An Exponential Function Approach to Parabolic Equations

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An Exponential Function Approach to Parabolic Equations

9229hc_9789814616386_tp.indd 1

15/7/14 9:00 am

SERIES ON CONCRETE AND APPLICABLE MATHEMATICS ISSN: 1793-1142 Series Editor:

Professor George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA

Published* Vol. 6

Topics on Stability and Periodicity in Abstract Differential Equations by James H. Liu, Gaston M. N’Guérékata & Nguyen Van Minh

Vol. 7

Probabilistic Inequalities by George A. Anastassiou

Vol. 8

Approximation by Complex Bernstein and Convolution Type Operators by Sorin G. Gal

Vol. 9

Distribution Theory and Applications by Abdellah El Kinani & Mohamed Oudadess

Vol. 10 Theory and Examples of Ordinary Differential Equations by Chin-Yuan Lin Vol. 11 Advanced Inequalities by George A. Anastassiou Vol. 12 Markov Processes, Feller Semigroups and Evolution Equations by Jan A. van Casteren Vol. 13 Problems in Probability, Second Edition by T. M. Mills Vol. 14 Evolution Equations with a Complex Spatial Variable by Ciprian G. Gal, Sorin G. Gal & Jerome A. Goldstein Vol. 15 An Exponential Function Approach to Parabolic Equations by Chin-Yuan Lin

*To view the complete list of the published volumes in the series, please visit: http://www.worldscientific/series/scaam

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Series on Concrete and Applicable Mathematics – Vol.15

An Exponential Function Approach to Parabolic Equations Chin-Yuan Lin National Central University, Taiwan

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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15/7/14 9:00 am

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Series on Concrete and Applicable Mathematics — Vol. 15 AN EXPONENTIAL FUNCTION APPROACH TO PARABOLIC EQUATIONS Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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To my wife and brothers, and in memory of my mother, Liu Gim

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Preface This book is intended for graduate students of mathematics, as well as for researchers. It requires a basic knowledge of both introductory functional analysis and elliptic partial differential equations of second order. However, for the convenience of the reader, some essential background results from elliptic partial differential equations are collected in the Appendix. As is revealed by its contents, the book consists of nine chapters. Chapter 1 is on the existence of exponential functions of linear or nonlinear, time-independent operators, and Chapter 2 is on the same subject but with time-dependent operators. Chapters 3 through 6 are on applications of Chapters 1 and 2 to initial-boundary value problems for parabolic partial differential equations. Chapter 7 is on the associated elliptic equations with Chapters 3 through 6. Chapter 8 is on an extension of Chapter 2, whose results are applied to solve more general initial-boundary value problems in Chapter 9. The author is indebted to Professors Jerome A. Goldstein and Gisele R. Goldstein at the University of Memphis, and to the late Professor Sen-Yen Shaw from the National Central University, for their teaching. Finally, the author wishes to thank his wife and son for their support and encouragement.

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Contents Preface

vii

Chapter 1. Existence Theorems for Cauchy Problems 1. Introduction 2. Main Results 2.1. Linear Homogeneous Equations 2.2. Nonlinear Homogeneous Equations 2.3. Linear Nonhomogeneous Equations 2.4. Nonlinear Nonhomogeneous Equations 3. Examples 4. Some Preliminary Results 5. Difference Equations Theory 6. Proof of the Main Results 6.1. The Nonlinear Homogeneous Case 6.2. The Linear Homogeneous Case 6.3. The Linear Nonhomogeneous Case 6.4. The Nonlinear Nonhomogeneous Case

1 1 5 5 6 7 9 10 19 23 25 25 30 34 35

Chapter 2. Existence Theorems for Evolution Equations (I) 1. Introduction 2. Main Results 3. Examples 4. Some Preliminary Estimates 5. Proof of the Main Results 6. Difference Equations Theory

37 37 39 41 54 60 66

Chapter 3. Linear Autonomous Parabolic Equations 1. Introduction 2. Main Results 3. Proof of One Space Dimensional Case 4. Proof of Higher Space Dimensional Case

75 75 77 77 79

Chapter 4. Nonlinear Autonomous Parabolic Equations 1. Introduction 2. Main Results 3. Proof of One Space Dimensional Case 4. Proof of Higher Space Dimensional Case

83 83 85 86 89

Chapter 5. Linear Non-autonomous Parabolic Equations 1. Introduction 2. Main Results

95 95 97

ix

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CONTENTS

3. Proof of One Space Dimensional Case 4. Proof of Higher Space Dimensional Case

99 103

Chapter 6. Nonlinear Non-autonomous Parabolic Equations (I) 1. Introduction 2. Main Results 3. Proof of One Space Dimensional Case 4. Proof of Higher Space Dimensional Case

109 109 111 112 115

Chapter 7. The Associated Elliptic Equations 1. Introduction 2. Main Results 3. Proof of Linear Case 4. Proof of the Nonlinear Case

119 119 121 121 123

Chapter 8. Existence Theorems for Evolution Equations (II) 1. Introduction 2. Main Results 3. Some Preliminary Estimates 4. Proof of the Main Results

129 129 132 134 141

Chapter 9. Nonlinear Non-autonomous Parabolic Equations (II) 1. Introduction 2. Main Results 3. Proof of One Space Dimensional Case 4. Proof of Higher Space Dimensional Case

145 145 148 149 151

Appendix 1. Existence of a Solution 2. Apriori Estimates 3. Hopf Boundary Point Lemma 4. Interpolation Inequality 5. Sobolev Embedding Theorem

155 155 158 159 160 160

Bibliography

161

Index

163

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CHAPTER 1

Existence Theorems for Cauchy Problems 1. Introduction In this chapter, linear and nonlinear Cauchy problems, together with their associated nonhomogeneous problems, will be studied. Those problems will be solved with the aid of elementary difference equations. The obtained results will be illustrated by solving simple, initial-boundary value problems for parabolic, partial differential equations with time-independent coefficients. Further illustrations of solving more general, parabolic partial differential equations with time-independent coefficients will be given in Chapters 3 and 4. Let constants ω ∈ R and M ≥ 1. Consider the linear Cauchy problem d u(t) = Bu(t), dt u(0) = u0

t>0

(1.1)

in a real Banach space (X, · ), where u is a function from [0, ∞) to X, and B : D(B) ⊂ X −→ X is an unbounded linear operator. Here recall • A real Banach space is a complete, real normed vector space equipped with a norm. For example, the real vector space R of real numbers over the field of itself, equipped with the norm of the usual function | · | of absolute value, is a real Banach space. Another example is the real Banach space (C[0, 1], · ∞ ) of all continuous, real-valued functions on [0, 1], equipped with supremum norm · ∞ , where f ∞ ≡ sup |f (x)|

if f ∈ C[0, 1].

x∈[0,1]

• One example of an unbounded linear operator is the first order ordinary differential operator S in the real Banch space (C[0, 1], · ∞ ), where S : D(S) ⊂ C[0, 1] −→ C[0, 1], d defined by (Sf )(x) ≡ dx f (x) for f in D(S), the set of all real, continuously differentiable functions on [0, 1].

To solve the linear Cauchy problem (1.1), let the simple case be considered first where X = R, and B = b, a real number. In this case, the unique solution is given by u(t) = etb u0 , 1

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1. EXISTENCE THEOREMS FOR CAUCHY PROBLEMS

where the exponential function etb can be represented by this limit t etb = lim (1 − b)−n . n→∞ n Thus the Cauchy problem (1.1) might be solved, if the quantity t −n B) u0 n can be defined for each t > 0 and for each n = 1, 2, . . ., and has a limit as n −→ ∞. Here I is the identity operator. But this will be true under suitable assumptions on the operator B, as the following describes it. The linear Cauchy problem (1.1) will be solved under the assumption that the operator B satisfies both the range condition (B1) and the dissipativity condition (B2) or satisfies, more generally, the mixture condition (B3). (B1) For each small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the range of (I − λB) contains the closure D(B) of D(B). (B2) For each v ∈ D(B), the inequality (I −

v ≤ v − λ(B − ω)v is true for each λ > 0. (B3) For each x ∈ D(B) and for each small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the quantity (I − λB)−1 x is singlevalued, and the inequality is true: (I − λB)−n x ≤ M (1 − λω)−n x,

n = 1, 2, . . . .

Here I is the identity operator. Under both the (B1) and the (B2) or under, more generally, the (B3), the quantity (I − λB)−n v, n = 1, 2, . . . will be well-defined for each small λ > 0 satisfying λω < 1 and for each v ∈ D(B). It is the first of our three purposes in this chapter to show, using the difference equations theory [28], that the limit t −n B) v n exists for bounded t ≥ 0. From this, it will follow that, if u0 lies in D(B) with Bu0 ∈ D(B) and if B is additionally a closed operator, then lim (I −

n→∞

t −n B) u0 n→∞ n is the unique solution of the linear Cauchy problem (1.1), in the sense that u(t) is the unique continuously differentiable function of t > 0 satisfying (1.1). Here B is said to be a closed operator, if the condition u(t) ≡ lim (I −

xn −→ x and Bxn −→ y

for xn ∈ D(B)

implies x ∈ D(B) and y = Bx. For example, the first order ordinary differential operator S, mentioned above, is a closed operator.

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1. INTRODUCTION

3

The second purpose of this chapter is to study the nonlinear analogue of (1.1), the nonlinear Cauchy problem in the real Banach space X d u(t) ∈ Au(t), t > 0 dt (1.2) u(0) = u0 , where A : D(A) ⊂ X −→ X, is a nonlinear, multi-valued operator with domain D(A). The D(A) is not necessarily a subspace of X, a case different from that with the linear Cauchy problem (1.1), and one example of such an A is the nonlinear, multi-valued function f : D(f ) ⊂ R −→ R, defined by ⎧ 2 ⎪ ⎨x + 1, if 0 < x < 2; f (x) = [−1, 1], if x = 0; ⎪ ⎩ 2 −x − 1, if −2 < x < 0; in the real Banach space R. Here the domain D(f ) = (−2, 2) of f is not a subspace of R, and the function value f (0) of f at x = 0 is not a single number but a set [−1, 1]. To solve the nonlinear Cauchy problem (1.2), let its approximate problem, a difference equation, be examined first [14, Page 9], [8, Page 72] u (t) − u (t − ) ∈ Au (t), u (0) = u0 .

t > 0,

(1.3)

Here > 0 is very small. This approximate problem (1.3) becomes the equation u (t) = [I − A]−1 u (t − ),

t > 0,

(1.4)

u (0) = u0 , if the quantity

(I − A)−1 can be defined. Here I is the identity operator. Thus when = equation (1.4) is readily seen to have the solution

t n

for n ∈ N, the

u (t) = (I − A)−1 (I − A)−1 · · · (I − A)−1 u0 ≡ (I − A)−n u0 , provided that the quantity

(I − A)−n u0 can be defined for each t > 0 and for each n = 1, 2, . . .. Therefore, the nonlinear Cauchy problem (1.2) might be solved, if the limit exists, as n −→ ∞, of the quantity t (I − A)−n u0 = (I − A)−n u0 . n But this will be true under suitable assumptions on the nonlinear operator A, as the following describes it. The nonlinear Cauchy problem will be solved under the similar assumption that A satisfies both the range condition (A1) and the dissipativity condition (A2). (A1) For each small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the range of (I − λA) contains the closure D(A) of D(A).

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(A2) For each v, w ∈ D(A), the inequality v − w ≤ (v − w) − λ(x − y) is true for each λ > 0 and for all x ∈ (A − ω)v, y ∈ (A − ω)w. As is the case with B, the quantity (I − λA)−n v,

n = 1, 2, . . . ,

under the conditions (A1) and (A2), will be well-defined for each small λ > 0 satisfying λω < 1 and for each v ∈ D(A). Using the difference equations theory again [28], it will be shown that the limit lim (I −

n→∞

t −n A) v n

exists for bounded t ≥ 0. Different from the case with the linear Cauchy problem (1.1), the quantity t u(t) ≡ lim (I − A)−n u0 n→∞ n for u0 ∈ D(A) will be only interpreted as a limit solution of the nonlinear Cauchy problem (1.2). However, u(t) will be a strong solution if A is what we call embeddedly quasi-demi-closed; this will be explained in Section 2. The third (final) purpose of this chapter is to study the simple nonhomogeneous equations associated with (1.1) and (1.2), respectively: d u(t) = Bu + f0 , dt u(0) = u0

t>0

(1.5)

and d u(t) ∈ Au + f1 , dt u(0) = u0 .

t>0

(1.6)

Here f0 and f1 are two elements in X. The results of this chapter will be used in Chapters 3 and 4, where the corresponding operators B or A are second order, elliptic differential operators, and the corresponding Cauchy problems are parabolic, initial-boundary value problems. Since the quantities lim (I −

n→∞

t −n t B) u0 and lim (I − A)−n u0 n→∞ n n

are similar to the ordinary, real-valued, exponential function of t eta = lim (1 − n→∞

t −n a) , n

a ∈ R,

the title of this book is explained. Finally, we organize the rest of this chapter as follows, which consists of five other sections. Section 2 states the main results, and Section 3 illustrates the main results by simple examples. Section 4 gives some preliminary results, and Section 5 presents a basic theory of difference equations. The last section, Section 6, proves the main results, using the results in Sections 4 and 5.

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2. Main Results 2.1. Linear Homogeneous Equations. With regard to the linear Cauchy problem (1.1), we have Theorems 2.1, 2.2, and 2.3. Theorem 2.1 (A classical solution). Let the linear unbounded operator B satisfy both the range condition (B1) and the dissipativity condition (B2) or satisfy, more generally, the mixture condition (B3). Then the limit t S(t)x ≡ lim (I − B)−n x n→∞ n t = lim (I − νB)−[ ν ] x ν→0

exists for each x ∈ D(B) and for bounded t ≥ 0. This limit S(t)x is also continuous in t ≥ 0 for x ∈ D(B), but Lipschitz continuous in t ≥ 0 for x ∈ D(B). Furthermore, if u0 lies in D(B) with Bu0 ∈ D(B), and if B is additionally a closed operator, then the function t u(t) ≡ S(t)u0 = lim (I − B)−n u0 n→∞ n t = lim (I − νB)−[ ν ] u0 ν→0

is the unique solution of the linear Cauchy problem (1.1). Theorem 2.2 (Regularity of solution). Following Theorem 2.1, d u(t) = Bu(t) = BS(t)u0 dt = S(t)(Bu0 ) is continuous in t ≥ 0 for u0 ∈ D(B) with Bu0 ∈ D(B), is Lipschitz continuous in t ≥ 0 for u0 ∈ D(B 2 ), and is differentiable in t ≥ 0 for u0 ∈ D(B 2 ) with d u(t) in t can be obtained iteratively. B 2 u0 ∈ D(B). More regularity of dt From the proofs of Theorems 2.1 and 2.2 in Subsection 6.2, it follows readily that Theorem 2.3. The results in Theorems 2.1 and 2.2 are still true, if the range condition (B1) and the mixture condition (B3) are replaced by the weaker conditions (B1) and (B3) below, respectively, provided that the initial conditions x ∈ D(B), Bu0 ∈ D(B), and B 2 u0 ∈ D(B), are changed to the initial conditions x ∈ D(B), Bu0 ∈ D(B), and B 2 u0 ∈ D(B), respectively. Here (B1) For each small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the range of (I − λB) contains D(B). (B3) For each x ∈ D(B) and for each small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the quantity (I − λB)−1 x is singlevalued, and the inequality is true: (I − λB)−n x ≤ M (1 − λω)−n x,

n = 1, 2, . . . .

Remark. Theorems 2.1 and 2.2 are the Hille-Yosida Theorem [9, 10, 14, 16, 17, 30, 31, 39], if the linear operator B is densely defined and satisfies the dissipativity condition (B2) and the stronger range condition (B1) :

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(B1) For each small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the range of (I − λB) equals X. In the present case, the section B D(B) of B on the Banach space D(B), B D(B) : D(B) ⊂ D(B) −→ D(B), is not necessarily well-defined, as B D(B) x = Bx for x ∈ D(B) may lie outside of D(B). Therefore, Theorems 2.1 and 2.2 will not follow from applying the HilleYosida theorem to the section B D(B) . 2.2. Nonlinear Homogeneous Equations. With regard to the nonlinear Cauchy problem (1.2), we have Theorems 2.4, 2.5, and 2.6. Theorem 2.4 ((Existence of a limit) Crandall-Liggett theorem [6, 30]). Let the nonlinear, multi-valued operator A satisfy both the range condition (A1) and the dissipativity condition (A2). Then the limit t U (t)x ≡ lim (I − A)−n x n→∞ n t = lim (I − νA)−[ ν ] x ν→0

exists for each x ∈ D(A) and for bounded t ≥ 0. This limit U (t)x is also continuous in t ≥ 0 for x ∈ D(A), but Lipschitz continuous in t ≥ 0 for x ∈ D(A). In order to state the next theorem, Theorem 2.5, concerning a limit solution and a strong solution, we need to make two preparations. The first preparation is for a limit solution. Let T > 0, u0 ∈ D(A), and n ∈ N be large. Consider the discretization of (1.2) on [0, T ) ui − νAui ui−1 , ui ∈ D(A), where ν =

T n

(2.1)

satisfies ν < λ0 for which νω < 1,

i = 1, 2, . . . , n,

and ui will exist uniquely by the range condition (A1) and the dissipativity condition (A2) (see Section 4). Putting ti = iν and defining the Rothe functions [12, 32] χn (0) = u0 ; for t ∈ (ti−1 , ti ]; ui − ui−1 un (t) = ui−1 + (t − ti−1 ) ν for t ∈ [ti−1 , ti ],

χn (t) = ui

(2.2)

it will follow (see Section 6) that lim sup un (t) − χn (t) = 0; n→∞

dun (t) ∈ Aχn (t), un (0) = u0 dt for almost every t.

(2.3)

Here the last equation has values in B([0, T ]; X), the real Banach space of bounded functions from [0, T ] to X.

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The other preparation is for a strong solution. Let (Y, .Y ) be a real Banach space, into which the real Banach space (X, .) is continuously embedded. For example, the space (C[0, 1], · ∞ ) is continuously embedded into the real Banach space (L2 (0, 1), · 2 ) of Lebesgue square integrable functions on (0, 1) by the identity mapping. The nonlinear, multi-valued operator A is said to be embeddedly quasi-demiclosed with respect to X and Y , if it satisfies the embedding condition (A3). (A3) If xn ∈ D(A) −→ x and if yn ≤ M0 for some yn ∈ Axn and for some positive constant M0 , then x ∈ D(η ◦ A), the domain of η ◦ A, (that is, η(A(x)) exists), and |η(ynl ) − z| −→ 0 for some subsequence ynl of yn , for some z ∈ η(Ax), and for each η ∈ Y ∗ ⊂ X ∗ , the real dual space of Y . Theorem 2.5 ((A limit or strong solution) [20]). Following Theorem 2.4, if u0 ∈ D(A), then the function u(t) ≡ U (t)u0 = lim (I − n→∞

t −n A) u0 n

t

= lim (I − νA)−[ ν ] u0 ν→0

is a limit solution of the nonlinear Cauchy problem (1.2) on [0, T ] and then on [0, ∞), in the sense that it is also the uniform limit of un (t) on [0, T ], where un (t) satisfies (2.3) on [0, T ] and T > 0 is arbitrary. Furthermore, if A is embeddedly quasi-demi-closed, that is, if A satisfies the embedding condition (A3), then u(t) is a strong solution of (1.2) in Y , that is, u(t) satisfies (1.2) in Y for almost every t. It follows readily from the proofs of Theorems 2.4 and 2.5 in Subsection 6.1 that Theorem 2.6. The results in Theorems 2.4 and 2.5 are still true, if the range condition (A1) is replaced by the weaker range condition (A1) below, provided that the initial condition x ∈ D(A) is changed to the condition x ∈ D(A). Here (A1) For each samll 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the range of (I − λA) contains D(A). 2.3. Linear Nonhomogeneous Equations. With regard to the nonhomogeneous, linear Cauchy problem (1.5), we have Theorems 2.7, 2.8, and 2.9. Here the element f0 in (1.5) will be conditioned by (F0): (F0) (y + λf0 ) lies in the range of (I − λB) whenever y ∈ D(B) and 0 < λ < λ0

with λ0 ω < 1.

It is readily seen that, if f0 ∈ D(B), then f0 satisfies the condition (F0). Theorem 2.7 (A classical solution). Let a nonlinear operator ˜ : D(B) ˜ ⊂ X −→ X B be defined by ˜ = Bv + f0 Bv

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˜ ≡ D(B), where B is the B in Theorem 2.1, and f0 ∈ X satisfies the for v ∈ D(B) condition (F0). Then the limit t ˜ −n ˜ S(t)x ≡ lim (I − B) x n→∞ n t ˜ −[ ν ] x = lim (I − ν B) ν→0

˜ need not satisfy both (A1) exists for each x ∈ D(B) and for bounded t ≥ 0. Here B and (A2), but it does so, if B is dissipative. ˜ This limit S(t)x is also continuous in t ≥ 0 for x ∈ D(B), but Lipschitz continuous in t ≥ 0 for x ∈ D(B). Furthermore, if u0 lies in D(B) with Bu0 , f0 ∈ D(B), and if B is additionally a closed operator, then the function t ˜ −n ˜ B) u0 u ˜(t) ≡ S(t)u 0 = lim (I − n→∞ n ˜ −[ νt ] u0 = lim (I − ν B) ν→0

is the unique solution of the nonhomogeneous, linear Cauchy problem (1.5). Theorem 2.8 (Regularity of solution). Follow Theorem 2.7 and use the S(t) in Theorem 2.1. It follows that d ˜u ˜ S(t)u ˜ u ˜(t) = B ˜(t) = B 0 dt ˜ 0) = S(t)(Bu = S(t)(Bu0 + f0 ) is continuous in t ≥ 0 for u0 ∈ D(B) with Bu0 , f0 ∈ D(B), is Lipschitz continuous in t ≥ 0 for u0 ∈ D(B 2 ) and for f0 ∈ D(B), and is differentiable in t ≥ 0 for d u ˜(t) u0 ∈ D(B 2 ) and for f0 ∈ D(B) with B 2 u0 , Bf0 ∈ D(B). More regularity of dt in t can be obtained iteratively. From the proofs of Theorems 2.7 and 2.8 in Subsection 6.3, it follows readily that Theorem 2.9. The results in Theorems 2.7 and 2.8 are still true, if the range condition (B1), the mixture condition (B3), and the condition (F0) are replaced by the weaker conditions (B1) , (B3) , and (F 0) below, respectively, provided that the initial conditions x ∈ D(B), Bu0 ∈ D(B), and B 2 u0 ∈ D(B) and the conditions f0 ∈ D(B) and Bf0 ∈ D(B) are changed to the initial conditions x ∈ D(B), Bu0 ∈ D(B), and B 2 u0 ∈ D(B) and the conditions f0 ∈ D(B) and Bf0 ∈ D(B), respectively. Here (B1) For each small 0 < λ < λ0 , where λ0 is λ0 ω < 1, the range of (I − λB) contains (B3) For each x ∈ D(B) and for each small positive number satisfying λ0 ω < 1, the valued, and the inequality is true:

some positive number satisfying D(B). 0 < λ < λ0 , where λ0 is some quantity (I − λB)−1 x is single-

(I − λB)−n x ≤ M (1 − λω)−n x,

n = 1, 2, . . . .

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(F 0) (y + λf0 ) lies in the range of (I − λB) whenever y ∈ D(B) and 0 < λ < λ0

with λ0 ω < 1.

2.4. Nonlinear Nonhomogeneous Equations. With regard to the nonhomogeneous, nonlinear Cauchy problem (1.6), we have Theorems 2.10, 2.11, and 2.12. Here the element f1 in (1.6) will be conditioned by either (F1) or weaker (F2): (F1) (y + λf1 ) lies in the range of (I − λA) whenever y ∈ D(A) and 0 < λ < λ0

with λ0 ω < 1.

(F2) (y + λf1 ) lies in the range of (I − λA) whenever y ∈ D(A) and 0 < λ < λ0

with λ0 ω < 1.

˜ ⊂ Theorem 2.10 (Existence of a limit). Let a nonlinear operator A˜ : D(A) X −→ X be defined by ˜ = Av + f1 Av ˜ for v ∈ D(A) ≡ D(A), where A is the A in Theorem 2.4, and f1 ∈ X satisfies the condition (F1). Then the limit ˜ (t)x ≡ lim (I − t A) ˜ −n x U n→∞ n ˜ −[ νt ] x = lim (I − ν A) ν→0

exists for each x ∈ D(A) and for bounded t ≥ 0. Here A˜ satisfies both (A1) and (A2). ˜ (t)x is also continuous in t ≥ 0 for each x ∈ D(A), but Lipschitz This limit U continuous in t ≥ 0 for x ∈ D(A). Theorem 2.11 (A limit or strong solution). Following Theorem 2.10, if u0 ∈ D(A), then the function ˜ (t)u0 = lim (I − t A) ˜ −n u0 u˜(t) ≡ U n→∞ n ˜ −[ νt ] u0 = lim (I − ν A) ν→0

is a limit solution of the nonlinear Cauchy problem (1.6) on [0, T ] and then on [0, ∞), in the sense that it is also the uniform limit of un (t) on [0, T ], where un (t) ˜ and T > 0 is arbitrary. satisfies (2.3) on [0, T ] with A replaced by A, Furthermore, if A is embeddedly quasi-demi-closed, that is, if A satisfies the embedding condition (A3), then u˜(t) is a strong solution of (1.6) in Y , that is, u ˜(t) satisfies (1.6) in Y for almost every t. It follows readily from the proofs of Theorems 2.10 and 2.11 in Subsection 6.4 that Theorem 2.12. The results in Theorems 2.10 and 2.11 are still true if the range condition (A1) is replaced by the weaker range condition (A1) below, provided that the initial condition x ∈ D(A) is changed to the condition x ∈ D(A), and that the condition (F1) is changed to the condition (F2). Here (A1) For each samll 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1, the range of (I − λA) contains D(A).

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3. Examples Three examples will be considered. The first one is about a linear, nonhomogeneous, parabolic, initial-boundary value problem of space dimension one, and the second one is about its analogue of higher space dimensions. The last example concerns a nonlinear, nonhomogeneous, parabolic, initial-boundary value problem of space dimension one. More complex examples will be a subject of other chapters. Example 3.1. Solve for u = u(x, t): ut (x, t) = uxx (x, t) + f0 (x), (x, t) ∈ (0, 1) × (0, ∞); ux (0, t) = β0 u(0, t),

ux (1, t) = −β1 u(1, t);

(3.1)

u(x, 0) = u0 (x); where β0 and β1 are two positive constants, and ∂ ∂ u, ux (x, t) ≡ u; ∂t ∂x ∂2 uxx (x, t) ≡ u. ∂x2 Solution. Define the linear operator ut (x, t) ≡

F : D(F ) ⊂ C[0, 1] −→ C[0, 1] by F v = v for v ∈ D(F ) ≡ {w ∈ C 2 [0, 1] : w (j) = (−1)j βj w(j), j = 0, 1}. It will be shown that F is a closed operator satisfying both the dissipativity condition (B2) and the range condition (B1). As a result, the following two cases are true: Case 1: f0 (x) ≡ 0. [25] In this case, we have, from Theorem 2.1, that (3.1) has a unique solution given by t u(t) = lim (I − F )−n u0 , n→∞ n if u0 ∈ D(F ) satisfies F u0 = u0 ∈ D(F ). More smoothness of u(t) in t follows from Theorem 2.2, if we further restrict u0 . Case 2: otherwise. In this case, similar results are true if f0 (x) satisfies the requirements in Theorems 2.7 and 2.8. For example, if f0 , F u0 ∈ D(F ), then, by Theorem 2.7, the equation (3.1) has a unique classical solution t v(t) = lim (I − F˜ )−n u0 . n→∞ n ˜ ˜ Here F w = F w + f0 for w ∈ D(F ) = D(F ). We now begin the proof, which is composed of three steps. Step 1. (F satisfies the dissipativity condition (B2).) Let v1 and v2 be in D(F ), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ (0, 1), v1 − v2 ∞ = |(v1 − v2 )(x0 )|;

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(v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0. Here x0 ∈ {0, 1} is impossible, due to the boundary conditions in D(F ). For, if x0 = 0 and v1 − v2 ∞ = (v1 − v2 )(0), then (v1 − v2 ) (0) = β0 (v1 − v2 )(0) > 0, so (v1 − v2 )(0) cannot be the positive maximum. This contradicts (v1 − v2 )(0) = v1 − v2 ∞ . Other cases can be treated similarly. The dissipativity condition (B2) is then satisfied, as the calculations show: (v1 − v2 )(x0 )(F v1 − F v2 )(x0 ) ≤ 0; v1 − v2 2∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(F v1 − F v2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(F v1 − F v2 )∞ for all λ > 0. Step 2. From the theory of ordinary differential equations [5], [24, Corollary 2.13, Chapter 4], the range of (I − λF ), λ > 0, equals C[0, 1], so F satisfies the range condition (B1). Step 3. (F is a closed operator.) Let vn ∈ D(F ) converge to v, and F vn to w. Then, there is a positive constant K, such that vn ∞ ≤ K; F vn ∞ = vn ∞ ≤ K. This, together with the interpolation inequality [1], [13, page 135], implies that vn ∞ and then vn C 2 [0,1] are uniformly bounded. Hence it follows from the Ascoli-Arzela theorem [33] that a subsequence of vn and then itself converge to v . That v ∈ D(F ) and F vn = vn converges to v = F v will be true, whence F is closed. For, by uniform covergence theorem [2], x vn (y) dy + vn (0) vn (x) = y=0

converges to v (x) =

x

w(y) dy + v (0),

y=0

and vn (j) = (−1)j βj vn (j) converges to v (j) = (−1)j βj v(j),

j = 0, 1;

so v ∈ D(F ) and F v = v = w by the fundamental theorem of calculus [2]. The proof is complete.

