'

vL = 0(e),

v = vL(t,e)

and

vL(a) = -Y/pl

0 > p(t,y)vL > kl-k.

Suppose first that the reduced problem uR p1uR(a) > A.

is the solution

with the properties:

is such that

uR(a) -

We then define the following pair of functions:

a(t,e) = uR(t) - vL(t,e) - eMm 1, cMml,

S(t,e) = uR(t) +

where M > lu"j. equalities.

The function

clearly satisfies the required in-

Q

It is easy to verify that the function

a(a,s) - p1a'(a,e) < A

a

satisfies

a(b,e) > B; moreover,

and

ea" - p(t,a)a'2 - g(t,a) = euR - ev' + [P(t,uR)-p(t,a)]uR2 + 2puRvL -

PvL2 + g(t,uR) - g(t,a)

> -eM + k1vL' + m(vL+ cMm 1) - kvL, - pvj2

= mvL + (k1-k-pvL)vL > 0,

by definition of k1

and

Suppose finally that

vL.

uR(a) - p1uj(a) < A.

the following pair of functions

a(t,e) = uR(t) - cMm1, S(t,e) = uR(t) + vL(t,e) + eMm

1,

In this case, we define

80

V.

with

k1 = k

QUADRATIC SINGULAR PERTURBATION PROBLEMS

in the definition of

fies the required inequalities.

vL.

The function a

clearly satis-

It is easy to verify that the function

satisfies S (a,c) - p1s' (a,c) > A

and 0 (b,c) > B; moreover,

[p(t,Q)-p(t,uR)]uR' 2 + 2puRvL + pvL'

p(t,s)61 2 + g(t,A) - ea"

2

+ g(t,Q) - g(t,uR) - evL - EuR > m(vL + Mm I) - kvL + k1vL - EM > 0. Theorem 5.9 now follows from Theorem 2.3.

Such a strong result is not possible if the reduced solution (RR) is not strongly stable, but only weakly stable.

uR of

However, if we im-

pose an additional assumption, we can obtain the following two theorems. Only the first theorem will be proved, since the proof for the second theorem is similar. Theorem 5.10. u = uR(t)

Assume that the reduced problem (RR) has a solution

of class

C(2)([a,b])

which is weakly stable and (Iq)-stable.

Then, if p(t,y)[A - uR(a) + p1uR'(a)] > 0 in

R(uR) f1 [a,a+i], there exists an

the problem (RP5) has a solution

c0 > 0

y = y(t,e)

such that for in

0 < e < co,

satisfying

[a,b]

ly(t,E) - uR(t)l < vL(t,E) + ccl/(2q+l)

where

vL

Proof:

is given in Theorem 3.4 with

u

replaced by

We give the proof only for the case that

such that

uR(a) - p1uj(a) < A. and

p > 0

in

uR R(uR)

uR.

is

for

(10)-stable and a < t < a+d.

The other cases can be proved in a similar manner. Define, for

a < t 0, the functions

a(t,E) = uR(t) - EMm 0(t,c) = uR(t) + vL(t,e) + E(M+1)m 1,

where M >

I

and

vL(t,e) _ (e/mpl)1/2IuR(a)-pluR(a)-AIexp[-(m/e)1/2(t-a)]

is the solution of

ev" - mvL, such that vu.") = P (uR(a)-p1uR(a)-A).

Then, as in the proof of the previous theorem, these functions can be shown to satisfy the required inequalities.

5.3.

Robin Problems:

Theorem 5.11. u = uR(t)

and

Boundary Layer Phenomena

Assume that the reduced problem (RR) has a solution

of class

which is weakly stable and (II)-stable,

C(2)([a,b])

Then, if

u'R' > 0.

uR(a) - pluR(a) < A in

81

p(t,y) > 0

and

R(uf) fl [a,a+d], there exists an

such that for

co > 0

y = y(t,e)

the problem (RP5) has a solution

in

[a,b]

0 < c < e0,

satisfying

0 < Y(t,e) - uR(t) < vL(t,e) + cel/n where

vL

is given in Theorem 3.5 with

u

replaced by

uR.

A result analogous to Theorem 5.11 can be obtained if the reduced solution

is weakly stable and (III n)-stable, and we leave it to the

uR

reader to provide details.

Similar results can be obtained, if we make similar assumptions with respect to the solution

of the reduced equation (R) which, in

u = u(t)

general, does not satisfy any of the given boundary conditions.

We state

only two representative results, and we omit proofs, since these are combinations of the proofs of Theorems 5.7, 5.8 and 5.9. Assume that the reduced equation (R) has a solution

Theorem 5.12. u = u(t) also in

of class

which is locally strongly stable and

C(2)([a,b])

(Iq)-stable or

R+(u) fl [b-6,b]

(Iq)-stable.

or in

Assume also that

R _(u) fl [b-6,b].

p(t,y)[B-u(b)] > 0

Then, if

kl = k - LP111uR(a) - p1ui(a) - Al > 0, there exists an a solution

e0 > 0

y = y(t,e) -cel/(2q+1)

(a)

if

such that for in

[a,b]

< Y(t,e) _ u(t) < vL(t,e) + WR(t,e) + ccl/(2q+1)

u(a) - plu'(a) < A cel/(2q+1)

(b)

-vL(t,e) -

and

and

u(b) < B;

-wR(t,e) - cel/(2q+1) < Y(t,e) _ u(t) < vL(t,e) + Ce1/(2q+1), if

(d)

u(b) < B;

< y(t,e) - u(t) < WR(t,e) + cel/(2q+1)

if u(a) - plu'(a) > A (c)

0 < e < co, the problem (RP5) has

satisfying

u(a) - plu'(a) < A

and

u(b) > B; and

-vL(t,e) - wR(t,e) - ccl/(2q+1) < y(t,e) - u(t) < cel/(2q+1)

if u(a) - plu' (a) > A

and

u(b) > B.

82

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

is given in Theorem 5.9 and

Here, vL

replaced by

uL

is given in Theorem 5.1, with

Assume that the reduced equation (R) has a solution

Theorem 5.13.

of class

u = u(t)

wR

u.

which is locally weakly stable and also

C(2)([a,b])

(Iq)-stable or (Iq)-stable.

Assume also that

p(t,y)[A - u(a) + plu'(a)] > 0 in

or in

R+(u) fl [a,a+d]

R _(u) fl

[a,a+6], and that

R _(u) fl

[b-d,b].

p(t,y) [B - u(b)] > 0 in

R+(u)

e0 > 0

or in

[b-6,b]

fl

such that for

y = y(t,e)

in

satisfying

[a,b]

-ccl/(2q+1) < Y(t,E)

(a)

-v

(b)

- u(t) < VL + WR + cel/(2q+1)

u(a) - plu'(a) < A

if

L

-

and

-

and

ccl/(2q+l) < y(t,c)

_WR

u(b) < B;

cel/(2q+1) < y(t,e) - u(t) < WR + ccl/(2q+1)

if u(a) - plu' (a) > A (c)

Then there exists an

0 < e < e0, the problem (RP5) has a solution

u(b) < B;

- u(t) < vL + cel/(2q+l),

-

if -v

(d)

u(a) - plu'(a) < A

L

if Here, vL

- w

R

-

and

u(b) > B; and

cc1/(2q+1) < y(t,c) - u(t) < cel/(2q+1),

u(a) - plu'(a) > A

and

is given in Theorem 3.4 and

u(b) > B.

wR

is given in Theorem 3.1.

We expect analogous results for the Robin problem (RP6) as well. We state only the following result, leaving it to the reader to formulate others.

Theorem 5.14. tion

u = u(t)

(I0)-stable.

Assume that the reduced problem (RL) (or (RR)) has a soluof class Let

L =

C(2)([a,b])

max lp(t,y)j.

which is strongly stable and also

Then if

R(u)

k2 = k - Lp21IuL(b) + p2uL(b) - BI < 0, (or if

kl = k - Lp111uR(a) - pluR(a) - Al > 0), there exists an

such that for in

[a,b]

0 < e < E0, the problem (RP6) has a solution

satisfying

E0 > 0

y = y(t,e)

5.4.

Interior Layer Phenomena

83

Iy(t,e) - uL(t)I <_ vR(t,e) + eMm

1,

(or

Iy(t,c) - uR(t)I < vL(t,c) + cMm 1).

Here vR(t,c) = E(P2k2)-l1uL(b) + P2uL' (b)

and

vL(t,e)

-

Blexp[-k2(b-t)e-l]

is given in Theorem 5.9.

Interior Layer Phenomena

§5.4.

We turn now to the study of the interior crossing phenomena exhibited by the solutions of the problems (DP3), (RP5) and (RP6).

The theory for

these problems differs from that discussed in Chapter IV because of two main factors.

Firstly, as we have remarked earlier, the reduced equation

p(t,u)u'2 + g(t,u) = 0

may have singular solutions

(R)

us = constant, which are the envelopes of

the one-parameter family of solutions of (R).

Secondly, since (R) is a

quadratic function of u', there are two one-parameter families of solutions which can intersect one another within

[a,b]

with unequal slopes.

This situation can be seen from the reduced equation of Example (E12), namely,

1 -u'2=0, whose solutions are

(R 12)

u = t + c

and

u = -t + c.

Let us first discuss the Dirichlet problem eyn = P(t,y)y'2 + g(t,y),

a < t < b, (DP3)

y(a,c) = A,

y(b,e) = B.

Let the reduced problem

p(t,u)u'2 + g(t,u) = 0,

a < t
u(a) = A and the reduced problem p(t,u)u'2 + g(t,u) = 0,

u(b) = B

tR < t < b, (RL)

84

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

have, respectively, solutions

u = uL(t)

tL > tR, uL(t0) = un(to), and

uL(t0)

and

u = uR(t)

such that

at a point

u'R(t0)

t0

in

(tR,tL) We define the reduced path

u0(t)

uL(t),

a < t < to,

uR(t),

to < t < b,

and we state a result due to Haber and Levinson [27] who studied a more general problem.

Theorem 5.15 (Haber and Levinson).

is strongly stable in

p(t0,Q)w

2

[a,b]

f

u = u0(t)

and that

> 0, if uL(to) < w < uR(t0)

+ g(to,Q)

(*)

< 0, if where

Assume that the reduced path

a = uL(t0) = uR(to).

u''(to) < w < uL(to),

Then there exists an

0 < e < e0, the problem (DP3) has a solution

co > 0

y = y(t,e)

such that for in

[a,b]

satisfying y(t,e) = u0(t) + 0(e).

The following corollary holds if

Ip(t,y)I > v > 0

in

R(uL,uR),

since the inequality (*) is automatically satisfied.

Assume that the reduced path

Corollary 5.15.

stable in

[a,b]

in

Ip(t,y)I > v

u = u0(t)

and that there is a constant R(uL,uR).

v > 0

is strongly

such that

Then the conclusion of Theorem 5.15 is

valid.

If the reduced path

u0

is only weakly stable in

[a,b], it is

still possible to prove results analogous to Theorem 5.15 provided that u0

is also (Iq)- or (IIn)-stable.

We have the following two theorems,

which can be proved in the same manner as Theorems 3.9 - 3.10. Theorem 5.16. and

Assume that the reduced path

(Iq)-stable in

0 < e < e0

[a,b].

u = u0(t)

Then there exists an

the problem (DP3) has a solution

is weakly stable

c0 > 0

y = y(t,e)

such that for in

[a,b]

satisfying y(t,E) = u0(t) + 0(vl(t,e)) + 0(el/(2q+l)), where

by

vI

is as defined in Theorem 3.9 with

IuR(to) - uL(to)I

Iu'(t+)-u'(t0)I

replaced

interior Layer Phenomena

5.4.

Assume that the reduced path

Theorem 5.17.

and (IIn)-stable in in

uR > 0

and

0 < a <

> 0

in

(a,tL),

Then there exists an

uL(to) < uR(t0).

the problem (DP3) has a solution

e0

eo > 0

y = y(t,e)

cel/nI

0 < y(t,e) - u0(t) < vI(t,e) +

by

uL

satisfying

[a,b]

where

is weakly stable

u = u0(t)

Assume also that

[a,b].

(tR,b)

such that for in

85

is as defined in Theorem 3.10 with

v1

- u'(ta)I

Iu'(to)

replaced

IuR(to) - uL(to)I If the function

[a,b], then an analogous

is (III n)-stable in

u0

result holds, provided that

uL < 0

in

(a,tL), uR < 0

in

and

(tR,b)

uL(t0) > uR(t0) . uL

We suppose next that the functions

t2

(tR,b), respectively, with

in

and

intersect a third

uR

of the reduced equation (R) at points

u

solution

in

t1

and

(a,tL)

The next two theorems

tL < tR.

relate to the reduced path

fuL(t), u = ul(t) =

u(t)

,

uR(t),

Theorem 5.18.

a < t < tl, tI < t < t2, t2 < t < b.

Assume that the reduced problems (R

C(2)([tl,t2])

and

(Iq)-stable in

stable and that for

and

0 < e < e0

[a,b].

uL(tl) = u(tl),

at points

u'(t2) # uR(t2)

Assume also that the reduced path

(a,b).

(R) and (R ) have R C(2)([a,ti]),

),

of Lclass

C(2)([t2,b]), respectively, such that

uL(t1) # u'(tl), u(t2) = uR(t2) in

u = uR(t)

and

u = uL(t), u = u(t)

solutions

tl < t2 is strongly

u = ui(t)

Then there exists an

the problem (DP3) has a solution

e0 > 0

y = y(t,e)

such in

satisfying

[a,b]

Iy(t,E) - ul(t)I < vl(t,E) + V2(t,E) + CE1/(24+1)

where

and

v1

Theorem 5.19.

v2

Assume that the reduced path

and (IIn)-stable in u" > 0 uR'(t2).

in

are as defined in Theorem 4.21.

[a,b].

(tl,t2), uR > 0

Then there exists an

problem (DP3) has a solution

Assume also that in

and

(t2,b)

co > 0

u = ul(t)

uL > 0

in

(a,ti),

uL(ti) < u'(tl), u'(t2) <

such that for

y = y(t,c)

is strongly stable in

[a,b]

1/ 0 < y(t,E) - ul(t) < vl(t,E) + v2(t,E) + CCn,

0 < E < E0

satisfying

the

86

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

where

and

v1

are as defined in Theorem 4.22.

v2

If the reduced path

is strongly stable and (III n)-stable in

ul

[a,b], then the corresponding result is valid, provided that (a,tl), u" < 0

in

in

(tl,t2), u'R' < 0

uL < 0

in

and

(t2,b), uL(tl) > u'(tl)

u'(t2) > uR(t2).

Suppose now that the reduced path and

that is, uI(t1) = u'(tl) function

u = ul(t)

is a singular solution of (R).

u

is of class

C(1)[a,b],

This is the case if the

u'(t2) = uR(t2).

Then results analogous to

Theorems 5.18 and 5.19 hold, and are indeed their corollaries. Theorem 5.20.

Assume that the reduced path

C(1)([a,b]), weakly stable and an

e0 > 0

ever

u = u1(t)

(Iq)-stable in

Then there exists

[a,b].

such that the problem (DP3) has a solution

0 < e < e0.

In addition, for

t

in

[a,b]

is of class

when-

y = y(t,e)

we have that

(Y(t,e) - ul(t)l < ce1/(2q+1) Theorem 5.21.

Assume that the reduced path

uL > 0

in

(a,tl), u" > 0

there exists an y = y(t,E)

(tl,t2)

and

Assume also that

[a,b].

u" > 0

in

(t2,b).

Then

such that the problem (DP3) has a solution

e0 > 0

whenever

in

is of class

u = ul(t)

C 1)( [a,b]), weakly stable and (IIn)-stable in

0 < e < e0.

In addition, for

in

t

[a,b]

we

have that 0 < Y(t,e) - ul(t) < cel/n

We note that if the

C(1)-path

u1

is (III)-stable

suit corresponding to Theorem 5.21 holds, if

addition, if uI(tl) and

u'(t2) # uR(t2)

u'(tl)

and

ul < 0

u'(t2) = uR(t2)

in

or

Let us next give some results for the reduced path

u(t),

a < t < t2,

uR(t),

t2 < t < b.

u2 (t) These results can be applied to the path

{uL(t). a < t < tl, u(t),

In

uL(tl) = u'(tl)

then the obvious results are valid and may be ob-

tained directly from Theorems 5.18 and 5.19.

u3 (t)

then the re(a,b).

tl < t < b,

5.4.

Interior Layer Phenomena

87

by making the change of variable

t -

a + b - t.

Since the function

is not required to satisfy the boundary condition at pate that solutions of (DP3) described by

u

t = a, we antici-

will exhibit boundary

u2

layer behavior there, as well as interior layer behavior at

t = t2.

The

results which follow are a combination of our previous results on boundary layers and interior layers, and so their proofs are omitted. Assume that the reduced problems (R) and (RR) have solu-

Theorem 5.22. tions

u = uR(t)

(Iq)- or (Iq)-stable in

R+(u,uR) n [a,a+d], if

such that for

y = y(t,c)

[a,b], and that

in

[a,b]

0 < c < c0

C(2)([t2,b]),

is strongly stable

p(t,y) > v > 0

p(t,y) < -v < 0

u(a) < A, and

and

v.

in

in

R (u,uR) n

Then there exists an

the problem (DP3) has a solution

satisfying

-ccl/(2q+l) < Y(t,c) - u2(t) < wL(t,E) + v2(t,E) + c£1/(2q+l)

(a)

if

C(2)([a,t2]) u = u2(t)

u(a) > A. for a positive constant

[a,a+d], if

e0 > 0

of class

Assume also that the path

respectively. and

and

u = u(t)

u(a) < A

(b)

u'(t2) < uR(t2);

-v2(t>E) - csl/(2q+l) < Y(t,c) - u2(t) < wL(t,E) + u(a) < A

if

(c)

and

and

cEl/(2q+l)

u'(t2) > u' (t2);

-wL(t,E) - ce1/(2q+l) < Y(t,E) - u2(t) < v2(t,e) + Cel/(2q+1) if

u(a) > A

and

u'(t2) < uR(t2);

and (d)

-wL(t,c) - v2(t,E) - ccl/(2q+l) < y(t,c) - u2(t) < cel/(2q+l) if

where

wL

u(a) > A

u'(t2) > uR(t2),

and

is as defined in the conclusion of Theorem 5.1 and ck-llu'(t2)

v2(t,E) =

-

uR(t2)Iexp[-kc-lIt-t2I].

2

Theorem 5.23.

Assume that the reduced path

stable and (IIn)- or (IIn)-stable in p(t,y) > v > 0

in

R+(u,uR)

uR'(t2), u" > 0

in

(a,t2)

an

c0 > 0

y = y(t,E)

such that for in

[a,b]

[a,b].

u = u2(t)

for a positive constant and

uR > 0

0 < c < c0

in

is strongly

Assume also that

(t2,b).

u(a) < A,

v, u'(t2) < Then there exists

the problem (DP3) has a solution

satisfying

0 < Y(t,e) - u2(t) < wL(t,E) + v2(t,c) + ccn, l/

88

QUADRATIC SINGULAR PERTURBATION PROBLEMS

V.

wL

where

and

are as defined in the conclusion of Theorem 5.22.

v2

Assume that the reduced path

Theorem 5.24.

stable and (III n)- or (IIIn)-stable in

p(t,y) < -v < 0

R (u,uR) n [a,a+6]

in

u'(t2) > uR(t2), u" < 0 exists an solution

in

such that for

e0 > 0

y = y(t,e)

in

[a,b]

is strongly

Assume also that

for a positive constant

and

(a,t2)

u = u2(t)

[a,b].

uR < 0

in

(t2,b).

u(a) > A, v,

Then there

the problem (DP3) has a

0 < e < e0 satisfying

-wL(t,e) - v2(t,e) - ce1/n < y(t,e) - u2(t) < 0, where

wL

and

v2

are as defined in the conclusion of Theorem 5.22. u2

If the reduced path

that is, if

is weakly stable and of class

C(1)([a,b]),

u'(t2) = uR(t2), then the results corresponding to Theorems

5.22 - 5.24 are valid mutatis mutandis.

We note that if

u'(t2) = uR(t2),

v2(t,e) = 0.

then

Lastly we state our results on the interior crossing phenomena for solutions of the Robin problem (RP6).

The results relating to the prob-

lem (RP5) are left to the reader to formulate. Theorem 5.25.

Assume that the reduced problems (R ) and (R ) have soluR L and u = uR(t) of class C(2)([a,tL]) and C(2)([tR,b]),

u = uL(t)

tions

tL > tR, uL(t0) = uR(to)

respectively, such that uR(t0)

at a point

u = u0(t)

in

to

(= e)

and

uL (to) #

Assume also that the reduced path

(tR,tL).

( UL(t),

a < t < to,

Sl uR(t),

t0 < t < b,

is strongly stable and (Iq)-stable in

[a,b], and that

r > 0, for

uL(t0) < w < uR(t0),

< 0, for

uR(t0) < w < uL(t0).

P(t0,(Y)w2 + g(t0,a)

(*) (1`

Then there exists an

co >0 such that for

(RP6) has a solution

y = y(t,c)

in

[a,b]

0 < e < e0

the problem

satisfying

y(t,e) = u0(t) + 0(el/(2q+1))

We note that if

jp(t,y)T > 0

in

R(uL,uR), then the inequality

(*) is automatically satisfied.

Theorems 5.16 - 5.21 hold for the problem (RP6) mutatis mutandis, and so we proceed directly to a consideration of the analogs of Theorems 5.22 - 5.24 for the reduced path

Interior Layer Phenomena

5.4.

J

u = u2 (t) _

89

a < t < t2,

u(t),

UR(t), t2 < t < b.

Assume that the reduced problems (R) and (RR) have solu-

Theorem 5.26. u = u(t)

tions

and

u = uR(t)

C(2)([t2,b]), respectively.

(Iq)-stable in

strongly stable and

C(2)([a,t2])

of class

and class u = u2(t)

Assume also that the path

is

u(a) - plu'(a) = A

[a,b], and that

or

(u(a) - plu'(a) - A)[p(a,u(a)){pll(A-u(a))}2 + g(a,u(a))] > 0. Then there exists an has a solution

such that for

£0 > 0 in

y = y(t,£)

0 < c < £0

the problem (RP6)

satisfying

[a,b]

ly(t,£) - U2(t)l < vL(t,e) + v2(t,c) + c£1/(2Q+1), where

vL

placed by

u, and

and (IIn)-stable in

and

in in

u" R> 0

0 < £ < £0

re-

is as defined in the conclusion of Theorem 5.22.

v2

Assume that the reduced path

Theorem 5.27.

p(t,y) > 0

uR

is as defined in the conclusion of Theorem 5.9 with

[a,b].

is strongly stable

u = u2(t)

Assume also that

u(a) - plu'(a) < A,

R+(u,uR) 11 [a,a+6], u'(t2) < uR(t2), u., > 0

Then there exists an

(t2,b).

the problem (RP6) has a solution

in

(a,t2)

such that for

£0 > 0 y = y(t,£)

in

[a,b]

satisfying 0 < y(t,£) - u2(t) < vL(t,£) + v2(t,£) + cc1/n where

vL

and

Theorem 5.28.

v2

are as defined in the conclusion of Theorem S.27.

Assume that the reduced path

stable and (III n)-stable in

there exists an has a solution

c0 > 0

in

(a,t2)

in

and

such that for

y = y(t,c)

in

[a,b]

u = u2(t)

is strongly

Assume also that

p(t,y) < 0

u(a) - plu'(a) > A.

u'(t2) > uR(t2), u" < 0

[a,b].

R-(u,uR) u'R' < 0

0 < £ < c0

[a,a+6],

f1

in

(t2,b).

Then

the problem (RP6)

satisfying

-vL(t,£) - v2(t,£) - c£1/n < y(t,£) - u2(t) < 0, where

vL

and

v2

are as defined in the conclusion of Theorem 5.26.

90

V.

We note that if the path

QUADRATIC SINGULAR PERTURBATION PROBLEMS

u2

is

C(1)-smooth, that is, if

u'(t2)

uR(t2), and weakly stable, then the results corresponding to Theorems 5.26 and 5.28 hold mutatis mutandis with

v2 = 0.

Notes and Remarks 5.1.