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Example 3.2. Solve for u = u(x, t): ut (x, t) = u(x, t) + f0 (x), (x, t) ∈ Ω × (0, ∞); ∂ u(x, t) + β2 u(x, t) = 0, ∂n ˆ u(x, 0) = u0 (x);

x ∈ ∂Ω;

(3.2)

where Ω is a bounded, smooth domain in RN , and N ≥ 2 is a positive integer; ∂2 ∂ x = (x1 , x2 , . . . , xN ), u = N i=1 ∂xi u, and ut = ∂t u; ∂Ω is the boundary of Ω, and ∂∂nˆ u is the outer normal derivative of u; β2 is a positive number. Solution. Define the linear operator G : D(G) ⊂ C(Ω) −→ C(Ω) by Gv = v for v ∈ D(G) ≡ {w ∈ C 2+μ (Ω) :

∂w + β2 w = 0 ∂n ˆ

for x ∈ ∂Ω}.

Here 0 < μ < 1, is a constant. It will be shown that G satisfies both the dissipativity condition (B2) and the weaker range condition (B1) . Consequently, the following two cases are true: Case 1: f0 ≡ 0. [25] In this case, we have, from Theorem 2.3, that the quantity t u(t) = lim (I − G)−n u0 n→∞ n exists if u0 ∈ D(G). If u0 ∈ D(G2 ), further estimates will be derived in order for u(t) to the unique solution of (3.2). More smoothness of u(t) in t will then follow from Theorem 2.3 again, if we impose more restrictions on u0 . Case 2: otherwise. In this case, it will be shown that if f0 ∈ C μ (Ω), then f0 satisfies the weaker condition (F 0) . Thus, Theorem 2.9 assures the existence of the limit t ˜ −n u0 u(t) = lim (I − G) n→∞ n ˜ −[ νt ] u0 = lim (I − ν G) ν→0

˜ satisfies the for u0 ∈ D(G), where the corresponding nonlinear operator G dissipativity condition (A2) and the weaker range condition (A1) , defined ˜ ≡ D(G). Further estimates will be derived ˜ = Gv + f0 for v ∈ D(G) by Gv under additional assumptions on u0 and f0 , so that the u(t) is in fact a unique classical solution. We now begin the proof, which consists of eight steps. Step 1. (G satisfies the dissipativity condition (B2).) Let v1 and v2 be in D(G), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ Ω, v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 )(x0 ) = 0,

(the gradient of (v1 − v2 ));

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(v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0. Here x0 ∈ ∂Ω is impossible, due to the boundary condition in D(G). For, if x0 ∈ ∂Ω and v1 − v2 ∞ = (v1 − v2 )(x0 ), then ∂ (v1 − v2 )(x0 ) > 0 ∂n ˆ by the Hopf boundary point lemma [13]. But this is a contradiction to ∂ (v1 − v2 )(x0 ) = −β2 (v1 − v2 )(x0 ) < 0. ∂n ˆ The case where x0 ∈ ∂Ω and v1 − v2 ∞ = −(v1 − v2 )(x0 ) is similar. The dissipativity condition (B2) is then satisfied, as the calculations show: (v1 − v2 )(x0 )(Gv1 − Gv2 )(x0 ) ≤ 0; v1 − v2 2∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(Gv1 − Gv2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(Gv1 − Gv2 )∞ for all λ > 0. Step 2. From the theory of linear, elliptic partial differential equations [13], the range of (I −λG), λ > 0, equals C μ (Ω), so G satisfies the weaker range condition (B1) on account of C μ (Ω) ⊃ D(G). Step 3. It will be shown that ui C 3+η (Ω) , 0 < η < 1, is uniformly bounded if u0 ∈ D(G2 ), where ui = (I − νG)−i u0 is that in the discretized equation (2.1) in which A is replaced by G. Let u0 ∈ D(G) for a moment. By the dissipativity condition (B2) or using Lemma 4.2 in Section 4, we have ui − ui−1 ∞ ≤ Gu0 ∞ , Gui ∞ = (3.3) ν which, together with relation ui − u 0 =

i

(uj − uj−1 ),

j=1

yields a uniform bound for ui ∞ . Hence, a uniform bound exists for ui C 1+λ (Ω) for any 0 < λ < 1, on using the proof of (4.1) in Chapter 5. (Alternatively, it follows that uiW 2,p (Ω) is uniformly bounded for any p > 2, on using the Lp elliptic estimates [37]. Hence, so is ui C 1+η (Ω) = (I − νG)−i u0 C 1+η (Ω) , 0 < η < 1,

(3.4)

as a result of the Sobolev embedding theorem [1, 13].) This, applied to the relation Gui = (I − νG)−i (Gu0 ),

(3.5)

shows the same thing for Gui C 1+η (Ω) , if Gu0 ∈ D(G), that is, if u0 ∈ D(G2 ). Therefore, ui C 3+η (Ω) is uniformly bounded if u0 ∈ D(G2 ), on employing the Schauder global regularity theorem [13, page 111].

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Step 4. (Existence of a solution) The result in Step 3, together with the AscoliArzela theorem [33], implies that, on putting i = [ νt ], a subsequence of ui and then itself, converge to u(t) as ν −→ 0, with respect to the topology in C 3+λ (Ω) for any 0 < λ < 1. Consequently, as in (6.4), (6.5), and (6.6) in Section 6, we have eventually du(t) t = Bu(t) = lim (I − G)−n (Gu0 ); n→∞ dt n u(0) = u0 . Thus u(t) is a solution. Step 5. (Uniqueness of a solution) Let v(t) be another solution. Then, by the first and second derivative tests, we have, for x0 ∈ Ω, u(t) − v(t)∞ = |[u(t) − v(t)](x0 )|; [u(t) − v(t)](x0 ) = 0; [u(t) − v(t)](x0 ) [u(t) − v(t)](x0 ) ≤ 0. Thus it follows that d u(t) − v(t)2∞ dt d = [u(t) − v(t)]2 (x0 ) dt d = 2[u(t) − v(t)](x0 ) [u(t) − v(t)](x0 ) dt = 2[u(t) − v(t)](x0 )[Gu(t) − Gv(t)](x0 ) ≤ 0. This implies u(t) − v(t)∞ ≤ u(0) − v(0)∞ = u0 − u0 ∞ = 0, from which uniqueness of a solution results. Step 6. If f0 is in C μ (Ω), then f0 satisfies (F 0) . This is because (y + λf0 ) is in C μ (Ω) for y ∈ D(G), but C μ (Ω) is contained in the range of (I − λG), λ > 0, by Step 2. Step 7. (Further estimates under additional assumptions on f0 (x) and u0 ) We ˜ with assume additionally that f0 is in C 2+μ (Ω), and that u0 is in D(G) ˜ ˜ 0 = u0 + f0 ∈ D(G). Gu Let, for v0 ≡ u0 ∈ D(G), ˜ −i u0 vi = (I − ν G) ˜ and ui is be that in the discretized equation (2.1), in which A is replaced by G replaced by vi . For convenience, we also define ˜ 0. v−1 = (I − ν G)v ˜ is readily seen to satisfy the weaker range condition (A1) and the dissipaHere G tivity condition (A2). Hence, as in Step 3, applying (A2) or using Lemma 4.2 in

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Section 4 results in a uniform bound for vi − vi−1 ∞ ν and vi C 1+η (Ω) ;

˜ i ∞ = Gv 0 < η < 1,

i = 1, 2, . . . .

˜ 0 = Gu ˜ 0 ∈ D(G), ˜ vi satisfies On the other hand, because of Gv vi − ν[ vi + f0 (x)] = vi−1 , i = 0, 1, . . . ;

x ∈ Ω,

∂vi + β2 vi = 0, x ∈ ∂Ω, ∂n ˆ i = −1, 0, 1, . . . . Hence, it follows, on letting wi = vi for i = 0, 1, . . ., that wi − λ[ wi + f0 (x)] = wi−1 ,

x ∈ Ω,

i = 1, 2, . . . ; ∂f0 ∂wi +[ + β2 f0 ] = −β2 wi , ∂n ˆ ∂n ˆ i = 0, 1, . . . .

x ∈ ∂Ω,

Here

vi − vi−1 , λ This induces a nonlinear, dissipative operator wi − f0 (x) =

i = 0, 1, . . . .

˜ ˜˜ ⊂ C(Ω) −→ C(Ω), ˜ : D(G) G ˜ = v + f (x) for ˜ defined by Gv 0 ˜ ˜ v ∈ D(G) ∂w ∂f0 +[ + β2 f0 ] ∂n ˆ ∂n ˆ = −β2 w, x ∈ ∂Ω}.

≡ {w ∈ C 2+μ (Ω) :

Thus, as in Step 3 again, applying dissipativity condition (A2) or using Lemma 4.2 in Section 4 yields a uniform bound for ˜˜ wi + f0 (x)∞ = Gw i ∞ wi − wi−1 ∞ ; = λ wi C 1+η (Ω) , 0 < η < 1, i = 0, 1, . . . , from which so is yielded for vi C 3+η (Ω) ,

i = 0, 1, . . .

by the Schauder global regularity theorem [13, page 111]. Step 8. That, on putting i = [ νt ], u(t) = limν→0 vi is a unique calssical solution follows as in Steps 4, and 5. The proof is complete.

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Example 3.3. Solve for u = u(x, t): ut (x, t) = uxx (x, t) + f1 (x), (x, t) ∈ (0, 1) × (0, ∞); ux (0, t) ∈ (−1)j βj (u(j, t)),

j = 0, 1;

(3.6)

u(x, 0) = u0 (x); where β0 and β1 are maximal monotone graphs in R × R. This is the nonlinear analogue of the problem in Example 3.1. Here a monotone graph β is a subset of R × R that satisfies (y2 − y1 )(x2 − x1 ) ≥ 0

for yi ∈ β(xi ),

i = 1, 2.

This β is a maximal monotone graph, if it is not properly contained in any other monotone graph. In this case, for any λ > 0, (I + λβ)−1 : R −→ R is single-valued and non-expansive, as readily checked [3]. Here non-expansiveness means |(I + λβ)−1 x − (I + λβ)−1 y| ≤ |x − y| for x, y ∈ R. Solution. Define the nonlinear operator H : D(H) ⊂ C[0, 1] −→ C[0, 1] by Hv = v for v ∈ D(H) ≡ {w ∈ C 2 [0, 1] : w (j) ∈ (−1)j βj (w(j)),

j = 0, 1}.

It will be shown that H satisfies both the dissipativity condition (A2) and the range condition (A1). In consequence, the following two cases are true: Cases 1: f1 (x) ≡ 0. In this case, it follows from Theorem 2.4 that the quantity t H)−n u0 n t = lim (I − μH)−[ μ ] u0

u(t) = lim (I − n→∞ μ→0

exists if u0 ∈ D(H). H will be further shown to satisfy the embedding condition (A3) of embeddedly quasi-demi-closedness, so, by Theorem 2.5, u(t) for u0 ∈ D(H) is not only a limit solution but also a strong solution of (3.3). This u(t) will also satisfy the middle equation in (3.6). Case 2: otherwise. In this case, similar results are true if f1 (x) satisfies the requirements in Theorems 2.10 and 2.11. For instance, if u0 ∈ D(H) and f1 ∈ C[0, 1], then f1 satisfies the condition (F1). So, by Theorem 2.11, the equation (3.3) has a strong solution v(t) = lim (I − n→∞

t ˜ −n H) u0 . n

˜ = D(H). ˜ = Hw + f1 for w ∈ D(H) Here Hw

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We now begin the proof, which is comprised of four steps. Step 1 (H satisfies the dissipativity condition (A2).) Let v1 and v2 be in D(H), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ [0, 1], v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0. Here x0 ∈ {0, 1} is possible, due to the boundary conditions in D(H). For, if x0 = 0 and v1 − v2 ∞ = (v1 − v2 )(0), then the monotonicity of β0 and the positivity of (v1 − v2 )(0) implies (v1 − v2 ) (0) ≥ 0. From this, there must (v1 − v2 ) (0) = 0 because if (v1 − v2 ) (0) > occurs, then (v1 − v2 )(0) cannot be the positive maximum. Other cases can be treated similarly. The dissipativity condition (A2) is then satisfied, as the calculations show: (v1 − v2 )(x0 )(Hv1 − Hv2 )(x0 ) ≤ 0; v1 − v2 2∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(Hv1 − Hv2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(Hv1 − Hv2 )∞ for all λ > 0. Step 2. (H satisfies the range condition (A1).) [35, 15] Let h ∈ C[0, 1] be given, and let μ > be small such that 0 < μ < [log(3)]−1 . It will be shown that the equation u − μ2 u = h (3.7) has a solution u ∈ D(H). From the theory of ordinary differential equations [5, 24], the equation (3.7) has the general solution 1

1

u = ae μ x + be− μ x + up ; 1 x 1 up = − sinh[ (x − y)]h(y) dy, μ y=0 μ where a and b are two arbitrary constants, and the particular solution up is obtained from the variation of constants formula [5, 24]. We will choose suitable a and b, so that the corresponding u lies in D(H). The boundary conditions u (0) ∈ β0 (u(0)) and u (1) ∈ −β1 (u(1)) in D(H), together with the existence of (I + μβj )−1 , j = 0, 1 : R −→ R, require a and b to satisfy b = (I + μβ0 )−1 (2a) − a; 1

(I + μβ1 )−1 {2be− μ 1 1 1−y −(1 − y) + [cosh( ) − sinh( )]h(y) dy} μ y=0 μ μ

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1

= ae μ + be− μ −

1 μ

1

sinh( y=0

1−y )h(y) dy. μ

It follows that a and b meet the requirement if a is a fixed point of the nonlinear map T : R −→ R, where, for x ∈ R, 2

T x = −e− μ [(I + μβ0 )−1 (2x) − x] 1−y 1 − μ1 1 )h(y) dy sinh( + e μ μ y=0 1

1

+ e− μ (I + μβ1 )−1 {2e− μ [(I + μβ0 )−1 (2x) − x] 1 1 1−y 1−y + ) − sinh( )]h(y) dy}. [cosh( μ y=0 μ μ This T has a unique fixed point by the Banach fixed point theorem, because, for x1 , x2 ∈ R and 0 < μ < [log(3)]−1 , T is a strict contraction: 2

|T x2 − T x1 | ≤ 9e− μ |x2 − x1 | < |x2 − x1 |. Here the non-expansiveness of (I + μβj )−1 ,

j = 0, 1 : R −→ R

was used. Step 3. (H satisfies the embedding condition (A3) of embeddedly quasi-demiclosedness.) [20] Let vn ∈ D(H) converge to v in C[0, 1], and let Hvn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (0, 1) = (L2 (0, 1))∗ , η(Hv) exists and |η(Hvn ) − η(Hv)| −→ 0. Here (C[0, 1]; · ∞ ) is continuously embedded into L2 (0, 1); · ). Since vn ∞ and Hvn ∞ are uniformly bounded, so is vn C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 [0, 1] to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (0, 1), whence, by Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (0, 1), 1 (vn − v )η dx| |η(Hvn ) − η(Hv)| = | 0

−→ 0. Therefore H satisfies the embedding condition (A3). Step 4. (u(t) satisfies the middle equation in (3.6).) Consider the discretized equation ui −νHui = ui−1 , ui ∈ D(H),

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where i = 1, 2, . . ., ν > 0 satisfies νω < 1, and ui = (I − νH)−i u0 exists uniquely by the range condition (A1) and the dissipativity condition (A2) (see Section 4). On putting i = [ νt ], it follows that t

lim ui = lim (I − νH)−[ ν ] u0 = u(t).

ν→0

ν→0

On the other hand, by utilizing the dissipativity condition (A2), we have ui − ui−1 ui ∞ = Hui ∞ = ∞ ν ≤ Hu0 ∞ . This, combined with the relation ui − u0 =

i

(uj − uj−1 ),

j=1

yields a bound for ui∞ . Those, in turn, result in a bound for ui C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge to a limit in C 1 [0, 1], as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (3.6), as ui does so. The proof is complete. 4. Some Preliminary Results Let the nonlinear, multi-valued operator A in Section 1 satisfy the range condition (A1) and the dissipativity (A2). Let Dμ be the range of (I − μA) where μ ∈ R. For x ∈ Dμ , let Jμ be such that Jμ x = (I − μA)−1 x. In this section, some properties of Jμ will be explored, with the aid of which a recursive inequality will be established. Then, by using a basic theory of elementary difference equations in Section 5, we solve this inequality in Section 6. Once this inequality is solved, the proof of the main results follows. Lemma 4.1. Let μ > 0 be such that μω < 1. Then the function Jμ = (I − μA)−1 : Dμ −→ D(A) is single-valued, and for x, y ∈ Dμ , the inequality Jμ x − Jμ y ≤ (1 − μω)−1 x − y is true. Proof. Let x, y ∈ Dμ and let v ∈ Jμ x, w ∈ Jμ x. Then v = w, so Jμ is single-valued. This is because v − μAv x and w − μAw x, from which (v − w) −

μ [(A − ω)v − (A − ω)w] 0. 1 − μω

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By virtue of the dissipativity condition (A2), we have v − w ≤ 0, giving v = w. Similarly, let u = Jμ y, and the desired inequality follows. This is because v − μAv x and u − μAu y, whence (v − u)−μ(1 − μω)−1 [(A − ω)v − (A − ω)u]

(1 − μω)−1 (x − y). Using the dissipativity condition (A2), it follows that v − u ≤ (1 − μω)−1 x − y. Lemma 4.2. Let μ > 0 be such that μω < 1. Then, for n ∈ N and x ∈ D(A)∩Dμn where Dμn is the range of (I − μA)n , the inequalities Jμ x − x ≤ μ(1 − μω)−1 |Ax|; Jμn x − Jμn−1 x ≤ μ(1 − μω)−n |Ax|; nμ(1 − μω)−n |Ax|, if ω ≥ 0; Jμn x − x ≤ nμ|Ax|, if ω ≤ 0; are true, where |Ax| ≡ inf{y : y ∈ Ax}. Proof. Let y ∈ Ax. Then x − μy = x − μy ∈ Dμ , so by Lemma 4.1, we have x = Jμ (x − μy), Jμ x − x = Jμ x − Jμ (x − μy) ≤ (1 − μω)−1 y. Since y ∈ Ax is arbitrary, the first desired inequality follows. The second and third inequalities are proved by repeated use of Lemma 4.1. For, Jμn x − Jμn−1 x ≤ (1 − μω)−1 Jμn−1 x − Jμn−2 x ≤ · · · ≤ μ(1 − μω)−n |Ax|; and Jμn x − x ≤ Jμn x − Jμn−1 x + Jμn−1 x − Jμn−2 x + · · · + Jμ x − x ≤ (1 − μω)−(n−1) Jμ x − x + (1 − μω)−(n−2) Jμ x − x + · · · + Jμ x − x

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4. SOME PRELIMINARY RESULTS

≤

nμ(1 − μω)−n |Ax|, nμ|Ax|,

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21

if ω ≥ 0; if ω ≤ 0.

The proof is complete.

Lemma 4.3 (Nonlinear resolvent identity). Let λ, μ > 0 be such that λω, μω < 1. Then for x ∈ Dλ , the quantity λ−μ μ x+ Jλ x λ λ lies in Dμ , and the identity μ λ−μ Jλ x = Jμ ( x + Jλ x) λ λ is true. Proof. Let x ∈ Dλ , for which there is an x0 ∈ D(A) and a y0 ∈ Ax0 , such that x0 − λy0 = x. Hence x0 = Jλ x, and μ λ−μ λ−μ μ x+ Jλ x = (x0 − λy0 ) + x0 λ λ λ λ = x0 − μy0 ∈ Dμ . Consequently,

λ−μ μ Jλ x). Jλ x = x0 = Jμ ( x + λ λ Here the functions Jλ and Jμ are single-valued by Lemma 4.1.

Lemma 4.4. Let λ ≥ μ > 0 be such that λω, μω < 1. Then, for x ∈ Dλ ∩ Dμ , (1 − λω)Aλ x ≤ (1 − μω)Aμ x holds. Here Aλ denotes the operator λ−1 (I − Jλ ). Proof. [8] The result follows from this calculation x − Jλ x x − Jμ x Jμ x − Jλ x ≤ + λ λ λ 1 μ μ λ−μ ≤ Aμ x + Jμ x − Jμ ( x + Jλ x) λ λ λ λ μ λ−μ x − Jλ x (1 − μω)−1 ≤ Aμ x + λ λ λ μ λ−μ = Aμ x + (1 − μω)−1 Aλ x. λ λ Here Lemmas 4.1 and 4.3 were used. Aλ x =

Remark 4.5. By the assumption (A1), the range of (I − λA) contains D(A) for small 0 < λ < λ0 . Thus it follows from Lemma 4.4 that, for x ∈ D(A), the limit x − Jλ x λ→0 λ→0 λ exists and can equal ∞. This will be used in Chapter 2. lim Aλ x = lim

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Lemma 4.6. Let λ ≥ μ > 0 be such that λω, μω < 1. Then for x ∈ Dλm ∩ Dμn and n, m ∈ N, the inequality am,n ≤ γαam−1,n−1 + γβam,n−1 is true, where am,n ≡ Jμn x − Jλm x, γ ≡ (1 − μω)−1 ; μ α ≡ , β ≡ 1 − α. λ Proof. Using Lemmas 4.3 and 4.1, it follows that μ λ−μ m Jλ x) am,n = Jμn − Jμ ( Jλm−1 x + λ λ μ λ−μ m Jλ x) ≤ (1 − μω)−1 Jμn−1 x − ( Jλm−1 x + λ λ μ λ − μ n−1 Jμ x − Jλm x} ≤ (1 − μω)−1 { Jμn−1 x − Jλm−1 x + λ λ = γαam−1,n−1 + γβam,n−1 . Lemma 4.7. Let α, β > 0 be such that α + β = 1, and let positive integers n, m satisfy n ≥ m. Then the inequality m n j n−j (m − j) ≤ (nα − m)2 + nαβ) α β j j=0 is true. Proof. The proof will be divided into three steps. Step 1. Using the Schwartz inequality, we have m n n j n−j n j n−j (m − j) ≤ |m − j| α β α β j j j=0 j=0

⎛ ⎞ 12 ⎛ ⎞ 12 n n n n ≤⎝ αj β n−j ⎠ ⎝ αj β n−j (m − j)2 ⎠ . j j j=0 j=0

(4.1)

Step 2. The relations are true: n n j n−j = (α + β)n ; α β j j=0 n n jαj β n−j = αn(α + β)n−1 ; j j=0 n n 2 j n−j j α β = α2 n(n − 1)(α + β)n−2 + αn(α + β)n−1 . j j=0 The first relation is the binomial theorem, the second follows from the differentiation of the first, with respect to α, and the third is the result of differentiating the second, with respect to α. Step 3. The relations in Step 2, together with α + β = 1, applied to the right side of (4.1), complete the proof.

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The material of this section is taken from Crandall-Liggett [6, pages 268-271]. 5. Difference Equations Theory We now introduce a basic part of the theory of difference equations [28]. Let {bn} = {bn }n∈{0}∪N = {bn }∞ n=0 be a sequence of real numbers. For such a sequence {bn }, we further extend it by defining bn = 0, if n = −1, −2, . . .. The set of all such sequences {bn}’s will be denoted by S. Thus, if {an } ∈ S, then 0 = a−1 = a−2 = · · · . Define a right shift operator E : S −→ S by E{bn } = {bn+1 } Similarly, define a left shift operator E

#

for {bn } ∈ S.

: S −→ S by

#

E {bn } = {bn−1 }

for {bn } ∈ S.

For c ∈ R and c = 0, define the operator (E − c)∗ : S −→ S by (E − c)∗ {bn } = {cn

n−1 i=0

bi } ci+1

for {bn} ∈ S. Here the first term on the right side of the equality, corresponding to n = 0, is zero. One more definition is that define, for {bn} ∈ S, (E − c)i∗ {bn } = [(E − c)∗ ]i {bn}, E i# {bn } = (E # )i {bn },

i = 1, 2, . . . ;

i = 1, 2, . . . ;

0

(E − c) {bn } = {bn}. Based on those definitions above, we derive the following results. It will be seen from below that (E − c)∗ acts approximately as the inverse of (E − c). Lemma 5.1. Let {bn } and {dn } be in S. Then the following are true: (E − c)∗ (E − c){bn } = {bn − cn b0 }; (E − c)(E − c)∗ {bn} = {bn }; (E − c)∗ {bn } ≤ (E − c)∗ {dn },

if c > 0 and {bn} ≤ {dn }.

Here {bn } ≤ {dn } means bn ≤ dn for n = 0, 1, 2, . . .. Proof. The proof follows from straightforward calculations.

Proposition 5.2. Let ξ, c ∈ R be such that c = 1 and c = 0. Let, be in S, the n ∞ ∞ three sequences {n}∞ n=0 , {c }n=0 , and {ξ}n=0 of real numbers. Then the following identities are true: n cn 1 (E − c)∗ {n} = { − 2 + 2 }; d d d ξ ξcn ∗ (E − c) {ξ} = { − }; d d

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n n−i c }. (E − c) {c } = { i i∗

n

Here d = 1 − c and i = 0, 1, 2, . . .. Proof. Since, by definition, ∗

(E − c) {n} = {c

n

n−1 j=0

= {c

n−1

j } cj+1

[c−1 + 2c−2 + · · · + (n − 1)c−(n−1) ]}

d −1 [c + c−2 + · · · + c−(n−1) ]}, dc the first identity follows, using the formula for a finite geometric series. Because (E − c)∗ {ξ} = {ξcn (c−1 + c−2 + · · · + c−n )}, the second identity follows. Finally, we use mathematical induction to prove the third identity. This identity is true for i = 0, 1, due to the calculations

n }; (E − c)0 {cn } = {cn } = {cn−0 0 = {cn−1 (−1)c

(E − c)∗ {cn } = {cn

n−1 j=0

cj

} = {ncn−1 }

cj+1

n n−1 }. c ={ 1 Hence, by assuming it holds for i = k < n, we shall show that it continues to hold for i = k + 1 < n. The calculations (E − c)(k+1)∗ {cn } = (E − c)∗ (E − c)k∗ {cn }

n n−k ∗ c } = (E − c) { k n−1 cj−k j n k = {c } cj+1 j=0 = {c

n−k−1

j }, k

n−1

j=0

together with the standard combinatorics identity [4, page 79] or [27, page 52]

n+1 n r+1 r = + ···+ + r+1 r r r for r, n ∈ N and n ≥ r, imply that the third identity holds for i = k + 1.

Proposition 5.3. Let ξ, c ∈ R be such that c = 1 and cξ = 0. Let, be in S, n ∞ n ∞ the three sequences {nξ n }∞ n=0 , {ξ }n=0 , and {(cξ) }n=0 of real numbers. Then the identities are true: ξn nξ n cn ξ n 1 − 2 + 2 ) }; (E − cξ)∗ {nξ n } = {( d d d ξ

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6. PROOF OF THE MAIN RESULTS

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cn ξ n 1 ξn ) }; (E − cξ)∗ {ξ n } = {( − d d ξ

n (cξ)n−i }. (E − cξ)i∗ {(cξ)n } = { i Here d = 1 − c and i = 0, 1, 2, . . .. Proof. Observe that the third identity in the Proposition 5.3 is the same as that in the Proposition 5.2, so no proof is needed for it. Thanks to the calculations (E − cξ)∗ {nξ n } = {(cξ)n

n−1 j=0

= {ξ

n−1

(E − c)∗ n};

(E − cξ)∗ {ξ n } = {(cξ)n

n−1 j=0

= {ξ

n−1

jξ j } (cξ)j+1

ξj } (cξ)j+1

(E − c)∗ (1)},

the first and second identities follow from applying Proposition 5.2.

The material of this section is taken from our article [21]. 6. Proof of the Main Results Within this section, it suffices to consider ω ≥ 0, in view of Lemma 4.2. This will be seen from the following proofs. The nonlinear homogeneous case will be proved first, because, once this is done, the remaining cases become clearer. 6.1. The Nonlinear Homogeneous Case. We shall use the notions of difference equations introduced in Section 5. For a doubly indexed sequence {ρm,n } = {ρm,n }∞ m,n=0 ∈ S × S of real numbers, define E1 {ρm,n } = {ρm+1,n }; E2 {ρm,n } = {ρm,n+1 }. Thus, E1 is the right shift operator acting on the first index m, and E2 is the right shift operator acting on the second index n. It is readily seen that E1 and E2 commute: E1 E2 {ρm,n} = E2 E1 {ρm,n }. Theorems 2.4 and 2.5 will be proved after the following Lemma 6.1 and Proposition 6.2. Lemma 6.1. Under the assumptions in Lemma 4.6, the inequalities are true: {am,n } ≤ (αγ(E2 − βγ)∗ )m {a0,n } +

m−1

(γα(E2 − γβ)∗ )i {(γβ)n am−i,0 }

i=0

{am,n } ≤ (γβ + γαE1# )n {am,0 }

for n ≥ m;

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1. EXISTENCE THEOREMS FOR CAUCHY PROBLEMS n n n−i i i# β α E1 {am,0 } i i=0 n n n−i i n α am−i,0 } for m ≥ n. = {γ β i i=0

= γn

Proof. From Lemma 4.6, we have E1 (E2 − βγ){am,n } ≤ {αγam,n }, ∗

so applying (E2 − βγ) to both sides of the above and using Lemma 5.1, we readily derive {am+1,n } = E1 {am,n} ≤ αγ(E2 − βγ)∗ {am,n } + (γβ)n E1 {am,0 }. This recursive relation will give the first inequality immediately. On the other hand, if applying E1# instead to both sides, we have, on using Lemma 5.1, {am,n+1 } = E2 {am,n } ≤ (γβ + γαE1# ){am,n }. This recursive relation will easily deliver the second inequality.

Proposition 6.2. Under the assumptions in Lemma 4.6 with λ0 > λ ≥ μ, the inequality is true: am,n ≤ [(nμ − mλ) + (nμ − mλ)2 + nμ(λ − μ)]γ n |Ax| + γ n (1 − λω)−m (nμ − mλ)2 + nμ(λ − μ)|Ax| for n, m ≥ 0, where |Ax| = inf{y : y ∈ Ax}. Proof. The proof will be divided into two cases. Case 1: n ≥ m. By applying Lemmas 4.2, 5.1, and 6.1, and by making repeated use of Proposition 5.3 and carefully grouping terms in which tedious but not difficult calculations are involved, we have, for n ≥ m, {a0,n } ≤ {nμγ n |Ax|}; nγ n 1 mγ n 1 ((E2 − βγ)∗ )m {nγ n } = { m m − m+1 m α γ α γ m−1 n β n−i 1 + (m − i) m γ n }; m+1−i i α γ i=0 m−1

(γα(E2 −γβ)∗ )i {(γβ)n am−i,0 }

i=0

≤ {γ n (1 − λω)−m

n i n−i αβ (m − i)λ|Ax|}. i

m−1

i=0

This, combined with Lemma 4.7, gives the desired result. Case 2: m ≥ n. Applying Lemmas 4.2, 5.1, and 6.1, we also have, for m ≥ n,

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{am,n } −m

≤ {γ (1 − λω) n

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27

n n n−i i β α [(n − i) + (m − n)]λ|Ax|} i i=0

≤ {γ n (1 − λω)−m (mλ − nμ)|Ax|} after some calculations. Cases 1 and 2 complete the proof.

Proposition 6.3. Under the assumptions in Lemma 4.6 with λ0 > λ ≥ μ, the inequality, if ω = 0, is true for n, m ≥ 0: am,n ≤ [(nμ − mλ) + 2 (nμ − mλ)2 + nμ(λ − μ)]|Ax|. Proof. By letting ω = 0 in Proposition 6.2, the result follows.

We are now ready for Proof of Theorem 2.4: Proof. We divide the proof into three steps, where Step 1 consists of two cases. Step 1. (The existence of U (t)x for x ∈ D(A)) Case 1: x ∈ D(A). From Proposition 6.2, it follows that J nt x − J m t x, n

the am,n with μ = n ≥ m −→ ∞. Thus

t n

≤ λ =

t m

m

< λ0 , converges to zero for finite t ≥ 0, as

t −n A) x n for x ∈ D(A) is a Cauchy sequence in X, so the limit J nt x = (I − n

lim (I −

n→∞

t −n A) x n

exists for finite t ≥ 0. Case 2: x ∈ D(A). Let xl ∈ D(A) be such that xl −→ x as l −→ ∞. Then, using Lemma 4.1, we have, for nt ≤ mt < λ0 , n n n m J nt x − J m t x ≤ J t x − J t xl + J t xl − J t xl n

m

n

n

n

m

m + J m t xl − J t x m

m

t ≤ (1 − ω)−n x − xl + J nt xl − J m t xl n m n t + (1 − ω)−m xl − x, m and this, together with Case 1, implies, for finite t ≥ 0, 0 ≤ lim sup J nt x − J m t x n,m→∞

n

m

≤ 2eωt xl − x for all l. Hence, letting l −→ ∞, we deduce n m lim J nt x − J m t x = lim sup J t x − J t x

n,m→∞

n

m

n,m→∞

= 0,

n

m

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that is, (I − nt A)−n x for x ∈ D(A) is a Cauchy sequence in X. Therefore, the limit U (t)x ≡ lim (I − n→∞

exists for finite t ≥ 0. Step 2. On setting μ = λ =

t n

t −n A) x n

and m = [ μt ] in Proposition 6.2, it follows from

the Case 2 above that, for x ∈ D(A),

t

U (t)x = lim (I − μA)−[ μ ] x. μ→0

Here [a] for each a ∈ R is the greatest integer that is less than or equal to a. Step 3. (The continuity and Lipschitz continuity of U (t)x, [6, page 272], [30, page 136]) For x ∈ D(A), let xl ∈ D(A) be such that xl −→ x as l −→ ∞. Then as in the Case 2 above, we have, for μ ≤ λ < λ0 , Jμn x − Jλm x ≤ (1 − μω)−n x − xl + Jμn xl − Jλm xl + (1 − λω)−m xl − x. Hence, on setting n = [ μt ], m = [ λτ ] in the inequality in Proposition 6.2 and letting μ, λ −→ ∞, where t, τ ≥ 0, it follows that U (t)x − U (τ )x ≤ (eωt + eωτ )xl − x + |t − τ |[2eωt + eω(t+τ ) ]|Axl | for all l. The continuity of U (t)x in t results, if we let t −→ τ first and then l −→ ∞ next. However, if x ∈ D(A), then the Lipschity continuity of U (t)x is a consequence of setting xl = x for all l and letting l −→ ∞. Next, Proof of Theorem 2.5: Proof. We divide the proof into five steps. Step 1. By Lemma 4.1, the ui in (2.1) satisfies ui − ui−1 ≤ (1 − νω)−i νv0 , where ν = Tn satisfies ν < λ0 , (1 − νω)−i is uniformly bounded by some K > 0 for all large n ∈ N and i = 1, 2, . . . , n, and v0 ∈ Au0 is such that the new element u−1 ≡ u0 − νv0 is defined. This is because ui − ui−1 = Jν ui−1 − Jν ui−2 . Step 2. (2.3) is an immediate consequence of Step 1 and (2.2). Step 3. By letting μ = λ = nt and m = [ μt ] in Proposition 6.2, it is readily seen from the proof of Theorem 2.4 that lim (I −

n→∞

t t −n A) u0 = lim (I − μA)−[ μ ] u0 . μ→0 n

Here for each a ∈ R, [a] is the greatest integer that is less than or equal to a.

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Step 4. Claim that u(t) is the uniform limit of un (t) on [0, T ], where T > 0 is arbitrary. For each t ∈ [0, T ), we have t ∈ [ti−1 , ti ) for some i, so i − 1 = [ νt ]. Thus it follows from Step 3 that, for each t ∈ [0, T ), ui−1 = (I − νA)−(i−1) u0 t

= (I − νA)−[ ν ] u0 converges, as ν = Tn −→ 0. But the pointwise convergence of ui−1 is the same as that of un (t), on account of Step 1 and (2.2). Hence, the Ascoli-Arzela theorem [33] will prove that u(t) is the uniform limit of un (t) on [0, T ], as n −→ ∞. This is because Step 1 applied to (2.2) gives un (t) − un (τ ) ≤ Kv0 |t − τ | for t, τ ∈ [0, T ], so un (t) is equi-continuous in C([0, T ]; X), the real Banach space of continuous functions from [0, T ] to X. Step 5. (Concerning a strong solution) Let vn (t) ∈ Aχn (t) be such that (2.3) gives dun (t) = vn (t) dt for t ∈ (ti−1 , ti ]. Integrating (2.3) yields that, for each φ ∈ Y ∗ ⊂ X ∗ , t φ(v n (τ )) dτ φ(un (t) − u0 ) = 0 t ∈ φ( Aχn (τ ) dτ ) =

0

t

φ(Aχn (τ )) dτ,

0

where supt∈[0,T ] vn (t) ≤ K by Step 1. Since un (t) −→ u(t) uniformly for bounded t and A is embeddedly quasi-demi-closed, we have that φ(vn (t)) converges to φ(v(t)) through some subsequence for some v(t) ∈ Au(t). It then follows from the Lebesgue convergence theorem that t φ(u(t) − u0 ) = φ(v(τ )) dτ 0 t v(τ ) dτ ), = φ( 0

t

so u(t) − u0 = 0 v(τ ) dτ in Y . On employing the Radon-Nikodym type theorem [30, pages 10-11], [33], we see at once that du(t) = v(t) ∈ Au(t) in Y dt for almost every t; u(0) = u0 .

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6.2. The Linear Homogeneous Case. Define the operator B as in Section 1, and use the same quantities Jμn x, Jλm x, μ, λ, β, α as in Section 4 but with the operator B replacing A. Letting am,n = Jμn x − Jλm x for μ ≤ λ < λ0 and for x ∈ D(B), it follows from the resolvent identity in Lemma 4.3 that am,n = Jμ [Jμn−1 x − (αJλm−1 x + βJλm x)] = Jμ (αam−1,n−1 + βam,n−1 ). This is the same as E1 (E2 − Jμ β){am,n } = Jμ α{am,n }

(6.1)

if used are the right shift operators defined below. The right shift operators E1 , E2 and the left shift operator E1# are as in Subsection 6.1 but act on sequences in X. Thus E1 , E2 , E1# : S × S −→ S × S; S

is the set of all sequences in X;

E1 {ρm,n} = {ρm+1,n }, and

E1# {ρm,n }

E2 {ρm,n } = {ρm,n+1 },

= {ρm−1,n } for {ρm,n }∞ m,n=0 ∈ S × S.

Here each sequence in S has value zero for negative integer indices. Theorems 2.1 and 2.2 will be proved after the following Lemma 6.4 and Proposition 6.5. Lemma 6.4. The recursive realations for am,n are true: {am,n } = αm Jμm (E2 − βJμ )m∗ {a0,n } +

m−1

(αJμ (E2 − Jμ β)∗ )i (Jμ β)n {am−i,0 }

i=0

for n ≥ m; {am,n } = [Jμ (β + αE1# )]n {am,0 } n n n−i i i# β = Jμn α E1 {am,0 } i i=0 n n n−i i n β α am−i,0 } for m ≥ n. = {Jμ i i=0 Proof. Applying (E2 − Jμ β) and E1# , respectively, to (6.1), the relations for am,n become {am+1,n } = E1 {am,n } = αJμ (E2 − βJμ )∗ {am,n } + E1 (Jμ β)n {am,0 }; {am,n+1 } = E2 {am,n } = Jμ (β + αE1# ){am,n}. Here, in the way similar to Section 5, (E2 − βJμ )∗ : S −→ S is defined by (E2 − βJμ )∗ {bn } ≡ {

n−1 i=0

Jμn−(i+1) β n−(i+1) bi }

(6.2)

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31

for a sequence {bn }∞ n=0 ∈ S, and is readily seen to commute with E1 and satisfy (E2 − βJμ )∗ (E2 − βJμ ){bn } = {bn − Jμn β n b0 }; (E2 − βJμ )(E2 − βJμ )∗ {bn } = {bn }.

Thus, (E2 − βJμ )∗ acts approximately as the inverse of (E2 − βJμ ), and (E2 − βJμ )m∗ ≡ ((E2 − βJμ )∗ )m and E1i# ≡ (E1# )i for m ∈ N ∪ {0} are defined in an obvious way. The results follow from the recursive relations in (6.2).

Thus, as in Proposition 6.2, we have the estimate Proposition 6.5. am,n

(nμ − mλ)2 + nμ(λ − μ)]γ n |Bx| + M 3 γ n (1 − λω)−m (nμ − mλ)2 + nμ(λ − μ)|Bx|

≤ M 3 [(nμ − mλ) +

for n, m ≥ 0, where |Bx| = inf{y : y ∈ Bx} = Bx. Proof. The proof will be divided into two cases. Case 1: n ≥ m. Simple calculations show (E2 − βJμ )2∗ {a0.n } n−1 i 1 −1 β n−(i1 +1) Jμn−i0 −1 β i1 −(i0 +1) a0,n , = i1 =0

i0 =0

and repeating such calculations yields immediately (E2 − βJμ )m∗ {a0,n } ={

n−1

β n−(im−1 +1) · · ·

im−1 =0

×

i 1 −1

i 2 −1

β i2 −(i1 +1) (6.3)

i1 =0

Jμn−i0 −m β i1 −(i0 +1) a0,n }.

i0 =0

On account of (6.3), it follows that αm Jμm (E2 − βJμ )m∗ a0,n ≤ M (γα)m

n−1 im−1 =0

×

i 1 −1

β n−(im−1 +1) · · ·

i 2 −1 i1 =0

γ n−i0 −m β i1 −(i0 +1) a0,n

i0 =0 3

≤ M (γα)m (E2 − γβ)m∗ nγ n μ|Bx|,

β i2 −(i1 +1)

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where used were the mixture condition (B3) and the inequality a0,n ≤

n

Jμi x − Jμi−1 x

i=1

≤ M γ n−1 nJμ x − x by (B3) ≤ M 2 γ n nμ|Bx|. Similarly, it is readily deduced that

m−1

(αJμ (E2 − Jμ β)∗ )i (Jμ β)n am−i,0

i=0

≤

m−1

M (γα(E2 − γβ)∗ )i (γβ)n am−i,0

i=0

≤

m−1

M 3 (1 − λω)−m (γα)i (E2 − γβ)i∗ (m − i)λ|Bx|.

i=0

Hence, the result follows if the proof of Proposition 6.2 is applied to the first inequality in Lemma 6.4. Case 2: m ≥ n. The second inequality in Lemma 6.4 yields, on employing the mixture condition (B3), am,n = M γ n

n

β n−i αi am−i,0

i=0

≤ M 3 γ n(1 − λω)−m

n

β n−i αi (m − i)λ|Bx|.

i=0

Therefore, it follows again from the proof of Proposition 6.2 that am,n ≤ M 3 γ n (1 − λω)−m (mλ − nμ)|Bx|.

Cases 1 and 2 complete the proof. We are now in a position to do Proof of Theorem 2.1:

Proof. We divide the proof into three steps. Step 1. (The existence, continuiy, and Lipschitz continuity of S(t)x) Thanks to Proposition 6.5, it follows from the proof of Theorem 2.4 that S(t)x ≡ lim (I − n→∞

t −n B) x n

exist for each x ∈ D(B) and for bounded t ≥ 0. The continuity and Lipschitz continuity of S(t)x in t ≥ 0, respectively, for x ∈ D(B) and x ∈ D(B), also follows from that proof, where the mixture condition (B3) was used. Step 2. (The existence of a solution) Replace the nonlinear, multi-valued operator A by the linear operator B, and replace and ∈ by =, in the equations

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(2.1), (2.2), and (2.3). It follows from Steps 1 to 5 in the proof of Theorem 2.5 that t S(t)x = lim (I − B)−n x n→∞ n t = lim (I − μB)−[ μ ] x for x ∈ D(B); μ→0

t

lim ui−1 = lim (I − νB)−[ ν ] u0

ν→0

(6.4)

ν→0

= S(t)u0 for each t ∈ [ti−1 , ti ); t Bχn (τ ) dτ ; un (t) − u0 = τ =0

and that un (t) converges uniformly in t ∈ [0, T ], to u(t) ≡ S(t)u0 . Here u0 ∈ D(B) and Bu0 ∈ D(B). Because B is linear, we obtain ui − ui−1 Bui = ν (I − νB)−1 − I = (I − νB)−(i−1) [ ]u0 ν −i = (I − νB) (Bu0 ). Hence, on employing the closedness of B and the mixture condition (B3), there result Bui or Bχn (t) −→ S(t)Bu0 = Bu(t); t

Bui = Bχn (t) ≤ M (1 − νω)−[ ν ] Bu0 , uniformly bounded. Consequently, the Lebesgue dominated convergence theorem [33] implies that the last equation in (6.4) converges to, as n −→ ∞, t Bu(τ ) dτ. (6.5) u(t) − u0 = τ =0

Since Bu(t) = S(t)Bu0 is continuous in t for Bu0 ∈ D(B) by Step 1, the fundamental theorem of calculus applied to (6.5) yields du(t) = Bu(t) = S(t)Bu0 , dt u(0) = u0 .

t>0

(6.6)

Thus u(t) is a solution of (1.1). Step 3. (Uniqueness of a solution, [17, pages 481-482], [14, page 83]) Let v(t) be another solution of (1.1). Then it will follow that v(t) = S(t)u0 , proving uniqueness. For, by using Step 1 and the mixture condition (B3), the calculations, for nt < λ0 and for x ∈ D(B), S(t)x t n t B) x + (I − B)−n x n n t t ≤ S(t)x − (I − B)−n x + M (1 − ω)−n x n n for all large n

≤ S(t)x − (I −

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−→ M etω x

as n −→ ∞

show that S(t) is a bounded operator on D(B). Hence the product rule for differentiation can be used to give, for 0 ≤ s ≤ t, d [S(t − s)v(s)] ds = (−1)S(t − s)Bv(s) + S(t − s)

d v(s) ds

= 0. Thus S(t − s)v(s) is constant in s, from which v(t) = S(t)v(0) = S(t)u0 results, on setting s = t and s = 0, respectively. Finally, Proof of Theorem 2.2: Proof. Following the proof of Theorem 2.1, the results are a consequence of applying Step 1 in the proof of Theorem 2.1 to (6.6). 6.3. The Linear Nonhomogeneous Case. Use the quantities in Subsection 6.2: Jμn x = (I − μB)−n x; λ ≥ μ > 0, λω < 1;

Jλm x = (I − λB)−m x; α, β > 0, α + β = 1.

Here x ∈ D(B). We begin Proof of Theorem 2.7: Proof. We divide the proof into five steps. ˜ satisfies the range condition (A1). For, Step 1. As a nonlinear operator, B the equation ˜ = w, u − λBu ˜ = D(B), is the same as the equation where w is a given element in D(B) u − λBu = w + λf0 , and f0 satisfies the condition (F0). Here λ < λ0 . Step 2. Since the linear operator B satisfies the dissipativity condition (B2) or ˜ −1 w is single-valued the mixture condition (B3), it follows from Step 1 that (I − λB) for w ∈ D(B), and that ˜ −1 w1 ˜ −1 w2 − (I − λB) (I − λB) = (I − λB)−1 (w2 − w1 ) for w1 , w2 ∈ D(B). Here 0 < λ < λ0 . Thus, a corollary of this is that, if B is ˜ dissipative, then so is B. Step 3. Let a ˜m,n = J˜μn x − J˜λm x for μ ≤ λ < λ0 and for x ∈ D(B). Thanks to Step 2 and the resolvent identity in Lemma 4.3, we have a ˜m,n = Jñ x − J˜μ (αJ˜m−1 x + β J˜m x)] μ

λ

λ

am−1,n−1 + Jμ β˜ am,n−1 . = Jμ α˜

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Thus Lemma 6.4 and Proposition 6.5 and then Step 1 in the proof of Theorem 2.1 can be carried over to show the existence, continuity, and Lipschitz continuity of t ˜ −n ˜ x S(t)x ≡ lim (I − B) n→∞ n t ˜ −[ ν ] x. = lim (I − ν B) ν→0

˜ ia a nonlinear operator, we have, for u0 ∈ D(B) with Step 4. Although B Bu0 ∈ D(B), for 0 < ν < λ0 , and for f0 ∈ D(B), ˜ i = ui − ui−1 Bui + f0 = Bu ν 1 ˜ −1 ui−1 − 1 (I − ν B) ˜ −1 ui−2 = (I − ν B) ν ν 1 = (I − νB)−1 (ui−1 − ui−2 ) ν 1 = (I − νB)−(i−1) (u1 − u0 ) ν ˜ 0 ) = (I − νB)−[ νt ] (Bu0 + f0 ) = (I − νB)−i (Bu −→ S(t)(Bu0 + f0 ) as ν −→ 0, as is the case for B. Hence, Step 2 in the proof of Theorem 2.1 can be carried over ˜ to obtain a classical solution u ˜(t) ≡ S(t)u 0 to the linear, nonhomogeneous equation (1.5). Step 5. (Uniqueness of a solution) Let v1 and v2 be two solutions of the equation (1.5). Then, by substraction, d ˜ 1 − Bv ˜ 2 (v1 − v2 ) = Bv dt = B(v1 − v2 ), t > 0; (v1 − v2 )(0) = 0, so ˜ v1 ≡ v2 ≡ u ˜(t) ≡ S(t)u 0 by uniqueness of a solution for equation (1.1).

We next present Proof of Theorem 2.8: Proof. In view of d ˜ u˜(t) = S(t)(Bu0 + f0 ), u˜(t) = B dt a consequence of Step 4 in the above proof of Theorem 2.7, the proof is an immediate consequence of the proof of Theorem 2.2. 6.4. The Nonlinear Nonhomogeneous Case. The proof of Theorems 2.10 and 2.11 will follow at once from the proof of Theorems 2.4 and 2.5. This is because the nonlinear operator A˜ is readily seen to satisfy the range condition (A1) and the dissipativity condition (A2). The material of this section is based on our articles [20, 21].

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CHAPTER 2

Existence Theorems for Evolution Equations (I) 1. Introduction In this chapter, nonlinear evolution equations, which extend those in Chapter 1, will be investigated. In a way similar to Chapter 1, those equations will be solved again with the aid of elementary difference equations. The obtained results will be applied to solve simple, initial-boundary value problems for parabolic, partial differential equations with time-dependent coefficients. More applications to solving more general, parabolic partial differential equations with time-dependent coefficients will be given in Chapters 5 and 6. Let (X, · ) be a real Banach space with the norm · , and let T > 0 and ω be two real constants. Consider the nonlinear evolution equation du(t) ∈ A(t)u(t), dt u(s) = u0 ,

0 ≤ s < t < T,

(1.1)

where A(t) : D(A(t)) ⊂ X −→ X is a nonlinear, time-dependent, and multi-valued operator. Since A(t) depends on t and is multi-valued, this equation extends those in Chapter 1. To solve the evolution equation (1.1), let its approximate equation, a difference equation, be looked at first [8, Page 72] u (t) − u (t − ) ∈ A(t)u (t), u (s) = u0 .

0 ≤ s < t < T,

(1.2)

Here > 0 is very small. This approximate equation (1.2) is reduced to the equation u (t) = [I − A(t)]−1 u (t − ),

0 ≤ s < t < T,

u (s) = u0 ,

(1.3)

if the quantity J (t) ≡ [I − A(t)]−1 can be defined. Here I is the identity operator. Thus when = equation (1.3) is readily seen to have the solution

t−s n

u (t) = [I − A(s + n)]−1 [I − A(s + (n − 1))]−1 · · · [I − A(s + )]−1 u0 ≡

n

[I − A(s + i)]−1 u0 ,

i=1 37

for n ∈ N, the

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2. EXISTENCE THEOREMS FOR EVOLUTION EQUATIONS (I)

provided that the quantity n n J (s + i)u0 ≡ [I − A(s + i)]−1 u0 i=1

i=1

can be defined for each 0 ≤ s < t < T and for each n = 1, 2, . . .. As a result, the evolution equation (1.1) might be solved by taking the limit, as n −→ ∞, of the quantity n J (s + i)u0 . i=1

But this will be true under suitable assumptions on the evolution operator A(t), as the following describes it. Equation (1.1) will be solved under the set of hypotheses, namely, the dissipativity condition (H1), the range condition (H2), and the time-regulating condition (HA) or (HA) . (H1) For each 0 ≤ t ≤ T , A(t) is dissipative in the sense that u − v ≤ (u − v) − λ(g − h) for all u, v ∈ D(A(t)), g ∈ (A(t) − ω)u, h ∈ (A(t) − ω)v, and for all λ > 0. (H2) The range of (I − λA(t)), denoted by E, is independent of t and contains D(A(t)) for all t ∈ [0, T ] and for small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1. (HA) There is a continuous function f : [0, T ] −→ R, of bounded variation, and there is a nonnegative function L on [0, ∞) with L(s) bounded for bounded s, such that, for each 0 < λ < λ0 , we have {Jλ (t)x − Jλ (τ )y : 0 ≤ t, τ ≤ T, x, y ∈ E} = S1 (λ) ∪ S2 (λ). Here S1 (λ) denotes the set: {Jλ (t)x − Jλ (τ )y : 0 ≤ t, τ ≤ T, x, y ∈ E, Jλ (t)x − Jλ (τ )y ≤ L(Jλ (τ )y)|t − τ |}, while S2 (λ) denotes the set: {Jλ (t)x − Jλ (τ )y : 0 ≤ t, τ ≤ T, x, y ∈ E, Jλ (t)x − Jλ (τ )y ≤ (1 − λω)−1 [x − y (Jλ (τ ) − I)y )]}. λ Observe that (HA) is reduced to (H1) when S1 (λ) = ∅ and t = τ . (HA) There is a continuous function f : [0, T ] −→ R, of bounded variation, ˜ on [0, ∞) × [0, ∞) with L(s ˜ 1 , s2 ) and there is a nonnegative function L bounded for bounded s1 , s2 , such that, for each 0 < λ < λ0 , we have + λ|f (t) − f (τ )|L(Jλ (τ )y)(1 +

{Jλ (t)x − Jλ (τ )y : 0 ≤ t, τ ≤ T, x, y ∈ E} = S1 (λ) ∪ S2 (λ). Here S1 (λ) denotes the set: {Jλ (t)x − Jλ (τ )y : 0 ≤ t, τ ≤ T, x, y ∈ E, Jλ (t)x − Jλ (τ )y ˜ ≤ L(J λ (τ )y, y)|t − τ |}, while S2 (λ) denotes the set: {Jλ (t)x − Jλ (τ )y : 0 ≤ t, τ ≤ T, x, y ∈ E, Jλ (t)x − Jλ (τ )y

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≤ (1 − λω)−1 [x − y (Jλ (τ ) − I)y )]}. λ Again, (HA) becomes (H1), as is the case with (HA), when S1 (λ) = ∅ and t = τ . It is to be noted that, with (H1) and (H2) assumed, the quantity ˜ + λ|f (t) − f (τ )|L(J λ (τ )y, y)(1 +

Jλ (t)x ≡ (I − λA(t))−1 x, in S1 (λ) or S2 (λ) is readily seen to exist and be single-valued for x ∈ E. The purpose of this chapter is to show that the limit U (t, s)x ≡ lim

n

n→∞

J t−s (s + i

i=1

n

t−s )x n

(1.4)

ˆ ˆ will be = D(A(s)). This limit U (t, s)x for x = u0 ∈ D(A(s)) exists for x ∈ D(A(s)) only intepreted as a limit solution to equation (1.1), but it will be a strong one if A(t) satisfies additionally an embedding property [20] of embeddedly quasi-demiclosedness (see Section 2). Here the quantity (Jλ (τ ) − I)x λ will exist by (H1) and (H2) and can equal ∞ [7, 8], because of Lemma 4.4 and Remark 4.5 in Chapter 1. Further, the set |A(τ )x| ≡ lim λ→0

ˆ D(A(t)) ≡ {x ∈ D(A(t)) : |A(t)x| < ∞} clearly contains the domain D(A(t)) of A(t) and so, is called the generalized domain for A(t) [7, 38]. The results of this chapter will be used in Chapters 5 and 6, where the corresponding evolution operators A(t)’s are second order, elliptic differential operators with time-dependent coefficients, and the corresponding evolution equations are parabolic boundary value problems with time-dependent boundary conditions. In addition to this section, there are five more sections in this chapter. Section 2 states the main results, and Section 3 presents some simple examples. Section 4 obtains some preliminary estimates, and Section 5 proves the main results. Finally, Section 6 examines the difference equations theory in our papers [21, 22, 23], whose results, together with those in Section 4, were used in Section 5 to prove the main results in Section 2. Section 6 is a technical section with tedious calculations, so it is placed at the end of the chapter. The material of this chapter is based on our article [25]. 2. Main Results With regard to the evolution equation (1.1), we have three theorems. Theorem 2.1 (Existence of a limit [25]). Let the nonlinear operator A(t) satisfy the dissipativity condition (H1), the range condition (H2), and the time-regulating condition (HA) or (HA) . Then the limit U (s + t, s)u0 ≡ lim

n→∞

n

t J nt (s + i )u0 n i=1

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2. EXISTENCE THEOREMS FOR EVOLUTION EQUATIONS (I) [t]

= lim

μ→0

μ

Jμ (s + iμ)u0

i=1

ˆ = D(A(s)) where s, t ≥ 0 and 0 ≤ (s + t) ≤ T . exists for u0 ∈ D(A(s)) This limit U (s + t, s)u0 is also continuous in t ≥ 0 for u0 ∈ D(A(s)), but is ˆ Lipschitz continuous in t ≥ 0 for u0 ∈ D(A(s)). Remark. Theorem 2.1 is the Crandall-Pazy theorem [8], if S1 (λ) = ∅, if D(A(t)) ≡ D is independent of t while the range of (I − λA(t)) need not be, and if the E in (HA) is changed to D. In order to state next theorem, Theorem 2.2, concerning a limit solution and a strong solution, we need to make two preparations. As is the case with the nonlinear, autonomous operator A in Chapter 1, the first preparation is for a limit solution. Thus, consider the discretization of (1.1) on [0, T ] ui − A(ti )ui ui−1 , ui ∈ D(A(ti )), where n ∈ N is large, and 0 < < λ0 is such that s ≤ ti = s + i ≤ T for each i = 1, 2, . . . , n. Here it is to be noticed that, for u0 ∈ E, ui exists uniquely by hypotheses (H1) and (H2). ˆ Let u0 ∈ D(A(s)), and construct the Rothe functions [12, 32] by defining χn (s) = u0 ,