The theory developed above for the problems (DP 3)' (RP.) and (RP6)

can accommodate, with minor modification, the more general problem in which

p(t,y), g(t,y), A

and

p(t,y,c) = p(t,y,0) + 0(1),

A(e) = A(0) + 0(1) 5.2.

and

B

are functions of

a

satisfying

g(t,y,e) = g(t,y,0) + o(l),

B(c) = B(0) + 0(1).

Surprisingly little work has been done on the Dirichlet problem (DP3).

Motivated by a brief discussion in Section 5 of the paper

by Dorr, Parter and Shampine [20], Howes has developed a reasonably coherent boundary layer theory for (DP3) in [39].

He has also dis-

cussed a corresponding interior layer theory in [39) and [38] which includes not only the interior crossing phenomena treated above but also shock layer phenomena.

The classic interior crossing

result of Haber and Levinson [27] has been extended in several directions by Vasil'eva [87], O'Malley [74] and Howes [43].

The

Robin problems (RP5) and (RP6) have been considered by Vasil'eva [87], Macki (62), Searl [84] and Howes [40].

Chapter VI

Superquadratic Singular Perturbation Problems

56.1.

Introduction

In previous chapters we have presented fairly comprehensive results for boundary value problems involving the ey" = f(t,Y,Y'),

differential equation

a < t < b,

subject to the fundamental restriction: f(t,y,z)

= O(lzj2)

as

IzI - oo.

It is natural for us to ask if similar results can be extended to these

f

boundary value problems when f(t,y,z)

= O(Izln)

as

is subjected to the restriction:

(zI - oo

for

n > 2.

A partial and somewhat negative answer was given many years ago by Vishik and Liusternik [90] for the Dirichlet boundary value problem Ey" = f(t,y,y'),

Y(a,e) = A,

a < t < b,

(6.1) (6.2)

Y(b,E) = B.

They showed that if f = p(t,Y)Y'n + fl(t,Y,Y'), where

n > 2, Ip(t,y)I > v > 0 f1(t,Y,z) = O(Izln-1)

as

and

Izl

..

then every solution of (6.1), (6.2), if it exists (cf. (E14) below), satisfies

91

92

y'(a,e) = 0(1)

VI.

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

and

y'(b,e) = 0(1)

as E- 0+.

[Note that these authors actually considered the Cauchy problem (6.1), for 0 < n < 2, together with the initial conditions

y(a,c) = A, y'(a,e) =

This Cauchy problem is equivalent to the above Dirichlet

C/ey, C,y > 0.

problem, as the reader can see by solving the simple example ey" _ -vy',

y(l,e) = B, where

y(0,e) = A,

v > 0,

A short calculation shows that here

y = 1

and

A < B.

C - v(B - A).

However,

n = 2, they showed that this equivalence is obtained only if k/e , k > 0. Also compare this with our discussion at the y'(a,e) ee

when

beginning of Chapter I.]

Thus, in contrast to the solutions for similar problems studied in

previous chapters, Vishik and Liusternik's problem does not exhibit a boundary layer characteristic! is the following. f(t,u,u') = 0

u(a) # A

and

Let

u = u(t)

Another way of formulating their result be a solution of the reduced equation

which does not satisfy either boundary condition, that is, u(b) # B, then there is no solution

y = y(t,e)

of (6.1),

(6.2) satisfying lim+ y(t,e) = u(t) e+0

for

a < t < b.

This implies that for arbitrary choices of A

and

B, the Dirichlet prob-

lem (6.1), (6.2) has no solution for all sufficiently small values of

Therefore, the Dirichlet problem (6.1), (6.2), where n > 2 f = 0(jy'jn)

as

C.

and

ly'l + W is not, in general, a well-posed problem,

since we are interested in existence of solutions for small values of

E.

These remarks are best illustrated by the classic counterexample due to Coddington and Levinson [14] (see also [23], [94]): Ey" = -Y'

-

Y(O,e) = A,

y'3 S f(Y'),

0 < t < 1, (E14)

Y(l,e) = B.

By quadratures we obtain the general solution (x+c1)/e

y(t,e) = ±e arc sin (e where

cl, c2

)

+ c2,

are arbitrary constants.

To choose these constants to

satisfy the boundary conditions, we run into a difficulty. ficiently small values of

ly(t,E) - Al = 0(c),

c, in fact, if

For all suf-

0 < e < 21A - BJ/a, we have

6.2.

A Dirichlet Problem

and so

y(l,e)

cannot be equal to

solution, unless solution

93

B = A.

B, unless

B = A.

Thus there is no

latter case, we obtain the constant

In this

y(t,E) a A = B.

For this counterexample, note that f(Y') = O(Iy'13), as

I Y') + 00,

and that the only real solution of the reduced equation u(t) = constant.

This reduced solution

both boundary conditions, unless (E14) as

f(u') = 0

u(t) a constant

is

cannot satisfy

A = B. and so there is no solution to

c + 0', unless A = B, which is in agreement with the result

of Vishik and Liusternik.

The above results seem to indicate that there is very little that one can do with Dirichlet boundary value problems.

However, we hasten to

point out that, fortunately, the Robin problem for this class of differential equations turns out to be well-posed and solvable.

That this is

so should not be surprising, in view of our discussion of it in Chapter V, and also in view of the result of Vishik and Liusternik, which implies that the solution

y(t,e)

for the Robin boundary value problem should

satisfy

Y'(a,E) = 0(1) = pll(y(a,E) - A) and

Y'(b,e) = 0(1) = p21(Y(b,e) - B). and

This will be the case if y(a,e)

y(b,e)

are

0(1)

as

e - 0+.

Thus, solutions of such problems have uniformly bounded derivatives at the endpoints.

§6.2.

A Dirichlet Problem To study these superquadratic boundary value problems in detail, we

first consider the following class of Dirichlet problems ey" = h(t,y)f(t,y,y') a F(t,y,y'),

a < t < b (DP4)

y(a,c) = A,

y(b,E) = B.

Here the function the function in a domain

f

h(t,y)

is of the same type as in Chapter III, while

is continuous and satisfies

V0(u)

and for all

fined in Chapter III, where

z

in 1R.

u = u(t)

f(t,y,z) > p > 0 The domain

V0(u)

for

(t,y)

is as de-

is a solution of the reduced equation

94

VI.

h(t,u) = 0,

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

a < t < b.

(R4)

The stability properties of

stable, then

are as given in Definitions 3.1, 3.3

u(t)

and 3.3, with m replaced by

For example, if

mu-1.

hy(t,y) > mu-1 > 0

for

(t,y)

in

u(t)

V0(u).

is (I0)-

We also assume

that

f(t,y,z) = 0(Izln)

We note that for

lzi - - for

as

n > 2.

n = 2, the results of Chapter III for the Dirichlet prob-

lem apply directly to the problem (DP4), provided that the stability con-

m

stant

mu-1.

is replaced by

The following two theorems give the basic results for the Dirichlet problem (DP4). Theorem 6.1.

Assume that the reduced equation (R4) has an (Iq)- (or (IIn)-)

stable solution u(b) = B for

u = u(t)

u" > 0

and

of class

in

C(2)([a,b])

such that

Then there exists an

(a,b).

the problem (DP4) has a solution

0 < e < e0

u(a) = A,

co > 0

y = y(t,e)

such that in

[a,b)

satisfying u(t) < y(t,c) < u(t) + ce1/p.

where on

p = 2q + 1

m, loll

Proof:

and

n), and

(or

c

is a known positive constant depending

p.

Define for

t

in

[a,b]

and

c > 0

a(t,e) = u(t), a(t,e) = u(t) + r(e), where

r(e) = (eym

1)1/p,

for

y > p! M and M > lu"I.

a < a, a(a,e) < A < a(a,e), a(b,c) 0).

It only remains to show that

Clearly,

ca" > F(t,a,a')

ce" < F(t,a,s').

We have

F(t,a,S') - ea" = h(t,a)f(t,a,a') - call p-1

aih(t,u)rl/j! + ayh(t,)rP/P!]f(t,a,s') -

[h(t,u) +

cull

j=1

> (m)J-Ieym 1/P!)u - em > 0,

since y > p! M. (t,g)

belongs to

= f(t,e) = u(t) + e(a - u(t)), 0 < e < 1, and

Here V0(u)

Consequently, a

and

for all sufficiently small a

c, say

0 < c < co.

satisfy all of the required inequalities.

The proof will follow from Theorem 2.1 if we can show that whenever

6.3.

Robin Problems:

Boundary Layer Phenomena

and a < y < $

ey" = F(t,y,y') then

ly'(t,e)I < N

fies

u'(a) < y'(t,c) < u'(b).

and a(t,e) < y(t,e)

a'(b,e) = u'(b).

This is because

imply that

y(b,c) = u(b) = a(b,c)

and

J c [a,b],

Indeed, the solution

J x

on

on

95

satis-

y'(a,s) > a'(a,e) = u'(a), while

a(t,c) < y(t,c)

imply that

u'(a) < y'(t,e) < u'(b)

Therefore

y(t,c)

y(a,c) = u(a) = a(a,c)

y'(b,c) < y" > 0

since

for

a < y < 5. Theorem 6.2 is the "concave" version of Theorem 6.1. Theorem 6.2.

Assume that the reduced equation (R4) has an

stable solution and

u(b) = B for

0 < c < s0

u = u(t)

u" < 0

in

of class (a,b).

C(2)([a,b])

Then there exists an

the problem (DP4) has a solution

(Iq)- or (III n)-

such that

u(a) = A,

such that

e0 > 0

y = y(t,e)

in

[a,b]

satisfying u(t) - ccI/p < y(t,e) < u(t), where the constants Simply let

Proof:

c

and

y + -y

p

are as defined in Theorem 6.1.

and apply Theorem 6.1 to the transformed prob-

lem.

We remark that since it is assumed that domain

form

VO(u)

in which the functions

h

and

u(a) =.A f

V0(u) = {(t,y): a < t < b, ly - u(t)l < S}, where

arbitrarily small (but fixed) positive constant.

and

u(b) = B, the

are defined is of the d > 0

is an

Consequently, for such

problems (Iq)-, (IIn)- and (IIIn)-stability are essentially equivalent to their "tiled" counterparts.

§6.3.

Robin Problems:

Boundary Layer Phenomena

We turn now to the following classes of Robin problems cy" = h(t,y)f(t,y,y'),

y(a,e) - ply'(a,c) = A,

a < t < b, y(b,c) = B,

and

ey" = h(t,y)f(t,y,y'),

a < t < b, (RP8)

y(a,s) - ply'(a,c) = A, where

pl

and

p2

y(b,c) + P2y'(b,c) = B,

are positive constants.

The analogs of Theorems 6.1

and 6.2 for these problems follow easily from our discussion of the prob-

96

VI.

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

lems (RP1) and (RP2) in Chapter III, and so we omit the proofs of the next two theorems.

Assume that the reduced equation (R4) has an

Theorem 6.3.

plu'(a) < A, u(b) = B such that for

e0 > 0

y = y(t,e)

in

of class

u = u(t)

stable solution

[a,b]

and

u" > 0

0 < e < e0

such that

C(2) ([a,b])

in

(Iq)- or (IIn)-

u(a) -

Then there exists an

(a,b).

the problem (RP7) has a solution

satisfying

u(t) < y(t,E) < u(t) + VL(t,e) + cel/P

Here

is as defined in Theorem 6.1, and

c

Theorems 3.4 and 3.5 for

p = 2q+1

and

is as defined in

vL(t,e)

p = n, respectively.

The following result refers to the analogous problem (RP8). Assume that the reduced equation (R4) has an (Iq)- or (II

Theorem 6.4.

of class

u = u(t)

stable solution

plu'(a) < A, u(b) + p2u'(b) 0

y = y(t,e)

in

[a,b]

and

0 < e < e0

such that

C(2)([a,b])

u" > 0

in

(a,b).

n)-

u(a) -

Then there exists

the problem (RP8) has a solution

satisfying

U(t) < y(t,e) < U(t) + VL(t,e) + vR(t,e) + Cel/p where

c

and

vL

3.7 and 3.8 for

are as in Theorem 6.3 and p = 2q + 1

If the function u

and

vR

is as defined in Theorems

p = n, respectively.

is (Iq)- or (III n)-stable and satisfies the re-

verse inequalities, then results analogous to Theorems 6.3 and 6.4 hold true.

Tiis can be seen by using the change of variable

y - -y

and

applying Theorems 6.3 and 6.4, respectively.

§6.4.

Interior Layer Phenomena It is now an easy matter to consider the occurrence of interior layer

behavior for the problems (DP4), (RP7) and (RP8).

The situation described

in the following theorems arises most frequently when two solutions of the reduced equation (R4) intersect at a point

t0

slopes, as we have discussed earlier in Chapter III.

in

(a,b)

with unequal

We omit the straight-

forward proofs.

Theorem 6.5.

Assume that the reduced equation (R4) has an (Iq)- or (IIn)-

stable solution

u = u(t)

of class

C(2)([a,b]), except at the point

t0

Interior Layer Phenomena

6.4.

in

(a,b)

where

u'(t0) < u'(t0) and

u(a) = A, u(b) = B an

e0 > 0

u" > 0

in

ju"(tt)I < -.

and in

Assume also that Then there exists

(a,t0) U (t0,b).

0 < e < £0

such that for

y = y(t,e)

97

the problem (DP4) has a solution

satisfying

[a,b]

u(t) < y(t,c) < u(t) + vl(t,e) + ceI/p, where

c

3.10 for

is a positive constant and p = 2q+1

is defined in Theorems 3.9 and

vI

p = n, respectively.

and

It is clear that a result analogous to Theorem 6.5 can be obtained if the solution

u

is (Iq)- (or (III n)-) stable, and

u'(tp) > u'(t+)

and

u" < 0

in

(a,t0) U (t0,b).

We leave its exact formulation to the reader. In the same manner we can prove the next two results which deal with 'interior crossing' phenomena for the Robin problems (RP7) and (RPB), when the reduced solution

Theorem 6.6.

(a,b)

where

is (Iq)- (or (III n)-) stable and

concave.

there exists an

and

and

in

ju"(t±)I < -.

u" > 0

such that for

e0 > 0

y = y(t,c)

C(2)([a,b]), except at the point

of class

u'(t-) < u'(t4.)

u(a) - plu'(a) < A, u(b) = B

a solution

u

Assume that the reduced equation (R4) has an (Iq)- or (IIn)-

stable solution u = u(t) in

Similar

is (Iq)- (or (IIn)-) stable and convex.

u

results can be obtained when

in

(a,t0) U (t0,b).

0 < £ < £0

t0

Assume also that Then

the problem (RP7) has

satisfying

[a,b]

u(t) < y(t,£) < UM + vL(t,e) + v1(t,£) + ceI/p where

c

and

vL

are as given in Theorem 6.3 and

vI

is as given in

Theorem 6.5. Theorem 6.7.

Assume that the reduced equation (R4) has an (Iq)- or (IIn)-

stable solution in

(a,b)

u = u(t)

where

C(2)([a,b]), except at the point

of class

u'(t0) < u'(t+)

and

u(a) - plu'(a) < A, u(b) + p2u'(b) 0

(RP8) has a solution

y = y(t,e)

Iu"(t0)I < -.

and

such that for in

[a,b]

u" > 0

t0

Assume also that in

0 < £ < e0

(a,t0) U (t0,b).

the problem

satisfying

u(t) < y(t,e) < u(t) + VL(t,e) + vI(t,e) + vR(t,e) + ceI/p, where 6.4.

c

and

vI

are as given in Theorem 6.6 and

vR

is as in Theorem

98

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

VI.

A General Dirichlet Problem

§6.5.

We now return to the general equation given at the beginning of this chapter, namely, ey" = f(t,y,y'),

a < t < b.

In the sequel it is assumed that f(t,y,z) = 0(Izln)

IzI -- for n> 2.

as

Let us first give two definitions of stability for the solution of the reduced equation f(t,u,u') = 0.

(R)

They are obvious extensions of earlier ones. A solution

Definition 6.1. to be (I

q)_1

u = u(t)

of the reduced equation (R) is said

(IIn)- or (III n)-stable in

[a,b]

if the respective inequal-

ities in Definitions 3.1-3.3 hold for the function in

h(t,y) = f(t,y,u'(t))

V0(u) = {(t,y): a < t < b, Iy - u(t)I < 6}. A solution

Definition 6.2.

u = u(t)

of (R) is said to be stable in

if there is a positive constant

[a,b]

k

such

Ify,(t,y,y')l > k > 0

that ly'

in

V1(u) = {(t,y,y'): a < t < b, Iy - u(t)I < S,

- u'(t)l < d}.

With these stability properties we can now discuss the Dirichlet

problem ey" = f(t,y,y'),

a < t < b, (DP5)

y(a,E) = A,

y(b,E) = B,

and we obtain the same results as Theorems 6.1 and 6.2. Theorem 6.8.

Assume that the reduced equation (R) has an (Iq)- or (II

stable solution u(b) = B

u = u(t)

u" > 0

and

0 < e < e0

for

in

of class

C(2)([a,b])

such that

Then there exits an

[a,b].

the problem (DP5) has a solution

e0 > 0

y = y(t,e)

n)-

u(a) = A,

such that in

[a,b]

satisfying u(t) < y(t,e) < u(t) + CE: 1/p,

where

p = 2q + 1

on

lu"I

m,

Proof:

and

(or

n)

and

c

is a known positive constant depending

p.

We proceed as in the proof of Theorem 6.1 by defining the same

bounding pair

6.5.

A General Dirichlet Problem

a(t,e) = u(t),

where

r(c) _ (ey m

99

B(t,e) = u(t) + r(e), 1)1/p

for y > p! Iu"I.

Then it is clear that

a < B, a(a,e) = A < B(a,e), a(b,e) = B < B(b,e), call = Cu" > f(t,a,a') = 0, and

c6" < f(t,B,B').

This last inequality follows by virtue of our

stability assumption and our choice of y, that is, p-1

f(t,6,6') - e6" = f(t,u,u') +

2'.'f(t,u,u')B1/j!

E

j=l

3

+ 8Yf(t,&,u')$P/p! - cull

m(eym 1)/P! - elu"l > 0.

Finally we can show by arguing as in the proof of Theorem 6.1 that the function

f

satisfies a generalized Nagumo condition with respect to

a

B, and so the conclusion of Theorem 6.8 follows from Theorem 2.1.

and

The next result is the concave version of Theorem 6.8. Theorem 6.9.

Assume that the reduced equation (R) has an

stable solution u(b) = B for

and

u = u(t)

u" < 0

in

of class

such that

C(2) ([a,b])

Then there exists an

[a,b].

the problem (DP5) has a solution

0 < e < e0

(Iq)- or (III n)-

u(a) = A,

such that

e0 > 0

y = y(t,e)

in

[a,b]

satisfying u(t) - ce1/p < y(t,s) < u(t), where the constants

c

and

are as defined in Theorem 6.8.

p

A stronger result than the above two results can be obtained, if the reduced solution Theorem 6.10. u = u(t)

u(t)

Assume that the reduced equation (R) has a stable solution

of class

satisfying

C(2)([a,b])

Then there exists an has a solution

is stable in the sense of Definition 6.2.

such that for

e0 > 0

y = y(t,c)

in

[a,b]

and

u(a) = A 0 < e < e0

u(b) = B.

the problem (DP5)

satisfying

(y(t,e) - u(t)j < cc,

where Proof:

c

is a known positive constant depending on We suppose first that

f

I

< -k < 0.

f

and

u.

Clearly, there exists an

y R. > 0

such that

Ify(t,y,y')I < £

in

V1(u).

Setting

z = y - u(t)

in the proof of Theorem 4.1, we are first led to the problem

as

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

VI.

.100

ez" = f(t,u+z, u' + z') - cu"

= fy[.]z + fy["]z' - eu", z(b,e) = 0,

z(a,e) = 0, where

denote the appropriate intermediate points, and

and

then to the problem ez" + kz' + iz = -cu" z(b,c) = 0.

z(a,c) = 0, If

0 < c < k2/4R, one of the two negative roots of the corresponding

auxiliary equation

cA2 + kA + R = 0

is

A = -R/k + 0(e).

Then the

function

r(t,e) = eycI(exp[-A(b-t)] - 1) is a solution of cr" + kr' + tr = -cy. It has the following properties: 0 < r < cc

r(b) = 0, for some

and

-cc < r' < 0,

c > 0.

We define the bounding pair a(t,c) = u(t) - r(t,c), s(t,e) = u(t) + r(t,e),

and we need only verify that inequality for

s

call > f(t,a,a'), since the differential

follows by symmetry and the other required inequali-

ties clearly hold true.

By Taylor's Theorem, however, we obtain

ca" - f(t,a,a') = cu" - cr" +

-c u" J

- er" - kr' - Rr

c(Y - Iu") > 0 by choosing y >

I u" I.

Finally, it is not difficult to see that

for any solution of

cy" = f(t,y,y')

the theorem follows from Theorem 2.1.

y'(t,c) - u'(t) = 0(c)

which satisfies

fy, > k > 0, then we would use the bounding functions

a(t,e) = u(t) - r(t,e), s(t,e) = u(t) + r(t,c),

a < y < $, and so

We remark that in the proof, if

A General Robin Problem

6.6.

101

where P(t,e) = ey&-1(exp[A(a - t)]

56.6.

- 1).

Boundary and Interior Layer Phenomena

A General Robin Problem:

We now turn our attention to the nonlinear Robin problem a < t < b,

ey" = f(t,y,y'),

(RP9)

y(a,E) - Ply'(a,E) = A, It turns out that if

pl > 0

y(b,E) + P2y'(b,e) = B. and

p2 > 0, and if appropriate stability

assumptions hold, then these problems will have solutions, irrespective

f with respect to

of the growth of

y'.

To formulate these stability

assumptions, let us suppose that the reduced problem a < t < b,

f(t,u,u') = 0,

(RR)

u(b) + p2u'(b) = B,

has a smooth solution solution of (RP9).

uR

u = uR(t)

which we will use to approximate the

Since, in general, uR(a) - p1u.(a) # A, we require The first requirement is, of course,

to have two stability properties.

that

uR

is stable in the sense of Definition 6.2, that is,

fy(t,y,y') < -k < 0

(6.3)

in the region O1(uR) _ {(t,y,y'): a < t < b, Iy'

Iy - uR(t)I < 6

- uR(t)I < 6).

The second requirement is new and its motivation can be seen from the stability results for the following class of initial value problems eY" = f(t,y,y'),

a < t < b, (IVP)

y(a,c) = uR(a),

Note that here

y'(a,e) = p11(uR(a) - A)

y(a,e) - p1y'(a,e) = A.

It is known (cf. [61], [87])

that the solution of (IVP) is uniformly close to the reduced solution uR

such that either

for all

uR(a) _

or

(uR(a) - Qf(a,uR(a),A) > 0, for all

A

in

(uj(a),Q

or

[E,uR(a)).

Therefore, the second require-

102

VI.

ment is that if

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

uR(a) - p1uj(a) # A, then (6.4)

(uR(a) - P1uR(a) - A)f(a,uR(a),A) < 0, A in

for all

(uj(a), p1l(uR(a) - A)]

or in

[p11(uR(a) - A), uR(a)).

This inequality provides us with the required "boundary layer stability" of the function

uR, and for nonlinear functions

f, it serves to define

the admissible boundary layer jumplA - uR(a) + pluR(a)I. Finally, we must ensure that at

uR

approximates the solution of (RP9)

t = b, and so we assume that

P2fy(b,uR(b),uR(b)) - fy(b,UR(b),uA(b)) 0 0. Theorem 6.11. u = uR(t)

(6.5)

Assume that the reduced problem (RR) has a solution

of class

has a solution

satisfying the relations (6.3) - (6.5).

C(2)([a,b])

Then there exists an

such that for

e0 > 0

y = y(t,e)

in

[a,b]

0 < e < e0

the problem (RP9)

satisfying

Y(t,e) = uR(t) + 0(c.(p1k)-lexp[-k(t-a)/e]) + 0(e), k = IA - uR(a) + p1uR'(a)l.

where Proof:

The bounding functions are defined in the usual manner and using

the standard techniques, it is not difficult to verify that the required inequalities are indeed satisfied.

must also verify that if y = y(t,c)

In order to apply Theorem 2.3, we is a solution of

which lies between the lower and upper solutions, then formly bounded.

This is true, however, since

ey" = f(t,y,y') y'(t,e)

is uni-

y'(t,e) = uj(t) +

0(Rpl1 exp[-k(t-a)/e]) + 0(c); see [40] for complete details.

We remark that if, instead, the reduced equation smooth solution

u = uL(t)

which satisfies

f = 0

has a

uL(a) - pluL(a) = A, then a

result analogous to Theorem 6.11 can be obtained mutatis mutandis. Lastly we formulate a result for the Robin problem (RP9) which displays angular interior layer behavior which is similar to that discussed by Haber and Levinson [27] for the Dirichlet problem (cf. Chapter V). However, we must proceed with care as the

following example shows.

The problem is

0
whose unique solution is simply and

(E15)

y'(l,e) = 1,

u = uR(t) = t - 1

y(t,c) = t.

Nevertheless, u = uL(t) _ -t

are stable solutions of the corresponding reduced

6.6.

A General Robin Problem

problems which intersect at

103

t0 = 1/2

and satisfy the Haber-Levinson

condition that

1-A4>0

uL=-1<X < 1=uR.

for

Unfortunately, there is no solution of (E15) which is close to the angular path

maxi-t,t-1}

in

(0,1).

In order to formulate the correct result, we proceed as follows. First of all, let us assume that the reduced problem f(t,u,u') = 0, (RL)

u(a) - plu'(a) = A,

and the reduced problem (RR) have, respectively, solutions

and u = uR(t)

[a,t0]

[t0,b], where

in

u = uL(t)

in

a < t0 < b, uL(tO) = uR(tO)

(= a) and VL = uL(t0) # uR(t0) = VR Let

u(t)

uL(t),

a < t < t0,

uR(t),

t0 < t < b.

We naturally want

u

to be stable in the sense of Definition 6.2, that

is,

fy(t,y,y) >k > 0

in

[a,t0]

(6.6)

and

y,(t,y,y') < -k < 0 in the region

D1(u).

in

[t0,b]

(6.7)

At the crossing point

to, we must further require

that (vR - vL)f(to,a,)') > 0

for all

A

(6.8)

strictly between

and

VR.

Now in view of Example (E15),

"IL

we must add a further condition, namely, fy(t,y,y') > m > 0 in the region

D1(u).

(6.9)

With these assumptions, we can formulate the follow-

ing result.

Theorem 6.12. tions

Assume that the reduced problems (R ) and (R ) have soluL R of class C(2)([a,t0]), C(2)([t0,b]), res-

u = uL(t), u = uR(t)

pectively, which satisfy the relations (6.6) - (6.9).

Then there exists

104

an

SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS

VI.

0 < E < EO

such that for

e0 > 0

in

y = y(t,e)

[a,b]

the problem (RP9) has a solution

satisfying vLI(2k)-lexp[-klt-tol/e])

y(t,E) = u(t) + 0(EIVR

+ 0(E).

The proof proceeds in the now familiar manner by defining the

Proof:

appropriate bounding functions.

solution y = y(t,e)

of

It is also not difficult to see that any which lies between the bound-

Ey" = f(t,y,y')

ing functions is such that Y'(t,c) = u'(t) + 0(IVR VLI/2 exp[-kjt-t0I/E]) + 0(E), and so we can apply Theorem 2.3 to obtain the required result.

A Comment

§6.7.

We conclude this chapter with the remark that under certain circumstances (cf.

[44]) solutions of the Dirichiet problem (DP5) exist and In particular, if

exhibit boundary and shock layer behavior. 0(IY'In-1)

as

f(t,y,Y') = p(t,Y)Y'n +

and if path

vanishes at a value of

p (t,y)

in

t

in

Iy'I

-

-.

for all

[a,b]

y

or along a

[a,b] xIR1, then the result of Vishik and Liusternik pro-

hibiting boundary layer behavior does not apply.

As a simple illustra-

tion, consider the problem Eyit = -ty'3 = f(t,Y'),

y(0,E) = 0,

0 < t < 1, (E16)

y(l,e) = B > 0.

Its solution, which can be found by quadratures, satisfies

lim y(t,E) =B for 0<6
as

ly'l . -, for fixed

t

t = 0. in

Note that

(0,1], but

f(t,y') =

f(O,y') = 0.

The

interested reader should consult [44] for a detailed discussion of such "superquadratic" problems.

Notes and Remarks

105

Notes and Remarks 6.1.

As in the other chapters we can allow the righthand side the boundary data (that is, p1, p2, A

and

f

B) to depend on

and a

in

a regular manner. 6.2.

The results here are due to Howes in a series of papers (cf. [40], [42], [44]).

Recently Deshpande [19] has systematically considered

the existence and nonexistence of solutions of the Dirichlet problem (6.1), (6.2) as functions 6.3.

e - 0+

for certain classes of superquadratic

f.

Superquadratic boundary value problems of the type just discussed arise in many applied contexts involving capillarity and/or surface tension effects (see, for example, [50], [78], [79] and Chapter VIII below).

Studies of related problems on unbounded t-intervals

can be found in [50], [65], [58] and [42].

Chapter VII

Singularly Perturbed Systems

§7.1.

Introduction

In this chapter we turn our attention to some vector boundary value problems which may be regarded as vector analogs of the scalar problems. However, as the reader will see, our results for vector problems are very incomplete, especially in comparison with the scalar theory.

The study

of singularly perturbed vector second-order equations is in its infancy, and we hope that this chapter will serve to draw attention to some of the associated difficulties.

Indeed, many fundamental questions concerning

vector equations or systems have yet to be raised, let alone answered.

§7.2.

The Semilinear Dirichlet Problem Let us begin with the semilinear problem ey" = H(t,y),

a < t < b,

(S1) y(a,e) = A,

where

y, A

and

vectors), and on

H

[a,b] X 1RN.

y(b,c) = B, B

are

is an

N-vectors (interpreted throughout as column N-vector function which is defined and smooth

The assumed smallness of

e > 0

prompts us, as usual,

to look first for solutions of the reduced system

(RI)

H(t,u) = 0.

Let us assume, for simplicity, that this reduced system has the trivial solution

u = 0

of conditions on

in

H

[a,b].

In order to get some feeling for the types

which guarantee the existence of solutions of (S1)

106

7.2.

The Semilinear Dirichlet Problem

107

that are close to zero almost everywhere in

[a,b], we first look at the

simple linear system

ey" = Ay,

a < t < b, y(b,c) = B,

y(z,c) = A,

where A

(E17)

is a constant, symmetric (NxN)-matrix.

If A

is positive

definite, then it is similar to a diagonal matrix with positive entries m1,...,mN.

Thus the scalar theory (cf. Example (El) in Chapter III) im-

plies that, under this assumption, Example (E17) has a solution y = y(t,c) for all

e > 0

which satisfies

lim+ y(t,c) = 0

where

for

in

t

[a+d,b-S],

Indeed, the reader should have no difficulty showing

0 < 6 < b-a.

that each component of the solution estimate in

(7.1)

y(t,c)

satisfies the more precise

[a,b]:

Yi(t,c) = O(IAilexp[-(m i/c)1/2(t-a)])

+ O(IBileXp[-(mi/c)I'2(b-t)]), for

i = 1,...,N.

Generalizing to the system (S1), we may anticipate

that if there is a positive constant

matrix of H

evaluated at

YTJOY > mIIYII2 for all

y

in

m

such that

JO(t), the Jacobian

y = 0, satisfies the inequality (7.2)

[a,b],

in IRN, then the problem (S1) will have a solution which satis-

fies the limiting relation (7.1), provided ently small.

IIAII

and

IIBII

are suffici-

This restriction (7.2) is precisely the condition that the

coefficient matrix of the corresponding linearized system sould be positive definite.

To consolidate the above heuristic ideas, let us first define the following region in gtN+1

V = {(t,y): a < t < b, 11Y11 < d(t)}. Here in

d(t)

is a smooth positive function such that

[a,a + 6/2], d(t) ° IIBII + 6

[a + 6, b - d], where

6 > 0.

in

[b - 6/2, b]

d(t) = IIAII + 6 and

d(t)

=_ 6

the reduced solution of (R1) may be regarded as the vector analog of Definition 3.1.

in

The following definition of stability for

108

VII.

Definition 7.1.

The zero solution

said to be norm-stable

u E 0

SINGULARLY PERTURBED SYSTEMS

of the reduced system (R1) is

[a,b]) if there is a positive constant m

(in

such that

YTJ(t,y)y > m1Iy1I2 where to

for

(t,y)

in

(7.3)

D,

is the Jacobian matrix of H with respect

J(t,y) _ (8H/8y)(t,y)

Y.

The condition (7.3) is obviously stronger than the condition (7.2).

We will see that if u ° 0 small

e > 0

is norm-stable, then for all sufficiently

there exists a solution of the semilinear problem (S1) sat-

isfying the limiting relation (7.1).

This is the content of the first

theorem (cf. [54], [41]). Theorem 7.1.

Assume that the zero function is a norm-stable solution of

the reduced system (R1) in that for

IIy(t,E)II

Then there exists an

the problem (S1) has a solution

0 < e < e0

co > 0

y = y(t,c)

such

of

satisfying

C(2)([a,b])

class

[a,b].

IIAIIexp[-(mE-1)1/2(t-a)] + IIBIIeXp[-(me-1)1/2(b-t)]. (7.4)

Proof:

The result is a consequence of Theorem 2.4 if we can construct

a scalar function P = P(t,y,e) = IIyII - y(t,E)

with

p

satisfying (2.5), or equivalently with

Ey" < YTH(t,y)/IIyII

whenever

y

satisfying

y = IIyII

(Refer to the discussions immediately following Theorem 2.4). Define

y = y(t,E) = IIAIIexp[-(ma-1)1/2(t-a)] + IIBIIexp[-(ma-1)1/2(b-t)].

Then y

satisfies

IIyII < y, is in

Ey" = my.

Note that the set

V for all suffficiently small

(t,y), with

E, say

From the Mean Value Theorem and (7.3) it follows that for

yTH(t,y)/I IrI I where have

(t,g)

=

a < t < b,

0 < e < E0. (t,y)

in

D,

yTJ(t, )y/I Iyj I _ ml Iyl I,

is the intermediate point, and so, whenever y = IIyII

we

The Semilinear Dirichlet Problem

7.2.

109

YTH(t,Y)/IIYII - Ey" > gym, - ey" = 0.

We conclude from Theorem 2.4 that for solution

y = y(t,e)

of class

0 < e < c0, the problem (SI) has a such that

C(2)([a,b])

p(t,y(t,e),e) < 0,

that is, IIy(t,E)II < Y(t,E) It is possible to prove a more general result than Theorem 7.1, in which a limiting relation of the form (7.4) is still valid (cf. [41]). We simply replace the inequality (7.3) by the condition that there exist a smooth, real-valued function

h = h(t,z)

such that, for

(t,y)

in

V,

(YT/IIiiI)H(t,r) > h(t,IIrlI), and

h

satisfies:

h(t,0) = 0; 8 (t,0) > m > 0,

1E h(a,s)ds > 0 0

for

for

a < t < b;

0 < C < IIAII,

-

and fn

h(b,s)ds > 0

for

0 < n < IIBIJ.

0

(See Remark 3.3 at the end of Chapter III and Example 8.21 in Chapter VIII.)

Thus far we have concentrated on the bounds for the norm of the solution of (S1), and in so doing, we have ignored the behavior of individual components as

H

side

[68], [69]).

(cf.

e -

0+.

Let us now seek conditions on the righthand

which will allow the study of each component of the solution We first define the

N

regions in IR:

Ui = {yi: Iyil <_ di(t)),

where each in

di

is a smooth positive function such that

[a,a + 6/2], di(t) a IBiI + 6

[a + 6, b - 6].

in

[b - 6/2,b], and

u = 0, not in the sense that

sense

that

in

in

H(t,0) = 0, but in the much stronger

Hi(t,Y1,...,Yi-1'0'yi+l"..'YN) = 0, t

di(t) = 6

Then we assume that the reduced system (RI) has a solu-

tion

for

di(t) a IAii + 6

[a,b]

and for all

yj

in

V.

i = 1,...,N, (j

i).

(7.5)

In other words, we

are only studying the class of problems (S1) such that (7.5) has a solution

yi a 0.

The assumption (7.5) has the effect of decoupling the

110

SINGULARLY PERTURBED SYSTEMS

VII.

vector second-order system into

N

second-order scalar equations, so that

the methods of Chapter III can be applied to estimate each component. Following Definition 3.1, we now make the following definition of stability for the reduced solution.

The zero solution of the reduced system (R1) (in the

Definition 7.2.

sense of (7.5)) is said to be componentwise-stable if there exist positive constants

mi

such that 1 mi > 0, for

in

t

[a,b]

and

yi

(7.6)

in V.

The condition (7.6) is clearly the system analog of the scalar condition (cf. Definition 3.1), and it allows us to establish the next theorem (cf. [68], [69]).

Assume that the equations (7.5) have a componentwise-stable

Theorem 7.2.

solution

ui = 0.

Then there exists an

the problem (S1) has a solution

e0 > 0

y = y(t,c)

0 < e < c0

such that for

of class

C(2) ([a,b])

satis-

fying lAiI.xp[-(mic-1)1/2(t-a)]

lyl(t,s)l < Proof: 3

IB.Iexp[-(mic-1)1/2(b-t)]

The proof of this result is very similar to the proof of Theorem For definiteness, let us suppose

!, and will follow from Theorem 2.4.

that

+

and

Ai > 0

Bi < 0, for some

For such

analogously.

Ail

and

i; the other cases can be handled

Bi, we define the pairs of functions

ai(t,c) = Bi exp[-(mlc-1)1/2(b-t)] and Bi(t,e) = Ai exp[-(m ic-1)1/2(t-a)].

The functions

ai

and

Bi

are solutions of

ai < Bi, ai(a,e) < Ai < Bi(a,c)

[a,b] x [ai(t,c),Bi(t,c)]

[a,b] x Vi

(for fixed

for bounding functions

6 > 0) when aj, Bj

[a,b] x [aj(t,c),Bj(t,c)]

= miv

Clearly

ai(b,c) < Bi < Bi(b,e); moreover,

and

the region

cv

is contained in the region is sufficiently small.

c

Similarly,

(j # i) defined analogously the region [a,b] x Vj, when

is contained in

c

is suf-

ficiently small.

Finally we show that

ai

satisfies the required differential inequal-

ity; the verification for

Bi

is similar and is omitted.

from the Mean Value Theorem that for

yj

in

Vj

(i # i)

It follows

The Semilinear Robin Problem

7.3.

E0

iii

H = miai - Hi(t,Y1,...,0,...,YN) (aHi/ayi)(t,Y1,.... 0ai,...,yN)ail

where

Now since

0 < 0 < 1.

and if

of course that

ai < 0a. <

we know that

(j # i), then we know that

a. < yj < 0j

is in

V.,

Dj, provided

0 < c < e0.

is sufficiently small, say

a

Oai

is in

yj

Consequently

we may apply (7.6) to conclude that ea

Hi(t,Y1,...,ai,...,yN) 2-'i'i - miai = 0, It follows from Theorem 2.4 that for

as required.

the problem (S1) has a solution

a < t < b

Z(t,e)

0 < c < e0

and

satisfying

ai <

yi(t,c) < Si, that is, exp[_(mic-1)1/2(t-a)]

B. exp[-(m ie-1)1/2(b-t)] < yl(t,c) < Ai

As was the case with Theorem 7.1, this result can be improved if certain integral conditions are used.

We refer the interested reader to

O'Donnell [69] and also to Example 8.22 in Chapter VIII.

The Semilinear Robin Problem

§7.3.

It is possible to obtain similar results for the following Robin problem

a < t < b,

ey" = _H(t,y),

(S )

Py(a,e) - Y'(a,c) = A,

Qy(b,c) + Y'(b,e) = B,

Q

(NxN)-matrices.

where

and

P

are constant

scalar Robin problem, we assume that

P

and

Q

2

By analogy with the are positive semi-

definite in the sense that there exist nonnegative scalars

p

and

q

such that YTPY ? PH Y112, for any

y

in IRN.

YTQY ? gllYu 2,

If we assume, for simplicity, that

[a,b], then we might ask under what conditions on

H(t,O) = 0

in

H will the problem

(S2) have a solution which is close to the zero vector in

[a,b].

The

results just found for the Dirichlet problem (S1) suggest that one sufficient condition is the existence of a scalar function

h = h(t,z)

with

112

VII.

SINGULARLY PERTURBED SYSTEMS

the following properties:

h(t,O) - 0 for

t

in

[a,b];

(7.7)

(ah/az)(t,0) > m > 0 J (7.8)

yTH(t,y)/IIYII > h(t,IIyII), for

in the region

(t,y)

b - [a,b] x {y

is a small positive constant.

in 1RN: IIyII < d}, where

Now we may recall from the results of

Chapter III that under the condition (7.7), the Robin problem for h(t,z) in

has a nonnegative solution

z(t,e)

also be close to zero, if we can show that result of the inequality (7.8).

ez" _

which is asymptotically zero

Consequently, the norm of a solution

[a,b].

d

of (S2) will

y(t,e)

IIy(t,c)II < z(t,e), as a

The precise statement is contained in

the following theorem. Theorem 7.3. in

[a,b]

Assume that the reduced system (R1) has the solution

and that there exists a scalar function

for

(t,z)

in

[a,b] x {z: Izi < 6}, satisfying (7.7) and (7.8).

there exists an a solution

and continuously differentiable with respect to

t, z

with respect to

u = 0

h = h(t,z), continuous

e0 > 0

such that for

0 < e <.c 0

z,

Then

the problem (S2) has

satisfying

y = y(t,c)

IIy(t,e)II < (Cm 1)112IIAIIeXP[-(me-1)1/2(t-a)}

+ (em Proof:

The proof is an easy application of Theorem 2.5. and

(a,b]

where

1)1/2IIBIIeXP[-(me 1)1/2(b-t)].

e > 0, define the scalar function

z = z(t,c)

For

t

in

p(t,y) = IIYII - z(t,c),

is the nonnegative solution of the scalar problem in

(a,b) : ez" = h(t,z), pz(a,e) - z'(a,C) = IIAII,

qz(b,c) + z'(b,c) = IIBII.

Now,

pp(a) - p'(a) = PIIy(a)II - pz(a) - (YT(a)/IIy(a)II)y'(a) + z1(a) = PIIy(a)II - pz(a) + (yT(a)/IIy(a)II)A -

yT(a)P(y(a)/IIy(a)II) + z1(a)

PIIy(a)II - pz(a)+ (YT(a)/IIy(a)II)A - PIIy(a)II + z1(a)

7.3.

The Semilinear Robin Problem

113

= -IIAII + (), T(a)/Ilr(a)II)A < 0, since

-yT(a)Py(a) < pIIY(a)II2

(YT(a)/IIY(a)II)A < IIAII

using the definiteness of we show that

by the definiteness of

P, and

by the Cauchy-Schwarz inequality. Q, it follows that

ez" < (yT/IIYII)H(t,y)

whenever

Similarly,

qp(b) + p'(b) < 0.

Finally

z = IIYII; indeed,

(yT/IIYII)H(t,Y) - ez"

> h(t,IIYII) - ez" h(t,z) - ez" = 0, by our choice of the function

solution

z, provided

a

is sufficiently small, say

Thus Theorem 2.5 tells us that the problem (S2) has a smooth

0 < e < e0.

y = y(t,c)

satisfying

p(t,y(t,e)) < 0

[a,b], that is,

in

IIY(t,e)II < z(t,e) < (em 1)1/2I IAIlexp[_(me-1)1/2(t_a)] +

(CM1)1/2IIBIIeXp[-(me-1)1/2(b-t)],

by virtue of Theorem 3.4.

We note that if the reduced system H(t,u) = 0 zero solution

has a smooth, non-

u = u(t), and if the inequalities (7.3) and (7.7), (7.8)

are modified accordingly, then results analogous to Theorems 7.1 and 7.3 are valid.

These modified conditions allow us to derive an estimate for

Ily(t,e) - u(t)II

in

0+.

[a,b]

as

a -

It is also possible to obtain

componentwise bounds on solutions of the Robin problem (S2), in much the same way we obtained bounds in Theorem 7.2 on solutions of the Dirichlet problem (Sl).

Indeed, for the Robin problem (S2) the assumptions (7.5)

and (7.6) reduce to the simpler assumptions that for

t

in

and

[a,b]

i = 1,...,N Hi(t,Yl,...,Yi_1,0,y.

(7.5a)

11...,yN) = 0

and

(8Hi/ayi)(t,Y1,...,yN) > mi > 0,

for

Iyil < 6.

(7.6a)

That is to say, we need only verify (7.6a) along the zero solution.

With

these ideas, we leave it to the interested reader to formulate for (S2) a result which is analogous to Theorem 7.2.

114

VII.

SINGULARLY PERTURBED SYSTEMS

The Quasilinear Dirichlet Problem

97.4.

We turn finally to an examination of the existence and the asymptotic behavior of solutions of the quasilinear vector problem ey" = F(t,y)y' + g(t,y), y(a,c) = A, Here

tinuous in

y(b,c) = B.

is a continuous

F

a < t < b,

(NXN)-matrix-valued function and

g

is a con-

N-vector-valued function, and each is continuously differentiable

y, on

[a,b] x1RN.

Depending on the properties of

F

and

tions of (S3) can exhibit a variety of asymptotic behavior as

g, soluc-* 0*;

indeed, as we have already noted in Chapter IV, the scalar form of (S3) is already fairly complicated.

The principal difficulty in studying the

system (S3) arises from the coupling of the first-order derivatives in the righthand side.

It is perhaps not surprising then that we must treat

this problem under some rather restrictive conditions on

F.

We will

study (S3) in the same manner as the semilinear problem (S 1), by first

considering norm-bound estimates on its solutions, followed by componentIn order to apply

wise estimates, in the spirit of O'Donnell's work [68]. O'Donnell's techniques we must assume that

F

is a diagonal matrix, that

is, we assume (S3) is a weakly coupled system in which the derivative of the i-th component appears only in the i-th equation.

Norm-bound results

can also be obtained for systems which are not necessarily weakly coupled; however, the estimates on the norm are

usually much cruder than the cor-

responding estimates on the individual components. Motivated by the scalar theory of Chapter IV, let us begin by considering solutions of the two reduced problems in

(a,b)

F(t,u)u' + g(t,u) = 0,

u(a) = A

(RL)

F(t,u)u' + g(t,u) = 0,

u(b) = B,

(RR)

and

which are stable in the sense of Definitions 7.3, 7.4 respectively.

We

first define the regions

D(uL) _ {(t,y): a < t < b, IIy-uL(t)II < dL(t)} and

D(uR) = {(t,y): a < t < b, 11y-uR(t)II < dR(t)}. Here

dL

is a smooth positive function such that

dL(t) = IIB-uL(b)II + 6

The Quasilinear Dirichlet Problem

7.4.

for

in

t

[b-6/2,b]

dL(t) F 6

and

a smooth positive function such that in

and

[a,a+6/2]

A solution

Definition 7.3.

for

dR(t) E 6

115

for

t

in

[a,b-S], while

dR(t) E IIA-uR(a)II + S t

u = uL(t)

in

in

(t,y)

dR

is

t

[a+S,b].

of the reduced problem (RL) is

said to be norm-stable if there exists a positive constant for

for

k

such that

V(uL)

zTF(t,y)w > kzTw, z, w

for all

in IR

N

A solution

Definition 7.4.

u = uR(t)

of the reduced problem (RR) is

said to be norm-stable if there exists a positive constant for

in

(t,y)

k

such that

V(uR)

zTF(t,y)w < -kzTw,

in e.

z, w

for all

These definitions are obvious extensions of Definitions 4.1 and 4.2 for the scalar analog of (S3).

They imply that the matrix

tive definite (negative definite) along layer at

t = b (t = a).

uL

F

is posi-

(uR) and within the boundary

This is a rather strong restriction on

F; how-

ever, this can be slightly weakened by means of certain integral condi[47] and Example 8.24 in the next chapter).

tions (cf.

We can now state

a basic result for the quasilinear problem (S3). Theorem 7.4. tion

Assume that the reduced problem (RL) has a norm-stable soluof class

u = uL(t)

such that for

0 < E < c0

C(2)([a,b]).

Then there exists an

the problem (S3) has a solution

e0 > 0

y = y(t,c)

satisfying

IIY(t.E) - h(t)II < IIB - uL(b)Ilep[-k1E-1(b-t)] + KE, where

K

Proof:

is a known positive constant

0 < k1 < k.