C n (s) = A(s);

χn (t) = ui ,

C n (t) = A(ti )

for t ∈ (ti−1 , ti ], and un (s) = u0 ; t − ti−1 for t ∈ (ti−1 , ti ] ⊂ [s, T ].

un (t) = ui−1 + (ui − ui−1 )

ˆ It will follow (see Section 5) that, for u0 ∈ D(A(s)), we have lim

sup un (t) − χn (t) = 0,

n→∞ t∈[0,T ]

un (t) − un (τ ) ≤ K3 |t − τ |,

(2.1)

ˆ where t, τ ∈ (ti−1 , ti ], and that, for u0 ∈ DA(s)), dun (t) ∈ C n (t)χn (t), dt un (s) = u0 ,

(2.2)

where t ∈ (ti−1 , ti ]. Here the last equation has values in B([s, T ]; X), the real Banach space of all bounded functions from [s, T ] to X. The other preparation is for a strong solution, as is the case with the nonlinear, autonomous operator A in Chapter 1. Let (Y, .Y ) be a real Banach space,

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into which the real Banach space (X, .) is continuously embedded. Assume additionally that A(t) satisfies the embedding condition of embeddedly quasi-demiclosedness : (HB) If tn ∈ [0, T ] −→ t, if xn ∈ D(A(tn )) −→ x, and if yn ≤ M0 for some yn ∈ A(tn )xn and for some positive constant M0 , then η(A(t)x) exists and |η(ynl ) − z| −→ 0 for some subsequence ynl of yn , for some z ∈ η(A(t)x), and for each η ∈ Y ∗ ⊂ X ∗ , the real dual space of Y . Theorem 2.2 (A limit or a strong solution [25]). Following Theorem 2.1, if ˆ then the function u0 ∈ D(A(s)), u(t) ≡ U (t, s)u0 = lim

n→∞

n

J t−s (s + i

i=1

n

t−s )u0 n

[ t−s μ ]

= lim

μ→0

Jμ (s + iμ)u0

i=1

is a limit solution of the evolution equation (1.1) on [0, T ], in the sense that it is also the uniform limit of un (t) on [0, T ], where un (t) satisfies (2.2). Furthermore, if A(t) satisfies the embedding property (HB), then u(t) is a strong solution in Y , in the sense that d u(t) ∈ A(t)u(t) in Y dt for almost every t ∈ (0, T ); u(s) = u0 is true. The strong solution is unique if Y ≡ X. It will be readily seen from the proof of Theorems 2.1 and 2.2 that Theorem 2.3. The results in Theorems 2.1 and 2.2 are still true if the range condition (H2) is replaced by the weaker range condition (H2) below, provided that ˆ ˆ the initial conditions u0 ∈ D(A(s))(⊃ D(A(s))) and u0 ∈ D(A(s)) = D(A(s))(⊃ D(A(s))) are changed to the condition u0 ∈ D(A(s)). Here (H2) The range of (I − λA(t)), denoted by E, is independent t and contains D(A(t)) for all t ∈ [0, T ] and for small 0 < λ < λ0 , where λ0 is some positive number satisfying λ0 ω < 1. ˆ And in this case, the set D(A(s)) is not well-defined, but the set ˜ D(A(s)) ≡ {x ∈ D(A(s)) : |A(s)x| < ∞} coincides with D(A(s)). 3. Examples Three examples will be considered, which are time-dependent analogues of Examples 3.1, 3.2, and 3.3 in Chapter 1. The first one is about a linear, timedependent, nonhomogeneous parabolic boundary value problem of space dimension one, and the second one is about its analogue of higher space dimensions. The last

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example concerns a nonlinear, time-dependent, nonhomogeneous parabolic boundary value problem of space dimension one. More complex examples will be a subject of other chapters. Example 3.1. Solve for u = u(x, t): ut (x, t) = uxx (x, t) + f0 (x, t), (x, t) ∈ (0, 1) × (0, T ); ux (0, t) = β0 u(0, t),

ux (1, t) = −β1 u(1, t);

(3.1)

u(x, 0) = u0 (x); ∂ ∂ u, ux(x, t) ≡ ∂x u, where T, β0 , and β1 are three positive constants, and ut (x, t) ≡ ∂t 2 ∂ and uxx (x, t) ≡ ∂x2 u, respectively. Here f0 (x, t) is jointly continuous in x, t, and satisfies, for x ∈ [0, 1], t, τ ∈ [0, T ], and ζ(t), a function in t of bounded variation,

|f0 (x, t) − f0 (x, τ )| ≤ |ζ(t) − ζ(τ )|. Solution. Define the time-dependent operator F (t) : D(F (t)) ⊂ C[0, 1] −→ C[0, 1] by F (t)v = v + f0 (x, t) for v ∈ D(F (t)) ≡ {w ∈ C 2 [0, 1] : w (j) = (−1)j βj w(j),

j = 0, 1}.

It will be shown [25] that F (t) satisfies the four conditions, namely, the dissipativity condition (H1), the range condition (H2), the time-regulating condition (HA), and the embedding condition (HB). As a result, the quantity n t t u(t) = lim [I − F (i )]−1 u0 n→∞ n n i=1 t

[ν ] = lim [I − νF (iν)]−1 u0 ν→0

i=1

ˆ (0)), then this u(t) is ˆ (0)), on using Theorem 2.1. If u0 ∈ D(F exists for u0 ∈ D(F not only a limit solution to the equation (3.1), but even a strong one by Theorem 2.2. In the latter case, u(t) also satisfies the middle equation in (3.1). Under additional assumptions on u0 and f0 (x, t), we will make further estimates, so that u(t) for u0 ∈ D(F (0)) with F (0)u0 = (u0 + f0 (x, 0)) ∈ D(F (0)) is, in fact, a unique classical solution. We now begin the proof, which is composed of eight steps. Step 1 It is readily verified as in solving Example 3.1, Chapter 1 that F (t) satisfies the dissipativity condition (H1). Step 2. From the theory of ordinary differential equations [5], [24, Corollary 2.13, Chapter 4], the range of (I − λF (t)), λ > 0, equals C[0, 1], so F (t) satisfies the range condition (H2). Step 3. (F (t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C[0, 1], i = 1, 2, and let v1 = (I − λF (t))−1 g1 ; v2 = (I − λF (τ ))−1 g2 ;

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where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ(v1 − v2 ) = λ[f0 (x, t) − f0 (x, τ )] + (g1 − g2 ), and so v1 − v2 ∞ ≤ g1 − g2 ∞ + λ max |f0 (x, t) − f0 (x, τ )| x∈[0,1]

≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|, proving the condition (HA). This is because, as in solving Example 3.1, Chapter 1, the maximum principle applies, that is, there is an x0 ∈ [0, 1] such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ (0, 1), (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0, and that, for x0 ∈ {0, 1}, (v1 − v2 ) (0) ≤ 0 or ≥ 0, according as (v1 − v2 )(0) > 0 or < 0; (v1 − v2 ) (1) ≥ 0 or ≤ 0, according as (v1 − v2 )(1) > 0 or < 0. Here the boundary conditions in D(F (t)) make x0 ∈ {0, 1} impossible. Step 4. (F (t) satisfies the embedding condition (HB).) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(F (tn )) converge to v in C[0, 1], and F (tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (0, 1) = (L2 (0, 1))∗ , η(F (t)v) exists and |η(F (tn )vn ) − η(F (t)v)| −→ 0. Here (C[0, 1]; · ∞ ) is continuously embedded into L2 (0, 1); · ). Since vn ∞ and F (tn )vn ∞ are uniformly bounded, so is vn C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Hence, by Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 [0, 1] to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (0, 1), whence, by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (0, 1), |η(F (tn )vn ) − η(F (t)v)| −→ 0, because 1 | [(vn − v ) + (f0 (x, tn ) − f0 (x, t))]η dx| 0 1

≤|

0

−→ 0.

(vn − v )η dx| + |

0

1

[f0 (x, tn ) − f0 (x, t)]η dx|

Therefore F (t) satisfies the embedding condition (HB). ˆ (0)) satisfies the middle equation in (3.1).) Consider Step 5. (u(t) for u0 ∈ D(F the discretized equation ui −νF (ti )ui = ui−1 , (3.2) ui ∈ D(F (ti )),

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ˆ (0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that where u0 ∈ D(F ν < λ0 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νF (tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νF (0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νF (tk )]−1 u0

k=1 n

n→∞

t t F (k )]−1 u0 n n

[I −

k=1

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4, we have ui ∞ ; ui + f0 (x, ti )∞ = F (ti )ui ∞ ui − ui−1 = ∞ ; ν are uniformly bounded. Those, in turn, result in a bound for uiC 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1 [0, 1] to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (3.1), as ui does so. Step 6. (Further estimates of ui under additional assumptions on f0 (x, t), where u0 ∈ D(F (0)) with F (0)u0 = (u0 + f0 (x, 0)) ∈ D(F (0))) We assume additionally that Dt f0 (x, t) and f0 (x, t) exist and are continuous in x, t, and that Dt f0 (x, t) satisfies, for x ∈ [0, 1], t, τ ∈ [0, T ], |Dt f0 (x, t) − Dt f0 (x, τ )| ≤ |ζ(t) − ζ(τ )|. Here Dt f0 (x, t) is the partial derivative of f0 (x, t) with respect to t, and f0 (x, t) is the second partial derivative of f0 (x, t) with respect to x. Because of F (0)u0 ∈ D(F (0)), the ui in Step 5 satisfies ui − ν[ui + f0 (x, ti )] = ui−1 , ui (0)

= β0 ui (0),

ui (1)

From this, it follows, on letting vi =

i = 0, 1, . . . ;

= −β1 ui (1),

ui −ui−1 ν

for i = 0, 1, . . ., that

vi − ν[vi + g(x, ν, ti )] = vi−1 , vi (0)

= β0 vi (0),

vi (1)

i = −1, 0, 1, . . . .

i = 1, 2, . . . ;

= −β1 vi (1),

i = 0, 1, . . . ;

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where, with ti−1 = ti − ν, g(x, ν, ti ) = g(x, ν, ti , ti−1 ) f0 (x, ti ) − f0 (x, ti−1 ) . ν Here, for convenience, we also define =

v−1 = v0 − ν[v0 + g(x, ν, t0 )]; t−1 = 0; for which g(x, ν, t0 ) = g(x, ν, 0) = 0. Thus, either from Corollary 4.3 or from the proof of Proposition 4.1 and from both the results in Proposition 4.2 and the proof of Proposition 4.2 in Section 4, we have vi − vi−1 vi + g(x, ν, ti )∞ = ∞ , i = 0, 1, . . . ; ν is uniformly bounded, whence so are ui − ui−1 vi C 2 [0,1] = C 2 [0,1] ν = ui + f0 (x, ti )C 2 [0,1] , i = 0, 1, . . . ; ui C 4 [0,1] ,

i = 0, 1, . . . ,

as in Step 5. This is because those vi ’s above, i = −1, 0, 1, . . ., satisfy the conditions (C1), (C2), and (C3) in Corollary 4.3, that is, the conditions ((4.3) or (4.4)), ((4.5) or (4.6)), and ((4.7) or (4.8)) in Section 4. A proof of it follows from applying the maximum principle argument in Step 3 and the fact that the quantity ui −uνi−1 ∞ in Step 5 is bounded. Step 7. (Existence of a solution) Now that, from Step 6, ui C 4 [0,1] , i = 2, 3, . . . , is uniformly bounded, it follows from the Ascoli-Arzela theorem [33], as in Step 5, that a subsequence of ui and then itself, through the discretized equation (3.2), converge in C 3 [0, 1] to the limit u(t), as ν −→ 0. Therefore u(t) is a classical solution. Step 8. (Uniqueness of a solution) This proceeds as in Step 5 in the proof of Example 3.2, Chapter 1. The proof is complete. Example 3.2. Solve for u = u(x, t): ut (x, t) = u(x, t) + f0 (x, t), (x, t) ∈ Ω × (0, T ); ∂ u(x, t) + β2 (x, t)u(x, t) = 0, ∂n ˆ u(x, 0) = u0 (x).

x ∈ ∂Ω;

(3.3)

Here T > 0, and Ω is a bounded, smooth domain in RN ; N ≥ 2 is a positive integer, ∂2 ∂ and x = (x1 , x2 , . . . , xN ); u = N i=1 ∂x2 u, and ut = ∂t u; ∂Ω is the boundary of Ω, i

and ∂∂nˆ u is the unit, outer, normal derivative of u; f0 (x, t) is in C μ (Ω), 0 < μ < 1, for all t, jointly continuous in x, t, and satisfies, for x ∈ Ω, t, τ ∈ [0, T ], and ζ(t), a function in t of variation, |f0 (x, t) − f0 (x, τ )| ≤ |ζ(t) − ζ(τ )|;

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and finally, β2 (x, t) is in C 1+μ (Ω) for all t, jointly continuous in x, t, and satisfies, for x ∈ Ω, t, τ ∈ [0, T ], and δ, M0 , positive constants, β2 (x, t) ≥ δ; |β2 (x, t) − β2 (x, τ )| ≤ M0 |t − τ |. Solution. Define the time-dependent operator G(t) : D(G(t)) ⊂ C(Ω) −→ C(Ω) by G(t)v = v + f0 (x, t) for v ∈ D(G(t)) ≡ {w ∈ C 2+μ (Ω) :

∂w + β2 (x, t)w = 0 for x ∈ ∂Ω}. ∂n ˆ

Here 0 < μ < 1, is a constant. It will be shown [25] that G(t) satisfies the four conditions, namely, the dissipativity condition (H1), the weaker range condition (H2) , the time-regulating condition (HA), and the embedding condition (HB). As a result, the quantity n

u(t) = lim

n→∞

i=1

[ νt ]

= lim

ν→0

[I −

t t G(i )]−1 u0 n n

[I − νG(iν)]−1 u0

i=1

ˆ ˆ exists for u0 ∈ D(G(0)), then this u(t) is on using Theorem 2.1. If u0 ∈ D(G(0)), not only a limit solution to the equation (3.1), but even a strong one by Theorem 2.2. In the later case, u(t) also satisfies the middle equation in (3.1). Under additional assumptions on u0 and f0 (x, t), we will make further estimates, so that u(t) for u0 ∈ D(G(0)) with G(0)u0 = ( u0 + f0 (x, 0)) ∈ D(G(0)) is, in fact, a unique classical solution. We now begin the proof, which is composed of eight steps. Step 1 It is readily verified as in solving Example 3.2, Chapter 1 that G(t) satisfies the dissipativity condition (H1). Step 2. From the theory of linear, elliptic partial differential equations [13], the range of (I − λG(t)), λ > 0, equals C μ (Ω), so G(t) satisfies the weaker range condition (H2) because of C μ (Ω) ⊃ D(G(t)) for all t. Step 3. (G(t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C μ (Ω), i = 1, 2, and let v1 = (I − λG(t))−1 g1 ; v2 = (I − λG(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ (v1 − v2 ) = λ[f0 (x, t) − f0 (x, τ )] + (g1 − g2 ),

x ∈ Ω;

∂(v1 − v2 ) + β2 (x, t)(v1 − v2 ) = −(β2 (x, t) − β2 (x, τ ))v2 , ∂n ˆ

x ∈ ∂Ω;

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so the condition (HA) follows: v1 − v2 ∞ ≤ g1 − g2 ∞ + λ max |f0 (x, t) − f0 (x, τ )| x∈[0,1]

≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|; or 1 v1 − v2 ∞ ≤ M0 v2 ∞ |t − τ |. δ This is because, as in proving the dissipativity condition (H1) in Step 1, the maximum principle argument applies, that is, there is an x0 ∈ Ω such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ Ω, D(v1 − v2 )(x0 ) = 0; (v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0, and that, for x0 ∈ ∂Ω, ∂(v1 − v2 ) (x0 ) ≥ 0 or ≤ 0 according as (v1 − v2 )(x0 ) > 0 or < 0. ∂n ˆ Step 4. (G(t) satisfies the embedding condition (HB).) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(G(tn )) converge to v in C(Ω), and G(tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (Ω) = (L2 (Ω))∗ , η(G(t)v) exists and |η(G(tn )vn ) − η(G(t)v)| −→ 0. Here (C(Ω); · ∞ ) is continuously embedded into L2 (Ω); · ). Since vn ∞ and G(tn )vn ∞ are uniformly bounded, so is vn C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.1) in Chapter 5. (Alternatively, uniformly bounded are vn W 2,p (Ω) for any p ≥ 2 and then vn C 1+λ (Ω) for any 0 < λ < 1, on using the Lp elliptic estimates [37] and the Sobolev embedding theorem [1, 13].) Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 (Ω) to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (Ω), whence, by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (Ω), |η(G(tn )vn ) − η(G(t)v)| −→ 0, because

|

Ω

≤|

Ω

[( vn − v)η + (f0 (x, tn ) − f0 (x, t))]η dx|

( vn − v)η dx| + |

−→ 0.

0

1

[f0 (x, tn ) − f0 (x, t)]η dx|

Therefore G(t) satisfies the embedding condition (HB). ˆ satisfies the middle equation in (3.3).) Consider Step 5. (u(t) for u0 ∈ D(G(0)) the discretized equation ui −νG(ti )ui = ui−1 , (3.4) ui ∈ D(G(ti )),

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ˆ where u0 ∈ D(G(0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that ν < λ0 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νG(tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νG(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν ]

[I − νG(tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t G(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4, we have ui∞ ; ui + f0 (x, ti )∞ = G(ti )ui ∞ ui − ui−1 ∞ ; = ν are uniformly bounded, whence so is ui C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.1) in Chapter 5. (Alternatively, those, in turn, result in a bound for ui W 2,p (Ω) for any p ≥ 2, by the Lp elliptic estimates [37]. Hence, a bound exists for ui C 1+η (Ω) for any 0 < η < 1, as a result of the Sobolev embedding theorem [1, 13].) Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1+μ (Ω) to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (3.3), as ui does so. Step 6. (Further estimates of ui under additional assumptions on f0 (x, t), where u0 ∈ D(G(0)) with G(0)u0 = ( u0 + f0 (x, 0)) ∈ D(G(0))) There are three additional assumptions. One is that the (2+μ)-th derivative of f0 (x, t) with respect to x exists and is jointly continuous in x, t. The second is that the first partial derivative Dt f0 (x, t) of f0 (x, t) with respect to t exists and is jointly continuous in x, t, and satisfies, for x ∈ Ω, t, τ ∈ [0, T ], |Dt f0 (x, t) − Dt f0 (x, τ )| ≤ |ζ(t) − ζ(τ )|. The third is that β2 (x, t) is twice continuously differentiable in t, or weakly satisfies, for x ∈ ∂Ω, τ > 0, and t, t + τ, t + 2τ ∈ [0, T ], β2 (x, t + 2τ ) − 2β2 (x, t + τ ) + β2 (x, t) | ≤ M0 , τ2 the second difference quotient of β2 (x, t) in t being bounded. |

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Because of G(0)u0 ∈ D(G(0)), the ui in Step 5 satisfies ui − ν[ ui + f0 (x, ti )] = ui−1 ,

x ∈ Ω,

i = 0, 1, . . . ; ∂ ui (x) + β2 (x, ti )ui (x) = 0, x ∈ ∂Ω, ∂n ˆ i = −1, 0, 1, . . . . From this, it follows, on letting wi =

ui −ui−1 ν

for i = 0, 1, . . ., that

wi − ν[ wi + g(x, ν, ti )] = wi−1 ,

x ∈ Ω,

i = 1, 2, . . . ; ∂wi + β2 (x, ti )wi ∂n ˆ β2 (x, ti ) − β2 (x, ti−1 ) ui−1 , =− ν i = 0, 1, . . . ;

x ∈ ∂Ω,

where, with ti−1 = ti − ν, g(x, ν, ti ) = g(x, ν, ti , ti−1 ) f0 (x, ti ) − f0 (x, ti−1 ) , ν Here, for convenience, we also define =

i = 0, 1, . . . .

w−1 = w0 − ν[ w0 + g(x, ν, t0 ))]; t−1 = 0; for which g(x, ν, t0 ) = g(x, ν, 0) = 0. Hence, either from Corollary 4.3 or from the proof of Proposition 4.1 and from both the results in Proposition 4.2 and the proof of Proposition 4.2 in Section 4, we have wi − wi−1 wi + g(x, ν, ti )∞ = ∞ , i = 0, 1, . . . ; ν wi C 1+η (Ω) , 0 < η < 1, i = 0, 1, . . . ; i−1 = ui + f0 (x, ti ); hence, are uniformly bounded, as in Step 5, where wi = ui −u ν so is ui C 3+η (Ω) , i = 0, 1, . . .

by the Schauder global regularity theorem [13, page 111]. This is because those wi ’s, i = −1, 0, 1, . . ., satisfy the conditions (C1), (C2), and (C3) in Corollary 4.3, that is, the conditions ((4.3) or (4.4)), ((4.5) or (4.6)), and ((4.7) or (4.8)) in Section 4, for which employed were both the maximum principle argument in Step 3 and the boundedness of ui −uνi−1 ∞ in Step 5. Step 7. (Existence of a solution) Now that, from Step 6, ui C 3+η (Ω) , i = 2, 3, . . . , is uniformly bounded, it follows from the Ascoli-Arzela theorem [33], as in Step 5, that a subsequence of ui and then itself, through the discretized equation (3.4), converge in C 3+μ (Ω) to the limit u(t), as ν −→ 0. Therefore u(t) is a classical solution. Step 8. (Uniqueness of a solution) This proceeds as in Step 5 in the proof of Example 3.2, Chapter 1.

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The proof is complete. Example 3.3. Solve for u = u(x, t): ut (x, t) = uxx (x, t) + f1 (x, t), (x, t) ∈ (0, 1) × (0, T ); ux (0, t) ∈ (−1)j βj (u(j, t)),

j = 0, 1;

(3.5)

u(x, 0) = u0 (x); where T > 0, and β0 and β1 are maximal monotone graphs in R × R; f1 (x, t) is jointly continuous in x, t, and satisfies, for x ∈ [0, 1], t, τ ∈ [0, T ], and ζ(t), a function in t of bounded variation, |f1 (x, t) − f1 (x, τ )| ≤ |ζ(t) − ζ(τ )|. This is the nonlinear analogue of the problem in Example 3.1. Here a monotone graph β is a subset of R × R that satisfies (y2 − y1 )(x2 − x1 ) ≥ 0

for yi ∈ β(xi ),

i = 1, 2.

This β is a maximal monotone graph , if it is not properly contained in any other monotone graph. In this case, for any λ > 0, (I + λβ)−1 : R −→ R is single-valued and non-expansive [3]. Solution. Define the time-dependent operator H(t) : D(H(t)) ⊂ C[0, 1] −→ C[0, 1] by H(t)v = v + f1 (x, t) for v ∈ D(H(t)) ≡ {w ∈ C 2 [0, 1] : w (j) ∈ (−1)j βj (w(j)),

j = 0, 1}.

It will be shown [25] that H(t) satisfies the four conditions, namely, the dissipativity condition (H1), the range condition (H2), the time-regulating condition (HA), and the embedding condition (HB). As a result, the quantity u(t) = lim

n

n→∞

i=1

[ νt ]

= lim

ν→0

[I −

t t H(i )]−1 u0 n n

[I − νH(iν)]−1 u0

i=1

ˆ ˆ on using Theorem 2.1. If u0 ∈ D(H(0)), exists for u0 ∈ D(H(0)), then this u(t) is not only a limit solution to the equation (3.5), but even a strong one by Theorem 2.2. In the latter case, u(t) also satisfies the middle equation in (3.5). Under additional assumptions on f1 (x, t) and β, we will make further estimates, so that u(t) for u0 ∈ D(H(0)) with H(0)u0 ∈ D(H(0)) is, in fact, a unique classical solution. We now begin the proof, which is composed of eight steps. Step 1 It is readily verified as in solving Example 3.3, Chapter 1 that H(t) satisfies the dissipativity condition (H1).

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Step 2. That the range of (I −λH(t)), λ > 0, equals C[0, 1] follows immediately as in solving Example 3.3, Chapter 1. Hence, H(t) satisfies the range condition (H2). Step 3. (H(t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C[0, 1], i = 1, 2, and let v1 = (I − λH(t))−1 g1 ; v2 = (I − λH(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ(v1 − v2 ) = λ[f1 (x, t) − f1 (x, τ )] + (g1 − g2 ), so v1 − v2 ∞ ≤ g1 − g2 ∞ + λ max |f0 (x, t) − f0 (x, τ )| x∈[0,1]

≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|, proving the condition (HA). This is because, as in solving Example 3.3, Chapter 1, the maximum principle applies, that is, there is an x0 ∈ [0, 1] such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ (0, 1) (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0, and that, for x0 ∈ {0, 1}, (v1 − v2 ) (0) ≤ 0 or ≥ 0, according as (v1 − v2 )(0) > 0 or < 0; (v1 − v2 )(1) ≥ 0 or ≤ 0, according as (v1 − v2 )(1) > 0 or < 0. Here the boundary conditions in D(H(t)) make x0 ∈ {0, 1} impossible. Step 4. (H(t) satisfies the embedding condition (HB).) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(H(tn )) converge to v in C[0, 1], and H(tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (0, 1) = (L2 (0, 1))∗ , η(H(t)v) exists and |η(H(tn )vn ) − η(H(t)v)| −→ 0. Here (C[0, 1]; · ∞ ) is continuously embedded into L2 (0, 1); · ). Since vn ∞ and H(tn )vn ∞ are uniformly bounded, so is vn C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Hence, by Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 [0, 1] to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (0, 1), whence, by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (0, 1), |η(H(tn )vn ) − η(H(t)v)| −→ 0, because

|

0

1

[(vn − v )η + (f1 (x, tn ) − f0 (x, t))]η dx|

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≤|

n 0

(vn − v )η dx| + |

−→ 0.

0

1

[f1 (x, tn ) − f1 (x, t)]η dx|

Therefore H(t) satisfies the embedding condition (HB). ˆ Step 5. (u(t) for u0 ∈ D(H(0)) satisfies the middle equation in (3.5).) Consider the discretized equation ui −νH(ti )ui = ui−1 , (3.6) ui ∈ D(H(ti )), ˆ i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that where u0 ∈ D(H(0)), ν < λ0 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νH(tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νH(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νH(tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t H(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4, we have ui ∞ ; ui + f1 (x, ti )∞ = H(ti )ui ∞ ui − ui−1 ∞ ; = ν are uniformly bounded. Those, in turn, result in a bound for uiC 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1 [0, 1] to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (3.5), as ui does so. Step 6. (Further estimates of ui under additional assumptions on f1 (x, t) and β, where u0 ∈ D(H(0)) with H(0)u0 ∈ D(H(0))) We make two additional assumptions. One is that Dt f1 (x, t) and f1 (x, t) exist and are continuous in x, t, and that Dt f1 (x, t), satisfy, for x ∈ [0, 1], t, τ ∈ [0, T ], |Dt f1 (x, t) − Dt f1 (x, τ )| ≤ |ζ(t) − ζ(τ )|. Here Dt f1 (x, t) is the first partial derivative of f1 (x, t) with respect to t, and

f1 (x, t) is the second partial derivative of f1 (x, t) with respect to x.