In order to simplify the proof, let us introduce the new depen-

dent variable

v = y-uL(t), in terms of which the problem (S3) becomes

the problem

EV" = f(t,V,V',E), v(a,E) = 0, where

and

v(b,c) = B - uL(b),

116

SINGULARLY PERTURBED SYSTEMS

VII.

f(t,v,v',e) = F(t,v + uL(t))v'

+ F(t,v + uL(t))uL(t) + g(t,v + uL(t)) - euL(t). The antitipcated application of Theorem 2.4 prompts us to define, for 0 < e < k2/(4L), the function

and

a < t < b

p(t,v) = IIvil-IlB-uL(b)Ilexp[al(b-t)] - evk-I(exp[A2(a-t)] - 1). -ke-1

Here (cf. the proof of Theorem 4.1), al -

A2 -

and

roots of the quadratic

eA2 + kA + £

Definition 7.3, and

is a positive constant such that

k

[a,b] x {n: I Inl I_ dL(t)};

in

the matrix norm

is defined by

The positive constant

will be determined later.

v

are the

is the positive constant in

(k

9., for (t,n)

III (af/av) (t,n,0,0)

-Rk-1

IIIGllI2

supJIG?I1

II?lI = 1}),

If we assume that

Y=

I

IvI

l

and

y' = vTy'/11v11'

for

y(t,e) = IIB-uL(b)Ilexp[al(b-t)] + ev1t-1(exp[A2(a-t)] - 1),

then Theorem 2.4 is applicable, if we can show that (vT/llvll)f(t,v,v',c)

- ey" > 0

(a,b)).

(in

Expanding by the Mean Value Theorem gives us

(vT/I lvI I)f - ey" (vT/IIvii)[f(t,0,0,0) Y

=

+ F(t,v+uL(t))v'

+ (af/av)(t,r1,0,0)v - euL(t)] - ey"

(vT/Ilvll)F(t,v + uL(t))v'

+ (vT/Ilvll)(3f/av)(t,n,o,o)v - eL - ey", since

f(t,0,0,0) - 0.

point and

L =_

max

Here

(t,n,0,0)

IIuL(t)Il.

Now if

is the appropriate intermediate a

is sufficiently small, say

[a,b]

0 < e < CO, then the point

(t,v + uL(t))

and so the norm-stability of uL ity

belongs to the region

V(uL),

allows us to continue with the inequal-

7.4.

The Quasilinear Dirichlet Problem

(vT/I

117

lvIl)f - ey" > kvTv'/I IvI I - 9I Ivl I - eL - EY" -ka1IIB - uL(b)Ilexp[Xl(b-t)] -ka2cvL-lexp[X2(a-t)] -RIIB - uL(b)Ilexp[al(b-t)] -9.evR-1exp[a2(a-t)] + ev - eL

- ex IIB - uL(b)IIeXp[Xl(b-t)] -

ex2cvR-1exp[J' 2(a-t)]

= 0,

if we set

v = L, since

cX

+ kai + R = 0, i = 1,2.

Therefore it follows

from Theorem 2.4 that the problem (Si), and hence the original problem (S3) have, respectively, C(2)-solutions that

v = v(t,c), y = y(t,e), such

p(t,v(t,e)) < 0, that is,

IIv(t,e)II = IIY(t,e) - h(t)II < IIB - uL(b)IIexp[-klc-1(b-t)] + Kc in [a,b], 0 < k1 < k

for

K = L£-1(exp[X2(a-b)] - 1).

and

The companion result for a boundary layer at

t = a

follows easily

from Theorem 7.4 and Definition 7.4 via the change of variable

t - a+b-t.

We leave its precise formulation to the reader. For classes of problems such as (S3), it is often advantageous to seek componentwise-bounds, rather than norm-bounds, on the solutions.

In

order to accomplish this, let us assume in what follows that the matrix F

is diagonal, say

tions

fi.

cY

F(t,y) = diag{f 1(t,y),...,fN(t,y)}

for smooth func-

Then the system (S3) can be written in component form as

= fi(t,Y)y! + gi(t,Y),

a < t < b, (S4)

Yi(a,c) = Ai,

Y1(b,c) = Bi.

Since the righthand side of the i-th equation depends only on

y

!

and

i

does not depend on

y!

(j # i), we say that the quasilinear system is

weakly coupled.

Let us now look for solutions of (S4) which exhibit boundary layer behavior at t = b

t = a; analogous results for boundary layer behavior at

then follow in the usual manner.

With our first assumption that

118

VII.

the reduced problem (RR) has a solution

SINGULARLY PERTURBED SYSTEMS

u = u(t)

of class

C(2) ([a,b]),

we define the regions i = 1,...,N,

V. = {yi: lyi - ui(t)l < di(t)},

is a smooth positive function such that

where each

d i

ui(a)l + 6

for

with

and

[a,a+6/2]

in

t

a small positive constant.

6

di(t) - 6

for

di(t) = JAi in

t

[a+6,b],

The second assumption is that

u

additionally satisfies the reduced differential equation in the following strong sense, namely, for

i = 1,...,N

fi(t,yui)ui + gi(t,yui) = 0, for all j # i.

(7.9)

(t,yui) - (t'yl' ...Ayi-1'ui'yi+1'" ''N )

with

in

yj

Dj,

A solution of the reduced equations (7.9) will be called a strong

reduced solution, to distinguish it from the reduced solution of the reduced system (RR).

As was the case with the semilinear problem (S1) (cf.

(7.5)), the second assumption is precisely the condition which allows us to decouple the system, and thereby apply the scalar theory of Chapter IV to the problem (S4).

Lastly we require

to be stable in the follow-

u

ing sense.

A strong reduced solution

Definition 7.5.

u = u(t)

lem (RR) is said to be componentwise-stable (in

[a,b]) if there are

such that

ki

positive constants

of the reduced prob-

fi(t,y) < -ki < 0, N for all

(t,y)

in the region V _ [a,b] x

V

11

i=l

With this notion of stability, we have the following result (cf. [68]). Theorem 7.5.

Assume that the reduced problem (RR) has a componentwise-

stable strong solution exists an solution

E0 > 0

u = u(t)

such ;hat for of class

y = y(t,E)

Proof:

0 < ki < ki

and each

Then there

C(2)([a,b]).

the problem (S4) has a satisfying, for

i = 1,...,N,

KiE, Ki

is a known positive constant.

It is enough to consider just the i-th component.

for definiteness, that tions

0 < e < c0 C(2)([a,b])

lyi(t,c) - ui(t)I < IAi -

where

of class

We assume,

ui(a) > Ai, and so we define the bounding func-

The Quasilinear Dirichlet Problem

7.4.

119

ai(t,E) = ui(t) - wi(t,e) - Wi(t,e) and Si(t,E) = ui(t) + Wi(t,E), where

wi(t,E) = (ui(a)-Ai)exp[Ai(t-a)]

and

Wi(t,c) = eviRi1(exp[ui(t-b)]-1),

in order to apply the "componentwise" version of Theorem 2.4. and

ai - -kic-1

Eat + kiA + Ri

ui -

-Rikil

0 <e < k2(4Ri)-1), Ri

(for

Here

are the negative roots of the quadratic is a positive constant

such that ((afi/Byi)(t,y)ul + (Bgi/Byi)(t,y)I < Ri, for

(t,y)

V, and

in

max

vi

Iu (t)I.

Clearly

ai < Si, ai(a,c) <

[a,b]

ai(b,c) < Bi < 8.(b,c), and so it remains to verify

Ai < Si(a,c)

and

that

satisfy the differential inequalities in

ail Si ca

> fi(t,Yai)ai + gi(t,yai),

The vector

yvi

aj < y. < Sj

is equal to

cs

< fi(t,ysi)S1 + gi(t,y81).

(y1, ....yi-1'v1,yi+1'" .,yN), where

for appropriate bounding functions

a., S., j # i, that is,

yj = uj(t) + 0(IAj-uj(a)Iexp[-kjc-1(t-a)] + 0(c), and so for

c

(a,b):

sufficiently small, say

0 < c < c0.

yj

satisfies the required inequality, as the verification for analogously.

is in

We only verify that Si

V.

ai

proceeds

Differentiating and expanding via the Mean Value Theorem,

we have ca

- fi(t,yai)ai - gi(t,yai)

= CO - eaiwi

-

cp1(Wi + Evl)

- fi(t,Yui)ui - gi(t,yui)

+ [(afi/ayi)(t,nai)ui + (agi/ayi)(t,nai)](wi + Wi) + fi(t,yai)[aiwi + ui(Wi + eviRil)], nai = (yl' ..., yi-1,ui + 0(ai-ui)'yi+1" ... N)' 0 < 0 < 1, is the appropriate intermediate point. Since, by assumption, fi(t'yui)ui +

where

gi(t'yui) ' 0 ity

and

fi(t,yai) < -ki < 0, we can continue with the inequal-

120

SINGULARLY PERTURBED SYSTEMS

VII.

eai - fi(t,yai)ai - gi(t,yi)

> - evi - ex2wl - eui(W1 + v

ii

- tiwi - Ii(Wi + evitil) + evi - kiaiwi

- ku(W. +

evltil)

= 0,

eai + kiai + ti = 0

owing to the fact that

and

We conclude from Theorem 2.4 that for has a solution

of class

y = y(t,e)

yi(t,e) < di(t,e)

eui + kiwi + fi = 0.

0 < e < e0

C(2)([a,b])

the problem (S4)

satisfying

ai(t,e) <

[a,b], that is,

in

-(ui(a) - Ai)exp[a1(t-a)] - Kie < yi(t,e) - ui(t) < Kie, Ki = viti1(exp[ui(a-b)] - 1).

for

The above result can, of course, be improved by appealing to componentwise integral conditions of the type mentioned at the end of Chapter IV [68] and Example 8.25).

(cf.

As noted before, the complementary theorem

involving a solution of the reduced problem (RL) and a boundary layer at follows from Theorem 7.5 by making the usual change of variable

t = b

t + a + b - t.

Finally it is possible to combine these two results into a "hybrid" theorem which can be proved in exactly the same manner as Theorem 7.5. It involves a solution

u = U(t)

F(t,u)u' + g(t,u) = 0,

of the reduced problem

a < t < b, (R2)

ui(a) = Ai

(1 < i < M),

ui(b) = Bi

(M+l < i < N),

as well as the regions Di ={ Yi: IYi - U1(t)I < di(t)}, where in

d.

is a smooth positive function such that

[b-6/2,b] and

di(t) = 6

di(t) = IAi - U1(a)I + 6 for

i = M+1,...,N.

in

in

[a,b-6]

[a,a+6/2]

A solution

U

for and

di(t) 5 IB1-U1(b)I + 6

i = 1,...,M, and di(t) = 6

in

[a+6,b]

of (R2) is then a strong solution if,

in addition, it satisfies the system (1 < i < N) on

(a,b)

fi(t,YUl)U! + gi(t,yui) = 0, for all

(t,yui) =

(t'yl'..''yi-1'ui'Yi+i ...,YN)

with

yj

in

Vj, j # i.

Notes and Remarks

121

This strong solution is said to be componentwise-stable if there are ki (1 ki > 0

i = 1,...,M

and

fi(t,y) < -ki < 0 Theorem 7.6.

Assume that the reduced problem (R2) has a componentwiseu = U(t)

stable strong solution exists an tion

i = M+1,...,N.

for

such ;hat for

e0 > 0

y = y(t,c)

of c l ass

of class

C(2) ([a,b]).

0 < c < e0

C (2) ([a, b ])

Then there

the problem (S4) has a solu-

sat i s fy i ng

IYi(t,s)-Ui(t)I < IB1-Ui(b)IeXp[-kie-1(b-t)] + Kie

(1 < i < M)

IYi(t,e)-Ui(t)I < IAi-Ui(a)Iexp[-T iE-1(t-a)] + Kie

(M+l < i < N),

and

where

0 < ki < ki

and

Ki

is a known positive constant.

Notes and Remarks 7.1.

The theory of this chapter applies, with obvious modifications, to problems in which the righthard sides and boundary data depend regularly on

7.2.

c.

It is possible to extend the scalar theory of interior layer phenomena, discussed in Chapters III and IV, to the semilinear and quasilinear systems considered in this chapter, if the appropriate reduced paths are componentwise-stable.

The interested reader can

consult the papers of O'Donnell [68], [691 for details and many examples. 7.3.

In our discussion of the semilinear problems (SI) and (S2) we assumed that either

hZ > m > 0

h(t,IIyII)) or

(for

h

(8Hi/ayi) > mi > 0

such that for

(yT/IIyII)H(t,y) >

i = 1,...,N.

Since these

are "scalar" conditions, we can easily apply the theory of Chapter III on higher-order stability conditions (cf. Definitions 3.1-3.6) to (S1) and (S2). regard.

The papers [41] and [46] are relevant in this

122

VII.

7.4.

The conditions (7.5) and (7.9), which guarantee that a reduced solu-

SINGULARLY PERTURBED SYSTEMS

tion is a "strong" solution, deserve a brief comment.

It turns out

that these conditions are not invariant, under even a linear change of variables.

In other words, it may be possible to transform a

system like (SI) or (S4), not originally having any reduced solution satisfying (7.5) or (7.9), respectively, into a new system for which these conditions obtain. 7.5.

Theorem 7.1 is due originally to Kelley [54] (cf. also [41]), Theorems 7.2, 7.5 and 7.6 are due to O'Donnell [68], [69], Theorem 7.3 is due to Howes [46] and Theorem 7.4 is due originally to Chang [12], who used a "diagonalization" method of approach.

Earlier work

on related problems includes the papers of Levin and Levinson [61], Levin [59], [60], Harris [32], Hoppensteadt [37], Chang and Coppel [13], Howes and O'Malley [47], as well as the monograph of Vasil'eva and Butuzov [88].

Additional references may be found in the mono-

graphs of Wasow [93] and O'Malley [75], and in O'Malley's long survey article [73].

Chapter VIII

Examples and Applications

Part I - SCALAR PROBLEMS Examples of Semilinear Problems and Applications

§8.1.

Example 8.1.

Consider the Dirichlet problem

ey" = (y - u(t))2q+1, y(-l,c) = A, where

-1 < t < 1,

y(l,e) = B,

is a nonnegative integer.

q

If the function

u(t), defined for

-1 < t < 1, is twice continuously differentiable or has a bounded second derivative, then by Theorem 3.1, for sufficiently small chlet problem has a solution lim+ y(t,e) = u(t)

y = y(t,e)

e > 0, the Diri-

which satisfies

in

(8.1)

e->0

where

0 < d < 1.

Moreover, the behavior of the solution

boundary layers at

t = -1

and/or

t = 1

u(-l) # A

(if

y(t,e)

in the

and/or

u(1) # B) can be described by means of the layer functions given in the conclusion of Theorem 3.1. If we choose

u(t) _ Itl = max{-t,t}, then the reduced solution is

not differentiable at

t = 0.

In this situation the reduced solution

is best regarded as the union of the stable path and the stable path

u+(t) = t

in

[0,1].

3.9 to deduce the existence of a solution fies the limiting relation (8.1). y

is also obtained from this theorem.

123

u

[-1,0]

We can then apply Theorem y = y(t,e)

t = 0

which also satis-

of the reduced paths

Note that, as

larger, the thickness of the angular layer at pondingly.

in

The precise behavior of the solution

in a neighborhood of the crossing point

u_, u+

u _(t) = -t

t = 0

q

becomes

increases corres-

124

VIII.

Example 8.2.

Let us consider next the related problem

ey" = (y

- UM)

y(-l,E) = A,

where in

2n,

-1
y(l,e) = B,

is a positive integer and

n

(-1,1).

EXAMPLES AND APPLICATIONS

u

is a convex function, say

u" > 0

The difference between this example and the last is that the

right hand side is now raised to an even power. that there exists a solution

y(t,e)

In this case, to ensure

satisfying the relation (8.1), we

need to restrict the sign of both boundary layer 'jumps' B - u(l); indeed, we must require that indicated in Theorem 3.2.

A > u(-l)

and

and

A - u(-1)

B > u(l), as

If any of these inequalities is not satisfied,

then our theory does not apply, and one can show that the Dirichlet problem has no solution of bounded

As an illustration, if

t-variation as

u(t) = t2

c > 0, the problem has a solution

a - 0+; cf. [76].

then for sufficiently small

y = y(t,E)

satisfying

y(t,e) > t2

and

lim+ y(t,e) = t2

in

E

provided A > 1 Example 8.3.

and

B > 1.

As our third example we take the problem

0 < t < 1,

ey" = y - y3 = h(Y), y(0,E) = A,

Y(l,E) = B.

The reduced equation u3 = -1, and since is stable.

h(u) = 0 h'(0) > 0

has three solutions while

ul = 1, u2 = 0

h'(±l) < 0, we see that only

and

u2

Applying the integral condition (cf. Remark 3.3)

iE

(s-s3)ds > 0

provided

0 < JEI < f,

0

we find that if

as

CAI < f and

IBI < T2, then the problem has a solution

e - 0+ such that lim+ y(t,e) = 0 E+

in

[6,1-d],

by virtue of O'Malley's result [76]. The problem also has solutions exhibiting what is termed spike layer behavior, in that the solutions are asymptotically zero except at regularly spaced points.

In a neighborhood of such a point the solution

8.1.

Examples of Semilinear Problems and Applications

has a spike of finite height which does not vanish as 8.1, 8.2.

u2 = 0

125

a - 0+; cf. Figures

This follows, again from a result of O'Malley [76], because

is a maximum point of the potential energy functional

'1'(Y) = -fA (s-s3)ds = (y4-A4)/4 - (y2-A2)/2

and T(J) = Y'(0) > 0

JAI < f), with

T.

not a maximum point of n > 2

plies that for each integer four solutions

y = y(t,e)

1im+ y(t,a) = 0

e

in

as

e -

(if

O'Malley's result im-

the problem (with

CAI, IBI < 72) has

satisfying

0+

[6,14],

y y

I

T2 B

A

J t

1/2

0

1/2

Figure 8.1

Spiked Solutions of Example 8.3 for n = 2.

1

126

VIII.

n

A

A

v .. 1/3 1/2 2/3

EXAMPLES AND APPLICATIONS

t

U 1/3 1/2 2/3

1

1

fl

A

"UU 1/3 1/2 2/3

t

uuu 1/3 1/2 2/3

1

Figure 8.2

Spiked Solutions of Example 8.3 for n - 3.

t 1

Examples of Semilinear Problems and Applications

8.1.

127

with the exception that

1+ y(ti,a) = T for ti = i/n The four solutions for the cases

(1 2

J181 Al, < I) also has for each integer

'Y, the problem (if

lim+ y(t,e) = 0

as

a -

0+

satisfying

in

a+0

with the exception that

r.

1+ Y(te) _

We consider next the related problem

Example 8.4.

0 < t < 1,

ey" = Y3 - y = g (y) , y(O,E) = A,

Since

g = -h

Y(l,e) = B.

we see now that the reduced solutions

are stable, while

u2 = 0

ul = 1

and

u3 = -1

Let us look at the function

is unstable.

ul

The integral conditions require that (cf. Remark 3.3)

first.

(s3 - s)ds > 0

for

E

between

1

and

A

or

B,

1

and a short calculation shows this inequality holds provided B > -1.

Consequently, if A,B > -1

y = yl(t,e)

as

1

and

then the problem has a solution

such that

e + 0+

li0 m+

A > -1

in

By symmetry we note that an analogous result holds for the reduced solution

u3 = -1.

Namely, if the boundary values A and

A, B < 1, then the problem has another solution 1iQ+ y3(t,e) = -1

y1, y3

B

satisfy such that

in

In particular, we note that if A = B = 0 three solutions:

y = y3(t,e)

and

y2

0.

then the problem has at least

VIII.

128

EXAMPLES AND APPLICATIONS

Finally let us hasten to point out that the problem has solutions which display discontinuous interior layer (shock layer) behavior (cf. Remark 3.4).

As an illustration, suppose that

A < -1

O'Malley [76] has shown that the problem has a solution

and

B > 1.

y = y(t,e)

Then as

satisfying

e + 0+

1-1

in

[6, Z - d],

1

in

[2 + d, 1-d],

lim+ Y(t,e) _

that is, y

transfers from

u3

to

which shrinks to zero as

t = 1/2

in a neighborhood of the point

u1

a - 0+.

He has also shown that when

A = B = 0, for example, the problem has for each nonnegative integer two solutions with limiting values and

-1.

which switch

1

n

n-times between

1

Thus, because in this example there are two stable reduced

solutions separated by an unstable one, we see that there is a countably infinite number of solutions. Application 8.1.

The following boundary value problem arises as a model

problem in the theory of nonpremixed combustion (cf. [97])

-1 < t < 1,

ey" = Y2 - t2 S h(t,y), Y(-',E) = Y(1,E) = 1.

Here

(assumed to be very small) is a ratio of diffusive effects to

a

the speed of reaction, and t = 0

is a distance coordinate, chosen so that

t

is the location of the flame, where the fuel and the oxidizer meet

each other and react.

The functions

y - t

and

y + t

represent the

mass fractions of fuel and oxidizer, respectively. In the limit of infinite reaction rate solutions

path

uI(t) = t

u(t) = Itl

and

u2(t) _ -t.

e = 0, we obtain the reduced

From these we form the stable

(known in combustion theory as the Burke-Schumann ap-

proximation [97]), that is, in

[-1,1] 2

2y (t,u(t)) > 0

2 > 0.

and ay

Theorem 3.10 then tells us that for sufficiently small ary value problem has a solution

y = y(t,e)

in

[-1,1]

e > 0

the bound-

satisfying

Y(t,E) = Itl + 0((e1/3/a)(1 + altl/e1/3)-2),

where a at

t = 0

is a known positive constant. is of order

c

1/3

Thus the thickness of the flame

Examples of Semilinear Problems and Applications

8.1.

129

The article of Williams [97] also discusses the same boundary value problem for more general differential equations of the form

ey = (y2

-

(8.2)

n > 1,

t2)n,

and m > n > 1.

syn = (Y+t)n(Y-t)m,

(8.3)

Using the theory of Chapter III, the reader should have no difficulty in seeing that the thickness of the flame at t = 0 in the case of model e1/(2n+1) (8.2) is of order and of order el/(m+n+l) in the case of model (8.3).

We turn now to a consideration of some related Robin problems. Example 8.5.

The first Robin problem is

eY" _ (y - u(t))2q+1,

-1
y(-l,e) - Y'(-l,c) = A,

where

q

Y(l,E) + Y'(l,e) = B,

is a nonnegative integer.

continuously differentiable in there is a solution

y = y(t,c)

lim+ y(t,e) = u(t) e+0

On the other hand, if

in

u

If the function

u

is, say, twice

[-1,1], then Theorem 3.7 tells us that as

c.+ 0+

such that

[-1,1].

(8.4)

is a function such as

ItI, then the same rela-

tion (8.4) still holds; however, there is an angular layer at whose thickness is of order

cl/(2q+2)

t = 0

(cf. Theorem 3.9).

In this example, we do not need any restriction on the boundary values, in order that the uniform limit (8.4) holds.

The next example

shows that restrictions are sometimes needed on the boundary values. Example 8.6.

Consider the Robin problem

cY"= (Y-t2)2,

0 < t < 1,

Y(O,e) - Y'(O,c) = A,

Y(l,e) + y'(l,e) = B.

Owing to the quadratic nature of the righthand side, we require that the reduced solution

u(t) = t2

satisfy

u(O) - u'(0) < A

that is, the boundary values must satisfy

values of A and

A > 0

and

and B > 3.

u(l) + u'(1) < B, For such

B, Theorem 3.8 states that for sufficiently small

c > 0, the problem has a solution satisfying

130

VIII.

lim+ y(t,e) = t2

in

EXAMPLES AND APPLICATIONS

[0,1].

C-0

We consider next some Robin analogs of Examples 8.3 and

Example 8.7.

8.4, namely (8.5)

and

eyll = Y3 for

in

t

-y

(8.6)

(0,1), along with the boundary conditions

y(0,e) - y'(O,e) = A.

In the case of (8.5) we know that

y(l,e) + y'(l,e) = B.

u = 0

is the

only stable reduced solution, and so Theorem 3.7 tells us that for sufficiently small

e > 0, the Robin problem (8.5) has a solution satisfying

lim+ y(t,E) = 0 E+0

in

Similarly, since both

[0,1].

u1 = 1

and

u3 = -1

are stable reduced solutions

of (8.6), this same theorem states that for sufficiently small the Robin problem (8.6) has two solutions fying in

y1(t,e)

and

y2(t,e)

e > 0,

satis-

[0,1]

lim+ Y1(t,e)

and

1+ y2(t,E) = -1However we are unable to say at the present time whether there are other solutions of (8.6) switching between Example 8.4 using O'Malley's results.

1

and

-1

such as those found in

Such an occurrence would be very

interesting mathematically, and it would also have important consequences for several problems in catalytic reaction theory, the subject of the next application. Application 8.2.

Our final example in this section is an application of

our results to a class of boundary value problems arising in catalytic reaction theory (see, for example, [2; Chapter 3]).

The simplified phy-

sical problem involves an isothermal reaction A + B which is catalyzed in a pellet of length two.

An equation that describes the mass balance

between diffusion and reaction inside of the pellet is then Y" =

where

y

ID2R(Y),

0 < t < 1,

is the normalized concentration of the reactant

A, t

is the

8.1.

Examples of Semilinear Problems and Applications

131

(dimensionless) distance from the center of the pellet (t = 0) to the mouth (t = 1),

is the Thiele modulus which measures the effect of

4'

diffusion as opposed to reaction, and In particular,

t2

coefficient and

R(y)

k

is the reaction rate term.

k/D, where

is proportional to

the reaction rate constant.

R(y) = yn, for a nonnegative integer

is the diffusion

D

Let us assume that

n, that is, the reaction is nth

order, and that the catalytic reaction is diffusion-limited, that is, 2 ID

>> 1.

At the line of symmetry (t = 0) there is no flux, that is,

y'(O,e) = 0, while at the mouth of the pellet the correct boundary condiy(l,e) + Ey'(l,e) = 1, for a nonnegative constant

tion is

reciprocal is called the Sherwood number.

E

whose

The Sherwood number measures

the ability of the reactant to reach the pellet from the bulk flow.

In-

corporating all of this into the model, we arrive finally at the problem

ey = yn,

0 < t < 1,

-y'(0,e) = 0,

where

y(l,e) =

y'(l,e) = 1,

e = 4 2 << 1.

Let us consider first the case when uniformly accessible to the bulk flow.

E = 0, that is, the pellet is

Theorems 3.4 and 3.5 then tell

us that the boundary value problem has a nonnegative solution satisfying lim+ y(t,e) = 0

in

(8.7)

[0,1-6].

eQualitatively this means that owing to the large rate of reaction (or equivalently, small rate of diffusion), most of the reaction is confined to the pore mouth.

The concentration of reactant falls off to zero

exponentially (algebraically) for

n = 1 (n > 2), as we proceed into the

pellet.

Suppose we study now the more realistic case

t > 0, in which there

is some resistance to the transfer of reactant from the bulk to the pellet.

If the resistance is small, that is, if

have the situation described by (8.7). larger than

e, say

sufficiently small

E = 0(e), then we again

However if

E

is asymptotically

E = 0(1), then Theorems 3.7 and 3.8 state that for e > 0

the boundary value problem has a nonnegative

solution satisfying lim+ y(t,e) = 0 e+0

in

[0,1].

Qualitatively this means that in the presence of resistance to mass transfer, the concentration of reactant in the reaction zone near the pore

132

VIII.

mouth

t = 1

if

0(e)

vanishes as

n = 1, and

e + 0+.

EXAMPLES AND APPLICATIONS

More precisely, we have

y(l,e) = 0(el/(n+l))

y(l,c) =

if n > 2, by virtue of the

estimates in these theorems.

Examples of Quasilinear Problems and Applications

§8.2.

Example 8.8.

We begin with the simplest quasilinear problem

ey" = f(Y)Y',

a < t < b,

y(a,e) = y_,

where

f

y(b,e) = y+,

y_ < y < y+.

is continuous for uL = y_

solutions

y_ < y+,

uR = y+

and

Let us assume that the reduced

are stable, in the sense that

0 < f(yand let us define the functional F(y_), where

F

is an antiderivative of

f(y+) <

J[y] 2 fy_ f(s)ds = F(y) Then the theory of Chapter

f.

IV (refer also to Remark 4.3) asserts that if J[y+] < 0, that is, if F(y+) < F(y_), the problem has a solution

y = y(t,c)

as

e + 0+

satis-

fying lim+ y(t,e) = y+

On the other hand, if

in

[a+d,b].

F(y+) > F(y-), then

J[y+] > 0, and the theory

tells us that the solution of the problem satisfies lim+ y(t,e) = y

in

[a,b-d].

e+0

There remains the case when

F(y+) = F(y_).

The existence of bound-

ary layers is now precluded, and so we look for another type of limiting behavior as tion for all a(t,e) 2 y-

c + 0+.

e > 0 and

(Note that the boundary value problem has a soluby virtue of Theorem 2.1, which is unique, since

B(t,s) 2 y+

are lower and upper solutions, respectiv-

Because the reduced solutions are constants, the only type of

ely.)

behavior possible is that involving a shock layer connecting the states y-

and

y+

at some point

t0

in

(a,b).

In order to locate this

transition point, let us begin by noting that the solution satisfies y'(t,e) = const. exp[F(y(t,e))], and so

y'(t,e) > 0

since by assumption

(8.8)

y_ < y+.

the original differential equation in the form e(ln y')' = f(y).

Thus we can rewrite

8.2.

Examples of Quasilinear Problems and Applications

133

which in turn allows us to write the two equations Jt

C in y'(t0,e) - e in y'(a,e) = J 0 f(y(s,e))ds

(8.9)

a

and

e in y'(b,e) -

in y'(t0,e)

f(y(s,e))ds.

(8.10)

J t

But

y'(a,C) = y'(b,e)

by virtue of (8.8), since

F(y_) = F(y+), and so

adding (8.9) and (8.10) gives 1t

0 f(y(s,e))ds +

0 =

f(y(s,e))ds.

(8.11)

ft

a

0

Finally, since y_,

a < t < t0-d,

y+,

t0+d < t < b,

lim+ y(t,e)

(that is, there is a shock layer at

t0), if we take the limit as

a - 0+

of both sides of (8.11), we obtain from the Dominated Convergence Theorem the limiting relation

t

0=

! 0 f(y_)ds + a

j

b

t0

f(y+)ds = f(y-) (t0 - a) + f(y+) (b - t0).

It follows that the shock layer is located at

to = [f(y_)a - f(y+)b]/[f(y_) - f(y+)]. For example, if connecting at

y_

f(y) = -y, then in order for there to be a shock layer to

y+, we must have

y+ = -y_ > 0; the layer is located

t0 = (a+b)/2.

The relation

F(y_) = F(y+), known as the Rankine-Hugoniot shock

condition (cf. [17; Chapter 3], [55; Chapter 4]), arises in modelling compressible flows and chemically reacting flows, as shown in the next application.

Application 8.3.

This problem concerns the description of the one-

dimensional, steady-state flow pattern arising from the injection of a gas at supersonic velocity into a duct of uniform or diverging crosssectional area when a back pressure is applied.

Complications such as

the effect of viscous stresses on the duct walls is gas is assumed to be perfect and polytropic.

neglected, and the The time-independent laws

of conservation of mass, momentum and energy can be expressed in the

134

EXAMPLES AND APPLICATIONS

VIII.

following dimensionless form by referring all quantities to appropriate lengths, physical constants and upstream conditions (cf. [18]): d/dx(pyA) = 0,

(8.12) 2

y dx

(Yp)-1

+

(8.13)

dz (PT) = uP-1 a 2 dx

and

y dz + (Y-1)T[dx + y dz (ln A)] - Y(Y-1)UP-1(dx) wwpp-1

=

Here

r

8.14)

dx 2

is the dimensionless distance measured from the entrance of the

x

is the dimensionless velocity of the gas relative to the velo-

duct, y

city of sound, p between

is the density, y 5/3, T

and

1

is the adiabatic index with a value

is the dimensionless temperature, u

efficient of viscosity, and

Pr

is the co-

is the Prandtl number, taken equal to

Finally A = A(x) (A(0) = 1) is the dimensionless cross-sectional

3/4.

area of the duct relative to the area of the duct entrance.

Crocco [18] we have omitted terms of the form udA/dx

Following

and uPT1 dA/dx

By first neglecting the second-order

in (8.13) and (8.14), respectively.

stress terms in these equations we obtain easily two equations for isentropic flow (cf. [18]) Ay[1 - (Y21

y2)1/(Y-1) = const.

and

y

= 1 -

Y21) Y2

Upon substituting the expression

T(x) = 1 -

y-1

y2(x)

into (8.13) and

using (8.12) we obtain, after a straightforward calculation, an equation for the velocity

y

of the form

uY(POc0)-lAY d4 _ [(Y21) y-Y

1)ff

- d[ln A(1 -

Y21

y2)].

dx The quantities

p0. c0

are respectively an upstream reference density

and the upstream velocity of sound.

sionless term viscosity

u

py(p0c0)-1 = e

Let us now assume that the dimen-

is small, that is, the coefficient of

is small for fixed values of

y, p0

the singularly perturbed quasilinear problem in

and (0,1)

c0.

Then we have

8.2.

Examples of Quasilinear Problems and Applications

E Ay

2 da

=

L(Y21)

Y-Y I]d

dx

[In A(1 -

135

y-1) Y2)]2

(SL) Y(O,E) = Y_,

Y_ > Y+ > 0.

Y(l,e) = Y+>

(This is basically the same boundary value problem used by Pearson [77] in his numerical experiments with

A(x) = 1 + x2.)

The original physi-

cal problem can be restated now in terms of (SL) as follows: supersonic velocity

given a

at the entrance of the duct (x = 0), determine

y_

what subsonic velocity

at the end of the duct (x = 1) produces a

y+

supersonic-subsonic transition in the interior of the duct and also the location of this transition.

Let us consider first the case of a uniform duct, that is, A(x) - 1. The problem (SL) reduces to the simple form

0<x<1,

Edd = [(Y+1) -y2]ax, y(O,E) = Y_,

(SL0)

Y(l,c) = Y+

In order to find conditions under which (SL0) has a shock layer solution, we have only to apply the results of Example 8.8. tion

f(y) = (Y21

For

y > 0

is continuous and vanishes only at

-y-2

the func-

which can be regarded as a dimensionless critical velocity [17; 3].

Consequently, f(y) > 0

for

y > yc

and

f(y) < 0

for

(2/Y+l)1/2,

yc =

Chapter

0 < y < yc.

The Rankine-Hugoniot shock condition is

(+1 2

-1 - (Y+ 1) -1 Y_ + Y_ Y+ + Y+ 2

that is, 2

Y+Y- = Yc =

2 (Y+1)

,

which is known as Prandtl's relation (cf. [17; Chapter 3]). an initial supersonic velocity velocity

y+ = (Y+1) y_1

lim+ y(x,e) _ e+0

such that (SLO) has a solution satisfying

Y_,

0 < x < x0-d,

y+,

x0+d < x < 1.

Here

f (Y+) x0

f(Y+)-f(Y_)

Thus, given

y- > yc, there is a unique subsonic

Y_

Y_ + Y+

136

VIII.

EXAMPLES AND APPLICATIONS

is the location of the shock layer representing a supersonic-subsonic This formula for

transition.

allows us to conclude that if the

x0

supersonic inlet velocity is very large, then the major portion of the x0

flow is supersonic since

The shock layer sits

is close to unity.

close to the end of the duct.

We turn finally to a consideration of (SL) when the cross-sectional area of the duct increases in the downstream direction, that is, dA/dx > 0 for

()

0 < x < 1.

( +l

equation and

c = 0

u-u-1)du

=

we first obtain solutions of the reduced

d[ln A(1

-

Y-1

u2)]

satisfying

u1(0) = s

implicitly as

u2(1) = y+ ul(x)(1 -

Setting

(y-1

= y-(1 _

ui(x))1/(Y-1)

Y21

y2)1/(Y-1)/A(x)

and u2(x))I/(Y-1)

Y21

u2(x)(1 -

A(0) = 1.)

(Recall that

respectively.

(or

y-2

Not unexpectedly, these are the

in

[x2,1]

with

+l

y+y-I, has a solution

x0

y > yc = ((Y+l))1

ul(x) > yc

in

F(u1(x0)) = F(u2(x0)), for in the interval

, that is, Prandtl's relation must hold at

A

(x2,x1).

short calculation shows that this condition is equivalent to yc =

and

[0,x1]

Then our theory applies pro-

x1 > x2.

vided that the Rankine-Hugoniot equation F(y) =

Since 2

is positive (or negative) for

0 < y < yc), we must require that

0 < u2(x) < yc

y2)1/(Y-1)/A(x),

+

beginning of our discussion.

isentropic relations obtained at the f(y) _ (Y2I)

(i-i 2

= A(l)y+(l -

2

x0.

uI(x0)u2(x0)

We conclude

(Y+l)

that under these assumptions, the problem (SL) has a solution satisfying F

lim+ y(x,e) Ct

ul(x),

0 < x < x0-6,

u2(x)'

x0+d < x < 1.

The solution describes a supersonic-subsonic transition at the nonconstant states

u1(x)

and

between

x0

u2(x).

In order to illustrate this result, let us consider the case A(x) = 1 + x2, y- = 0.9129, y+ = 0.375 with this data and with the term

and y = 7/5.

The problem (SL),

added to the righthand side,

-ey dx was treated numerically by Pearson [77] whose results afford a means of

comparison with ours.

For y = 7/5

is slightly less than

y-, and so

and with value

yc

A(x) = 1 + x2 at a point

the critical velocity y_ > yc > y+.

the reduced solution x2

in

u2

y

=

(_(Y2 (Y+l))

1/2

c

With this choice of y+ assumes the critical

(0,1/2), that is, f(u2(x2)) = 0.

As

8.2.

Examples of Quasilinear Problems and Applications

u1, it is easy to see that

regards

137

ul(x) > yc; whence, f(u1(x)) > 0

in

Finally, one can show that the Rankine-Hugoniot equation

[0,1].

has a unique solution

F(ui(x0)) = F(u2(x0))

approximately

x0

in

(x2,l)

This compares well with the value

0.6.

which is

x0 = 0.634

ob-

tained numerically by Pearson for the slightly modified version of (SL) with

of order

c

Example 8.9.

10-8.

The reasoning employed in Example 8.8 extends to more gen-

eral problems of the form

a < t < b,

ey" = f(y)y' + g(y), y(a,e) = y-,

y(b,c) = y+,

provided that the boundary values are solutions of the reduced equation, For instance, we know from Example 8.8 that

that is, g(y-) = g(y+) = 0.

a = y- = -1, b = y+ = 1, f(y) = -y

for

solution connecting

y_

However, if

e + 0+.

to

y+

g = 0, this problem has a

and

across a shock layer at

t = 0, as

g(y) = 1 - y2, then this same result also holds.

Asymptotically the g-term has no effect on the behavior of solutions, provided that the f-term has the properties given above. Example 8.10.

Consider now the problem

0 < t < 1,

ey"=-Yny' +Y, y(O,e) = A,

where

y(l,e) = B,

When

is a positive real number.

n

n = 1, it is the classic

Lagerstrom-Cole model problem about which much has been written (cf. [55; Chapter 2], [20], [39]).

boundary layer at Bn)1/n

t = 0.

then the function

uR(t) = (n(t-1) +

is a strongly stable solution of the corresponding righthand

reduced problem.

Suppose now that

n

First, if n

two cases to consider. ent of

Let us look first for a solution with a

If Bn > n

is a natural number.

is even, then

There are

-yn, the coeffici-

y', is nonpositive throughout the layer, and so Theorem 4.1 implies

that the problem has a solution satisfying lim+ y(t,e) = uR(t) e+0

in

[d,l],

Second, if n

for all values of

A.

of A

is important.

and

uR(O)

is odd, then the relative position

If A > uR(O) = (Bn - n) 1/n

0 < A < uR(O), then the coefficient of

(8.15)

d > 0,

y'

or if

is again nonpositive in the

layer, and so the limit (8.15) also holds for such values of

A.

However,

138

VIII.

EXAMPLES AND APPLICATIONS

if A < 0, then we must apply the integral condition of Coddington and Levinson [14]

(cf. Remark 4.3) which allows values of A

n

(uR(O)

A <

-s ds < 0,

J

for which

< uD(0).

An easy calculation reveals that this inequality holds if

(B'-n)1/n.

JAI <

Thus for these values of A, the limiting relation (8.15) is also valid.

In summary, if n values of

is even there is a boundary layer at

A, while if

n

for all

t = 0

is odd there is a layer only if A >

A moment's reflection shows that these conclusions hold when

-(Bn-n)1/n,

Bn = n

(cf. Theorem 4.2).

Suppose next that 0 < Bn < n. Then the function uR(t) = (n(t-1) + Bn)1/n vanishes at tR = 1 - Bn/n in (0,1). If 0 < n < 1 then uR(tR) = uR(tR) = 0, that is, uR smoothly.

intersects the zero reduced solution

On the other hand, if n > 1

then

uR(tR) = °, and it is not

However, for the intermediate case

clear how to proceed.

uR(tR) = 1, and we can say a few words about this case. solution is clearly

n = 1,

The zero reduced

(I0)-stable (cf. Definition 4.4), and so the reduced

path

un(t) =

J0,

0 < t
t+B-1,

tR < t < 1,

If A > 0, then Theorem 4.12

is weakly stable (cf. Definition 4.5).

tells us that for sufficiently small

e > 0, the problem has a solution

satisfying in

Y(t,e) = u0(t)

[S,1].

lm+ i (Of course, if A = 0 Next, if

then this limit is assumed at

-1 < A < 0

duced solution

and

-A < 1-B

uL(t) = A + t

t = 0

as well.)

then the weakly stable lefthand re-

intersects the zero solution at

tR, and we have a situation described by Theorem 4.14.

tL = -A <

The problem has

a solution satisfying

Y(t,e) = Eio+

When

-A = 1-B

Finally, for

t+A,

0 < t < tL,

0

tL < t < tR,

,

t+B-1, tR < t < 1.

note that -1 < A < 0

y(t,e) = t+A = t+B-1 and

is the exact solution!

-A > 1-B, there are no angular crossings

8.2.

Examples of Quasilinear Problems and Applications

since

tL > tR.

solution as

uL

In this case, one can show (cf. [20], [39]) that the

c -+ 0

t0 = 1/2(1-B-A)

in

connects

uL

(tR,tL).

Finally, for

and

is strongly (weakly) stable in

with boundary layers at

uR

across a shock layer at A < -1

(A = -1) the function

[0,1], and so there are solutions

or with shock layers at

t = 1

uR, depending on the relative sizes of A

and

139

and

B.

joining

to

uL

See [20] or [39]

for all of the details.

The last phenomenon we discuss is the existence of boundary layers relative to the (I0)-stable zero function, when

If n

n

is a natural number.

B = 0, then Theorem 4.5 tells us that the problem

is even and

has a solution satisfying in

lim+ y(t,e) = 0

for all values of

[6,1]

A, with

However, if n

if A = 0.

S = 0

then by Theorem 4.5 we have for all values of A > 0

is odd,

B < 0, a solu-

and

tion satisfying lim+ y(t,e) = 0

in

E+0

For such values of

A, B

and/or

n, either

uL

or

uR

does not exist

or is unstable. Application 8.4.

For our final application of this section we consider a

catalytic reaction in a one-dimensional fixed-bed reactor packed with catalyst in the presence of axial diffusion.

for the dimensionless concentration

Ey"=y' +g(Y),

x

The boundary value problem

is then (cf. [80; Chapter 4])

0<x< 1,

Y(O,e) - p1Y'(O,e) = A, where

y

Y(l,E) = p2Y'(l,E) = B,

is the dimensionless axial coordinate, a

is the reciprocal of

the Peclet number (the diffusional analog of the Reynolds number), and g

is the reaction rate term, of the form

reaction.

The positive terms

pl = pl(E)

g(y) = yn

and

for an n-th order

p2 = p2(e)

are mass

transport coefficients of the type considered in Application 8.2. axial diffusion is weak then

a

If the

is small, and the Robin problem is sin-

gularly perturbed.

We consider only two cases.

First, if p2(e) = 0(E)

then Corollary

4.2 tells us that the problem has a solution with a boundary layer at x = 1, that is,

140

EXAMPLES AND APPLICATIONS

VIII.

lim+ y(x,e) = uL(x)

in

[0,1-6],

g+0 where

uL, the solution of the reduced problem

plu'(0) = A, is assumed to exist in

u' + g(u) = 0, u(0) -

Second, if p2 = 0(1)

[0,1].

or

larger, then Corollary 4.3 tells us that there is a solution which satisfies this limiting relation in all of g(y) = yn, for

n

[0,1].

Here we are taking

a natural number.

Depending on the nature of the function

g, O'Malley [75; Chapter 7]

has shown that this problem may have more than one solution.

Multipli-

city of solutions, which are steady states of the associated time-dependent partial differential equation, is often an important consideration in reactor design, since it implies that the reactor may be capable of operating in more than one stable regime. Example 8.11.

We conclude this section with a problem about which we

know very little, namely

-1 < t < 1,

cY" = ty, + Y 3 - Y,

= -1,

Y(-1,c) -

Clearly

uL

-1

and

uR a 1

Y(l,c) + Y'(l,e) = 1.

are solutions of the corresponding left-

and righthand reduced problems, respectively.

Since these functions are

(I0)-stable, we can apply Theorem 4.10 to deduce the existence of two solutions 1im

yl, y2

of the problem as

satisfying in

c - 0+

[-1,1]

Y1(t,e) = -1

and lim+ y2(t,e) = 1.

There are however other solutions of the reduced equation -- the zero function which is unstable and the one-parameter family of functions

2 2 1/2 ,

uc(t) = ct/(1 + c t ) 8.3.

The functions

uc

where

c

is a constant, as shown in Figure

are all capable of supporting boundary layers

at the endpoints, since the coefficient of

near

t = -1

origin.

y'

is negative (positive)

(t = 1), but they lose stability upon passing through the

We strongly suspect, though, that the

uc's do participate in

describing the asymptotic behavior of other solutions of the problem. It might be instructive to collect some numerical data in this direction.

8.3.

Examples of Quadratic Problems and Applications

141

..................... c < 0

-1

Figure 8.3

Solutions of the Reduced Equation of Example 8.11

58.3.

Examples of Quadratic Problems and Applications

Example 8.12.

Consider the problem

EY" = -Y12 - yy', Y(O,a) = A,

0 < t < 1,

y(l,c) = B,

treated by Dorr, Parter and Shampine [20; 55] in the case

A < B.

Our

first step is to examine the reduced equation

0 = u'(u'+u), and we find

that

uR(t) = Bel-t

uL(t) = A, uL(t) = Ae-t, uR(t) s B

and

solutions of the corresponding reduced problems. equation has no singular solutions. see that Ae

t

,

_ -2y' - y, and so

Setting

are four

Note that this reduced

f(t,y,y') _ -y'2 - yy', we

fy,(t,uL,ul) = y,[uL] _ -A, fy,[uL]

yf

fy,[uR]

-B

and

fY[uR] = Bel-t.

Thus the stability of these

142

EXAMPLES AND APPLICATIONS

VIII.

reduced solutions depends critically on the values of A

and

We

B.

distinguish the following cases. A,B > 0.

Case 1.

Ae_t

A, B, uL(t) =

Clearly for such

and

necessary for boundary layer behavior, that is, uL(1) = Ae For these values of

> B

and

A, B, it follows from Theorem 5.1

y = y(t,e)

that the problem has a solution

_E B

We then check the sign restrictions

are the stable reduced solutions.

uR(0) = B > A.

uR(t)

as

such that in

e + 0+

[0,1]

B - (B-A)exp[-Bt/e] < y(t,e) < B.

if

B > A > 0,

and

Ae-t

-

(Ae-1-B)exp[-Ae-1(1-t)/c] < y(t,e) < Ae-t + ey, if

where

y

is a known positive constant.

Ae-1 > B > 0,

We next check for the occurrence

of an angular crossing involving these reduced solutions. consider values of

ties fail simultaneously; namely, Ae-1 0

That is, we

for which the above boundary layer inequali-

A, B > 0

f(t0,uL(t0),w)

(-B,0), the theorem of Haber and Levinson [27]

We conclude that if

(cf. Corollary 5.20) applies.