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The other is that βj (x), j = 0, 1, satisfy, for x, τ, y1 , y2 ∈ R with τ > 0 and for some constant δ > 0, βj (x) − βj (y) } ≥ δ; x−y βj (x + τ y1 + τ y2 ) − βj (x + τ y1 ) max{|[ τ y2 βj (x + τ y1 ) − βj (x) − ]/τ |} τ y1 ≤ Mx,y1 ,y2 .

min{

Here Mx,y1 ,y2 is a positive number depending on x, y1 , y2 , and bounded for finite x, y1 , and y2 . That is, the first difference quotient of βj is strictly positive, and the second difference quotient like of βj is bounded by Mx,y1 ,y2 . Because of H(0)u0 ∈ D(H(0)), the ui in Step 5 satisfies ui − ν[ui + f1 (x, ti )] = ui−1 , ui (0)

∈ β0 (ui (0)),

ui (1)

i = 0, 1, . . . ;

∈ −β1 (ui (1)),

i = −1, 0, 1, . . . . ui −ui−1 ν


for i = 0, 1, . . ., that

vi − ν[vi + g(x, ν, ti )] = vi−1 , vi (0) ∈

i = 1, 2, . . . ;

β0 (ui (0)) − β0 (ui−1 (0)) vi (0), ui (0) − ui−1 (0) i = 0, 1, . . . ;

vi (1) ∈ −

β1 (ui (1)) − β1 (ui−1 (1)) vi (1), ui (1) − ui−1 (1) i = 0, 1, . . . ;

where, with ti−1 = ti − ν, g(x, ν, ti ) = g(x, ν, ti , ti−1 ) =

f1 (x, ti ) − f1 (x, ti−1 ) . ν

Here for convenience, we also define v−1 = v0 − ν[v0 + g(x, ν, t0 )]; t−1 = 0, for which g(x, ν, t0 ) = g(x, ν, 0) = 0. Hence, either from Corollary 4.3 or from the proof of Proposition 4.1 and the results in and the proof of Proposition 4.2 in Section 4, we have vi − vi−1 vi + g(x, ν, ti )∞ = ∞ , ν i = 0, 1, 2, . . . ; is uniformly bounded, whence so are ui − ui−1 C 2 [0,1] ν = ui + f1 (x, ti )C 2 [0,1] , i = 0, 1, 2, . . . ; vi C 2 [0,1] =

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ui C 4 [0,1] ,

i = 0, 1, 2, . . . ,

as in Step 5. This is because those vi ’s, i = −1, 0, 1, . . ., satisfy the conditions (C1), (C2), and (C3) in Corollary 4.3, that is, the conditions ((4.3) or (4.4)), ((4.5) or (4.6)), and ((4.7) or (4.8)) in Section 4, for which used were both the maximum principle argument in Step 3 and the boundedness of ui −uνi−1 ∞ in Step 5. Step 7. (Existence of a solution) Now that, from Step 6, ui C 4 [0,1] , i = 2, 3, . . . , is uniformly bounded, it follows from the Ascoli-Arzela theorem [33], as in Step 5, that a subsequence of ui and then itself, through the discretized equation (3.6), converge in C 3 [0, 1] to the limit u(t), as ν −→ 0. Therefore u(t) is a classical solution. Step 8. (Uniqueness of a solution) This proceeds as in Step 5 in the proof of Example 3.2, Chapter 1. The proof is complete. 4. Some Preliminary Estimates Within this section and Sections 5 and 6, we can assume, without loss of generality, that ω ≥ 0 where ω is the ω in the hypothesis (H1). This is because the case ω < 0 is the same as the case ω = 0. This will be readily seen from the corresponding proofs. To prove the main results, that is, Theorems 2.1 and 2.2 in Section 5, we need to make two preparations. One preparation is this section, and the other is Section 6. As is explained in Section 4 in Chapter 1, the two preparations will help to solve the associated recursive inequalityn Section 4. Once the inequality is solved, the proof of the main results follows. Section 6 is a technical section with tedious calculations, so it is placed at the end of the chapter. Proposition 4.1. Let A(t) satisfy the dissipativity condition (H1), the range condition (H2), and the time-regulating condition (HA) or (HA) , and let u0 be in D(A(s)) ⊂ E where 0 ≤ s ≤ T . Let 0 < < λ0 be so chosen that 0 < ω < 1, and let 0 ≤ ti = s + i ≤ T where i ∈ N. Then, under (HA), ui − u0 ≤ η i L(u0 )(i) + [η i−1 b1 + η i−2 b2 + · · · + ηbi−1 + bi ] and

ui − ui−1 ≤ [(ci ci−1 · · · c2 )L(u0 )

(4.1)

or (ci ci−1 · · · c3 )L(u1 ) or · · ·

or ci L(ui−2 ) or L(ui−1 )] + [(ci ci−1 · · · c1 )a0 + (ci ci−1 · · · c2 )d1 + (ci ci−1 · · · c3 )d2 + · · · + ci di−1 + di ]. Here ui =

i

J (tj )u0

exists uniquely

j=1

by the hypotheses (H1) and (H2); η = (1 − ω)−1 > 1; bi = ηv0 + η|f (ti ) − f (s)|L(u0 )(1 + v0 ),

(4.2)

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55

where v0 is any element in A(s)u0 ; ci = η[1 + L(ui−1 )|f (ti ) − f (ti−1 )|]; di = ηL(ui−1 )|f (ti ) − f (ti−1 )|; the right sides of (4.2) are interpreted as [L(u0 )] + [c1 a0 + d1 ]

for i = 1;

[c2 L(u0 ) or L(u1 )] + [c2 c1 a0 + c2 d1 + d2 ] for i = 2; . . ., and so on; and a0 = where u−1 is defined by

u0 − u−1 ,

u0 − v0 = u−1 ,

with v0 any element in A(s)u0 . Under (HA) , the results are similar, in which the quantities ˜ L(u j , uj−1 ),

j = 0, 1, . . . , (i − 1),

replace the above L(uj ). Proof. The proof below will be made under (HA), because it will be similar under (HA) . We will use the method of mathematical induction. Two cases will be considered, and for each case, we divide the proof into two steps. Case 1, where (4.1) is considered. Step 1. Claim that(4.1) is true for i = 1. This will follow from the arguments below. If (u1 − u0 ) ∈ S1 () (defined in Section 1), then u1 − u0 = J (t1 )u0 − J (s)(I − A(s))u0 ≤ L(u0 )|t1 − s|

(4.3)

≤ L(u0 ), which is less than or equal to the right side of (4.1) with i = 1. On the other hand, if (u1 − u0 ) ∈ S2 () (defined in Section 1), then u1 − u0 ≤ ηu0 − u0 + ηv0 + η|f (t1 ) − f (s)|L(u0 )(1 + v0 ),

(4.4)

which is less than or equal to the right side of (4.1) with i = 1. Here v0 is any element in A(s)u0 . Step 2. By assuming that (4.1) is true for i = i − 1, we shall show that it is also true for i = i. If (ui − u0 ) ∈ S1 (), then ui − u0 = J (ti )ui−1 − J (s)(I − A(s))u0 ≤ L(u0 )|ti − s|

(4.5)

= L(u0 )(i), which is less than or equal to the right side of (4.1) with i = i because of η i > 1.

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On the other hand, if (ui − u0 ) ∈ S2 (), then ui − u0 ≤ ηui−1 − u0 + bi

(4.6)

where η = (1 − ω)−1 , bi = ηv0 + η|f (ti ) − f (s)|L(u0 )(1 + v0 ). This recursive inequality, combined with the induction assumption, readily gives ui − u0 ≤ η{η i−1 L(u0 )(i − 1) + [η i−2 b1 + ηi−3 b2 + · · · + ηbi−2 + bi−1 ]} + bi = ηi L(u0 )(i − 1) + [η i−1 b1 + ηi−2 b2 + · · · + ηbi−1 + bi ], which is less than or equal to the right side of (4.1) with i = i because of (i − 1) ≤ i. Case 2, where (4.2) is considered. Step 1. Claim that (4.2) is true for i = 1. This follows from the Step 1 in Case 1, because there it was shown that u1 − u0 ≤ L(u0 ) or b1 , which, when divided by , is less than or equal to the right side of (4.2) with i = 1. Here a0 = v0 , in which a0 = (u0 −u−1 )/ and u−1 ≡ u0 −v0 . Step 2. By assuming that (4.2) is true for i = i − 1, we will show that it is also true for i = i. If (ui − ui−1 ) ∈ S1 (), then ui − ui−1 ≤ L(ui−1 )|ti − ti−1 | = L(ui−1 ).

(4.7)

This, when divided by , has its right side less than or equal to one of the right sides of (4.2) with i = i. If (ui − ui−1 ) ∈ S2 (), then ui − ui−1 ≤ (1 − ω)−1[ui−1 − ui−2 ui−1 − ui−2 + |f (ti ) − f (ti−1 )|L(ui−1 )(1 + )]. By letting ui − ui−1 , ci = (1 − ω)−1 [1 + L(ui−1 )|f (ti ) − f (ti−1 )|],

ai =

−1

di = L(ui−1 )(1 − ω)

|f (ti ) − f (ti−1 )|,

it follows that ai ≤ ci ai−1 + di . Here notice that u0 − v0 = u−1 ;

and

(4.8)

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u0 − u−1 = v0 . The above inequality, combined with the induction assumption, readily gives a0 =

ai ≤ ci {[(ci−1 ci−2 · · · c2 )L(u0 ) or (ci−1 ci−2 · · · c3 )L(u1 ) or · · · or ci−1 L(ui−3 ) or L(ui−2 )] + [(ci−1 ci−2 · · · c1 )a0 + (ci−1 ci−2 · · · c2 )d1 + (ci−1 ci−2 · · · c3 )d2 + · · · + ci−1 di−2 + di−1 ]} + di ≤ [(ci ci−1 · · · c2 )L(u0 ) or (ci ci−1 · · · c3 )L(u1 ) or · · · or ci L(ui−2 )] + [(ci ci−1 · · · c1 )a0 + (ci ci−1 · · · c2 )d1 + (ci ci−1 · · · c3 )d2 + · · · + ci di−1 + di ], each of which is less than or equal to one of the right sides of (4.2) with i = i. The induction proof is now complete.

Proposition 4.2. Under the assumptions of Proposition 4.1, the following are ˆ true if u0 is in D(A(s)) = {y ∈ D(A(s)) : |A(s)y| < ∞}: with (HA) assumed, ui − u0 ≤ K1 (1 − ω)−i (2i + 1) ≤ K1 e(T −s)ω (3)(T − s); ui − ui−1 ≤ K3 ; where the constants K1 and K3 depend on the quantities:

K1 = K1 (L(u0 ), (T − s), ω, |A(s)u0 |, KB ); K2 = K2 (K1 , (T − s), ω, u0 ); K3 = K3 (L(K2 ), (T − s), ω, u0 , |A(s)u0 |, KB ); KB is the total variation of f on [0, T ]. With (HA) assumed, the results are similar, in which the quantities ˜ ˜ L(u 0 , u−1) and L(K2 , K2 ) replace the above L(u0 ) and L(K2 ), respectively. Proof. The proof below is made with (HA) assumed, because it is similar with (HA) assumed. We divide the proof into two cases. Case 1, where u0 ∈ D(A(s)). It follows immediately from Proposition 4.1 that ui − u0 ≤ N1 (1 − ω)−i (2i + 1) ≤ N1 e(T −s)ω (3)(T − s);

ui − ui−1 ≤ N3 ;

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where the constants N1 and N3 depend on the quantities: N1 = N1 (L(u0 ), (T − s), ω, v0 , KB ); N2 = N2 (N1 , (T − s), ω, u0 ); N3 = N3 (L(N2 ), (T − s), ω, u0 , v0 , KB ); KB is the total variation of f on [0, T ]. Here the estimate in [8, Page 65] ci · · · c1 ≤ eiω eei +···+e1 was used, where ei = L(ui−1 )|f (ti ) − f (ti−1 )|. ˆ Case 2, where u0 ∈ D(A(s)). This involves two steps. μ Step 1. Let u0 = (I − μA(s))−1 u0 where μ > 0, and let ui =

i

J (tj )u0 ;

uμi =

j=1

i

J (tj )uμ0 .

j=1

As in [31, Lemma 3.2, Page 9], we have, by letting μ −→ 0, uμ0 −→ u0 ; ˆ here notice that D(A(s)) is dense in D(A(s)). Also it is readily seen that uμi

=

i

(I −

A(tk ))−1 uμ0

k=1

−→ ui =

i

(I − A(tk ))−1 u0

k=1

as μ −→ 0, since (A(t) − ω) is dissipative for each 0 ≤ t ≤ T . Step 2. Since uμ0 ∈ D(A(s)), Case 1 gives uμi − uμ0 ≤ N1 (L(uμ0 ), (T − s), ω, v0μ , KB )(1 − ω)−i (2i + 1); uμi − uμi−1 ≤ N3 (L(N2 ), (T − s), ω, uμ0 , v0μ , KB ),

(4.9)

where N2 = N2 (N1 , (T − s), ω, uμ0 ), and v0μ is any element in A(s)(I − μA(s))−1 u0 . We can take v0μ = w0μ ≡

(Jμ (s) − I)u0 , μ

since w0μ ∈ A(s)(I − μA(s))−1 u0 . ˆ On account of u0 ∈ D(A(s)), we have lim

μ→0

(Jμ (s) − I)u0 = |A(s)u0 | < ∞. μ

Thus, by letting μ −→ 0 in (4.9) and using Step 1, the results in the Proposition 4.2 follow. The proof is complete. It follows immediately from the proof of Propositions 4.1 and 4.2 that

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Corollary 4.3. Let u0 ∈ D(A(s)) and let ui , i = 1, 2, . . . , satisfy the difference relation, where 0 < < λ0 , ui − A(ti )ui ui−1 ,

i = 1, 2, . . . .

Then the conclusions in Propositions 4.1 and 4.2 are still true, if we do not assume that A(t) satisfies the dissipativity condition (H1), the range condition (H2), and the time-regulating condition (HA) or (HA) , but assume that ui satisfies either the three conditions (C1), (C2), and (C3) (or equally ( (4.3) or (4.4)), ( (4.5) or (4.6)), and ( (4.7) or (4.8))) or the three conditions (C1) , (C2) , and (C3) : (C1) u1 − u0 L(u0 ), or ≤ ηv0 + η|f (t1 ) − f (s)|L(u0 )(1 + v0 ). (C2)

ui − u0 ≤

L(u0 )(i), or ηui−1 − u0 + bi .

(C3) ui − ui−1 ⎧ ⎪ ⎨L(ui−1 ), or ≤ (1 − ω)−1 [ui−1 − ui−2 ⎪ ⎩ + |f (ti − f (ti−1 )|L(ui−1 )(1 +

ui−1 −ui−2 )].

Here v0 is any element in A(s)u0 ; u0 − v0 = u−1; η = (1 − ω)−1 ; bi = ηv0 + η|f (ti ) − f (s)|L(u0 )(1 + v0 ); functions f and L are as in (HA).

(C1)

u1 − u0 ˜ or L(u 0 , u−1 ), ≤ ˜ ηv0 + η|f (t1 ) − f (s)|L(u 0 , u−1 )(1 + v0 ). (C2)

ui − u0 ≤

˜ L(u 0 , u−1 )(i), ηui−1 − u0 + bi .

or

(C3) ui − ui−1 ⎧ ˜ ⎪ ⎨L(ui−1 , ui−2 ), or ≤ (1 − ω)−1 [ui−1 − ui−2 ⎪ ⎩ ˜ + |f (ti − f (ti−1 )|L(u i−1 , ui−2 )(1 +

ui−1 −ui−2 )].

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Here v0 is any element in A(s)u0 ; u0 − v0 = u−1 ; η = (1 − ω)−1 ; ˜ bi = ηv0 + η|f (ti ) − f (s)|L(u 0 , u−1)(1 + v0 ); ˜ are as in (HA) . functions f and L 5. Proof of the Main Results Proof of Theorem 2.1 will be done after those of Propositions 5.1 and 5.2 below. Using the preliminary estimates in Section 4, together with the difference equations theory in Section 6, it will be shown in Section 6 that Proposition 5.1. Under the assumptions of Proposition 5.2 with (HA) assumed, the inequality is true L(K2 )|nμ − mλ|, if S2 (μ) = ∅; am,n ≤ cm,n + sm,n + dm,n + fm,n + gm,n , if S1 (μ) = ∅; where am,n, cm,n , sm,n , fm,n , gm,n and L(K2 ) are defined in Proposition 5.2. With (HA) assumed, the results are similar, in which the quantities ˜ ˜ L(u 0 , u−1) and L(K2 , K2 ) replace the above L(u0 ) and L(K2 ), respectively. In view of this and Proposition 4.1, we are led to the claim: ˆ Proposition 5.2. Let x ∈ D(A(s)) where 0 ≤ s ≤ T , and let λ, μ > 0, n, m ∈ N, be such that 0 ≤ (s + mλ), (s + nμ) ≤ T , and such that λ0 > λ ≥ μ > 0 for which μω, λω < 1. If A(t) satisfies the dissipativity condition (H1), the range condition (H2), and the time-regulating condition (HA), then the inequality is true: am,n ≤ cm,n + sm,n + dm,n + em,n + fm,n + gm,n. Here am,n ≡

n

Jμ (s + iμ)x −

i=1

m

Jλ (s + iλ)x;

i=1

μ α ≡ ; β ≡ 1 − α; λ = 2K1 γ n [(nμ − mλ) + (nμ − mλ)2 + (nμ)(λ − μ)]; = 2K1 γ n (1 − λω)−m (nμ − mλ)2 + (nμ)(λ − μ);

γ ≡ (1 − μω)−1 > 1; cm,n sm,n

ρ(T ) n γ [(mλ)(nμ − mλ)2 δ2 m(m + 1) 2 λ ]}; + (λ − μ) 2 = L(K2 )γ n (nμ − mλ)2 + (nμ)(λ − μ);

dm,n = [K4 ρ(δ)γ n (mλ)] + {K4

em,n

fm,n = K1 [γ n μ + γ n (1 − λω)−m λ]; gm,n = K4 ρ(|λ − μ|)γ n (mλ);

(5.1)

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K4 = γL(K2 )(1 + K3 );

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δ > 0 is arbitrary;

ρ(r) ≡ sup{|f (t) − f (τ )| : 0 ≤ t, τ ≤ T, |t − τ | ≤ r} is the modulus of continuity of f on [0, T ]; and K1 , K2 , and K3 are defined in Proposition 4.2. If, instead of (HA), (HA) is assumed, the results are similar, in which the ˜ ˜ quantities L(u 0 , u−1) and L(K2 , K2 ) replace the above L(u0 ) and L(K2 ), respectively. Proof. The proof below will be made under (HA), because it will be similar under (HA) . We will use the method of mathematical induction and divide the proof into two steps. Step 2 will involve six cases. Step 1. (5.1) is clearly true by Proposition 4.2, if (m, n) = (0, n) or (m, n) = (m, 0). Step 2. By assuming that (5.1) is true for (m, n) = (m − 1, n − 1) or (m, n) = (m, n − 1), we will show that it is also true for (m, n) = (m, n). This is done by the arguments below. Using the nonlinear resolvent identity in [6], we have am,n = Ju (s + nμ)

n−1

Jμ (s + iμ)x

i=1

− Jμ (s + mλ)[α

m−1

Jλ (s + iλ)x + β

i=1

m

Jλ (s + iλ)x)].

i=1

Here α = μλ and β = λ−μ λ . Under the time-regulating condition (HA), it follows that, if the element inside the norm of the right side of the above equality is in S1 (μ), then, by Proposition 4.2 with = μ, n Jμ (s + iμ)x)|mλ − nμ| am,n ≤ L( (5.2) i=1 ≤ L(K2 )|mλ − nμ|, which is less than or equal to the right side of (5.1) with (m, n) = (m, n), where γ n > 1. If that element instead lies in S2 (μ), then, by Proposition 4.2 with = μ, am,n ≤ γ(αam−1,n−1 + βam,n−1 ) + γμ|f (s + mλ) − f (s + nμ)|L( n

n

Jμ (s + iμ)x)

i=1 n−1 i=1

Jμ (s + iμ)x − Jμ (s + iμ)x ] × [1 + μ ≤ [γαam−1,n−1 + γβam,n−1 ] + K4 μρ(|nμ − mλ|), i=1

(5.3)

where K4 = γL(K2 )(1 + K3 ) and ρ(r) is the modulus of continuity of f on [0, T ]. From this, it follows that proving the following relations is sufficient under the induction assumption: γαpm−1,n−1 + γβpm,n−1 ≤ pm,n ;

(5.4)

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γαqm−1,n−1 + γβqm,n−1 + K4 μρ(|nμ − mλ|) ≤ qm,n ;

(5.5)

where qm,n = dm,n , and pm,n = cm,n or sm,n or em,n or fm,n or gm,n . Now we consider six cases. Case 1: where pm,n = cm,n . Under this case, (5.4) is true because of the calculations, where bm,n = (nμ − mλ)2 + (nμ)(λ − μ) was defined and the Schwartz inequality was used: α[(n − 1)μ − (m − 1)λ] + β[(n − 1)μ − mλ] = (nμ − mλ); √ √ αbm−1,n−1 + βbm,n−1 = α αbm−1,n−1 + β βbm,n−1 1

1

≤ (α + β) 2 (αb2m−1,n−1 + βb2m,n−1 ) 2 ≤ {(α + β)(nμ − mλ)2 + 2(nμ − mλ)[α(λ − μ) − βμ] 1

+ [α(λ − μ)2 + βμ2 ] + (n − 1)μ(λ − μ)} 2 = bm,n . Here α + β = 1;

α(λ − μ) − βμ = 0;

α(λ − μ)2 + βμ2 = μ(λ − μ).

Case 2: where pm,n = sm,n . Under this case, (5.4) is true, as is with the Case 1, by noting that (1 − λω)−(m−1) ≤ (1 − λω)−m . Case 3: where qm,n = dm,n . Under this case, (5.5) is true because of the calculations: γαdm−1,n−1 + γβdm,n−1 + K4 μρ(|nμ − mλ|) ≤ {γα[K4 ρ(δ)γ n−1 (m − 1)λ] + γβ[K4 ρ(δ)γ n−1 (mλ)]} ρ(T ) n−1 2 γ [(m − 1)λ ((n − 1)μ − (m − 1)λ) δ2 (m − 1)m 2 + (λ − μ) λ ]} 2 ρ(T ) + γβ{K4 2 γ n−1 [(mλ) ((n − 1)μ − mλ)2 δ m(m + 1) 2 λ ]} + (λ − μ) 2 + K4 μρ(|nμ − mλ|)

+ γα{K4

= K4 ρ(δ)γ n [(α + β)(mλ) − αλ] ρ(T ) n γ {α[(nμ − mλ)2 + 2(nμ − mλ)(λ − μ) + (λ − μ)2 ](mλ − λ) δ2 m(m + 1) 2 + [α(λ − μ) λ − α(λ − μ)mλ2 ] 2 + β[(nμ − mλ)2 − 2(nμ − mλ)μ + μ2 ](mλ)

+ K4

m(m + 1) 2 λ ]} + K4 μρ(|nμ − mλ|) 2 ≤ K4 ρ(δ)γ n [(mλ) − μ] + K4 μρ(|nμ − mλ|) + [β(λ − μ)

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ρ(T ) n γ [(mλ)(nμ − mλ)2 δ2 m(m + 1) 2 λ − μ(nμ − mλ)2 ] + (λ − μ) 2 ≡ rm,n , + K4

where the negative terms [2(nμ − mλ)(λ − μ) + (λ − μ)2 ](−λ) were dropped, α2(nμ − mλ)(λ − μ) − β2(nμ − mλ)μ = 0, and [α(λ − μ)2 + βμ2 ](mλ) = (mλ)μ(λ − μ), which cancelled −α(λ − μ)mλ2 = −(mλ)μ(λ − μ); it follows that rm,n ≤ dm,n , since ≤

K4 μρ(|nμ − mλ|) K4 μρ(δ) ≤ K4 μρ(δ)γ n , 2

2

(nμ−mλ) K4 μρ(T ) (nμ−mλ) ≤ K4 μρ(T )γ n , δ2 δ2

if |nμ − mλ| ≤ δ; if |nμ − mλ| > δ.

Case 4: where pm,n = em,n . Under this case, (5.4) is true, as is with the Case 1. Case 5: where pm,n = fm,n. Under this case, (5.4) is true because of the calculations: γαfm−1,n−1 + γβfm,n−1 = γαK1 [γ n−1 μ + γ n−1 (1 − λω)−(m−1) λ] + γβK1 [γ n−1 μ + γ n−1 (1 − λω)−m λ] ≤ K1 [(α + β)γ n μ + (α + β)γ n (1 − λω)−m λ] = fm,n. Case 6: where pm,n = gm,n . Under this case, (5.4) is true because of the calculations: γαgm−1,n−1 + γβgm,n−1 ≤ K4 γ n ρ(|λ − μ|)α(m − 1)λ + K4 γ n ρ(|λ − μ|)β(mλ) ≤ K4 γ n ρ(|λ − μ|)(α + β)(mλ) = gm,n . Now the proof is complete.

We are ready for Proof of Theorem 2.1: ˆ Proof. For x ∈ D(A(s)), it follows from Proposition 5.2, by setting μ = nt ≤ √ t 2 λ = m < λ0 and δ = λ − μ, that, as n, m −→ ∞, am,n converges to 0 uniformly for 0 ≤ (s + t) ≤ T . Thus n t J nt (s + i )x lim n→∞ n i=1

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ˆ ˆ exists for x ∈ D(A(s)). This limit also exists for x ∈ D(A(s)) = D(A(s))), on following the familiar limiting arguments in Chapter 1 or Crandall-Pazy [8]. √ On the other hand, setting μ = λ = nt < λ0 , m = [ μt ] and setting δ 2 = λ − μ, it follows that [t]

n

μ t J nt (s + i )u0 = lim Jμ (s + iμ)u0 . lim n→∞ μ→0 n i=1 i=1

(5.6)

Now, to show the Lipschitz property, (5.6) and Crandall-Pazy [8, Page 71] will be used. From Proposition 4.2, it is derived that un − um ≤ un − un−1 + un−1 − un−2 + · · · + um+1 − um ≤ K3 μ(n − m) for un =

n

Jμ (s + iμ)x;

ˆ x ∈ D(A(s)); um =

i=1

m

Jμ (s + iμ)x,

i=1

where n = [ μt ], m = [ μτ ], t > τ and 0 < μ < λ0 . The proof is completed by making μ −→ 0 and using (5.6). Proof of Theorem 2.2 will be done after those of Propositions 5.3 and 5.4 below. With regard to Proposition 5.3, we need the following setup. Consider the discretization of (1.1) on [0, T ], ui − A(ti )ui ui−1 , ui ∈ D(A(ti )),

(5.7)

where n ∈ N is large, and 0 < < λ0 is such that s ≤ ti = s + i ≤ T for each i = 1, 2, . . . , n. Here, to be noticed is that, for u0 ∈ E, ui exists uniquely by the hypotheses (H1) and (H2). ˆ Let u0 ∈ D(A(s)), and construct the Rothe functions [12, 32] by defining χn (s) = u0 ,

C n (s) = A(s);

χn (t) = ui ,

C n (t) = A(ti )

(5.8)

for t ∈ (ti−1 , ti ], and

un (s) = u0 ; t − ti−1 for t ∈ (ti−1 , ti ] ⊂ [s, T ].

un (t) = ui−1 + (ui − ui−1 ) Since

ui − ui−1 ≤ K3 ˆ for u0 ∈ D(A(s)) by Proposition 4.2, it follows immediately that

ˆ Proposition 5.3. For u0 ∈ D(A(s)), we have that lim

sup un (t) − χn (t) = 0;

n→∞ t∈[0,T ]

un (t) − un (τ ) ≤ K3 |t − τ |,

(5.9)

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where t, τ ∈ (ti−1 , ti ], and that dun (t) ∈ C n (t)χn (t); dt un (s) = u0 , where t ∈ (ti−1 , ti ]. Here the last equation has values in B([s, T ]; X), the real Banach space of all bounded functions from [s, T ] to X. Proposition 5.4. If A(t) satisfies the assumptions in Theorem 2.1, then n

lim un (t) = lim

n→∞

n→∞

i=1

J t−s (s + i n

t−s )u0 n

[ t−s μ ]

= lim

μ→0

Jμ (s + iμ)u0

i=1

ˆ uniformly for finite 0 ≤ (s + t) ≤ T and for u0 ∈ D(A(s)). Proof. The asserted uniform convergence will be proved by using the AscoliArzela Theorem [33]. Pointwise convergence will be proved first. For each t ∈ [s, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ t−s ], the greatest integer that is less than or equal to t−s . That u converges is because, for each above t, i i

lim ui = lim

→0

→0

(I − A(tk ))−1 u0

k=1 n

= lim

n→∞

k=1

t − s −1 t−s A(s + k )] u0 [I − n n

(5.10)

by (5.6), which has the right side convergent by Theorem 2.1. Since

ui − ui−1 ≤ K3

ˆ for u0 ∈ D(A(s)), we see from the definition of un (t) that lim un (t) = lim ui

n→∞

→0

= lim

n→∞

n i=1

J t−s (s + i n

t−s )u0 n

for each t. On the other hand, due to

ui − ui−1 ≤ K3

again, we see that un (t) is equi-continuous in C([s, T ]; X), the real Banach space of all continuous functions from [s, T ] to X. Thus it follows from the Ascoli-Arzela ˆ some subsequence of un (t) and then itself theorem [33] that, for u0 ∈ D(A(s)), converge uniformly to some

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u(t) = lim

n→∞

n

J t−s (s + i

i=1

n

t−s )u0 n

∈ C([s, T ]; X).

This completes the proof. Here is Proof of Theorem 2.2:

Proof. That u(t) is a limit solution follows from Propositions 5.3 and 5.4. That u(t) is a strong solution under the embedding property (HB) follows as in the Step 5 for the proof of Theorem 2.5 in Chapter 1. 6. Difference Equations Theory In this section, Proposition 5.1 in Section 5 will be proved, using the theory of difference equations [28]. A basic part of this theory was introduced in Section 5 in Chapter 1, but we will collect it here for easy reference. This includes its Lemma 5.1, Proposition 5.2, and Proposition 5.3 which will be frequently referred to. In fact, we will apply them to derive the four lemmas below, after which Proposition 5.1 will be proved. We now review the basic part of difference equations theory. Let {bn} = {bn }n∈{0}∪N = {bn }∞ n=0 be a sequence of real numbers. For such a sequence {bn }, we further extend it by defining bn = 0 if n = −1, −2, . . . .. The set of all such sequences {bn}’s will be denoted by S. Thus, if {an } ∈ S, then 0 = a−1 = a−2 = · · · . Define a right shift operator E : S −→ S by E{bn } = {bn+1 }

for {bn } ∈ S.

Similarly, define a left shift operator E # : S −→ S by E # {bn } = {bn−1 }

for {bn } ∈ S.

For c ∈ R and c = 0, define the operator (E − c)∗ : S −→ S by (E − c)∗ {bn } = {cn

n−1 i=0

bi } ci+1

for {bn} ∈ S. Here the first term on the right side of the equality, corresponding to n = 0, is zero. One more definition is that define, for {bn} ∈ S, (E − c)i∗ {bn } = [(E − c)∗ ]i {bn}, E i# {bn } = (E # )i {bn },

i = 1, 2, . . . ;

i = 1, 2, . . . ;

0

(E − c) {bn } = {bn}. It will follow that (E − c)∗ acts approximately as the inverse of (E − c) in this sense (E − c)∗ (E − c){bn } = {bn − cn b0 }.

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Next we extend the above notions to doubly indexed sequences. For a doubly indexed sequence {ρm,n } = {ρm,n }∞ m,n=0 of real numbers, let E1 {ρm,n } = {ρm+1,n }; E2 {ρm,n } = {ρm,n+1 }. Thus, E1 and E2 are the right shift operators, which act on the first index and the second index, respectively. It is easy to see that E1 E2 {ρm,n } = E2 E1 {ρm,n } is true. Here are Lemma 5.1, Proposition 5.2, and Proposition 5.3 in Chapter 1, but we relabel them as Lemma 6.1, Proposition 6.2, and Proposition 6.3, respectively: Lemma 6.1. Let {bn } and {dn } be in S. Then the following are true: (E − c)∗ (E − c){bn } = {bn − cn b0 }; (E − c)(E − c)∗ {bn} = {bn }; (E − c)∗ {bn } ≤ (E − c)∗ {dn },

if c > 0 and {bn} ≤ {dn }.

Here {bn } ≤ {dn } means bn ≤ dn for n = 0, 1, 2, . . .. Proposition 6.2. Let ξ, c ∈ R be such that c = 1 and c = 0. Let, be in S, the n ∞ ∞ three sequences {n}∞ n=0 , {c }n=0 , and {ξ}n=0 of real numbers. Then the following identities are true: n cn 1 − 2 + 2 }; d d d ξcn ξ ∗ }; (E − c) {ξ} = { − d d

n n−i (E − c)i∗ {cn } = { c }. i (E − c)∗ {n} = {

Here d = 1 − c and i = 0, 1, 2, . . .. Proposition 6.3. Let ξ, c ∈ R be such that c = 1 and cξ = 0. Let, be in S, n ∞ n ∞ the three sequences {nξ n }∞ n=0 , {ξ }n=0 , and {(cξ) }n=0 of real numbers. Then the identities are true: ξn nξ n cn ξ n 1 − 2 + 2 ) }; d d d ξ n n n c 1 ξ ξ ) }; (E − cξ)∗ {ξ n } = {( − d d ξ

n (cξ)n−i }. (E − cξ)i∗ {(cξ)n } = { i (E − cξ)∗ {nξ n } = {(

Here d = 1 − c and i = 0, 1, 2, . . .. Before we prove the Proposition 5.1, we need the following four lemmas.

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Lemma 6.4. With (5.3) assumed, the two inequalities hold: {am,n } ≤ (αγ(E2 − βγ)∗ )m {a0,n } +

m−1

(γα(E2 − γβ)∗ )i {(γβ)n am−i,0 }

i=0

+

m

(γα)j−1 ((E2 − γβ)∗ )j {rm+1−j,n+1 }

(6.1)

for n ≥ m;

j=1

{am.n} ≤ {(βγ +

γαE1# )n am,0 }

+{

n−1

(βγ + γαE1# )j rm,n−j }

j=0

j n−1 n n n−k k n β α am−k,0 } + { γ j β j−i αi rm−i,n−j } = {γ k j=0 i=0

(6.2)

k=0

for m ≥ n. Here rm,n = K4 μρ(|nμ − mλ|). Proof. [22] The proof will be divided into two cases. Case 1: n ≥ m. From (5.3), we have E1 (E2 − βγ){am,n } ≤ {αγam,n } + E1 E2 {rm,n },

(6.3)

and both sides of this, when applied to by (E2 − βγ)∗ , yield E1 {am,n } ≤ αγ(E2 − βγ)∗ {am,n } + (βγ)n E1 {am,0 } + (E2 − βγ)∗ {rm+1,n+1 }. Here Lemma 6.1 was used. This recursive relation gives the desired result. Case 2: m ≥ n. On applying E1# to both sides of (6.3), it follows that E2 {am,n } ≤ (βγ + αγE1# ){am,n } + E2 {rm,n }. Here Lemma 6.1 was used again. This recursive relation delivers the desired result. The proof is complete. Lemma 6.5. The equality is true: for n ≥ m, ((E2 − βγ)∗ )m {nγ n} = {

nγ n 1 mγ n 1 − αm γ m αm+1 γ m m−1 n β n−i 1 + (m − i) m γ n }. γ i αm+1−i i=0

Here γ, α and β are defined in Proposition 5.2. Proof. This was proved in the proof of Proposition 6.2 in Chapter 1 and in [21]. Lemma 6.6. The equality is true: ((E − βγ)∗ )j {γ n } = {(

j−1 1 1 n n−i i n−j − α )γ } β αj αj i=0 i

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= {(

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69

n 1 n−i i n−j β α )γ } αj i=j

for j ≤ n ∈ N. Here γ, α and β are defined in Proposition 5.2. Proof. This is a consequence of repeatedly applying Proposition 6.3 and carefully grouping terms [22]. Lemma 6.7. The equality is true: for n ≥ m, (E − βγ)m∗ {n2 γ n } n2 (2m)n m(m − 1) m(1 + β) − m+1 + [ + ] αm α αm+2 αm+2 m−1 (m − j)(m − j − 1) (m − j)(1 + β) n β n−j }. [ + ] − αm−j+2 αm−j+2 j

= γ n−m {

j=0

Proof. [23] The proof will be divided into three steps. Step 1. Claim that (E − β)∗ {n2 } = {

n2 1 1 2n − 2 + 2 − 2 β n }. α α α α

From the definition of (E − β)∗ and Proposition 6.2, it follows that 1 2 n−1 + 3 + ···+ )} = (E − β)∗ {n} β2 β βn 1 n 1 = { − 2 + 2 β n }. α α α This, when differentiated with respect to β, proves the claim. Step 2. Claim that {β n (

(E − βγ)∗ {n2 γ n } = γ n−1 (E − β)∗ {n2 }. But this is an immediate consequence of the definition of (E − βγ)∗ . Step 3. Step 2, combined with Step 1, yields at once (E − βγ)∗ {n2 γ n } = {[

n2 γ n 1 1 1 2nγ n + 2 γ n − 2 (βγ)n ] }. − 2 α α α α γ

On applying (E − βγ)∗ to both sides and using Proposition 6.3, we obtain (E − βγ)2∗ {n2 γ n }

4n 2 2 2 2 n 1 n n n 1 n2 = {[ 2 − 3 + ( 4 + 3 ) − ( 4 + 3 )β − 2 β ]γ 2 }. α α α α α α α 1 γ This operation, when repeated, gives (E − βγ)3∗ {n2 γ n } 6n 6 3 6 3 n2 − 4 + ( 5 + 4 ) − ( 5 + 4 )β n 3 α α α α α

α 2 2 1 n n−2 n 1 n n−1 β −( 4 + 3) − 2 ]γ 3 }. β 1 α α α 2 γ

= {[

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Continuing in this way, we are led to, for m ∈ N, (E − βγ)m∗ {n2 γ n } n2 (2m)n χm m = [ m − m+1 + ( m+2 + m+1 ) α α α α

m χi i n 1 − β n−(m−i) ]γ n m , ( i+2 + i+1 ) α α γ m−i i=1

(6.4)

where χm satisfies χ0 = 0, χ1 = 0, χ2 = 2, and χm = χm−1 + 2(m − 1). This difference equation, when solved for χm , gives χm = m(m − 1). This, together with the substitution j = m − i, plugged into (6.4), completes the proof. Here is our Proof of Proposition 5.1: Proof. The proof below is made with (HA) assumed, because it is similar with (HA) assumed. The proof will be divided into two steps, where Step 2 involves two cases. Step 1. If S2 (μ) = ∅, then (5.2) is true, whence am,n ≤ L(K2 )|nμ − mλ|. This is the desired estimate. Step 2. If S1 (μ) = ∅, then the inequality (5.3) is true, from which so are the inequalities (6.1) and (6.2) as a consequence of Lemma 6.4. There are two cases to consider. Case 1: (6.1) holds. In this case, we have, by Proposition 4.2, a0,n ≤ K1 γ n (2n + 1)μ; am−i,0 ≤ K1 (1 − λω)−m [2(m − i) + 1]λ;

(6.5)

so Lemma 6.6 and the proof of Proposition 6.2 in Chapter 1 imply that the first two terms of the right side of the inequality (6.1) is less than or equal to {cm,n + sm,n + fm,n }. Thus, what remains to do is to estimate the third term, denoted by {tm,n}, of the right side of the inequality (6.1). Observe that, using the subadditivity of ρ, we have {tm,n } ≤

m

(γα)j−1 (E2 − γβ)j∗ K4 μ{ρ(|λ − μ|) + ρ(|nμ − mλ + jλ|)}

j=1

≤

m

(γα)j−1 (E2 − γβ)j∗ K4 μ{γ n ρ(|λ − μ|) + γ n ρ(|nμ − (m − j)λ|)}

j=1

≡ {um,n } + {vm,n}, where γ = (1 − μω)−1 > 1. It follows from Lemma 6.6 that {um,n } ≤ {K4 μγ n ρ(|λ − μ|)

m j=1

αj−1

n 1 n n−i i α} β αj i=1 i

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1 ≤ {K4 γ n ρ(|λ − μ|)μ m} = {K4 ρ(|λ − μ|)γ n (mλ)} α ≤ {gm,n}. To estimate {vm,n }, let, as in Crandall-Pazy [8, page 68], δ > 0 be given (1) (2) (1) and write {vm,n} = {Im,n } + {Im,n }, where {Im,n } is the sum over indices with (2) |nμ − (m − j)λ| < δ, but {Im,n } is the sum over indices with |nμ − (m − j)λ| ≥ δ. As a consequence of Lemma 6.6, we have n m 1 n n−i i (1) β {Im,n } ≤ {K4 μγ n ρ(δ) αj−1 j α} α i=j i j=1 1 ≤ {K4 ρ(δ)μγ n m } = {K4 ρ(δ)γ n mλ}. α On the other hand, we have (2) } ≤ K4 μρ(T ) {Im,n

m

(γα)j−1 (E2 − γβ)j∗ {γ n }

j=1

≤ K4 μρ(T )

m

(γα)j−1 (E2 − γβ)j∗ {γ n

j=1

[nμ − (m − j)λ]2 }, δ2

which will be less than or equal to {K4

ρ(T ) n m(m + 1) 2 λ ]}, γ [(mλ)(nμ − mλ)2 + (λ − μ) 2 δ 2

thus deriving the desired estimate {tm,n } = {um,n } + {vm,n } ≤ {gm,n } + {dm,n}. This is because of the following five calculations, where Lemmas 6.5, 6.6, and 6.7 were used. Calculation 1. [nμ − (m − j)λ]2 = n2 μ2 − 2(nμ)(m − j)λ + (m − j)2 λ2 . Calculation 2. m

(γα)j−1 (E2 − γβ)j∗ {γ n n2 }μ2

j=1

= γ n−1

m j=1

αj−1 {

n2 2jn j(j − 1) j(1 + β) − j+1 + [ j+2 + ] j α α α αj+2

(j − i)(j − i − 1) (j − i)(1 + β) n β n−i }μ2 [ + ] − j−i+2 j−i+2 α α i i=0 j−1

≤ γn

m n2 j(j − 1) j(1 + β) 2jn { + ]}μ2 , − 2 +[ 3 3 α α α α j=1

where the negative terms associated with

j−1 i=0

were dropped.

(6.6)

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Calculation 3. m (γα)j−1 (E2 − γβ)j∗ {γ n n}[2μ(m − j)λ](−1) j=1

=

m

(γα)j−1 {γ n−j [

j=1

+ m

n j − j+1 j α α

j−1 n n−i i−j−1 α (j − i)]}[2μ(m − j)λ](−1) β i i=0

(6.7)

n j γ { − 2 }[2μ(m − j)λ](−1), ≤ α α j=1 n

where the negative terms associated with =

m

j−1

were dropped;

i=0

γ n α−1 {−2(nμ)(mλ) + j[2nμλ +

j=1

2μ 2μλ (mλ)] − j 2 ( )}. α α

Calculation 4. m (γα)j−1 (E2 − γβ)j∗ {γ n }(m − j)2 λ2 j=1

= ≤

m j=1 m

j−1

(γα)

{γ

n−j

j−1 1 1 n n−i i [ j − j α ]}(m − j)2 λ2 β α α i=0 i

(6.8)

γ n α−1 (m2 − 2mj + j 2 )λ2 ,

j=1

j−1 where the negative terms associated with i=0 were dropped. Calculation 5. Adding up the right sides of the above three inequalities (6.6), (6.7), and (6.8), and grouping them as a polynomial in j of degree two, we obtain the following: The term involving j 0 = 1 has the factor 1 2 2 [n μ − 2(nμ)(mλ) + (mλ)2 ] = (mλ)(nμ − mλ)2 . α j=1 m

μ

The term involving j 2 has the factor μ2 2μλ λ2 = 0. − 2 + 3 α α α The term involving j has two parts, one of which has the factor 2nμ2 2nμλ 2μmλ 2mλ2 + − − = 0, 2 α α α α2 and the other of which has the factor m 1+β 1 m(m + 1) 2 ( 3 − 3 )jμ2 = (λ − μ) μ λ . α α 2 j=1

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Case 2: (6.2) holds. In this case, denote, by {hm,n + wm,n }, the right side of (6.2), and estimate {hm,n }. Thanks to (6.5), we derive {hm,n } ≤ {K1 γ n (1 − λω)−m

n

β n−k αk [2(m − k) + 1]λ}

k=0

= {K1 γ n (1 − λω)−m [2(mλ − nμ) + λ]}, which is less than or equal to {sm,n + fm,n }, the desired estimate. Next, to estimate {wm,n }, the quantity δ > 0 in Case 1 will be used. Owing to rm,n = K4 μρ(|nμ − mλ|), readily seen is {wm,n } ≤ {K4 γ n μ

j n−1

β j−i αi ρ(|(nμ − mλ) + (iλ − jμ)|)},

j=0 i=0

which will be written as (1) (2) {Wm,n } + {Wm,n }. (1)

(2)

Here {Wm,n } is the sum over indices with |(nμ−mλ)+(iλ−jμ)| < δ, while {Wm,n } is the sum over indices with |(nμ − mλ) + (iλ − jμ)| ≥ δ. An immediate consequence is this estimate (1) } ≤ {K4 γ n (nμ)ρ(δ)}, {Wm,n (2)

so we proceed to derive an estimate for {Wm,n }. Due to |(nμ − mλ) + (iλ − jμ)|2 ≥ 1, δ2 we obtain (2) {Wm,n } ≤ {K4 γ n μ

j n−1

β j−i αi ρ(T )

j=0 i=0

[(nμ − mλ) + (iλ − jμ)]2 }. δ2

This will yield (2) } ≤ {K4 γ n {Wm,n

n(n − 1) ρ(T ) μ}, [(nμ)(nμ − mλ)2 + (λ − μ) 2 δ 2

because of the calculations: [(nμ − mλ) + (iλ − jμ)]2 = (nμ − mλ)2 + 2(nμ − mλ)(iλ − jμ) + (iλ − jμ)2 ; (iλ − jμ)2 = j 2 μ2 − 2(jμλ)i + λ2 i2 ; μ

j n−1

β j−i αi (nμ − mλ)2 = (nμ)(nμ − mλ)2 ;

j=0 i=0

j

i=0

β j−i αi (iλ − jμ) = (αj)λ − jμ = 0;

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μ

j n−1

β j−i αi (iλ − jμ)2

j=0 i=0

=μ

n−1

[j 2 μ2 − 2(jμλ)(αj) + λ2 [α2 j(j − 1) + (αj)]]

j=0

=μ

n−1

[j 2 (μ − αλ)2 + j(λ2 α)(1 − α)]

j=0

= μ(λ − μ)

n−1

j = (λ − μ)

j=0

n(n − 1) 2 μ . 2

Since λ > μ and m ≥ n, it follows that (1) (2) } + {Wm,n } ≤ {dm,n }. {wm,n } = {Wm,n


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CHAPTER 3

Linear Autonomous Parabolic Equations 1. Introduction In this chapter, linear autonomous, parabolic initial-boundary value problems will be solved by utilizing the results in Chapter 1. The obtained solutions will be limit or strong ones under ordinary assumptions. But under stronger assumptions, they will be classical solutions. It is this case that will be demonstrated here. To see how a strong solution is derived, the reader is referred to Chapters 4, 6, and 9. Let be considered the linear, autonomous, parabolic, initial-boundary value problem with the Robin boundary condition ut (x, t) = a(x)uxx (x, t) + b(x)ux (x, t) + c(x)u(x, t), (x, t) ∈ (0, 1) × (0, ∞); ux (0, t) = β0 u(0, t),

(1.1)

ux (1, t) = −β1 u(1, t);

u(x, 0) = u0 (x). Here β0 and β1 are two positive constants, and a(x), b(x), and c(x) are real-valued, continuous functions on [0, 1]. Additional restrictions are that, for all x, c(x) is non-positive, and that, for all x, a(x) is greater than or equal to some positive constant δ0 . (1.1) will be written as the linear Cauchy problem d u(t) = F u(t), t > 0; dt (1.2) u(0) = u0 , where the linear operator F : D(F ) ⊂ C[0, 1] −→ C[0, 1] is defined by F v = a(x)v + b(x)v + c(x)v; v ∈ D(F ) ≡ {w ∈ C 2 [0, 1] : w (j) = (−1)j βj w(j),

j = 0, 1}.

It will be shown by using the theory in Chapter 1 that the equation (1.2) and then the equation (1.1), for u0 ∈ D(F ) with F u0 ∈ D(F ), have a unique classical solution given by t t u(t) = lim (I − F )−n u0 = lim (I − νF )−[ ν ] u0 . n→∞ ν→0 n The same result holds true for (1.1) with the Robin boundary condition replaced by the Dirichlet or the Neumann or the periodic one: • u(0, t) = 0 = u(1, t) (Dirichlet condition). • ux (0, t) = 0 = ux (1, t) (Neumann condition) [15]. 75

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3. LINEAR AUTONOMOUS PARABOLIC EQUATIONS

• u(0, t) = u(1, t), ux (0, t) = ux (1, t) (Periodic condition) [15]. In addition to (1.1), higher space dimensional analogue of (1.1) will be the other subject to be studied in this chapter. It takes the form ut (x, t) =

N

aij (x)Dij u(x, t) +

i,j=1

N

bi (x)Di u(x, t)

i

+ c(x)u(x, t), ∂ u(x, t) + β2 (x)u(x, t) = 0, ∂n ˆ u(x, 0) = u0 (x);

(x, t) ∈ Ω × (0, ∞);

(1.3)

x ∈ ∂Ω;

where Ω is a bounded, smooth domain in RN , and N ≥ 2 is a positive integer; N 2 ∂ ∂ x = (x1 , x2 , . . . , xN ), and Dij u = i,j=1 ∂x∂i ∂xj u; ut = ∂t u and Di u = ∂x u; ∂Ω is i the boundary of Ω, and ∂∂nˆ u is the outer normal derivative of u; β2 (x), for all x, is greater than or equal to some positive constant δ0 . Additional assumptions are: • 0 < μ < 1. • aij (x), bi (x), and c(x) are μ-Holder continuous functions on Ω. • β2 (x) is continuously differentiable on Ω, with its first partial derivatives μ-Holder continuous. N • δ0 |ξ|2 ≤ i,j=1 aij (x)ξi ξj ≤ δ10 |ξ|2 for all x, ξ. • c(x) ≤ 0 for all x. The corresponding linear Cauchy problem will be d u(t) = Gu(t), t > 0; dt (1.4) u(0) = u0 , in which the linear operator G : D(G) ⊂ C(Ω) −→ C(Ω) is defined by Gv =

N i,j=1

aij (x)Dij v +

N

bi (x)Di v + c(x)v;

i=1

∂ w + β2 (x)w = 0 on ∂Ω}. ∂n ˆ The theory in Chapter 1 will be employed again to prove that, for u0 ∈ D 2 (G), the quantity t u(t) = lim (I − G)−n u0 n→∞ n t = lim (I − νG)−[ ν ] u0 v ∈ D(G) ≡ {w ∈ C 2+μ (Ω) :

ν→0

is the unique classical solution for equation (1.4) or, equivalently, equation (1.3). The same result holds true for (1.3) with the Robin boundary condition replaced by the Dirichlet or the Neumann one: • u(x, t) = 0 on ∂Ω (Dirichlet condition). • ∂∂nˆ u(x, t) = 0 on ∂Ω (Neumann condition) [18].

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The rest of this chapter is organized as follows. Section 2 states the main results, and Sections 3 and 4 prove the main results. The material of this chapter is based on [25]. 2. Main Results Theorem 2.1. The linear operator F in (1.2) is a closed operator that satisfies both the dissipativity condition (B2) and the range condition (B1) in Chapter 1. As a result, it follows from Theorem 2.1, Chapter 1 that the equation (1.2) and then the equation (1.1) have a unique solution given by t −n F ) u0 n t = lim (I − νF )−[ ν ] u0 ,

u(t) = lim (I − n→∞ ν→0

if u0 ∈ D(F ) satisfies F u0 = [a(x)u0 + b(x)u0 + c(x)u0 ] ∈ D(F ). More smoothness of u(t) in t follows from Theorem 2.2, Chapter 1, if we further restrict u0 . The same results hold true for the equation (1.1) with the Robin boundary condition replaced by the Dirichlet or the Neumann or the periodic one. Remark. • In order for F u0 to be in D(F ), more smoothness assumptions should be imposed on the coefficient functions a(x), b(x), and c(x). • The condition c(x) ≤ 0 can be replaced by the condition c(x) ≤ ω for some positive constant ω. This is because, in that case, the operator (F − ωI) is instead considered. Theorem 2.2. The linear operator G in (1.4) satisfies both the dissipativity condition (B2) and the weaker range condition (B1) in Chapter 1, and hence Theorem 2.3 in Chapter 1 implies the existence of the quantity t −n G) u0 n t = lim (I − νG)−[ ν ] u0

u(t) = lim (I − n→∞ ν→0

for u0 ∈ D(G). In fact, this u(t), for u0 ∈ D2 (G), is a unique classical solution of equation (1.4) or, equivalently, equation (1.3). The same results are valid for the equation (1.3) with the Robin boundary condition replaced by the Dirichlet or the Neumann one. Remark. • The condition u0 ∈ D2 (G) requires more smoothness assumptions on the coefficient functions aij (x), bi (x), and c(x). • The condition c(x) ≤ 0 can be weakened to c(x) ≤ ω, ω > 0, because, in that case, it suffices to consider the operator (G − ωI). 3. Proof of One Space Dimensional Case Proof of Theorem 2.1: Proof. We now begin the proof, which is composed of three steps.

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Step 1. (F satisfies the dissipativity condition (B2).) Let v1 and v2 be in D(F ), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ (0, 1), v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0. Here x0 ∈ {0, 1} is impossible in the case of the Robin boundary condition. For, if x0 = 0 and v1 − v2 ∞ = (v1 − v2 )(0), then (v1 − v2 ) (0) = β0 (v1 − v2 )(0) > 0, and so (v1 − v2 )(0) cannot be the positive maximum. This contradicts (v1 − v2 )(0) = v1 − v2 ∞ . Other cases can be treated in the same token, where either x0 = 0

and (v1 − v2 )(0) = −v1 − v2 ∞

or x0 = 1. Reasoning simialr to the above applies to the case with the Dirichlet or periodic condition considered. However, x0 ∈ {0, 1} is possible under the Neumann condition. Nevertheless, it can be handled by the first and second derivative tests. The dissipativity condition (B2) is then satisfied, as the calculations show: (v1 − v2 )(x0 )(F v1 − F v2 )(x0 ) ≤ 0; v1 − v2 2∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(F v1 − F v2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(F v1 − F v2 )∞

for all λ > 0.

Step 2. From the theory of ordinary differential equations [5], [24, Corollary 2.13, Chapter 4], the range of (I − λF ), λ > 0, equals C[0, 1], so F satisfies the range condition (B1). Step 3. (F is a closed operator.) Let vn ∈ D(F ) converge to v, and F vn to w. Then, there is a positive constant K, such that vn ∞ ≤ K; F vn ∞ = a(x)vn + b(x)vn + c(x)vn ∞ ≤ K. This, together with the interpolation inequality [1], [13, page 135], implies that vn ∞ and then vn C 2 [0,1] are uniformly bounded. Hence it follows from the Ascoli-Arzela theorem [33] that a subsequence of vn and then itself converge to v . That v is in D(F ) and F vn converges to F v will be true, whence F is closed. For, by uniform covergence theorem [2], if the Robin boundary condition is considered, then x F vn − b(y)vn − c(y)vn dy + vn (0) vn (x) = a(y) y=0 x w(y) − b(y)v − c(y)v dy + v (0); converges to v (x) = a(y) y=0 vn (j) = (−1)j βj vn (j) converges to v (j) = (−1)j βj v(j),

j = 0, 1;

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and so v ∈ D(F ) and F v = w by the fundamental theorem of calculus [2]. The case with other boundary conditions is treated in the same way. The proof is complete. 4. Proof of Higher Space Dimensional Case Proof of Theorem 2.2: Proof. We now begin the proof, which consists of five steps. Step 1. (G satisfies the dissipativity condition (B2).) Let v1 and v2 be in D(G), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ Ω, v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 )(x0 ) = 0,


(v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0. Here x0 ∈ ∂Ω is impossible in the case of the Robin boundary condition. For, if x0 ∈ ∂Ω and v1 − v2 ∞ = (v1 − v2 )(x0 ), then ∂ (v1 − v2 )(x0 ) > 0 ∂n ˆ by the Hopf boundary point lemma [13]. But this is a contradiction to ∂ (v1 − v2 )(x0 ) = −β2 (x0 )(v1 − v2 )(x0 ) < 0. ∂n ˆ The case is similar, where x0 ∈ ∂Ω and v1 − v2 ∞ = −(v1 − v2 )(x0 ). Reasoning like the above applies to the case of the Dirichlet boundary condition. However, x0 ∈ ∂Ω is possible in the case of the Neumann boundary condition. Nonetheless, it can be handled again by the first and second derivative tests. This is because the tangential derivative ∂∂tˆ(v1 − v2 )(x0 ) is zero by v1 − v2 ∞ = (v1 − v2 )(x0 ),

x0 ∈ ∂Ω,

− v2 )(x0 ) = 0, implies the gradient of (v1 − v2 ) at x0 is which, together with zero. The dissipativity condition (B2) is then satisfied, as the calculations show: ∂ ∂n ˆ (v1

(v1 − v2 )(x0 )(Gv1 − Gv2 )(x0 ) ≤ 0; v1 − v2 ∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(Gv1 − Gv2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(Gv1 − Gv2 )∞

for all λ > 0.

Step 2. From the theory of linear, elliptic partial differential equations [13], the range of (I −λG), λ > 0, equals C μ (Ω), so G satisfies the weaker range condition (B1) on account of C μ (Ω) ⊃ D(G). Step 3. It will be shown that ui C 3+η (Ω) , 0 < η < 1, is uniformly bounded if u0 ∈ D(G2 ), where ui = (I − νG)−i u0 is that in the discretized equation (2.1) in which A is replaced by G.

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Let u0 ∈ D(G) for a moment. By the dissipativity condition (B2) or using Lemma 4.2 in Chapter 1, we have ui − ui−1 Gui ∞ = ∞ ≤ Gu0 ∞ , ν which, together with relation ui − u 0 =

i

(uj − uj−1 ),

j=1

yields a uniform bound for ui ∞ . Hence, a uniform bound exists for ui C 1+λ (Ω) for any 0 < λ < 1, on using the proof of (4.1) in Chapter 5. (Alternatively, it follows that uiW 2,p (Ω) is uniformly bounded for any p > 2, on using the Lp elliptic estimates [37]. Hence, so is ui C 1+η (Ω) = (I − νG)−i u0 C 1+η (Ω) ,

0 < η < 1,

as a result of the Sobolev embedding theorem [1, 13].) This, applied to the relation Gui = (I − νG)−i (Gu0 ), shows the same thing for Gui C 1+η (Ω) , if Gu0 ∈ D(G), that is, if u0 ∈ D(G2 ). Therefore, ui C 3+η (Ω) is uniformly bounded if u0 ∈ D(G2 ), on employing the Schauder global estimates [13]. Step 4. (Existence of a solution) The result in Step 3, together with the AscoliArzela theorem [33], implies that, on putting i = [ νt ], a subsequence of ui and then itself, converge to u(t) as ν −→ 0, with respect to the topology in C 3+λ (Ω) for any 0 < λ < 1. Consequently, as in (6.4), (6.5), and (6.6) in Section 6 of Chapter 1, we have eventually du(t) t = Bu(t) = lim (I − G)−n (Gu0 ) n→∞ dt n u(0) = u0 . Thus u(t) is a solution. Step 5. (Uniqueness of a solution) Let v(t) be another solution. Then, by the first and second derivative tests, we have, for x0 ∈ Ω, u(t) − v(t)∞ = |[u(t) − v(t)](x0 )|; [u(t) − v(t)](x0 ) = 0; [u(t) − v(t)](x0 ) [u(t) − v(t)](x0 ) ≤ 0. Thus it follows that d d u(t) − v(t)2∞ = [u(t) − v(t)]2 (x0 ) dt dt d = 2[u(t) − v(t)](x0 ) [u(t) − v(t)](x0 ) dt = 2[u(t) − v(t)](x0 )[Gu(t) − Gv(t)](x0 ) ≤ 0. This implies u(t) − v(t)∞ ≤ u(0) − v(0)∞ = u0 − u0 ∞ = 0,

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from which uniqueness of a solution results. The proof is complete.