0 < Ae-1 0, B < 0.

uR(t) = Bel-t

< t < 1. o-

are the stable reduced solutions.

see that only the boundary layer inequality

uL(t) = Ae-t

Since

uL(l) > B

obtains, and so

by Theorem 5.1 the problem has a solution satisfying

lim+ y(t,e) = Act in

[0,14].

E+O

In this case there are no interior crossings. Case 3.

This is the reflection of Case 2 and so the

A < 0, B > 0.

solution satisfies lim+ y(t,E) = B

e0

in

[6,1].

and

B < 0 < A we

8.3.

Examples of Quadratic Problems and Applications

A, B < 0.

Case 4.

reduced solutions

143

This is also a reflection, of Case 1, in which the and

uL(t) - A

uR(t) = Bel-t

are stable.

We see

easily that the problem has solutions with the following asymptotic behavior:

lim+ y(t,c) = A

in

if A > B,

[0,1-6],

e+0

lim+ y(t,E) = Bei-t e+0

in

if

[6,1],

Be > A,

and finally 0 < t < to,

A,

lim+ y(t,e) e+0 for

Be

(0,1).

Two cases remain:

first case, uL - A

R

since

uL < 0.

in this case.

A < 0, B = 0

(i)

and (ii) A = 0, B < 0.

is strongly stable; however, since

and f[0] = fy[0] - 0. U

Be < A < B,

t0 < t < 1,

,

in

t0 = ln(Be/A)

if

-

-

1-t

But

Thus

uL(1) = A < B

and

In the

B = 0, uR = uR = 0 does not intersect

uL

does not determine the asymptotic behavior

uL

Let us examine

uR - 0

more closely.

Clearly,

- 0

is

an upper solution of the problem, and a short calculation shows that

,.-1 -r+ a'%

.

1/2,-1

,

-

=

(A-' - (1-T(e))/e

is a lower solution for

and fying

B = 0

e+0+

- 7(1-T(e))e 1 /2,

of order

y(t,e) = 0

in

1-T(e) < t < 1,

el/2, provided that y > 0

We conclude from Theorem 2.2 that if A < 0

the problem has, for sufficiently small

a(t,s) < y(t,e) < 0 lim

1/2 )-1

T(e) > 0

is appropriately chosen.

1/2

in

e, a solution satis-

[0,1], that is,

[6,1].

Arguing in a similar manner, we see that in case (ii) the problem has a solution satisfying lim+ y(t,e) = 0

in

[0,1-6].

e+0

These two cases deserve to be given special care, since appropriate partial derivatives are neither strictly positive nor negative.

144

EXAMPLES AND APPLICATIONS

VIII.

The problem

Example 8.13.

0
eY"=-Y'2+Y,

y(l,e) = B,

y(O,e) = A.

illustrates how the existence of a singular reduced solution affects the asymptotic nature of solutions of a quadratic problem for certain values Its reduced equation

of the boundary data.

u'2 = u

is a Clairaut equa-

tion which has the regular solutions uL(t) = 1/4(2A1/2 - t)2,

uR(t) = 1/4(t + 2B 1/2 - 1)2

uL(t) = 1/4(2A1/2 + t)2,

uR(t) = 1/4(2B1/2 + 1 - t)2,

and

defined for

A, B > 0, as well as their envelope, the singular solution

us = 0; cf. Example (E13) of Chapter V.

Since

fy, _ -2y'

f(t,y,y') = -y'2 + y, we see that the functions

for

and

uL, uR

fy = 1,

are unstable,

while

fY l [uL (t) ] =

2A1/2

-t

>0

for t <

< 0

for

2A1/2,

t > 2A1/2

and

y,[uR(t)] = 1 - 2B

1/2

-

r < 0 t > 0

and uR

Consequently, uL

at

tL = 2A1/2

in

[0,1], since

Case 1.

Al/2,

and

t > 1-2B1/2

for

t < 1-2B

1/2

are strongly (weakly) stable if Al/2 > 1/2

B1/2 > 1/2

and

(A1/2 = 1/2)

0 < Al/2, B1/2 < 1/2

for

t {

(B1/2 = 1/2), respectively.

If

then these functions respectively lose stability tR = 1 - 2B1'2.

Finally, us = 0

is

(I0)-stable

fy = 1.

B1/2 > 1/2.

we see that since 1/4(2A1/2 - 1) 2 > B

Checking first for boundary layer behavior

fy,y, < 0, the inequalities required are and

uR(0) = 1/4(2B1/2 - 1) 2 > A.

uL(1) =

For these values

Theorem 5.1 tells us that the problem has solutions such that

lim y(t,e) = uL(t) and

in

[0,1-8]

if

1/4(2A1/2

-

1)2 > B

8.3.

Examples of Quadratic Problems and Applications

in

li m+ y(t,e) = uR(t) e+O

Next, if

uL(1) = 1/4(2A1/2 - 1)2 A.

if

[6,1]

145

(8.16)

uR(0) = 1/4(2B1/2 - 1)2 < A, uR angularly at t0 = A1/2 -

and

intersects

The existence of a solution satisfying

(0,1).

uL(t),

0 < t < to,

IuR(t),

t0 < t < 1,

(8.17) e4-0

follows from the theorem of Haber and Levinson. Case 2.

For these values of A

A,B < 0.

B, there is no regular

and

reduced solution which satisfies either of the boundary conditions, because

However, since

u = u'2 > 0.

us = 0

(I0)-stable and

is

us(0) > A, us(1) > B, we can apply Theorem 5.5 to conclude that the problem has a solution satisfying in

[0,1]

A exp[-t/el/2] + B exp[-(l-t)/e1/2] < Y(t,e) < 0. B > 0, A < 0.

Case 3.

uR(t) = 1/4(t + 2B1/2 - 1)2

Here

exists in

[0,1]; however, there is no regular reduced solution satisfying the lefthand boundary condition. [0,1], and since

in

uR(0) > A

On the other hand, if stability at

B1/2 > 1/2

If

0 < B1/2 < 1/2

tR = 1 - 2B1/2

uR

then

in

then we know that

uR

loses

(0,1), where it smoothly crosses the

singular solution, that is, uR(tR) = uR(tR) = 0. 5.27 tells us that for such

is strongly stable

we obtain the limiting relation (8.16).

A and

B

Since

A < 0

Theorem

the solution of the problem satis-

fies

lim

+,

Finally, if since

fy 5 1

Case 4.

0,

6 < t < 1-281/2

U (t),

1-281/2 < t < 1

y (te) =

B1/2 = 1/2

then

fy[uR(t)] < 0

and

for

t

in

[0,1]; however,

Theorem 5.2 guarantees that the solution satisfies (8.16).

A > 0, B < 0.

This case is the reflection of Case 3, in that

the statements made there apply with uL

for

(1-t), respectively.

0 < A1/2 < 1/2, uL

B, uR

and

t

replaced by

A,

We omit the details, except to note that

intersects

us = 0

smoothly at the point

2A1/2.

t

L

=

Case S.

uR

0 <

A1/2,

Bl/2 < 1/2.

lose stability at

For this last case, we note that

tL = 2A 1/2

and

uL

and

tR = 1-2B, 1/2 respectively, and

146

VIII.

us(0) < A

since

and

EXAMPLES AND APPLICATIONS

us(1) < B, there can be no boundary layer behavior

for these boundary values.

We distinguish however two types of interior

crossings : (i)

Al/2 + B1/2 < 1/2.

uL(tL) = uL(tL) = 0

In this case

0 < tL < tR < 1, that is,

and URN) = u'R(tk) = 0, and so Theorem 5.25 implies

that the solution satisfies

lim+ Y(t,E) =

uL(t),

0 < t < 2A1/2

0,

2A1/2 < t < 1-261/2

uR(t),

1-261/2 < t < 1.

e+0

(ii)

Al/2 + B1/2 > 1/2. Here tL > tR and so uL intersects uR B1/2 + 1/2 t0 = Al/2 in (0,1). The solution of the

angularly at

problem thus satisfies (8.17).

Finally, if Al/2 + B1/2 = 1/2 uniformly close to this function in Consider next the

Example 8.14.

cyfi = Y, 2 _ 2ty' + Y, y(-l,e) = A,

(cf.

up

in

u = 2tu' - u'2

c + 0+.

Clearly, uI

the parabola uI = 0

up(t) = t2

is

is the inflection locus

is a reduced solution, while

Another solution which passes through the origin is the uI(t) = 3/4t2, which is interesting in that

f(t,y,y') = y'2 - 2ty' + y), and so [-1,1]

as

-1 < t < 1,

[48; Chapter 3], [38]). is not.

uL = uR, and the solution is

problem

p-discriminant locus and the t-axis

parabola (for

[0,1]

y(l,e) = B.

For the reduced equation the

then

uI

fy,[ul(t)) = t

is locally strongly stable

The general parametric solution of

(cf. Definition 5.4).

the reduced equation is t = 2/3p + cp-2 2cp-1,

u = 2tp - p2 = 1/3p2 +

Suppose first that

0

Case (ii). 3f as p ;

we have

t-

0 0.

Case (i). -- < p < 0. as p by uR.

p # 0,

we have

p - --

As +oo

and

p - 0+ we have t -

As t -

we have t + - and u - +oo; while u - -oo. These solutions are denoted

t*

and

u _ t*2.

+oo

and

u + +oo; while

These curves end on the

8.3.

Examples of Quadratic Problems and Applications

p-discriminant locus and are denoted by Case (iii). while as

3

c < p < -.

p + +

with the curves them by

uR.

As

we have

uR

on

up

p + 3 T + and

t + +a*

147

uR.

we have u + +w.

and lie between

up

t + t*

and

u + t*2;

These curves form cusps and

ui.

We denote

The curves in these last three cases are shown in Figure

8.4.

If

c < 0

then the family of solution curves is obtained from the

curves of Cases (i) - (iii) by reflection; we call the corresponding solutions

uL, uL, uL.

Their graphs are shown in Figure 8.5.

uR

U

Figure 8.4

Solutions of u = 2tu' - u'2

for

c > 0.

VIII.

148

EXAMPLES AND APPLICATIONS

t

Figure 8.S

Solutions of u = 2tu' - u'2

Finally if

c = 0

for

then we obtain the curve

c < 0.

ul(t) = 3/4 t2.

stability of these solutions is determined as follows: fy'[UR(t)] < 0

for --< t < -;

fy'[uR(t)] < 0,

fy,[uR(t)] > 0

for

t* < t < ao;

The

8.3.

Examples of Quadratic Problems and Applications

fy,[uL(t)] > 0

-co< t <

for

fy,[uL(t)] < 0,

fyjuL(t)] 101

149

-- < t < -t*;

for

and finally fy,[ul(t)] = t

Thus the solutions weakly stable and

of the problem as

and

uL

uR

and

are strongly stable, uL

is locally strongly stable, while

u1

uR

and uL

and

We now investigate the behavior of the solution

are unstable.

A > 3/4.

Case 1.

fy[ui] = -2t.

and

by fixing A and varying

a - 0+ If

B > 3/4

then since

u1

are

uR

y = y(t,e)

B.

is locally strongly stable

(I0)-stable, Theorem 5.5 tells us that there are boundary layers at

both endpoints, that is, lim y(t,e) = 3/4 t2 E+O*

Next, for

0 0

in

in

and

u1 (0,1]

there is an angular layer at

(t*,l].

t2

uR

uR

For such

at a point

such that A, B

t2

there is

in

(t*,1).

we conclude from.Theorem 5.27 that and a boundary layer at

t = -1, that

is,

lim y(t,e)

f 3/4 t2,

If now

B = 0

un(t) _

t2 < t < 1.

uR(t) ,

C-01

-l+d < t < t2,

then the reduced path u1(t),

-1 < t < 0,

0
0,

is continuously differentiable, locally weakly stable and

(I0)-stable.

lim+ y(t,e) = u0(t)

Finally, if such that lim

(fy,[u0(t)] < 0),

Theorem 5.5 implies that

B < 0

uR(1) = B

in

[-1+5,1].

then there is a strongly stable reduced solution and

y(t,e) = uR(t)

uR(-1) < A. in

[-1+6,1].

uR

Theorem 5.1 implies that (8.18)

150

VIII.

Case 2.

A = 3/4.

The same results as in Case 1 are valid with the excep-

tion that there is no boundary layer at 0 < A < 3/4.

Case 3.

uL

such that

B > 3/4

intersects

uL

if

at a point

u1

Since

B > 0.

there is a reduced solution

fy,[uL(t)] > 0

there is an angular layer there.

boundary layer at

t = -1

For this range of A

L(-l) = A, with

then

EXAMPLES AND APPLICATIONS

-1 < t < -t*.

for

(-1,-t*), and

in

t1

u1(l) < B

If

there is also a

t = 1, that is, uL(t),

lim+ y(t,e) _ C-0

If

ul(t),

0 < B < 3/4

at a point

uR

then there is an angular crossing between

tI

in

(-1,-t*)

at a point

t2

in

lim+ y(t,e) _

e*0

B = 0, then

uL

lim+ y(t,e) _

-1 < t < tip

ul(t),

ti < t < t2,

B < 0

Finally, for (8.18), if uR

tI

in

C-0 A = 0.

Case 4.

by

uI = 0

-1 < t < tl'

tl
Otherwise, if

uR(-1) > A

then

uL

intersects

-1 < t < tit

luR(t) , ti < t < 1.

The same results as in Case 3 obtain, with

This is the final case.

stable reduced solution

since

(-1,-t*), and so

uL

intersects

uI

replaced uI

t = 0, that is, u1(0) = ul(0) = uI(0) = ul(0) = 0.

A < 0.

lim

in

tI

and with the important exception that

smoothly at Case S.

at a point

(-1,-t*), and so

IuL(t), lim+ y(t,e) _

u0

the solution is described by the limiting relation

uR(-l) < A.

at a point

u1

and

t2 < t < 1.

intersects

u0(t),

and u1

We conclude from Theorem 5.25 that

aL(t),

I uL(t),

uL

and an angular crossing between

(t*,1).

R(t),

If

ti < t <

y(t,c) = uL(t)

uL

such that

For such uL(-1) = A.

in

uL(l) < B, by virtue of Theorem 5.1.

A

there is a strongly If

B > 3/4

then (8.19)

Again, by the same theorem

8.3.

if

Examples of Quadratic Problems and Applications

0 B, then

uL(1) < B, then (8.19) is valid; whereas, if intersects

uL

If

at a point

uR

ul = 0

intersects

(8.20)

uL(1) < 0; whereas, for

at a point

t2

(t*,l), and so

in

(8.20) is valid with

uR

tion (8.19) holds if

uL(1) < B, while (8.18) holds if

Otherwise, if a point

t0

uL(l) > B in

(t*,l), and so

IuR(t) , t2 < t < 1.

B = 0, then again (8.19) is valid, if

u L(l) > 0, uL

in

t2

-1 < t < t2,

uL(t),

lim+ y(t,£) _

151

replaced by

and

0.

Finally, for

uR(-1) > A, then

B < 0

the rela-

uR(-1) < A.

intersects

uR

UL (-1,1), and so by Haber and Levinson's theorem

at

-1 < t < to,

f uL(t), lim+ y(t,e) = uR(t),

e - ro

Example 8.15.

t0 < t < 1.

Consider now the Robin problem

ey"=-Y'2+Y,

0
Y(O,e) = Y'(O,e) = A,

for various values of A

and

B.

It follows directly from the "Robin"

version of Lemma 2.1 that this problem has for each

solution y = y(t,e)

for all

e > 0

A, B

which satisfies in

a unique [0,1]

min{A,B,O} < y(t,e) < max{A,B,O}.

The differential equation is the one discussed in Example 8.13, and so we know that the reduced equation has the solutions their envelope

u(t) = 1/4(t+c) 2,

us = 0, and paths formed from finite combinations of

these curves, for example,

u(t)

0
(0, 1/4(t-c)2,

c < t < 1.

Clearly the reduced problems u = u'2,

u(0) - u'(0) = A,

(RL)

and

u = u'2,

u(1) = B,

each have two solutions:

(RR)

uL(t) = 1/4(t + 1 -

1/2 1/4(t + 1 + (1+4A))2 (if

A > -1/4) and

uL(t)

=

uR(t) = 1/4(t + 2B 1/2 - 1)2,

152

EXAMPLES AND APPLICATIONS

VIII.

u (t) - 1/4(t - 2B1/2 - 1)2

is strongly stable in

uL

and

uR

is strongly stable in

and

uR

are also

Of these, u

B > 0).

(if

R

stable, while

and

if

(1-2B1/2,l]

B > 0.

are un-

u

ifR A > 0

[0,(1+4A)1/2L- 1)

uL

Note that

(I0)-stable.

Let us first assume that

A > 3/4.

uL(l) > B, that is, if

[0,1], and so if

Then

is strongly stable in

uL

B < 0

or

B1/2 < 1/2(1+4A) 1/2 - 1,

satisfies

then the solution

y(t,e) = uL(t)

in

[0,1-6],

li0+

by virtue of Theorem 5.7.

strongly stable in

Next, if

B1/2 > 1/2

then

uR

is also

We consider for what values of A the solution

[0,1].

satisfies 1li+ y(t,e) = uR(t)

in

(8.21)

[0,1].

The relation (t) of Theorem 5.9 clearly holds if

uR(0) - uR(0) > A.

that is, if B1/2 < 1

-

or if

1/2(1+4A)1/2

and so for such values of

B1/2 > 1 +

A, (8.21) follows.

1/2(1+4A)1/2,

Suppose, however, that

uR(0) - uR(0) < A, that is, 1 - 1/2(1+4A)1/2 < B1/2 < 1 + 1/2(1+4A)1/2.

(8.22)

Then it is still possible to satisfy (t) if, in addition, A

is such that

-(uR(0) - A)2 + u R(O) > 0,

that is, if '12 1/2(1+4A) 1/2 B>

1/2 -1/2(1+4A) or if B<

1/2

Clearly the latter inequality is incompatible with (8.22), and so we see that (t) is satisfied if 1/2(1+4A)1/2 < B1/2 < 1

+

1/2(1+4A)1/2.

Thus, for this range of A, the relation (8.21) holds. We consider next an application of the "(RPS)" version of Theorem

Clearly uL

5.30. 2B1/2)

in

(0,1)

uL(1) < B,

intersects

uR

at the point

t0 = 1/2((1+4A)

1/2

if and only if uR(O) - uj(0) < A

and

-(uR(0)-A)2 + u R(O) < 0,

-

8.3.

Examples of Quadratic Problems and Applications

153

that is,

1/2(1+4A)1/2 - 1 < B <

1/2(1+4A)1/2.

For such values the solution satisfies uL(t),

0 < t < t0,

uR(t) ,

t0 < t < 1.

lim+ y(t,E) _

e*0

Our discussion has yet to involve the singular solution Since

is weakly stable and

us

A,B < 0

to see that

lim+ y(t,e) = 0 E+0

in

[0,1-6].

We also illustrate a smooth crossing. and so

intersects

uL

uL(tL) = uL(tL) = 0.

B = 0

0 < t <

uL(t),

smoothly at

us

lim+ y(t,E) = E*0

tL = (1+4A) 1/2 - 1, that is,

L,

0 < B < 1/4, then

tR = 1-281/2, and so for

0,

0 < t < tR,

uR (t),

t
0 < A < 3/4

1 < (1+4A)1/2 < 2,

the solution satisfies

Similarly, if

by the analog of Theorem 5.31. intersects

then

tL
,

0

0 < A < 3/4

If

smoothly at

u s

Thus for

lim+ y(t,E) c+O

Next, if

us - 0.

(I0)-stable, we apply Theorem 5.5 if

uR

A < 0

R

0 < B < 1/4, then there are three possibili-

and

ties: (i)

uL

intersects

(ii)

uL

and

(iii)

uL

intersects

uR

us, and then

intersect

us

us

intersects

uR;

at the same point, that is, uL = uR

uR.

Clearly (i), (ii) and (iii) obtain if and only if

tL < tR, tL = tR

and

tL > tR, respectively. 0 < A < 3/4

Finally, for us

smoothly at

tL

and

and

us > B.

B < 0

we know that

uL

intersects

Consequently the analog of Theorem

5.31 tells us that

uL(t),

0 < t < tL,

0,

tL
154

EXAMPLES AND APPLICATIONS

VIII.

We conclude this section with an application of the

Application 8.5.

theory to the model problem (cf. [95] and Application 8.2) Ey" = e0(l+ey)-ly'2 + Yr(1+ey),

0 < t < 1,

Y(l,E) + EY'(l,E) = 1,

-Y'(0,E) = 0,

which describes a chemical reaction accompanied by a change in volume. represents the dimensionless concentration of a gas

More precisely, y

A

undergoing an isothermal reaction on a flat plate catalytic surface, namely

A-+Bn

The variable

.

t

is the normalized distance from a plane

of symmetry to the edge of the plate (t = 1), and the positive constant As before, the parameter

n

is the stoichiometric factor of the reaction.

e

is the reciprocal of the square of the Thiele modulus, and the para(Note that

is the volume change modulus.

0 = 0

meter

8 z n-1

n = 1

in the absence of a volume change; cf. Application 8.2.)

plicity, the reaction is assumed to be of integral order

r > 1

or For simand

E

is again the reciprocal of the Sherwood number.

If n > 1, then to the reaction.

0 > 0, and there is an increase in the volume due

The coefficient of y'2

in the differential equation

is positive, and we can apply the quadratic theory if small.

Since

y

E

is sufficiently

is a concentration we are only interested in positive us = 0

solutions, and we see immediately that Since

solution of the reduced equation.

is the only nonnegative

is stable in the sense of

us

Definition 5.3 it follows from Theorem 5.19 that for positive values of E

the problem has a solution

satisfying in

y = y(t,e)

0 < Y(t,E) < E-lEl/2exp[-(l-t)/E1/2], if

[0,1]

r = 1,

and (l+p(1-t)/E1/(2r+2))-1/r,

E-lrp-lEl/(2r+2)

if

0 < Y(t,E) <

where

p = p(r)

r > 2,

is a known positive constant.

In conclusion, we

note that if

E = 0

(that is, if there is no

resistance to the transfer of reactant from the bulk flow to the surface at

t = 1), then Theorem 5.13 provides us with the estimates 0 < y(t,E) < exp[-(l-t)/E1/2],

if

r = 1,

and

0 < Y(t,E) <

where

pI = pI(r)

(l+pl(1-t)/E1/2)-1/r,

if

r > 2,

is again a known positive constant.

8.4.

8.4.

Examples of Superquadratic Problems and An Application

155

Examples of Superquadratic Problems and An Application

Example 8.16.

Consider the differential equation

£y" = (y-t2)2q+1(1+y'2)3/2 = F(t,y,y'), for

0 < t 0. F, and

is the only real zero of of Definition 6.1.

Since

u

The function

u(t) = t2

is clearly (Iq)-stable in the sense

u(O) - 0, u(l) = 1

and

u" __ 2, the Dirichlet

problem (cf. Example 8.1). cy" = F(t,y,y'), y(O,e) = 0,

0 < t < 1,

y(l,c) = 1,

has by virtue of Theorem 6.1 a solution fying in

y = y(t,e)

as

s

+

0+

satis-

[0,1]

t2 < y(t,£) < t2 + (2£)1/(2q+l).

u'(0) = 0

On the other hand, since

Theorem 6.3 tells us that the Robin

problem (cf. Example 8.5) ey" = F(t,y,y'), -y'(O,e) = A,

has a solution as

0 < t < 1,

y(l,e) = 1, a - 0+

satisfying in

[0,1] (2£)1/(2q+l)

t2 < y(t,e) < t2 + v(t,e) +

for any value of A > 0.

Here

v(t,e) = 0

if A = 0, and if A > 0,

then

v(t,e) = A£1/2 exp[-t/e1/2]

(q = 0)

and v(t,e) = AT-1ge1/(2q±2)(1 + Tt/£1/(2q±2))-i/q

(q > 1),

for

T = T(q) =

[A2gg2q+2/(q+1)]1/(29+2)

Application 8.6.

The superquadratic theory is often useful in solving

problems involving the form

the curvature of surfaces, since nonlinearities of (1 + y'2)3/2 - 'y'I3, as ly'l - -, arise naturally. As an

illustration, we note (cf. [3; Chapter 1] or [78]) that the elevation of the free surface of a liquid meeting a plane, vertical rigid wall (at

t = 0) is described by the problem

y

0 < t < L,

ey" = Y(i + y'2)3/2

let < w/2, y(L,e) = 0(e),

y'(0,e) = tan e, for p

EXAMPLES AND APPLICATIONS

VIII.