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CHAPTER 4

Nonlinear Autonomous Parabolic Equations 1. Introduction In this chapter, nonlinear autonomous, parabolic initial-boundary value problems will be solved by making use of the results in Chapter 1. The obtained solutions will be strong ones under suitable assumptions. Thus the linear cases in Chapter 3 will be extended to the nonlinear cases here. Let be considered first the case with space dimension equal to one, which is this nonlinear equation with the nonlinear Robin boundary condition ut (x, t) = α(x, ux )uxx (x, t) + g(x, u, ux ), (x, t) ∈ (0, 1) × (0, ∞); ux (0, t) ∈ β0 (u(0, t)),

ux (1, t) ∈ −β1 (u(1, t));

(1.1)

u(x, 0) = u0 (x). Here made are the assumptions: • β0 , β1 : R −→ R, are multi-valued, maximal monotone functions with 0 ∈ β0 (0) ∩ β1 (0). • α(x, p) and g(x, z, p) are real-valued, continuous functions of their arguments x ∈ [0, 1], z ∈ R, and p ∈ R. • α(x, p) is greater than or equal to some positive constant δ0 for all x ∈ [0, 1] and p ∈ R. • g(x, z, p) is monotone non-increasing in z for each x and p; that is, (z2 − z1 )[g(x, z2 , p) − g(x, z1 , p)] ≤ 0. • g(x, z, p)/α(x, p) is of at most linear growth in p, that is, for some positive, continuous function M0 (x, z), |g(x, z, p)/α(x, p)| ≤ M0 (x, z)(1 + |p|). (1.1) will be written as the nonlinear Cauchy problem d u(t) = Hu(t), dt u(0) = u0 ,

t > 0;

where the nonlinear, multi-valued operator H : D(H) ⊂ C[0, 1] −→ C[0, 1] is defined by Hv = α(x, v )v + g(x, v, v ); v ∈ D(H) ≡ {w ∈ C 2 [0, 1] : w (j) ∈ (−1)j βj w(j), j = 0, 1}. 83

(1.2)

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4. NONLINEAR AUTONOMOUS PARABOLIC EQUATIONS

It will be shown by using the theory in Chapter 1 that the equation (1.2) and then the equation (1.1), for u0 ∈ D(H), have a strong solution given by t H)−n u0 n→∞ n t = lim (I − νH)−[ ν ] u0 .

u(t) = lim (I − ν→0

The other case to be studied is with space dimension greater than one, which takes the form ut (x, t) = α0 (x, Du) u(x, t) + g0 (x, u, Du), (x, t) ∈ Ω × (0, ∞); ∂ u(x, t) + β(x, u) = 0, ∂n ˆ u(x, 0) = u0 (x).

x ∈ ∂Ω;

(1.3)

Here • Ω is a bounded, smooth domain in RN , and N ≥ 2 is a positive integer; N ∂ 2 • x = (x1 , x2 , . . . , xN ), and u = i=1 ∂x 2 u; i

∂ ∂ u, Di u = ∂x u, and Du = (D1 u, D2 u, . . . , DN u); • ut = ∂t i • ∂Ω is the boundary of Ω, and ∂∂nˆ u is the outer normal derivative of u; and additional assumptions are: • α0 (x, p) and g0 (x, z, p) are real-valued, continuously differentiable functions of their arguments x ∈ Ω, z ∈ R, and p ∈ RN , and their first partial derivatives are μ-Holder continuous, where 0 < μ < 1. • β(x, z) is twice continuously differentiable function of their arguments x ∈ Ω and z ∈ R, and its second partial derivatives are μ-Holder continuous. • α0 (x, p) is greater than or equal to some positive constant δ1 for all x ∈ [0, 1] and p ∈ R. • g0 (x, z, p)/α0 (x, p) is of at most linear growth in p, that is, for some positive, continuous function M1 (x, z),

|g0 (x, z, p)/α0 (x, p)| ≤ M1 (x, z)(1 + |p|). • g0 (x, z, p) is monotone non-increasing in z, that is, (z2 − z1 )[g0 (x, z2 , p) − g0 (x, z1 , p)] ≤ 0. • β(x, z) is strictly monotone increasing in z, that is, for some positive constant δ1 , βz (x, z) ≥ δ1 . The corresponding nonlinear Cauchy problem will be d u(t) = Ju(t), dt u(0) = u0 ,

t > 0;

in which the nonlinear operator J : D(J) ⊂ C(Ω) −→ C(Ω) is defined by Jv = α0 (x, Dv) v + g0 (x, v, Dv);

(1.4)

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∂ w + β(x, w) = 0 on ∂Ω}. ∂n ˆ The theory in Chapter 1 will be employed again to prove that, for u0 ∈ D(G), the quantity t u(t) = lim (I − G)−n u0 n→∞ n t = lim (I − νG)−[ ν ] u0 v ∈ D(J) ≡ {w ∈ C 2+μ (Ω) :

ν→0

is a strong solution for equation (1.4) or, equivalently, equation (1.3). The rest of this chapter is organized as follows. Section 2 states the main results, and Sections 3 and 4 prove the main results. 2. Main Results Theorem 2.1. The nonlinear operator H in (1.2) satisfies both the dissipativity condition (A2) and the range condition (A1) in Chapter 1. As a result, it follows from Theorem 2.4, Chapter 1 that the quantity t u(t) = lim (I − H)−n u0 n→∞ n t = lim (I − νH)−[ ν ] u0 ν→0

exists for u0 ∈ D(H). Furthermore, the H also satisfies the embedding condition (A3) in Chapter 1, of embeddedly quasi-demi-closedness, and hence, by Theorem 2.5 in Chapter 1, the u(t), for u0 ∈ D(H), is not only a limit solution but also a strong one for the equation (1.2) and then the equation (1.1). The u(t) will also satisfy the middle equation in (1.1). Theorem 2.1 is taken from [20]. Remark. • The condition (z2 − z1 )[g(x, z2 , p) − g(x, z1 , p)] ≤ 0 can be replaced by the condition (z2 − z1 )[g(x, z2 , p) − g(x, z1 , p)] ≤ ω(z2 − z1 )2 for some positive constant ω. This is because, in that case, the operator (H − ωI) is instead considered. Theorem 2.2. The nonlinear operator J in (1.4) satisfies the three conditions in Chapter 1: the dissipativity condition (A2), the weaker range condition (A1) , and the embedding condition (A3) of embeddedly quasi-demi-closedness. Consequently, Theorem 2.6 in Chapter 1 implies that the quantity t u(t) = lim (I − J)−n u0 n→∞ n t = lim (I − νJ)−[ ν ] u0 ν→0

exists for u0 ∈ D(G), and that this u(t), for u0 ∈ D(G), is not only a limit solution but also a strong one for the equation (1.4) or, equivalently, equation (1.3). The u(t) will also satisfy the middle equation in (1.3).

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Theorem 2.2 is taken from [19]. Remark. • The condition (z2 − z1 )[g0 (x, z2 , p) − g0 (x, z1 , p)] ≤ 0 can be weakened to (z2 − z1 )[g0 (x, z2 , p) − g0 (x, z1 , p)] ≤ ω(z2 − z1 )2 for some ω > 0, because, in that case, it suffices to consider the operator (J − ωI). 3. Proof of One Space Dimensional Case Proof of Theorem 2.1: Proof. We now begin the proof, which is composed of five steps. Step 1. (H satisfies the dissipativity condition (A2).) This is done as in solving Example 3.3, Chapter 1. Let v1 and v2 be in D(H), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ [0, 1], v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0. Here x0 ∈ {0, 1} is possible, due to the boundary conditions in D(H). For, if x0 = 0 and v1 − v2 ∞ = (v1 − v2 )(0), then the monotonicity of β0 and the positivity of (v1 − v2 )(0) implies (v1 − v2 ) (0) ≥ 0. From this, there must (v1 − v2 ) (0) = 0 because if (v1 − v2 ) (0) > occurs, then (v1 − v2 )(0) cannot be the positive maximum. Other cases can be treated similarly. The dissipativity condition (A2) is then satisfied, as the calculations show: (v1 − v2 )(x0 )(Hv1 − Hv2 )(x0 ) = (v1 − v2 )(x0 )[α(x0 , v1 )(v1 − v2 ) (x0 ) + g(x0 , v1 , v1 ) − g(x0 , v2 , v1 )] ≤ 0; v1 − v2 2∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(Hv1 − Hv2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(Hv1 − Hv2 )∞

for all λ > 0.

Step 2. The proof of Example 3.1 in Chapter 1 proved that, for each h ∈ C[0, 1] and for all λ = μ2 > 0 where 0 < μ < [log(3)]−1 , the equation v − λv = h, v (j) ∈ (−1)j βj (v(j)),

j = 0, 1

has a unique solution. Hence, the operator (I − λE)−1 : C[0, 1] −→ C 1 [0, 1] exists, where the operator E : D(E) ⊂ C[0, 1] −→ C[0, 1] is defined by Ev = v ;

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D(E) ≡ {w ∈ C 2 [0, 1] : w (j) ∈ (−1)j βj (u(j)), j = 0, 1}. It is also continuous because, for h1 , h2 ∈ C[0, 1]; vj = (I − λE)−1 hj , j = 1, 2, there results v2 − v1 ∞ ≤ h2 − h1 ∞ by the dissipativity of E, whence v2 − v1 ∞ ≤ 2h2 − h1 ∞ /λ; v2 − v1 ∞ ≤ K0 h2 − h1 ∞

for some constant K0 > 0.

Here the interpolation inequality was used [1], [13, page 135]: for each > 0, there is a positive C() such that, for v ∈ C 2 [0, 1], v ∞ ≤ v ∞ + C()v∞ . The above also shows that (I − λE)−1 is a compact operator, where the AscoliArzela theorem [33] was used. Step 3. (H satisfies the range condition (A1).) Let h ∈ C[0, 1] be given, and let λ = μ2 > 0 where 0 < μ < [log(3)]−1 . It will be shown that the equation u − λ[α(x, u, u )u + g(x, u, u )] = h u (j) ∈ (−1)j βj (u(j)),

(3.1)

j = 0, 1

has a solution. Consider the operator equation equation u = (I − λE)−1 W u, where (I − λE)−1 is from Step 2, and the operator W : C 1 [0, 1] −→ C[0, 1] is defined by Wu = u +

h − u + λg(x, u, u ) α(x, u )

for u ∈ C 1 [0, 1]. Solvability of this operator equation will complete the proof. Truncating W by defining, for each m ∈ N, W u, if uC 1 [0,1] ≤ m; Wm u = mu W ( u 1 ), if uC 1 [0,1] > m, C [0,1]

−1

1

it follows that (I − λE) Wm : C [0, 1] −→ C 1 [0, 1] is continuous, compact, and uniformly bounded for each m. Hence, the Schauder fixed point theorem [13] implies that (I − λE)−1 Wm um = um for some um . The proof is completed if um0 C 1 ≤ um0 for some m0 , because then, (I − λE)−1 W um0 = um0 . Assuming, to the contrary, that um C 1 > m for all m, we will seek a contradiction. It follows from the definition of Wm that um − λum = vm +

h − vm + λg(x, vm , vm ) , α(x, vm )

(3.2)

m where um ∈ D(E) and vm = umu . Using the relations which are consequences m C 1 of the first and second derivative tests,

um ∞ = |um (x0 )|; um (x0 )um (x0 ) ≤ 0

um (x0 ) = 0;

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for some x0 ∈ [0, 1], and multiplying (3.2) by um and evaluating it at x0 , it results that m )u2 (x0 ) − λum (x0 )um (x0 )]α(x0 , 0) 0 ≤ [(1 − um C 1 m = (h − vm )(x0 )um (x0 ) + λum (x0 )g(x0 , vm , 0) ≤ (h − vm )(x0 )um (x0 ). Therefore, vm ∞ ≤ h∞ . This, combined with (3.2), yields vm ∞ ≤ λ−1 K0 (h∞ + vm ∞ )

for some constant K0 , whence, on employing the above interpolation inequality again [1], [13, page 135], there results vm C 2 [0,1] ≤ K1 for constant K1 . But this is a contradiction to m = vm C 1 ≤ vm C 2 , if we let m −→ ∞. Step 4. (H satisfies the embedding condition (A3) of embeddedly quasi-demiclosedness.) [20] Let vn ∈ D(H) converge to v in C[0, 1], and let Hvn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (0, 1) = (L2 (0, 1))∗ , η(Hv) exists and |η(Hvn ) − η(Hv)| −→ 0. Here (C[0, 1]; · ∞ ) is continuously embedded into L2 (0, 1); · ). Since vn ∞ and Hvn ∞ are uniformly bounded, so is vn C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 [0, 1] to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (0, 1), whence, by Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (0, 1), 1 (Hvn − Hv)η dx| −→ 0 |η(Hvn ) − η(Hv)| = | 0

because

0

1

α(x, v )(v − +

−→ 0.

0

1

vn )η dx

1

+ 0

[α(x, v ) − α(x, vn )]vn η dx

[g(x, v, v ) − g(x, vn , vn )]η dx

Therefore H satisfies the embedding condition (A3). Step 5. (u(t) satisfies the middle equation in (1.1).) Consider the discretized equation ui −νHui = ui−1 , ui ∈ D(H), where i = 1, 2, . . ., ν > 0 satisfies ν < λ0 for which νω < 1, and ui = (I − νH)−i u0 exists uniquely by the range condition (A1) and the dissipativity condition (A2). On putting i = [ νt ], it follows that t

lim ui = lim (I − νH)−[ ν ] u0 = u(t).

ν→0

ν→0

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On the other hand, by utilizing the dissipativity condition (A2), we have Hui ∞ =

ui − ui−1 ∞ ≤ Hu0 ∞ . ν

This, combined with the relation ui − u0 =

i

(uj − uj−1 ),

j=1

yields a bound for ui∞ . Those, in turn, result in a bound for ui C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge to a limit in C 1 [0, 1], as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.1), as ui does so. The proof is complete. 4. Proof of Higher Space Dimensional Case Proof of Theorem 2.2: Proof. We now begin the proof, which consists of four steps. Step 1. (J satisfies the dissipativity condition (A2).) This will be proved as in solving Example 3.2 of Chapter 1. Let v1 and v2 be in D(J), and let v1 = v2 to avoid triviality. By the first and second derivative tests, there result, for some x0 ∈ Ω, v1 − v2 ∞ = |(v1 − v2 )(x0 )|; D(v1 − v2 )(x0 ) = 0,


(v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0. Here x0 ∈ ∂Ω is impossible, due to the boundary condition in D(J). For, if x0 ∈ ∂Ω and v1 − v2 ∞ = (v1 − v2 )(x0 ), then ∂ (v1 − v2 )(x0 ) > 0 ∂n ˆ by the Hopf boundary point lemma [13]. But this is a contradiction to ∂ (v1 − v2 )(x0 ) = −β2 (x0 )(v1 − v2 )(x0 ) < 0. ∂n ˆ The case is similar, where x0 ∈ ∂Ω and v1 − v2 ∞ = −(v1 − v2 )(x0 ). The dissipativity condition (A2) is then satisfied, as the calculations show: (v1 − v2 )(x0 )(Jv1 − Jv2 )(x0 ) = (v1 − v2 )(x0 )[α0 (x0 , Dv1 ) (v1 − v2 )(x0 ) + g0 (x0 , v1 , Dv1 ) − g0 (x0 , v2 , Dv1 )] ≤ 0; v1 − v2 ∞ = (v1 − v2 )(x0 )(v1 − v2 )(x0 ) ≤ [(v1 − v2 )(x0 )]2 − λ(v1 − v2 )(x0 )(Jv1 − Jv2 )(x0 ) ≤ v1 − v2 ∞ (v1 − v2 ) − λ(Jv1 − Jv2 )∞

for all λ > 0.

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Step 2. (J satisfies the weaker range condition (A1) .) Let h ∈ C μ (Ω) and λ > 0. It will be shown that the equation u − λ[α0 (x, Du) u + g0 (x, u, Du)] = h in Ω; (4.1) ∂u + β(x, u) = 0 on ∂Ω, ∂n ˆ has a solution, so that the range of (I − λJ) contains C μ (Ω) ⊃ D(J). To this end, use the method of continuity [13] by considering the family of equations, indexed by t ∈ [0, 1]: Lt u = tLu + (1 − t)L0 u = h

in Ω; ∂u ∂u + β(x, u)] + (1 − t)( + u) = 0 on ∂Ω; Nt u = t[ ∂n ˆ ∂n ˆ where L0 u = u − u; Lu = u − [α0 (x, Du) u + g0 (x, u, Du)]. Here note that λ = 1 was assumed, which is sufficient because λα0 and λg0 have the same properties as α0 and g0 have. By defining the set S = {t ∈ [0, 1] : Ltu = h in Ω and Nt u = 0 on ∂Ω for some u ∈ C 2+μ (Ω)}, it will be shown that S is open, closed, and not empty in [0, 1]. Hence S = [0, 1] and so, 1 ∈ S, proving that (4.1) is solved. It is readily seen from linear elliptic theory [13] that 0 ∈ S, so we proceed to verify that S is both open and closed. To show that S is open, let B1 = C 2+μ (Ω); B2 = C μ (Ω) × C 1+μ (∂Ω), and define the nonlinear operator G : D(G) ≡ B1 × [0, 1] −→ B2 ; G(u, t) = (Lt u − h, Nt u). By assuming t0 ∈ S, that is, assuming G(u0 , t0 ) = 0 for some u0 ∈ C 2+μ (Ω), an open neighborhood of t0 in S will exist as a consequence of the implicit function theorem [13]. Indeed, because G is continuously Frechet differentiable, the partial Frechet derivative Gu0 at (u0 , t0 ) : B1 −→ B2 exists and is given by Gu0 (v) = ([−t0 α0 − (1 − t0 )] v − t0 [(g0 )pi + (α0 )pi u0 ]Di v ∂v + [t0 βz + (1 − t0 )]v) + [t0 (1 − gz ) + (1 − t0 )]v, ∂n ˆ for v ∈ B1 . Here α0 = α0 (x, Du0 );

(α0 )pi = (α0 )pi (x, Du0 );

(g0 )z = (g0 )z (x, u0 , Du0 );

(g0 )pi = (g0 )pi (x, u0 , Du0 );

βz = βz (x, u0 ). Now that Gu0 is an invertible, linear operator by linear elliptic theory [13], where βz lies in C 1+μ (Ω), the implicit function theorem [13] implies the existence of an open neighborhood of t0 in S.

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That S is a closed set will be a result of some estimates. Let tk be a sequence in S that converges to t0 , and we will show t0 ∈ S. By the definition of S, there exists a sequence utk ∈ B1 such that Ltk utk = h in Ω; Ntk utk = 0 on ∂Ω,

(4.2)

whence, owing to the mean value theorem, utk − [tk α0 + (1 − tk )] utk − tk g0 = h in Ω; 1 ∂utk [tk = −tk β(x, 0) βz dθ + (1 − tk )]utk + ∂n ˆ 0

on ∂Ω.

(4.3)

Here the arguments of α0 and g0 were suppressed, and the bottom bracket will ∂u be, for convenience, written as [γtk (x)utk + ∂νtk ]. In order to estimate utk , the maximum principle arguments will be used. If the maximum value of |utk | occurs at some interior point in Ω, then the first and second derivative tests as in Step 1 will yield utk ∞ ≤ tk g(x, 0, 0)∞ + h∞ . Here taken into account was the monotone non-increasing assumption of g0 (x, z, p) in z. But if the maximum value occurs on the boundary, then utk ∞ = ±utk (x0 ) for some x0 ∈ ∂Ω. Considering the case utk ∞ = utk (x0 ) is sufficient, for which, ∂utk (x0 ) by the Hopf boundary point lemma [13], > 0. Here utk (x0 ) = 0 was ∂n ˆ assumed to avoid triviality. Since γtk (x) ≥ δ2 > 0 for some constant δ2 > 0 by the assumption that β(x, z) is stricly monotone increasing in z, there results, from (4.3), utk ∞ = utk (x0 ) ≤ [tk δ2 + (1 − tk )]−1 tk β(x, 0)∞ . To estimate utk further, rewrite (4.3) as

utk =

−tk g0 utk − h + ≡ F (x, utk , Dutk ) tk α0 + (1 − tk ) tk α0 + (1 − tk )

in Ω;

∂utk = −γtk utk − tk β(x, 0) ≡ H(x, utk ) on ∂Ω, ∂ν

(4.4)

from which utk has the integral representation [29] utk = − Z(x, y)H(y, utk ) dσy + Z(x, y)F (y, utk , Dutk ) dy. ∂Ω

Ω

Here Z(x, y) is the Green’s function of the second kind. Since [g0 (x, z, p)/α0 (x, p)] or F (x, z, p) is of at most linear growth in p and utk ∞ is uniformly bounded, we can differentiate the utk for (1 + μ) times (0 < μ < 1) in the above integral representation to obtain Dutk C μ (Ω) ≤ k Dutk ∞ + k. Here |Di Z(x, y)| ≤ k|x − y|1−n ; |Dij Z(x, y)| ≤ k|x − y|−n ;

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and Ω |x − y|n(μ−1) dx is finite for 1 > μ > 0 and for bounded y [13, page 159]. Thanks to the interpolation inequality [13, page 135], the estimate utk C 1+μ (Ω) ≤ k

(4.5)

is derived. This, combined with the Schauder global estimate [13], yields utk |C 2+μ (Ω) ≤ k. Therefore, it follows from the Ascoli-Arzela theorem [33] that a subsequence of utk converges to some u0 ∈ C 2+γ (Ω), where 0 < γ < μ. As a result, the equation (4.2) converges to Lt0 u0 = h in Ω; (4.6) Nt0 u0 = 0 on ∂Ω. Now u0 is not just in C 2+γ (Ω) but also in C 2+μ (Ω), from which t0 ∈ S and S is closed. Indeed, this is a result of the theory of linear elliptic partial differential equations [13], because u0 satisfies (4.6) and both the function h and the coefficient functions in (4.6) are in C μ (Ω). Step 3. (J satisfies the embedding condition (A3) of embeddedly quasi-demiclosedness.) [20] Let vn ∈ D(G(tn )) converge to v in C(Ω), and G(tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (Ω) = (L2 (Ω))∗ , η(G(t)v) exists and |η(G(tn )vn ) − η(G(t)v)| −→ 0. Here (C(Ω); · ∞ ) is continuously embedded into L2 (Ω); · ). Since vn ∞ and G(tn )vn ∞ are uniformly bounded, so is vn C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.5). (Alternatively, uniformly bounded are vn W 2,p (Ω) for any p ≥ 2 and then vn C 1+λ (Ω) for any 0 < λ < 1, on using the Lp elliptic estimates [37] and the Sobolev embedding theorem [1, 13].) Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 (Ω) to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (Ω), whence, by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (Ω), |η(G(tn )vn ) − η(G(t)v)| −→ 0, because

α0 (x, Dv)( v − vn )η dx + [α0 (x, Dv) − α0 (x, Dvn )] vn η dx Ω Ω + [g0 (x, v, Dv) − g0 (x, vn , Dvn )]η dx

−→ 0.

Ω

Therefore J satisfies the embedding condition (A3). Step 4. (u(t) for u0 ∈ D(J) satisfies the middle equation in (1.3).) Consider the discretized equation ui −νJui = ui−1 , ui ∈ D(J)), where i = 1, 2, . . . , ν > 0 is such that νω < 1, and ui = (I − νJ)−i u0

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exists uniquely by the weaker range condition (A1) and the dissipativity condition (A2). On putting i = [ νt ], it follows that t

lim ui = lim (I − νJ)−[ ν ] u0 = u(t).

ν→0

ν→0

On the other hand, by utilizing the dissipativity condition (A2), we have ui − ui−1 ∞ ≤ Ju0 ∞ , Jui ∞ = ν which, combined with the relation u i − u0 =

i

(uj − uj−1 ),

j=1

yields a bound for ui ∞ . It follows that uiC 1+λ (Ω) is uniformly bounded for any 0 < λ < 1, on using the proof of (4.5). (Alternatively, those, in turn, result in a bound for ui W 2,p (Ω) for any p ≥ 2, by the Lp elliptic estimates [37]. Hence, a bound exists for ui C 1+η (Ω) for any 0 < η < 1, as a result of the Sobolev embedding theorem [1, 13].) Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1+μ (Ω) to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.3), as ui does so. The proof is complete.

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CHAPTER 5

Linear Non-autonomous Parabolic Equations 1. Introduction In this chapter, linear non-autonomous, parabolic initial-boundary value problems will be solved by applying the results in Chapter 2. The obtained solutions will be limit or strong ones under ordinary assumptions. But under stronger assumptions, they will be classical solutions. It is this case that will be shown here. To see how a strong solution is yielded, the reader is referred to Chapters 6 and 9. Thus the linear autonomous cases in Chapter 3 shall be generalized to the linear non-autonomous cases here. The case with space dimension equal to one is the linear, non-autonomous equation with the time-dependent Robin boundary condition ut(x, t) = a(x, t)uxx (x, t) + b(x, t)ux (x, t) + c(x, t)u(x, t) + f0 (x, t), (x, t) ∈ (0, 1) × (0, T ); ux (0, t) = β0 (0, t)u(0, t),

ux (1, t) = −β1 (1, t)u(1, t);

(1.1)

u(x, 0) = u0 (x). This equation is different from that in Chapter 3, in that the t dependence in the coefficient functions is permitted. Here the six functions a(x, t), b(x, t), c(x, t), f0 (x, t), β0 (x, t), and β1 (x, t) are real-valued and jointly continuous in x ∈ [0, 1] and t ∈ [0, T ] where T > 0. Three more restrictions are made. One is that, for all x and t, c(x, t) is non-positive, and the second is that, for all x and t, the functions a(x, t), β0 (x, t), and β1 (x, t) are greater than or equal to some positive constant δ0 . The third is that the above six functions satisfy, for x ∈ [0, 1], t, τ ∈ [0, T ], and ζ(t), a function in t of bounded variation, |a(x, t) − a(x, τ )|,

|b(x, t) − b(x, τ )| ≤ |ζ(t) − ζ(τ )|;

|c(x, t) − c(x, τ )|,

|f0 (x, t) − f0 (x, τ )| ≤ |ζ(t) − ζ(τ )|;

|βj (x, t) − βj (x, τ )| ≤ M0 |t − τ |,

j = 0, 1.

Here M0 is a positive constant. (1.1) will be written as the evolution equation d u(t) = F (t)u(t), dt u(0) = u0 ,

0 < t < T;

where the time-dependent operator F (t) : D(F (t)) ⊂ C[0, 1] −→ C[0, 1] 95

(1.2)

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5. LINEAR NON-AUTONOMOUS PARABOLIC EQUATIONS

is defined by F (t)v = a(x, t)v + b(x, t)v + c(x, t)v + f (x, t); v ∈ D(F (t)) ≡ {w ∈ C 2 [0, 1] : w (j) = (−1)j βj (j, t)w(j), j = 0, 1}. It will be shown by using the theory in Chapter 2 that the equation (1.2) and then ˆ (0)), have a limit solution given by the equation (1.1), for u0 ∈ D(F n t t [I − F (i )]−1 u0 n→∞ n n i=1

u(t) = lim

t

[ν ] = lim [I − νF (iν)]−1 u0 . ν→0

i=1

ˆ (0)), this u(t) is even a strong solution. It becomes a unique classical If u0 ∈ D(F solution if more restrictions are imposed on u0 , a(x, t), b(x, t), c(x, t), f0 (x, t), and βj (x, t). The same results hold true for (1.1) with the Robin boundary condition replaced by the Dirichlet or the Neumann or the periodic one: • u(0, t) = 0 = u(1, t) (Dirichlet condition). • ux (0, t) = 0 = ux (1, t) (Neumann condition) [15]. • u(0, t) = u(1, t), ux (0, t) = ux (1, t) (Periodic condition) [15]. The other case with space dimension greater than one will be the equation, taking the form ut (x, t) = a0 (x, t) u(x, t) +

N

bi (x)Di u(x, t)

i=1

+ c(x)u(x, t) + f0 (x, t), ∂ u(x, t) + β2 (x, t)u(x, t) = 0, ∂n ˆ u(x, 0) = u0 (x).

(x, t) ∈ Ω × (0, T );

(1.3)

x ∈ ∂Ω;

Here Ω is a bounded, smooth domain in RN , and N ≥ 2 is a positive integer; ∂2 ∂ ∂ x = (x1 , x2 , . . . , xN ), and u = N i=1 ∂x2 u; ut = ∂t u and Di u = ∂xi u; ∂Ω is the i

boundary of Ω, and ∂∂nˆ u is the outer normal derivative of u. There are four additional assumptions. One is that the four functions a0 (x, t), bi (x, t), c(x, t), and f0 (x, t) are in C μ (Ω),

0 < μ < 1,

for all t ∈ [0, T ] where T > 0, and are jointly continuous in x ∈ Ω and t ∈ [0, T ]. The second is that the above four functions satisfy, for x ∈ Ω, t, τ ∈ [0, T ], and ζ(t), a function in t of bounded variation, |a0 (x, t) − a0 (x, τ )|, |c(x, t) − c(x, τ )|,

|bi (x, t) − bi (x, τ )| ≤ |ζ(t) − ζ(τ )|; |f0 (x, t) − f0 (x, τ )| ≤ |ζ(t) − ζ(τ )|.

The third is that, for all x and t, a0 (x, t), β2 (x, t) ≥ δ0 c(x, t) ≤ 0.

a positive constant;

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The fourth is that β2 (x, t) is in C 1+μ (Ω) for all t ∈ [0, T ] and satisfies, for x ∈ Ω, t, τ ∈ [0, T ], and some positive constant M0 , |β2 (x, t) − β2 (x, τ )| ≤ M0 |t − τ |. The corresponding evolution equation will be d u(t) = G(t)u(t), dt u(0) = u0 ,

0 < t < T;

(1.4)

in which the time-dependent operator G(t) : D(G(t)) ⊂ C(Ω) −→ C(Ω) is defined by G(t)v = a0 (x, t) v +

N

bi (x, t)Di v + c(x, t)v + f0 (x, t);

i=1

v ∈ D(G(t)) ≡ {w ∈ C 2+μ (Ω) :

∂ w + β2 (x, t)w = 0 ∂n ˆ

on ∂Ω}.