156

and

e2 = T/(pg)

Here

an arbitrarily large positive constant.

L

is the density of the liquid, g

is the gravitational constant, T

the coefficient of surface tension, and

is

is the contact angle, the

e

angle the surface makes with the wall, measured from the horizontal axis, y = 0.

If

is small then

T

is small, and the problem for the eleva-

c

tion is singularly perturbed.

Suppose first that the liquid is water, and therefore, that Since

0 < e < 1/2.

u E 0

is an (10)-stable reduced solution and

-u'(0) = 0 < tan 0, Theorem 6.3 tells us that the elevation y in

satisfies

[0,L)

0 < y(t,e) < (tan e)e1/2exp[-t/el/2] + 0(e). However, if the liquid is mercury, then the contact angle is negative (-1/2 < e < 0),

pression"

y

we now conclude that the "de-

-u'(0) = 0 > tan a

Since

satisfies in

[0,L]

0(e) + (tan e)el/2exp[-t/el/2] < y(t,e) < 0. Example 8.17.

Consider now the problem

eyV1 = -Y' -

y'3, 0 < t < 1,

pY(O,e) - y'(0,e) = A,

p > 0.

Y(l,e) = B,

(We saw earlier that the Dirichlet problem

f(u') = -u' - u'3

fu(0) = -1

we make the corresponding reduced solution satisfy

that is, u = uR(t) - B. y(t,e) E B

The reduced equation

as its only real solution, and since

u' = 0

has

ey" = -y' - y'3, y(0,e) = A,

if A # B.)

c - 0

y(l,e) = B, has no solution as

Suppose first that

is the solution.

for all values of

A

between

0

and

If A = 0

p = 0.

However, if A # 0

then

-A, A 0 0.

u(1) = B, then

Af(x) = -AA(1+a2) > 0

Consequently Theorem

all values of A the solution satisfies in

6.11 tells us that for

[0,1]

y(t,e) = B +

Suppose finally that

p > 0.

tion, while if A # pB

If A = pB

then

y(t,e) E B

is the solu-

then

(p5 - A)f(A) = -A(PB - A)(1 + A2) < 0,

for all values of

A

between

0

and

6.11 we see that for all values of A

pB - A, X j 0. and

B

Again from Theorem

8.4.

Examples of Superquadratic Problems and An Application

Y(t,e) = B + 0(elpB-Alexp[-t/E]) Example 8.18.

0 < t < 1,

Y(O,E) - y'(O,e) = A,

The reduced equation and

we make

u1

Y(l,e) + Y'(l,e) = B.

f(u') = u' - u'3 = 0

u3 = 0, which satisfy and

u1(t) = t + B -2

u2

u3(1) +

satisfy

and

However, if A < B - 3

then

y(t,c) = t + B - 2

and

1

u3

is the solution.

(B - 3 - A)A(1-A2) < 0

B - 2 - A, A # 1.

implies that the solution satisfies in

satisfy

If A = B - 3

first.

ul

(ul(0) - 1 - A)f(A)

between

A

Consider

Theorem 6.11

[0,1]

y(t,s) = u1(t) + 0(Z e(B-3-A)exp[-2t/E]).

Finally, if A > B - 3 and

B - 2 - A, A

then

6.11 it follows that for

(8.23)

(B - 3 - A)A(1 - A2) > 0

1, provided that

Thus

j = 1,2, that is,

for

B

ui = 1,

fu(0) = 1.

and

u2(t) = -t + B + 2, and we make

u1(0) - u1(0) = A, and so

for all values of

now has three solutions

fu,(±1) = -2

u3(0) - u3(0) = A, that is, u3 = A. then

[0,1].

Consider next the related problem

Ey" = Y' - y'3,

u2 = -1

in

157

B - 2 - A > 0.

for

A

between

1

Again from Theorem

the solution satisfies in

B - 3 < A < B - 2

[0,1]

y(t,c) = ul(t) + 0((c/k)IB-3-AIexp[-kt/e]), for a positive constant

k < 2.

The asymptotic behavior described by of that described by

is clearly a reflection

u2

B + 3 < A

In particular, if

ul.

then the relation (8.23) [(8.24)] obtains with the term

IB - 3 - Al

replaced by

IB + 3 - Al

Consider next the reduced solution y(t,e) E A

B - A < 1.

B - A > -1.

u3

is the solution, while if A B Consequently for

that the solution satisfies in

0

and

k < 1.

u1

[B + 2 < A < B + 3]

replaced by

u2

and

inside, the Landau symbols.

A.

then

If

A = B

then

(A-B)f(A) =

B - A, A # 0, provided

then this inequality obtains provided

B - 1 < A < B + 1 [0,1]

y(t,e) = A + 0((e/k)IB - Alexp[-kt/e]), for a positive constant

(8.24)

Theorem 6.11 tells us

EXAMPLES AND APPLICATIONS

VIII.

158

Summarizing to this point, we have estimates of the solution for all

values of A and

B - 2 < A < B -

B, except those satisfying

B + 1 < A < B + 2, for which Theorem 6.11 is inapplicable.

1

and

Thus we are

led to consider the angular paths

fA,

0 < t < t0,

u5(t) t+B-2,

(A,

-

u4(t) =

0 < t < t0,

llt

t0 < t < 1,

-t+B+2,

t0 < t < 1.

It follows directly that

t0 = A - B + 2

is in

(0,1)

if and only if

B - 2 < A 0

implies that for

B - 2 < A < B -

u4.

For

for

0 < A < 1, and so Theorem 6.12

aL = 0

oR = 1

and

we see

1

Y(t,s) = u4(t) + 0(Z(c/k)exp[-kIt - t0I/e]) in

[0,1], for a positive constant

B + 1 < A < B + 2

with

u4, t0

Finally, if A = B - 2

k < 1.

[A = B + 2]

[-t + B + 2], while if A = B - 1

A similar result holds for

replaced by

or

u5, t0.

then

lim+ y(t,e) = t + B - 2

A = B + 1, then

lim+ y(t,e) = 0.

None of these limits is a surprise; the convergence is, of course, uniform in

[0,1].

Example 8.19.

cyli = y

The solutions of the nonautonomous problem

- tY' - Y'3 a f(t,Y,Y'), -1
-y'(-l,e) = A.

y(l,c) = B,

exhibit the types of boundary and interior layer behavior encountered in

the two previous examples and more, for various choices of A Here the reduced equation

u = tu' + u'3

solutions are the one-parameter family of straight lines and their envelope, the singular solution fined for

B.

u(t) = ct + c3

u5(t) = ±2(-t)3/1/(3f), de-

t < 0.

Suppose first that

B = 2.

Then

uR(t) = t + 1

the reduced problem (RR) which is stable since in

and

is a Clairaut equation whose

[-1,1].

is a solution of

fyl[uR(t)] = -t - 3 < -2

In order to apply Theorem 6.11 we must determine for what

values of A (-uR(-1) - A)f(-1,uR(-1),A) = (-l-A)A(-l-A2) < 0

(8.25)

8.4.

Examples of Superquadratic Problems and An Application

for

A

between

uR(-1) = 1

and

-A, A

If A > -1

1.

then (8.25) A < -1.

obtains provided A < 0, while it clearly obtains for all

ally, if A = -1 that for all

then

y(t,c) = uR(t)

is the solution.

the solution satisfies in

A < 0

159

Fin-

We conclude

[-1,1]

lim+ y(t,c) = t + 1. e+0

Suppose next that

A = 0

which intersect at in

since

fy,[uL(t)] = -t > 4

uL 5 0

Then

is a solution

uR(t) = 2 t + s is a solution of (RR) The corresponding angular path

t0 = -1/4.

uL(t)

B = 5/8.

and

of the reduced problem (RL) and

[-1,-1/4], ul(t) = uR(t) in

< - 2

fy,[uR(t)] = -t -

and

[-1,-

u1(t) _

is globally stable

[-1/4,1]

in

4 in

To apply Theorem 6.12 we note that

[- 4, 1].

(aR - aL)f(- 4, O,A) > 0 for

A

between

0

and 2.

Consequently the solution satisfies in

[-1,1]

lim+ Y(t,e) = ul(t). C-0

We consider finally two applications of the "singular" analogs of Set

Theorems 6.11 and 6.12.

The reduced problem (RR) has

B = 10/27.

uR(t) = t/3 + 1/27, and it intersects the lower branch

the solution

us (= -2(-t)3/2/(3))

of the singular solution at the point

Since

us(t2) = uR(t2), and therefore the reduced path

us

is singular

u3(t) = us(t)

in

differentiable.

[-l,-1/3], u3(t) = uR(t)

in

[-1/3,1]

t2 = -1/3.

is continuously

It remains for us to determine the values of

A

for

which (-us(-l) - A)f(-l,us(-1),)L) < 0 for

A

between 1// and

-A, A # l/ v.

this inequality obtains for all

A < -1/yr3-.

A short calculation shows that We conclude that the solu-

tion satisfies limm+ Y(t,c) = u3(t) C-O

(If A = -1/v, then B = 26/27.

Then

y(t,e) = u3(t).)

uR(t) = 2t/3 + 8/27

Finally, let

A = 1/3

and

intersects the upper branch

us (= 2(-t)3/2/(3 /))

of the singular solution at

for this choice of A

and

t2 = -1/3.

However,

B, oL = us(-1/3) = -1/3 < 2/3 = uR(-1/3) = o'R,

in contrast to the previous case.

A short calculation shows that

VIII.

160

EXAMPLES AND APPLICATIONS

(OR - GL)f(-1/3,2/27,x) > 0 for

-1/3 < A < 2/3, and so we conclude that

I

lim+ y(t,e) _

us(t), -1/3 < t < 1.

uR(t),

Example 8.20.

We close this section with the problem -1 < t < 1,

ey" = y + ty' + yn = f(t,y,y'), y(-l,e) - y'(-l,c) = A, for

n m> 3

y(l,e) + y'(l,e) = B,

The function

an integer.

u = 0

is clearly (10)-stable in

the sense of Definition 6.1, and it is also locally strongly since

fyJ0] = t.

Suppose first that

n

is odd.

y'-stable

In order to proceed

we consider the two inequalities (u(-l) - u'(-1) - A)f(-1, u(-l),A) < 0, for

A

between

and

0

-A, A # 0, and

(u(1) + u'(1) - B)f(l,u(1),X) < 0, for for

A

between

and

0

B, A

At < 1, A # 0, since

values of

B # 0.

n

0.

The first inequality clearly obtains

is odd, while the second obtains for all

If A = B = 0

then

y(t,e) __ 0, and so we

by arguing as in the proof of Theorem 6.11 that if n then for all values of lim+ y(t,E) = 0 c+0

B

is odd and

JAI < 1,

the solution satisfies

in [-1,1].

On the other hand, if n

deduce

(8.26)

is even then these inequalities obtain for all

values of A > -1, A # 0, and

B > -1, B # 0, respectively.

have the limiting relation (8.26).

We again

Examples of Semilinear Systems and An Application

B.S.

161

Part II - VECTOR PROBLEMS

Examples of Semilinear Systems and An Application

58.5.

Example 8.21.

Let us illustrate the norm-bound theory of Chapter VII by

first considering the two-dimensional system in

(0,1)

EY1 = y1-y2-y1 = h1(y1,Y2),Y1(O,E) = A1,Y1(l,E) = B1,

EY'2 = Y2+yl-y2 = h2(y1,Y2),Y2(O,e) = A2,Y2(l,c) = B2.

The corresponding reduced system

h = (h1 h2)T = (0 0) T has the solution

u = 0, and it is clearly stable, in the sense of Definition 7.1, since the 1-3yi - 1

matrix

is positive definite, for

J(0,0)

J(y)

1 - 3y2

1

, the

2

Jacobian matrix. y2 (1-3y 2)

Finally, the quadratic form yTJ(y)y = yi(1-3yi) +

is positive definite, only for vectors

lyll < l/, and A2

and

B1, B2

11jg.

ly21 <

satisfying

A1,

(i = 1,2) Theorem

IBij < l1 r3

y = y(t,E)

as

e -

0+

[0,1] IIY(t,E)II_

for

satisfying

Y2

Consequently, for boundary values

IAit < 1/vr3-,

7.1 tells us that the problem has a solution satisfying in

)T

(y1

IIAllexp[-mt/e] + IIBII exp[-m(1-t)/,r],

m2 = min{1-3Ai, 1-3B2 }, i = 1,2. i

The restriction imposed on A

and

is rather severe; we can try

B

to relax it slightly by replacing the strong positive definiteness assumption with the weaker integral condition alluded to in Chapter VII.

A

short calculation shows that

YT h(Y)/I IYI I_ (I IYI I2 - I IYI I4)/I IYI I

=

11Y11-1,

I IYI I -

where we have used the simple inequality

y4 + y4 < (y2 + y2)2.

applying the reasoning in Example 8.3 to

Ilyll, we conclude that, in

fact, the solution

y = y(t,e)

1

2

1

2

Thus, by

found earlier actually exists and satis-

fies

lim+ y(t,e) = (0 0)T in [d, 1- &] for boundary values such that e+0 IIAII, IIBII < /. These bounds are sharper than the bounds IIAII,

IIBII < T obtained from the more restrictive definiteness condition. Example 8.22.

Consider now the problem in

(0,1)

VIII.

162

eyl =

EXAMPLES AND APPLICATIONS

= hl(Y1,Y2),Y1(O,e) = Al, Y1(1,e) = B1,

Y1(1-Yl)(1+Y2)

l) = h2(Yl,y2),Y2(O,e) = A2, Y2(1,E) = B2, ey2 = Y2(1-Y2)(i+Y2 in order to illustrate the componentwise theory.

The function

u = 0

is

clearly a solution of the reduced system which satisfies, in addition, hl(0,y2) = 0

for all

h2(y1,0) = 0

and

y2

for all

y1.

Moreover, it

is stable in the sense of Definition 7.2 since = 1 + y2 > 1

hi,Yl(O,y2)

and

h2,Y2(Y1,0) = 1 + yl > 1.

Finally we observe that

hl,yl(Y1.

2) = (i 2y1)(1+y2) > 0

and h2,y2(Yl,Y2)

_ (1-2Y2)(i+yi) > 0,

Therefore, for boundary values such that

for all values of yl,y2 < 1/2. Ai < 1/2

and

(i = 1,2), Theorem 7.2 states that the problem

B. < 1/2

has a solution

y = y(t,e)

lim+ y(t,e) _ (0 0)T

as

satisfying

a - 0+

(8.27)

in

We note that this bound on the boundary values can be improved by using the less restrictive integral conditions that 711

O

h1(s,A2 or B2)ds > 0

for all values of cf.

[69].

ni

012 h2(A1 or B1,s)ds > 0,

and

between

0

and

A.

For boundary values such that

or

Bi, ni # 0 (i = 1,2);

Ai,Bi < 3/2, i = 1,2, we de-

duce therefore the existence of a solution satisfying the limiting relation (8.27).

Application 8.7.

The scalar theory of Application 8.2 concerned itself,

of course, with the behavior of a single reactant undergoing mal catalytic reaction.

an isother-

Suppose however that we have a system of N

reactants, each component of which undergoes such a reaction, influenced by and influencing the other

N-i

components.

Then by arguing as in

Application 8.2 (cf. also [2; Chapter 5]), we see that the steady-state behavior of the concentrations can be governed by a boundary value problem of the form

8.5.

Examples of Semilinear Systems and An Application

0 < x < 1,

ey" = h(y),

PY(O,e) - Y'(O,e) = A,

T

Here an

Qy(l,e) + Y'(l,e) = B,

is the vector of normalized concentrations, h

Y = (yl " 'YN)

N-vector-valued function of y

tics, and

N x N

is

which represents the nonlinear kine-

is the normalized distance.

x

163

P, Q

are positive semidefinite

matrices which contain the various transfer coefficients between

the bulk flow and the solid phase, and

e2

is the reciprocal of the Thiele

modulus, assumed to be the same for each reaction. Suppose now that evaluated along

h(0) = 0

and that the Jacobian matrix

J

of h

is positive definite, in the sense that there is a

0

positive constant mI

for which

(8.28)

yT Jy > ml 2 I IYI I2 , y

for all

in 1RN.

YT h(Y)/I IYI I _ for

It follows from the Mean Value Theorem that m2

i IYI I .

sufficiently small.

IIYII

the problem has a solution

I IY(x,c) I for

Consequently Theorem 7.3 tells us that

y = y(x,e)

as

+ a - 0*

satisfying in

[0,1]

(,T/m) I JAI Iexp[-mx/,T] + (Fe/m) I IBI I exp[-m(l-x)/,r]

I

0 < m < ml.

As an illustration, consider the problem in

(0,1)

eYl = Yl(1-Y2) - (k-X) Y2 E hl(Y1,Y2), PY1(O,E) - Y11(0,E) = P,

Py1(1,e) + Y11(l,e) = P,

ey2 = -Yl(1-Y2) + kY2 ° h2(Y1,Y2),

where

k, A

and

p

-YZ(O,E) = 0,

are positive constants with

k > A.

YZ(l,E) = 0,

It is taken

from Aris's discussion [2; Chapter 5] of the pseudo-steady-state hypothesis in an enzyme reaction.

reduced equation matrix

J

of h

Clearly

u = 0

is the only solution of the

h = (h1 h2)T = (0 0)T, and the corresponding Jacobian evaluated along

0

is

-(k-a) -1 1

Since

k

k

1

is positive, a necessary and sufficient condition for

positive definite is that

J

to be

164

EXAMPLES AND APPLICATIONS

VIII.

(k-A+1)2 < 4k,

in which case inequality (8.28) obtains with m1 = {k+l - [(k-1) 2 + (1_A+1)2]1/2}/2. (Here we have used the results [26; Chapter 8] that a real symmetric

2 x 2

if

a11a22

a11 > 0

and

y = y(x,e)

I IY(x,c)I I

for

(a ..) i7

- a12 >

is positive definite if and only

0, and that

yT Ay = yT A*y

Thus, for such values of k

A* _ (A + AT)/2.) a solution

matrix

e + 0+

as

and

satisfying in

A

for

the problem has

[0,1]

<_

0 < m < ml.

Consider finally the symmetric problem in

Example 8.23.

(0,1)

eY1 = Y1-Y1/2-YlY2 S h1(Y1,Y2),Y1(O,e) = A1, Y1(l,e) = B11 eY'2 = Y2-YZ/2-YlY2 = h2(Y1,Y2),Y2(O,e) = A2,Y2(l,e) = B2,

such systems arise in the study of travelling wave solutions to nonlinear partial differential equations governing the dynamics of stratified fluids [4], [82]).

(cf.

h2(y1,0) all

_- 0

The reduced solution y1,y2, and

for all

yl,y2 < 1.

u = 0

satisfies

hl(0,y2) = 0,

hl,yl(0,y2) > 0, h2,y2(y1,0) > 0

for

Invoking the integral conditions for boundary layer be-

havior, we see that

f 1 hl(s,A2)ds > 0

for

< 3(1-A2),

A 1

0

while f 2 h2(Al,s)ds > 0

for

A2 < 3(1-Al),

0

and similarly for

B1,B2.

If A1,A2,B1,B2 < 3/4

layer inequalities obtain, and so (cf. solution as

e + 0+

then all of the boundary

[68], [69]) the problem has a

satisfying

lim+ y(t,E) = (0 0)T e+0

in

Recalling Example 8.3 we suspect that the system also has solutions with spikes. yl(y2)

This follows because in a subinterval of

is asymptotically zero, the

the uncoupled equation Example 8.3.

(0,1)

where

yl-equation (y2-equation) reduces to

ey" = y-y2/2, which has solutions like those of

More precisely, O'Malley's result [76] states that for each

Examples o£ Quasilinear Systems and An Application

8.6.

integer n > 2

165

the scalar problem 0 < t < 1,

ey" = y - y2/2,

(*)

y(O,e) = A,

A,B < 3/4,

y(l,e) = B,

has four solutions

y = y(t,e)

lim+ y(t,e) = 0

in

a - 0+

as

satisfying

[6,l-8],

e+0 except that lim+ y(tt,e) = 3

tR = R/n,

for

t = 1,...,n-1.

C-O

It follows from the theory of Chapter VII, for example by taking solutions of (*) for

n = 2

and n = 3, that the original vector problem has the

eight solutions pictured below, together with the eight solutions obtained from these by interchanging n = 2,3,...,p, p

for as

a . 0+

and

yl

y2.

Using the solutions of (*)

a prime, we can show the existence of solutions

having spikes of height, 3

at the

t-values

1/2, 1/3, 2/3,

1/5, 2/5, 3/5, 4/5,...,l/p, 2/p,...,(p-1)/p, provided that

Al, B1,

A2, B2 < 3/4.

Examples of Quasilinear Systems and An Application

§8.6.

Example 8.24.

We turn first to an illustration of a norm-bound result

for a problem with a solution exhibiting boundary layer behavior, namely CX" = F(X)X' + &(Y), y(O,e) = A.

Here

y = (y1 Y2)

0 < t < 1,

y(l,e) = B. T,

j

F =

11

J,

g = (y1+1

y2+1)T, and so this

y2 I'

problem can be regarded as a generalization of the Lagerstrom-Cole model problem (cf. Example 8.10) to two dimensions.

Let us examine under what

conditions there is a solution with a boundary layer at by assuming that the reduced problem has a stable solution in negative definite. at

2-vectors

We begin

F(uR)uR' + g(uR) = 0, uR(l) = B.

[0,1], that is, the matrix

F(uR(t))

must be

We also require that there is boundary layer stability

t = 0, in that the matrix

for all

t = 0.

C

F(uR(O) + E)

satisfying

must be negative definite

0 < II§II < IIA-uR(0)II.

166

VIII.

Yl

3

EXAMPLES AND APPLICATIONS

Y2

3

n

A

A B1

B2

Al

A2 t

t

1 2

1

Y2

Y1

3

3

B1

A

A2

1

t

01

1/2

t

1

Y1

Y2

3

3

n

n

A

n

n B1

B2

A2 t

1/2

t 0

Y1

A

A

n

B1

0

B2

0

1/3

Figure 8.6

Eight Solutions of Example 8.23

2/3 1

8.6.

Examples of Quasilinear Systems and An Application

The reduced problem has the solution (C D) = (B1-1 B2-1), and we see that t+C

1

1

t+D

167

uR(t) _ (t+C t+D)T, for is stable in

uR

provided

[0,1]

l

1 > 0, that is, provided

Il

CD > 1, or

and

Finally, boundary layer stability requires that

B1 B2 > B1 + B2 > 2.

El+C

C+D > 0

1

> 0, that is,

2+D

1

El + Chi + 2E 1E2 + E2 + DE2 > 0, for all

(Ei

(A1-B1+1)2 + (A2-B2+1)2.

IIA-uR(0)II

y(0,e) = A

Setting

polynomial.

0 < II

II <

(C,D) = uR(0)

whose radius

of the nontrivial zeros of the cubic

II§II

such a

E2 = tEl

satisfying

Thus the initial values

are restricted to a disk about

is less than the least norm

2)T

E

will satisfy

(1+t3)E1 =

d(t) = IIkHI = 1 + t2 I. The

-(C + 2t + Dt2), and we minimize

solution of this calculus problem then determines an upper bound for

IIA - WO) I for

I

C = D = 2, that is, B = (3 3)T, we obtain the minimum value

For

d(t), corresponding to

if A

lies in the disk of radius

has a solution

y = y(t,e)

in

limm+ y(t,e) = uR(t)

about

2

c - 0+

as

2

Thus Theorem 7.4 tells us that

tmin = 0.

(2 2)T, then the problem

satisfying (8.29)

[6,l].

This is a rather severe restriction on the size of the boundary layer Jump

It can be improved by replacing the definiteness

IIA - uR(0)II.

condition on~ F(uR(0) +

)

with the less demanding integral condition

that

T

F(uR(0) + )dE < 0,

I

0

for all

2-vectors

C

such that

0 < IICII < IIA - uR(0)II; cf. [47].

In our case this integral condition is equivalent to the requirement that

1 + 2CE1 + 4E1&2 + E2 + for all such

&.

Proceeding as before, we can show that for

the minimum value of

Thus if A

0,

d(t)

is 3.39, corresponding to

lies in the disk of radius 3.39 about

(

tmin =

C = D = 2 -0.291.

2 2)T, then the

EXAMPLES AND APPLICATIONS

VIII.

168

limiting relation (8.29) obtains again.

Even though this is an improve-

ment over the previous result, the estimate on near optimal. + uR(0)

for

Example 8.25.

IIA - h(O)II

is nowhere

We expect that boundary layer stability need only hold

on the actual solution path joining A

and

uR(0).

We consider now an example which illustrates the component-

wise boundary layer results, as well as the differences between this theory and the norm-bound theory, namely ey" = (1-Y1)Yi + ylgl(t,Y2),Yl(O,E) =

B1,

EYZ = (1-Y2)YZ + Y2g2(t,Yl),Y2(0,c) = 0,Y2(l,C) = B2, for

t

in

The lefthand reduced problem clearly has the solution

(0,1).

u1 = u2

0, which is stable because

i = 1,2.

Since

fi(yi) > 0

whenever

7.5 that the problem has a solution B1,B2 < 1, satisfying in

f1(0) = f2(0) > 0

for

fi(yi)

= 1-yi,

yi < 1, we conclude from Theorem

y = y(t,e)

c - 0+, for all

as

[0,1]

yi(t,c) = Bi exp[-k(1-t)/c], where

0 < k < min{l-B1, 1-B2).

The restriction on

can be relaxed by using appropriate integral

B

conditions, as was done in the scalar theory.

We require that

El (u1.(1)

- B1.)

fu for

Bi < !; < ui(l)

(l)

if

f .(Bsi)ds < 0,

(8.30)

1

Bi < ui(1)

or for

ui(1) <

< Bi

if

ui(1) < Bi.

For both components we see that

fi(Bsi)ds = J ui(1)

whenever

(1-s)ds =

-

2/2 > 0

0

0 < C < 2, and therefore (8.30) obtains for

_- < B. < 2.

A

theorem of O'Donnell [68] tells us that the solution satisfies

1+ y(t,c) = 0 for all

in

[0,1-8],

B1,B2 < 2.

Let us now compare this result with that obtained from the norm-bound theory.

This theory requires that

and that the inner product

F(O) > 0, for

F(y) = diag(l-y1, l-y2),

8.6.

Examples of Quasilinear Systems and An Application

T y

169

F(s)ds > 0,

-

10

y on paths between 0 and B satisfying 0 < IIyII < IIBII The stability condition is certainly satisfied since F(0) is the identity for all

matrix, while the boundary layer stability condition is

yT

diag{l-s1, 1-s2}(ds1 ds2)T > 0, I

0

that is, y2(1-yl/2) + y2(1-Y2/2) > 0, for all Thus, we must require that

IIBII.

y2(1-y2/2)

is positive whenever

Application 8.8.

y = (y1 y2)T, 0 < IIyII <

IIBII < 2, even though

yi(1-yl/2) +

yi,y2 < 2.

As an application of the quasilinear theory consider the A reactant fluid flows through

following system analog of Application 8.1.

a tubular reactor at a constant average speed

U, and we assume that

there is axial dispersion caused by turbulent mixing.

react isothermally, then the

N-vector

y

If N

species

of steady-state concentrations

satisfies a system of the form (cf. [80; Chapter 4])

ey"=Uy' +g(y), 0<x
Here

c

y(L,c) = yL.

is the reciprocal of the Peclet number, g

function which contains the kinetics of the reaction, L of the tubular reactor, and

yD

and

yL

N-vector-valued

is an

is the length

are the prescribed values of

the reactant concentrations at the inlet and outlet of the pectively.

Since

U

tube, res-

is positive we look for a solution u = u(x)

of

the lefthand reduced problem Uu' + g(u) = 0, Let us assume that

u(O) = y0.

u

exists in

[0,L]; then

is clearly stable, in

u

the sense of both Definition 7.4 and Definition 7.5. ary layer stability at

x = L

in both senses.

tells us that the original problem has a solution satisfying in

We also have bound-

Thus the norm-bound theory y = y(x,e)

as

[0,L]

IIy(x,e) - u(x)II < IIYL - u(l)II exp[-U(L-x)/e] + 0(e),

that is, there is a boundary layer at

x = L.

wise theory provides the sharper estimate, for

1
a - 0+

(8.31)

Moreover, the componentx

in

[0,L]

and

VIII.

170

EXAMPLES AND APPLICATIONS

Iyi(x,c) - ui(x)I < Ibi - ui(L)I exp[-U(L-x)/el + 0(e),

is the i-th component of

(8.32)

where

bi

x = L

is replaced with a zero-flux condition, y'(L,c) = 0, then we ex-

yL.

If the Dirichlet

pect that the solution is uniformly close to

in

u(x)

condition at

[0,L]; cf. Appli-

cation 8.4.

It is also possible to consider the more general situation in which each species

yi

in general, for

flows at a constant average speed i # j.

Ui > 0, with

The differential equation for

y

Ui ¢ Uj,

becomes

ey = diag{U1,...,UN}y' + g(y), and clearly the above theories apply mutatis mutandis.

In particular,

the estimate (8.31) obtains with

Ui, while each of

U

replaced by

min

1
U

replaced by

1 .

References

[1]

R. C. Ackerberg and R. E. O'Malley, Jr., Boundary Layer Problems Exhibiting Resonance, Studies in Appl. Math. 49 (1970), 277-295.

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F. A. Howes, Singularly Perturbed Semilinear Systems, Studies in Appl. Math. 61 (1979), 185-209.

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F. A. Howes, Singularly Perturbed Semilinear Robin Problems, Studies in Appi. Math. 67 (1982), 125-139.

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F. A. Howes and R. E. O'Malley, Jr., Singular Perturbations of SecondOrder Semilinear Systems, in Lecture Notes in Math., no. 827, Springer-Verlag, New York, 1980, pp. 131-150.

[48]

E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.

[49]

L. K. Jackson, Subfunctions and Second-Order Ordinary Differential Inequalities, Adv. in Math. 2 (1968), 308-363.

[50]

W. E. Johnson and L. M. Perko, Interior and Exterior Boundary Value Problems from the Theory of the Capillary Tube, Arch. Rational Mech. Anal. 29 (1968), 125-143.

[51]

H. B. Keller, Existence Theory for Multiple Solutions of a Singular Perturbation Problem, SIAM J. Math. Anal. 3 (1972), 86-92.

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W. G. Kelley, Second Order Systems with Nonlinear Boundary Conditions, Proc. Amer. Math. Soc. 62 (1977), 287-292.

[53]

W. G. Kelley, A Geometric Method of Studying Two Point Boundary Value Problems for Second Order Systems, Rocky Mtn. J. Math. 7 (1977), 251-263.

[54]

W. G. Kelley, A Nonlinear Singular Perturbation Problem for Second Order Systems, SIAM J. Math. Anal. 10 (1979), 32-37.

[5S]

J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.

[56]

H. 0. Kreiss and S. V. Parter, Remarks on Singular Perturbations with Turning Points, SIAM J. Math. Anal. 5 (1974), 230-251.

[57]

A. Lasota and J. A. Yorke, Existence of Solutions of Two Point Boundary Value Problems for Nonlinear Systems, J. Differential Equations 11 (1972), 509-518.

[58]

T. C. Lee, Boundary Value Problems for Second Order Ordinary Differential Equations and Applications to Singular Perturbation Problems on [a,b] c Rocky Mtn. J. Math. 6 (1976), 761-765.

[59]

J. J. Levin, Singular Perturbations of Nonlinear Systems of Differential Equations Related to Conditional Stability, Duke Math. J. 23 (1956), 609-620.

[60]

J. J. Levin, The Asymptotic Behavior of the Stable Initial Manifolds of a System of Nonlinear Differential Equations, Trans. Amer. Math. Soc. 85 (1957), 357-368.

[61]

J. J. Levin and N. Levinson, Singular Perturbations of Nonlinear Systems and an Associated Boundary Layer Equation, J. Rational Mech. Anal. 3 (1954), 247-270.

L62]

J. W. Macki, Singular Perturbations of a Boundary Value Problem for a System of Ordinary Differential Equations, Arch. Rational Mech. Anal. 24 (1967), 219-232.

[63]

B. J. Matkowsky, On Boundary Layer Problems Exhibiting Resonance, SIAM Rev. 17 (1975), 82-100; Errata, ibid. 18 (1976), 112.

[64]

R. von Mises, Die Grenzschichte in der Theorie der gewohnlichen Differentialgleichungen, Acta Univ. Szeged., Sect. Sci. Math. 12 (1950), 29-34.

[65]

A. D. Myshkis and G. V. Scherbina, A Limit Boundary Value Problem Not Satisfying S. N. Bernstein's Condition with Applications in the Theory of Capillary Phenomena, Differential Equations 12 (1976), 698-704.

[66]

M. Nagumo, her die Differentialgleichung y" = f(x,y,y'), Proc. Phys. Math. Soc. Japan 19 (1937), 861-866.

[67]

M. Nagumo, her das Verhalten der Integrale von ay" + f(x,y,y',X) = 0 fur A + 0, ibid. 21 (1939), 529-534.

[68]

M. A. O'Donnell, Boundary and Interior Layer Behavior in Singularly Perturbed Systems of Boundary Value Problems, Doctoral Diss., U.C. Davis, 1983.

[69]

M. A. O'Donnell, Boundary and Corner Layer Behavior in Singularly Perturbed Semilinear Systems of Boundary Value Problems, SIAM J. Math. Anal. 15 (1984), 317-332.

[70]

0. A. Oleinik and A. N. Zizina, On Boundary Value Problems for the Equation ey" = F(x,y,y') for Small a (in Russian), Mat. Sbornik 31 (1952), 707-717.

References

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[71]

F. W. J. Olver, Uniform Asymptotic Expansions and Singular Perturbations, in SIAM-AMS Proceedings, vol. 10, 1976, pp. 105-117.

[72]

R. E. O'Malley, Jr., A Boundary Value Problem for Certain Nonlinear Second Order Differential Equations with a Small Parameter, Arch. Rational Mech. Anal. 29 (1968), 66-74.

[73]

R. E. O'Malley, Jr., Topics in Singular Perturbations, Adv. in Math. 2 (1968), 365-470.

[74]

R. E. O'Malley, Jr., On Singular Perturbation Problems with Interior Nonuniformities, J. Math. Mech. 19 (1970), 1103-1112.

[75]

R. E. O'Malley, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.

[76]

R. E. O'Malley, Jr., Phase-Plane Solutions to Some Singular Perturbation Problems, J. Math. Anal. Appl. 54 (1976), 449-466.

[77]

C. E. Pearson, On Non-Linear Ordinary Differential Equations of Boundary Layer Type, J. Math. and Phys. 47 (1968), 351-358.

[78]

L. M. Perko, Boundary Layer Analysis of the Wide Capillary Tube, Arch. Rational Mech. Anal. 45 (1972), 120-133.

[79]

L. M. Perko, A Class of Superquadratic Boundary Value Problems of Capillary Type, ibid., 74 (1980), 355-377.

[80]

E. E. Petersen, Chemical Reaction Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.

[81]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967.

[82]

L. G. Redekopp and P. Weidman, Solitary Rossby Waves in Zonal Shear Flows and Their Interactions, J. Atmospheric Sci. 35 (1978), 790804.

[83]

J. Schroder, Operator Inequalities, Academic Press, New York, 1980.

[84]

J. W. Searl, Extensions of a Theorem of Erdelyi, Arch. Rational Mech. Anal. 50 (1973), 127-138.

[85]

Y. Tschen, Uber das Verhalten der Losungen einer Folge von Differentialgleichungen, welche im Limes ausarten, Compositio Math. 2 (1935), 378-401.

[86]

V. A. Tupchiev, On the Existence, Uniqueness and Asymptotic Behavior of the Solution of a Boundary Value Problem for a System of Differential Equations with a Small Parameter Multiplying the Highest Derivatives, Sov. Math. Sokl. 3 (1962), 302-305.

[87]

A. B. Vasil'eva, Asymptotic Behavior of Solutions of Problems for Ordinary Nonlinear Differential Equations with a Small Parameter Multiplying the Highest Derivatives, Russian Math. Surveys 18 (1963), 13-84.

[88]

A. B. Vasil'eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations (in Russian), Nauka, Moscow, 1973.

[89]

A. B. Vasil'eva and V. A. Tupchiev, Periodic Nearly-Discontinuous Solutions of Systems of Differential Equations with a Small Parameter in the Derivatives, Soc. Math. Dokl. 9 (1968), 179-183.

[90)

M. I. Vishik and L. A. Liusternik, Initial Jump for Nonlinear Differential Equations Containing a Small Parameter, Sov. Math. Dokl. 1 (1960), 749-752.

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V. M. Volosov, Averaging in Systems of Ordinary Differential Equations, Russian Math. Surveys 17 (1962), 1-126.

[92]

W. R. Wasow, Singular Perturbation of Boundary Value Problems for Nonlinear Differential Equations of the Second Order, Comm. Pure Appl. Math. 9 (1956), 93-116.

[93]

W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York, 1965.

[94]

W. R. Wasow, The Capriciousness of Singular Perturbations, Nieuw Arch. Wisk. 18 (1970), 190-210.

[95]

V. W. Weekman and R. L. Gorring, Influence of Volume Change on Gas-Phase Reactions in Porous Catalysts, J. Catalysis 4 (1965), 260-270.

[96]

D. Willett, On a Nonlinear Boundary Value Problem with a Small Parameter Multiplying the Highest Derivative, Arch. Rational Mech. Anal. 23 (1966), 276-287.

[97]

F. A. Williams, Theory of Combustion in Laminar Flows, Fluid Mech., vol. 3, 1971, pp. 171-188.

Ann. Rev.

Author Index

Ackerberg, R. C., 59

Fife, P. C., 35

Aris, R., 130, 162 ff.

Flaherty, J. E., 35 Friedrichs, K. 0., 133, 135

Batchelor, G. K., 155

Benney, D. J., 164

Gantmacher, F. R., 164

Bernfeld, S. R., 6, 13, 17

Gorring, R. L., 154

Boglaev, Yu. P., 35 Bris, N. I., 17, 35, 59 ff. Butuzov, V. F., 5, 122

Carrier, G. F., 5, 35 Chang, K. W., 59, 122 Coddington, E. A., 59, 92 Cohen, D. S., 59 Cole, J. D., 2, 4 ff., 59, 133, 137

Cook, L. P., 59

Haber, S., 61 ff., 84, 90, 102, 142, 145, 151 Habets, P., 5, 35, 59 Harris, W. A., 122 van Harten, A., 12

Hartman, P., 13 Heidel, J. W., 7 ff.

Hoppensteadt, F., 122 Howes, F. A., 17, 35, 59 ff., 63, 90, 102, 104 ff., 108 ff., 115, 121 ff., 137, 139, 146, 167

Coppel, W. A., 122

Courant, R., 133, 13S

Ince, E. L., 63, 146

Crocco, L., 134

Jackson, L. K., 6 Deshpande, M. A., 105

Johnson, W. E., 105

Dorr, F. W., 35, 59, 90, 137, 139, 141

Keller, H. B., 59 Kelley, W. G., 13, 108 ff., 122

Eckhaus, W., 5, 59 Erd6lyi, A., 5, 59, 92

Kevorkian, J., 2, 4 ff., 133, 137 Kreiss, H. 0., 59

177

178

AUTHOR INDEX

Lakshmikantham, V., 6, 13, 17 Lasota, A., 17 Lee, T. C., 105 Levin, J. J., 78, 101, 122 Levinson, N., 59, 61 ff., 78, 84, 90, 92, 101 ff., 122, 142, 145, 151

Wasow, W. R., 5, 39, 59, 62, 78, 92, 122

Weekman, V. W., 154 Weidman, P., 164 Weinberger, H. F., 9 Willett, D., 59 Williams, F. A., 128 ff.

Liusternik, L. A., 91, 104

Yorke, J. A., 17 Macki, J. W., 59, 90 Matkowsky, B. J., 59 von Mises, R., 59 Myshkis, A. D., 105 Nagumo, M., 6 ff., 17 O'Donnell, M. A., 109 if., 114, 118 ff., 162, 164, 168 Oleinik, 0. A., 59 Olver, F. W. J., 59 ff.

O'Malley, R. E., 1 if., 21, 35 ff., 39, 59 if., 90, 115, 122, 124 ff., 128, 140, 164, 167

Parter, S. V., 35, 59, 90, 137, 139, 141 Pearson, C. E., 135 ff. Perko, L. M., 105, 155

Petersen, E. E., 139, 169 Protter, M. H., 9

Redekopp, L. G., 164 Scherbina, G. V., 105 Schroder, J., 6, 13, 17 Searl, J. W., 59, 90 Tschen, Y., 59 Tupschiev, V. A., 35 Vasil'eva, A. B., 5, 35, 59, 78, 90, 101, 122

Vishik, M. I., 91, 104 Volosov, V. M., 36

Zizina, A. N., 59

Subject Index

A'priori bounds, 9 ff.

(Dirichlet problems)

for singularly perturbed scalar superquadratic equations, 91 ff., 101 ff., 155 ff.

Boundary layer behavior, 18 ff., 27 ff., 37 ff., 49 ff., 61 ff. , 76 ff. , 95 ff. , 101 ff. , 108 ff. , 115 ff.

for singularly perturbed vector semilinear equations, 106 ff., 161 ff.

Bounding (comparison) functions, 7 ff., 14 ff.

for singularly perturbed vector quasilinear equations, 114 ff.,

Burke-Schumann approximation, 128

165 ff.

Catalytic reactions one-dimensional, 130 ff., 154 ff.

Existence and comparison theorems for scalar Dirichlet problems, 7 ff.

N-dimensional, 162 ff.

for scalar Robin problems, 8

Clairaut equation, 63, 144, 158

for singularly perturbed scalar Dirichlet problems, 9 ff., 24 ff., 32 if., 43 ff., 56 ff., 70 ff., 84 ff., 94 ff., 96 ff., 98 if.

Convexity conditions, 3, 20 ff., 34, 94 ff.

Differential inequalities, theory of, 6 ff.

for singularly perturbed scalar Robin problems, 13, 28 ff., 32 ff., 51 ff., 58, 77 ff., 88 ff., 96 ff.,

Dirichlet problems

102 ff.

for scalar differential equations, 6 ff.

for vector Dirichlet problems, 15 ff.

for vector differential equations, 13 ff.

for a vector Robin problem, 17

for singularly perturbed vector Dirichlet problems, 108 ff.,

for singularly perturbed scalar semilinear equations, 18 ff.,

115 ff.

123 ff.

for a singularly perturbed vector Robin problem, 112 ff.

for singularly perturbed scalar quadratic equations, 61 ff., 141 ff.

Hessian matrix, 14 ff.

179

180

SUBJECT INDEX

Inflection locus, 146 Initial value (Cauchy) problems, 78, 92, 101 Inner expansion, 2, 4 Interior layer behavior, 32 if., 55 ff., 83 if., 96 if., 101 ff., 121 Invariant regions, 15 if.

Jacobian matrix, 14, 107 ff., 161 ff.

Matched asymptotic expansions, method of, 2, 4

Nagumo (growth) conditions, 7 ff., 13 ff.

Neumann problem, 36 Nonpremixed combustion, 128 Nozzle flow, one-dimensional, 133 ff.

[Robin problems]

for singularly perturbed scalar superquadratic equations, 95 ff., 155 ff.

for a singularly perturbed vector semilinear equation, 111 ff., 162 ff.

Sherwood number, 131 if., 154 Shock layer behavior, 35, 60, 90, 104, 128, 132 ff., 135 ff., 139 Singular solution, 63, 66, 75, 83, 86, 144, 153, 158 ff. Spike layer behavior, 124 ff., 164 if. Stability conditions, 21 ff., 28, 30 ff., 35 ff., 40 ff., 50, 59, 67 ff., 76, 94 ff., 98, 101 if., 103, 108, 110, 115, 118, 121 Stretched variable, 2, 4

Thiele modulus, 131 ff., 154, 163 Tubular reactor with axial diffusion

Oscillatory behavior, 19, 36 Outer expansion, 2 if. p-discriminant locus, 146 ff. Peclet number, 139, 169 Prandtl's relation, 135 ff. Pseudo-steady-state hypothesis, 163

Rankine-Hugoniot shock condition, 133, 135 if. Robin problems for scalar differential equations, 8 ff. for vector differential equations, 16 ff.

for singularly perturbed scalar semilinear equations, 27 ff., 129 ff.

for singularly perturbed scalar quasilinear equations, 49 ff., 139 ff.

for singularly perturbed scalar quadratic equations, 76 ff., 151 ff.

one-dimensional case, 139 ff. N-dimensional case, 169 ff. Two-variable asymptotic expansions, method of, 2, 4

Applied Mathematical Sciences cont. from page ii

36. Bengtsson/Ghil/Kallan: Dynamic Meterology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion. 39. Piccinini/Stampacchia/Vidossich: Ordinary Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.

43. Ockendon/Tayler: Inviscid Fluid Flows.

44. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. 45. Glashoff/Gustafson: Linear Optimization and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs. 46. Wilcox: Scattering Theory for Diffraction Gratings. 47. Hale et al.: An Introduction to Infinite Dimensional Dynamical Systems

-

Geometric

Theory.

48. Murray: Asymptotic Analysis.

49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol. I. 52. Chipot: Variational Inequalities and Flow in Porous Media.

53. Majda:

Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables.

54. Wasow: Linear Turning Point Theory. 55, Yosida: Operational Calculus: A Theory of Hyperfunctions. 56. Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applications.

Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences)

Nonlinear Singular Perturbation Phenomena: Theory and Applications

Multiple Scale and Singular Perturbation Methods (Applied Mathematical Sciences)

Singular Perturbation Methods for Ordinary Differential Equations (Applied Mathematical Sciences)

Singular Perturbation Methods for Ordinary Differential Equations (Applied Mathematical Sciences)

Singular Perturbation Theory

Singular Perturbation Theory

Singular Perturbation Theory

Perturbation Methods, Bifurcation Theory and Computer Algebra (Applied Mathematical Sciences)

Elliptic Functions and Applications (Applied Mathematical Sciences)

Nonlinear theory of shallow shells (Applied Mathematical Sciences)

Piecewise-smooth Dynamical Systems: Theory and Applications (Applied Mathematical Sciences)

Determinants and Their Applications in Mathematical Physics (Applied Mathematical Sciences)

Perturbation Methods in Non-Linear Systems (Applied Mathematical Sciences 8)

Applications of Centre Manifold Theory (Applied Mathematical Sciences)

Applications of Centre Manifold Theory (Applied Mathematical Sciences)

Symmetry and perturbation theory in nonlinear dynamics

Linear Turning Point Theory (Applied Mathematical Sciences)

Symmetry and Perturbation Theory in Nonlinear Dynamics

Nonlinear Filtering and Optimal Phase Tracking (Applied Mathematical Sciences)

Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering

Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering

Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering

Singular Perturbation Theory: Mathematical and Analytical Techniques with Applications to Engineering

Random Perturbation Methods: With Applications in Science and Engineering (Applied Mathematical Sciences)

Computational Homology (Applied Mathematical Sciences)

Periodic Motions (Applied Mathematical Sciences)

Computational Homology (Applied Mathematical Sciences)

Fluid Dynamics (Applied Mathematical Sciences)

Periodic Motions (Applied Mathematical Sciences)

Algebraic methods in nonlinear perturbation theory

Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences)

Nonlinear Singular Perturbation Phenomena: Theory and Applications

Multiple Scale and Singular Perturbation Methods (Applied Mathematical Sciences)

Singular Perturbation Methods for Ordinary Differential Equations (Applied Mathematical Sciences)

Singular Perturbation Methods for Ordinary Differential Equations (Applied Mathematical Sciences)

Singular Perturbation Theory

Singular Perturbation Theory

Singular Perturbation Theory

Perturbation Methods, Bifurcation Theory and Computer Algebra (Applied Mathematical Sciences)

Elliptic Functions and Applications (Applied Mathematical Sciences)

Nonlinear theory of shallow shells (Applied Mathematical Sciences)

Piecewise-smooth Dynamical Systems: Theory and Applications (Applied Mathematical Sciences)

Determinants and Their Applications in Mathematical Physics (Applied Mathematical Sciences)

Perturbation Methods in Non-Linear Systems (Applied Mathematical Sciences 8)

Applications of Centre Manifold Theory (Applied Mathematical Sciences)

Applications of Centre Manifold Theory (Applied Mathematical Sciences)

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