ˆ The theory in Chapter 2 will be employed again to prove that, for u0 ∈ D(G(0)), the quantity u(t) = lim

n

n→∞

i=1

[ νt ]

= lim

ν→0

[I −

t t G(i )]−1 u0 n n

[I − νG(iν)]−1 u0

i=1

is a limit solution for equation (1.4) or, equivalently, equation (1.3). If u0 ∈ ˆ D(G(0)), this u(t) is even a strong solution. Further assumptions on u0 , a0 (x, t), bi (x, t), c(x, t), f0 (x, t), and β2 (x, t) will ensure that u(t) is a unique classical solution. The same results hold true for (1.3) with the Robin boundary condition replaced by the Dirichlet or the Neumann one: • u(x, t) = 0 on ∂Ω (Dirichlet condition). • ∂∂nˆ u(x, t) = 0 on ∂Ω (Neumann condition) [18]. The rest of this chapter is organized as follows. Section 2 states the main results, and Sections 3 and 4 prove the main results. The material of this chapter is based on [25]. 2. Main Results Theorem 2.1. The time-dependent operator F (t) in (1.2) satisfies the four conditions, namely, the dissipativity condition (H1), the range condition (H2), the time-regulating condition (HA), and the embedding condition (HB) in Chapter 2. ˆ (0)), then As a result, it follows from Theorem 2.1, Chapter 2 that, if u0 ∈ D(F

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the equation (1.2) and then the equation (1.1) have a limit solution given by the quantity n t t [I − F (i )]−1 u0 n→∞ n n i=1

u(t) = lim

t

[ν ] = lim [I − νF (iν)]−1 u0 . ν→0

i=1

ˆ (0)), this u(t) is even a strong solution by Theorem 2.2. In that If u0 ∈ D(F case, u(t) also satisfies the middle equation in (1.1). It becomes a unique classical solution of (1.1) if the two conditions (S1) and (S2) below are satisfied, and if u0 is such that u0 ∈ D(F (0)) satisfies F (0)u0 = [a(x, 0)u0 + b(x, 0)u0 + c(x, 0)u0 + f0 (x, 0)] ∈ D(F (0)). (S1) a(x, t), b(x, t), c(x, t), and f0 (x, t) are the type of the function h(x, t) below. Dt h(x, t) and h(x, t) exist and are continuous in x, t, and Dt h(x, t) satisfies, for x ∈ [0, 1], t, τ ∈ [0, T ], |Dt h(x, t) − Dt h(x, τ )| ≤ |ζ(t) − ζ(τ )|. Here Dt h(x, t) is the partial derivative of h(x, t) with respect to t, and

h(x, t) is the second partial derivative of h(x, t) with respect to x. (S2) βj (x, t), j = 0, 1 are twice continuously differentiable in t, or weakly satisfy, for x ∈ {0, 1}, τ > 0, and t, t + τ, t + 2τ ∈ [0, T ], βj (x, t + 2τ ) − 2βj (x, t + τ ) + βj (x, t) | ≤ M0 , τ2 the second difference quotient of βj (x, t) in t being bounded. |

The same results hold true for the equation (1.1) with the Robin boundary condition replaced by the Dirichlet or the Neumann or the periodic one. Remark. • In order for F (0)u0 to be in D(F (0)), more smoothness assumptions should be imposed on the coefficient functions a(x, 0), b(x, 0), c(x, 0), and f0 (x, 0). • The condition c(x, t) ≤ 0 can be replaced by the condition c(x, t) ≤ ω for some positive constant ω. This is because, in that case, the operator (F (t) − ωI) is instead considered. Theorem 2.2. The time-dependent operator G(t) in (1.4) satisfies the four conditions, namely, the dissipativity condition (H1), the weaker range condition (H2) , the time-regulating condition (HA), and the embedding condition (HB) in Chapter 2. As a result, it follows from Theorem 2.1, Chapter 2 that, if u0 ∈ ˆ D(G(0)), then the equation (1.4) and then the equation (1.3) have a limit solution given by the quantity u(t) = lim

n→∞

n

[I −

i=1

t t G(i )]−1 u0 n n

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t

= lim

[ν ]

ν→0

[I − νG(iν)]−1 u0 .

i=1

ˆ If u0 ∈ D(G(0)), this u(t) is even a strong solution by Theorem 2.2. In that case, u(t) also satisfies the middle equation in (1.3). It becomes a unique classical solution of (1.3) if the two conditions (S3) and (S4) below are satisfied, and if u0 is such that u0 ∈ D(G(0)) satisfies G(0)u0 = [a0 (x, 0) u0 +

N

bi (x, 0)Di u0 + c(x, 0)u0 + f0 (x, 0)]

i=1

∈ D(G(0)). (S3) a0 (x, t), bi (x, t), c(x, t), and f0 (x, t) are the type of the function h(x, t) below. The (2 + μ)-th derivative of h(x, t) with respect to x exists and is jointly continuous in x, t. The partial derivative Dt h(x, t) of h(x, t) with respect to t exists and is jointly continuous in x, t, and satisfies, for x ∈ Ω, t, τ ∈ [0, T ], |Dt h(x, t) − Dt h(x, τ )| ≤ |ζ(t) − ζ(τ )|. (S4) β2 (x, t) is twice continuously differentiable in t, or weakly satisfy, for x ∈ ∂Ω, τ > 0, and t, t + τ, t + 2τ ∈ [0, T ], β2 (x, t + 2τ ) − 2β2 (x, t + τ ) + β2 (x, t) | ≤ M0 , τ2 the second difference quotient of β2 (x, t) in t being bounded. The same results are valid for the equation (1.3) with the Robin boundary condition replaced by the Dirichlet or the Neumann one. |

Remark. • The condition G(0)u0 ∈ D(G(0)) requires more smoothness assumptions on the coefficient functions a0 (x, 0), bi (x, 0), c(x, 0), and f0 (x, 0). • The condition c(x, t) ≤ 0 can be weakened to c(x, t) ≤ ω, ω > 0, because, in that case, it suffices to consider the operator (G(t) − ωI). 3. Proof of One Space Dimensional Case Proof of Theorem 2.1: Proof. The proof below assumes the Robin boundary condition, although it is similar with other boundary conditions considered. This proof will be composed of eight steps. Step 1 It is readily verified as in proving Theorem 2.1 of Chapter 3 and Example 3.1 of Chapter 1 that F (t) satisfies the dissipativity condition (H1). Step 2. From the theory of ordinary differential equations [5], [24, Corollary 2.13, Chapter 4], the range of (I − λF (t)), λ > 0, equals C[0, 1], so F (t) satisfies the range condition (H2). Step 3. (F (t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C[0, 1], i = 1, 2, and let v1 = (I − λF (t))−1 g1 ; v2 = (I − λF (τ ))−1 g2 ;

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where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ[a(x, t)(v1 − v2 ) + b(x, t)(v1 − v2 ) + c(x, t)(v1 − v2 )] = λ[(a(x, t) − a(x, τ ))v2 + (b(x, t) − b(x, τ ))v2 + (c(x, t) − c(x, τ ))v2 + (f0 (x, t) − f0 (x, τ ))] + (g1 − g2 ); v2 ∞

≤ δ0−1 [F (τ )v2 ∞ + b(x, τ )v2 ∞ + c(x, τ )v2 ∞ + f0 (x, τ )∞ ];

(v1 − v2 ) (0) − β0 (0, t)(v1 − v2 )(0) = [β0 (0, t) − β0 (0, τ )]v2 (0); (v1 − v2 ) (1) + β1 (1, t)(v1 − v2 )(1) = −[β1 (1, t) − β1 (1, τ )]v2 (1); so there holds, for some function L in the condition (HA), v1 − v2 ∞ ≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|L(v2 ∞ )[1 + F (τ )v2 ∞ ]; 1 v1 − v2 ∞ ≤ M0 v2 ∞ |t − τ |, δ0

or

thus proving the condition (HA), where v2 − g2 [I − λF (τ )]−1 g2 − g2 = . λ λ This is because of the interpolation inequality [1], [13, page 135] F (τ )v2 =

v2 ∞ ≤ v2 ∞ + C()v2 ∞

(3.1)

for each > 0 and for some constant C(), and because, as in solving Example 3.1, Chapter 1, the maximum principle applies; that is, there is an x0 ∈ [0, 1] such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ (0, 1), (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0, and that, for x0 ∈ {0, 1}, (v1 − v2 ) (0) ≤ 0 or ≥ 0, according as (v1 − v2 )(0) > 0 or < 0; (v1 − v2 ) (1) ≥ 0 or ≤ 0, according as (v1 − v2 )(1) > 0 or < 0. Step 4. (F (t) satisfies the embedding condition (HB).) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(F (tn )) converge to v in C[0, 1], and F (tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (0, 1) = (L2 (0, 1))∗ , η(F (t)v) exists and |η(F (tn )vn ) − η(F (t)v)| −→ 0. Here (C[0, 1]; · ∞ ) is continuously embedded into L2 (0, 1); · ). Since vn ∞ and F (tn )vn ∞ are uniformly bounded, so is vn C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Hence, by Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 [0, 1] to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (0, 1), whence, by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (0, 1), |η(F (tn )vn ) − η(F (t)v)| −→ 0,

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because

1 0

a(x, t)(v − vn )η dx +

1

+ 0

1

0

1 0

1

0

[b(x, t) − b(x, tn )]vn η dx

c(x, t)(v − vn )η dx +

+ 0

−→ 0.

1

101

[a(x, t) − a(x, tn )]vn η dx

b(x, t)(v − vn )η dx +

+

main4

0

1

[c(x, t) − c(x, tn )]vn η dx

[f0 (x, t) − f0 (x, tn )]η dx

Therefore F (t) satisfies the embedding condition (HB). ˆ (0)) satisfies the middle equation in (1.1).) Consider Step 5. (u(t) for u0 ∈ D(F the discretized equation ui −νF (ti )ui = ui−1 , (3.2) ui ∈ D(F (ti )), ˆ (0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that where u0 ∈ D(F νω < 1 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νF (tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νF (0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 in Chapter 2 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νF (tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t F (k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4 of Chapter 2, we have ui ∞ ; a(x, ti )ui + b(x, ti )ui + c(x, ti )ui + f0 (x, ti )∞ ui − ui−1 ∞ ; = F (ti )ui ∞ = ν are uniformly bounded. Those, in turn, result in a bound for uiC 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1 [0, 1] to a limit,

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as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.1), as ui does so. Step 6. (Further estimates of ui under the conditions (S1), (S2), and u0 ∈ D(F (0)) with F (0)u0 ∈ D(F (0))) Because of F (0)u0 ∈ D(F (0)), the ui in Step 5 satisfies ui − ν[a(x, ti )ui + b(x, ti )ui + c(x, ti )ui + f0 (x, ti )] = ui−1 , i = 0, 1, . . . ; ui (0) = β0 (0, ti )ui (0), ui (1) = −β1 (1, ti )ui (1), i = −1, 0, 1, . . . . ui −ui−1 for i = 0, 1, . . ., that ν b(x, ti )vi + c(x, ti )vi + g(x, ν, ti )] =


vi − ν[a(x, ti )vi + i = 1, 2, . . . ;

vi−1 ,

β0 (0, ti ) − β0 (0, ti−1 ) ui−1 (0); ν β1 (1, ti ) − β1 (1, ti−1 ) ui−1 (1), vi (1) + β1 (1, ti )vi (1) = − ν where, with ti−1 = ti − ν, vi (0) − β0 (0, ti )vi (0) =

g(x, ν, ti ) = g(x, ν, ti , ti−1 ) a(x, ti ) − a(x, ti−1 ) b(x, ti ) − b(x, ti−1 ) ui−1 + ui−1 ν ν c(x, ti ) − c(x, ti−1 ) f0 (x, ti ) − f0 (x, ti−1 ) ui−1 + . + ν ν Here, for convenience, we also define =

v−1 = v0 − ν[a(x, t0 )v0 + b(x, t0 )v0 + c(x, t0 )v0 + g(x, ν, t0 )]; t−1 = 0; for which g(x, ν, t0 ) = g(x, ν, 0) = 0. Thus, either from Corollary 4.3 or from the proof of Proposition 4.1 and the results in and the proof of Proposition 4.2 in Section 4 of Chapter 2, we have a(x, ti )vi + b(x, ti )vi + c(x, ti )vi + g(x, ν, ti )∞ vi − vi−1 ∞ , i = 0, 1, . . . ; = ν is uniformly bounded, whence so are ui − ui−1 C 2 [0,1] vi C 2 [0,1] = ν = a(x, ti )ui + b(x, ti )ui + c(x, ti )ui + f0 (x, ti )C 2 [0,1] , i = 0, 1, . . . ; ui C 4 [0,1] ,

i = 0, 1, . . . ,

as in Step 5. This is because those vi ’s above, i = −1, 0, 1, . . ., satisfy the conditions (C1), (C2), and (C3) in Corollary 4.3, that is, the conditions ((4.3) or (4.4)), ((4.5) or (4.6)), and ((4.7) or (4.8)) in Section 4 of Chapter 2. A proof of it follows from applying (3.1), the maximum principle argument in Step 3, and the fact that

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103

the quantity ui −uνi−1 ∞ in Step 5 is bounded. Here it is to be observed that g(x, ν, ti )∞ is uniformly bounded, and that ui−1 − ui−2 = vi−1 ν =

1 vi−1 − vi−2 [ − b(x, ti−1 )vi−1 a(x, ti−1 ) ν − c(x, ti−1 )vi−1 − g(x, ν, ti−1 )].

Step 7. (Existence of a solution) Now that, from Step 6, ui C 4 [0,1] , i = 2, 3, . . . , is uniformly bounded, it follows from the Ascoli-Arzela theorem [33], as in Step 5, that a subsequence of ui and then itself, through the discretized equation (3.2), converge in C 3 [0, 1] to the limit u(t), as ν −→ 0. Therefore u(t) is a classical solution. Step 8. (Uniqueness of a solution) This proceeds as in Step 5 in the proof of Example 3.2, Chapter 1. The proof is complete. 4. Proof of Higher Space Dimensional Case Proof of Theorem 2.2: Proof. The proof below is composed of eight steps. The proof is proceeded with the Robin boundary condition assumed, while it is similar if other boundary conditions are considered. Step 1 It is readily verified as in proving Example 3.2 of Chapter 1 and Theorem 2.2 of Chapter 3 that G(t) satisfies the dissipativity condition (H1). Step 2. From the theory of linear, elliptic partial differential equations [13], the range of (I − λG(t)), λ > 0, equals C μ (Ω), so G(t) satisfies the weaker range condition (H2) because of C μ (Ω) ⊃ D(G(t)) for all t. Step 3. (G(t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C μ (Ω), i = 1, 2, and let v1 = (I − λG(t))−1 g1 ; v2 = (I − λG(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ[a0 (x, t) (v1 − v2 ) +

N

bi (x, t)Di (v1 − v2 ) + c(x, t)(v1 − v2 )]

i=1 n = λ[(a0 (x, t) − a0 (x, τ )) v2 + ( (bi (x, t) − bi (x, τ ))Di v2 i=1

+ (c(x, t) − c(x, τ ))v2 + (f0 (x, t) − f0 (x, τ ))] + (g1 − g2 ), v2 ∞ ≤ δ −1 [G(τ )v2 ∞ +

N

bi (x, τ )Di v2 ∞

i=1

+ c(x, τ )v2 ∞ + f0 (x, τ )∞ ; ∂(v1 − v2 ) + β2 (x, t)(v1 − v2 ) = −(β2 (x, t) − β2 (x, τ ))v2 , ∂n ˆ

x ∈ ∂Ω;

x ∈ Ω;

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so there holds, for some function L in the condition (HA), v1 − v2 ∞ ≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|L(v2 ∞ )[1 + G(τ )v2 ∞ ]; or 1 v1 − v2 ∞ ≤ M0 v2 ∞ |t − τ | ≤ L(v2 ∞ )|t − τ |, δ where [I − λG(τ )]−1 g2 − g2 v2 − g2 = . G(τ )v2 = λ λ This is because, as in proving the dissipativity condition (H1) in Step 1, the maximum principle argument applies, that is, there is an x0 ∈ Ω such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ Ω, D(v1 − v2 )(x0 ) = 0; (v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0, and that, for x0 ∈ ∂Ω, ∂(v1 − v2 ) (x0 ) ≥ 0 or ≤ 0 according as (v1 − v2 )(x0 ) > 0 or < 0. ∂n ˆ Here, to derive, for some constants c1 and c2 , Dv2 ∞ ≤ v2 C 1+μ (Ω) (4.1) v2 − g2 ∞ + c2 v2 ∞ , ≤ c1 λ we used the integral representation of v2 , with the Green’s function Z(x, y) of the second kind [29], v2 = − Z(x, y)[−β2 (y, τ )v2 ] dσy ∂Ω

+ Ω

Z(x, y)a0 (y, τ )−1 [

v2 − g2 − bi (y, τ )Di v2 − c(y, τ )v2 ] dy, λ i=1 N

differentiated the above v2 for (1 + μ) times, with respect to the variable x, to obtain, for some constants d1 and d2 , v2 − g2 Dv2 C μ(Ω) ≤ d1 ∞ + d2 (Dv2 ∞ + v2 ∞ ), λ and then employed the interpolation inequality [1], [13, page 135] Dv2 ∞ ≤ Dv2 C μ (Ω) + C()v2 ∞ for each > 0 and for some constant C(). Here to be noticed were, for some constants b1 and b2 , |Di Z(x, y)| ≤ b1 |x − y|1−N ; |Dij Z(x, y)| ≤ b2 |x − y|−N ; |x − y|N (μ−1) dx is finite for 0 < μ < 1 [13, page 159]. Ω

Step 4. (G(t) satisfies the embedding condition (HB).) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(G(tn )) converge to v in C(Ω), and G(tn )vn ∞ be uniformly

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105

bounded. It will be shown that, for each η in the self-dual space L2 (Ω) = (L2 (Ω))∗ , η(G(t)v) exists and |η(G(tn )vn ) − η(G(t)v)| −→ 0. Here (C(Ω); · ∞ ) is continuously embedded into L2 (Ω); · ). Since vn ∞ and G(tn )vn ∞ are uniformly bounded, so is vn C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.1). (Alternatively, uniformly bounded are vn W 2,p (Ω) for any p ≥ 2 and then vn C 1+λ (Ω) for any 0 < λ < 1, on using the Lp elliptic estimates [37] and the Sobolev embedding theorem [1, 13].) Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 (Ω) to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (Ω), whence, by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (Ω), |η(G(tn )vn ) − η(G(t)v)| −→ 0, because a0 (x, t)( v − vn )η dx + [a0 (x, t) − a0 (x, tn )] vn η dx Ω

+

N Ω i=1

+

−→ 0.

Ω

bi (x, t)(Di v − Di vn )η dx +

N Ω i=1

[bi (x, t) − bi (x, tn )]Di vn η dx

c(x, t)(v − vn )η dx + [c(x, t) − c(x, tn )]vn η dx Ω Ω + [(f0 (x, tn ) − f0 (x, t))]η dx| Ω

Therefore G(t) satisfies the embedding condition (HB). ˆ Step 5. (u(t) for u0 ∈ D(G(0)) satisfies the middle equation in (1.3).) Consider the discretized equation ui −νG(ti )ui = ui−1 , (4.2) ui ∈ D(G(ti )), ˆ where u0 ∈ D(G(0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that νω < 1 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νG(tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νG(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 in Chapter 2 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

[ν ] k=1

[I − νG(tk )]−1 u0

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= lim

n→∞

n

[I −

k=1

t t G(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4 of Chapter 2, we have ui ∞ ; a0 (x, ti ) ui +

N

bj (x, ti )Dj ui + c(x, ti )ui + f0 (x, ti )∞

j=1

ui − ui−1 ∞ ; ν are uniformly bounded, whence so is ui C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.1). (Alternatively, those, in turn, result in a bound for uiW 2,p (Ω) for any p ≥ 2, by the Lp elliptic estimates [37]. Hence, a bound exists for ui C 1+η (Ω) for any 0 < η < 1, as a result of the Sobolev embedding theorem [1, 13].) Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1+μ (Ω) to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.3), as ui does so. Step 6. (Further estimates of ui under the conditions (S3), (S4), and u0 ∈ D(G(0)) with G(0)u0 ∈ D(G(0))) Because of G(0)u0 ∈ D(G(0)), the ui in Step 5 satisfies = G(ti )ui ∞ =

ui − ν[a0 (x, ti ) ui +

N

bj (x, ti )Dj ui

j=1

+ c(x, ti )ui + f0 (x, ti )] = ui−1 , ∂ ui (x) + β2 (x, ti )ui (x) = 0, ∂n ˆ From this, it follows, on letting wi = wi − ν[a0 (x, ti ) wi +

N

x ∈ ∂Ω,

ui −ui−1 ν

x ∈ Ω,

i = 0, 1, . . . ;

i = −1, 0, 1, . . . .

for i = 0, 1, . . ., that

bj (x, ti )Dj wi

j=1

+ c(x, ti )wi + g(x, ν, ti )] = wi−1 ,

x ∈ Ω,

i = 1, 2, . . . ;

β2 (x, ti ) − β2 (x, ti−1 ) ∂wi + β2 (x, ti )wi = − ui−1 , ∂n ˆ ν x ∈ ∂Ω, i = 0, 1, . . . ; where, with ti−1 = ti − ν, g(x, ν, ti ) = g(x, ν, ti , ti−1 ) =

N a0 (x, ti ) − a0 (x, ti−1 ) bj (x, ti ) − bj (x, ti−1 )

ui−1 + Dj ui−1 ν ν j=1

+

c(x, ti ) − c(x, ti−1 ) f0 (x, ti ) − f0 (x, ti−1 ) ui−1 + , ν ν

i = 0, 1, . . . .

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Here, for convenience, we also define w−1 = w0 − ν[a0 (x, t0 ) w0 +

N

bj (x, t0 )Dj w0

j=1

+ c(x, t0 )w0 + g(x, ν, t0 ))]; t−1 = 0; for which g(x, ν, t0 ) = g(x, ν, 0) = 0. Hence, either from Corollary 4.3 or from the proof of Proposition 4.1 and from both the results in Proposition 4.2 and the proof of Proposition 4.2 in Section 4 of Chapter 2, we have a0 (x, ti ) wi +

N

bj (x, ti )Dj wi + c(x, ti )wi + g(x, ν, ti )∞

j=1

wi − wi−1 ∞ , i = 0, 1, . . . ; ν wi C 1+η (Ω) , 0 < η < 1, i = 0, 1, . . . ; =

are uniformly bounded, as in Step 5, where ui − ui−1 wi = ν N = a0 (x, ti ) ui + bj (x, ti )Dj ui + c(x, ti )ui + f0 (x, ti ); j=1

hence, so is ui C 3+η (Ω) ,

i = 0, 1, . . .

by the Schauder global regularity theorem [13, page 111]. This is because those wi ’s, i = −1, 0, 1, . . ., satisfy the conditions (C1), (C2), and (C3) in Corollary 4.3, that is, the conditions ((4.3) or (4.4)), ((4.5) or (4.6)), and ((4.7) or (4.8)) in Section 4 of Chapter 2, for which employed were (4.1) and both the maximum principle argument in Step 3 and the boundedness of ui −uνi−1 ∞ in Step 5. Here it is to be observed that g(x, ν, ti )∞ is uniformly bounded, and that

ui−1 − ui−2 = wi−1 ν N wi−1 − wi−2 1 [ − = bj (x, ti−1 )Dj wj−1 a0 (x, ti−1 ) ν j=1 − c(x, ti−1 )wi−1 − g(x, ν, ti−1 )]. Step 7. (Existence of a solution) Now that, from Step 6, ui C 3+η (Ω) , i = 2, 3, . . . , is uniformly bounded, it follows from the Ascoli-Arzela theorem [33], as in Step 5, that a subsequence of ui and then itself, through the discretized equation (3.4), converge in C 3+μ (Ω) to the limit u(t), as ν −→ 0. Therefore u(t) is a classical solution. Step 8. (Uniqueness of a solution) This proceeds as in Step 5 in the proof of Example 3.2, Chapter 1. The proof is complete.

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CHAPTER 6

Nonlinear Non-autonomous Parabolic Equations (I) 1. Introduction In this chapter, nonlinear non-autonomous, parabolic initial-boundary value problems will be solved with the aid of the results in Chapter 2. The obtained solutions will be strong ones under suitable assumptions. Thus, by extending the nonlinear autonomous cases in Chapter 4 and the linear non-autonomous cases in Chapter 5, we shall consider the nonlinear, nonautonomous equation with the nonlinear Robin boundary condition, as well as its higher space dimensional analogue ut (x, t) = α(x, t, ux )uxx (x, t) + g(x, t, u, ux ), (x, t) ∈ (0, 1) × (0, T ); ux (0, t) ∈ β0 (u(0, t)),

ux (1, t) ∈ −β1 (u(1, t));

(1.1)

u(x, 0) = u0 (x). Here the assumptions are made: • β0 , β1 : R −→ R, are multi-valued, maximal monotone functions with 0 ∈ β0 (0) ∩ β1 (0). • α(x, t, p) and g(x, t, z, p) are real-valued, continuous functions of their arguments x ∈ [0, 1], t ∈ [0, T ], z ∈ R, and p ∈ R. Here T > 0. • α(x, t, p) is greater than or equal to some positive constant δ0 for all its arguments x, t, and p. • g(x, t, z, p) is monotone non-increasing in z for each x, t, and p; that is, (z2 − z1 )[g(x, t, z2 , p) − g(x, t, z1 , p)] ≤ 0. • g(x, t, z, p) is of at most linear growth in p, that is, for some positive, continuous function M0 (x, t, z), |g(x, t, z, p)| ≤ M0 (x, t, z)(1 + |p|). • The following are true for some continuous, positive functions N0 and N01 and for some continuous function ζ of bounded variation: |α(x, t, p) − α(x, τ, p)|/α(x, τ, p) ≤ |ζ(t) − ζ(τ )|N0 (x, t, τ ); |g(x, t, z, p) − g(x, τ, z, p)| ≤ |ζ(t) − ζ(τ )|N01 (x, t, τ, z)(1 + |p|). Here it is to be noted that assuming linear growth of g(x, t, z, p) in p is stronger than assuming that of g(x, z, p)/α(x, p) in p in Chapter 4. 109

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6. NONLINEAR NON-AUTONOMOUS PARABOLIC EQUATIONS (I)

(1.1) will be written as the nonlinear evoution d u(t) = H(t)u(t), dt u(0) = u0 ,

t ∈ (0, T );

(1.2)

where the nonlinear, multi-valued operator H(t) : D(H(t)) ⊂ C[0, 1] −→ C[0, 1] is defined by H(t)v = α(x, t, v )v + g(x, t, v, v ); v ∈ D(H(t)) ≡ {w ∈ C 2 [0, 1] : w (j) ∈ (−1)j βj w(j), j = 0, 1}. It will be shown by using the theory in Chapter 2 that the equation (1.2) and then ˆ the equation (1.1), for u0 ∈ D(H(0)), have a limit solution given by u(t) = lim

n

n→∞

i=1

[ νt ]

= lim

ν→0

I−

t t H(i )]−1 u0 n n

I − νH(iν)]−1 u0 .

i=1

ˆ If u0 ∈ D(H(0)), this u(t) is even a strong solution. The higher space dimensional analogue of (1.1) will be of the form ut (x, t) = α0 (x, t, Du) u(x, t) + g0 (x, t, u, Du), (x, t) ∈ Ω × (0, T ); ∂ u(x, t) + β(x, t, u) = 0, ∂n ˆ u(x, 0) = u0 (x);

x ∈ ∂Ω;

(1.3)

in which seven assumptions are made. • Ω is a bounded smooth domain in RN , N ≥ 2, and ∂Ω is the boundary of Ω. • n ˆ (x) is the unit outer normal to x ∈ ∂Ω, and μ is a real number such that 0 < μ < 1. • α0 (x, t, p) ∈ C 1+μ (Ω × RN ) is true for each t ∈ [0, T ] where T > 0, and is continuous in all its arguments. Furthermore, α0 (x, t, p) ≥ δ1 > 0 is true for all x, p, and all t ∈ [0, T ], and for some constant δ1 > 0. • g0 (x, t, z, p) ∈ C 1+μ (Ω × R × RN ) is true for each t ∈ [0, T ], is continuous in all its arguments, and is monotone non-increasing in z for each t, x, and p. • g0 (x, t, z, p) is of at most linear growth in p, that is, |g0 (x, t, z, p)| ≤ M1 (x, t, z)(1 + |p|) for some positive, continuous function M1 and for all t ∈ [0, T ]. • β(x, t, z) ∈ C 2+μ (Ω × R) is true for each t ∈ [0, T ], is continuous in all its arguments, and is strictly monotone increasing in z so that βz ≥ δ1 > 0 for the constant δ1 > 0.

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111

• The following are true for some continuous, positive functions N1 , N2 , N3 and for some continuous function ζ of bounded variation: |α0 (x, t, p) − α0 (x, τ, p)|/α0 (x, τ, p) ≤ |ζ(t) − ζ(τ )|N1 (x); |g0 (x, t, z, p) − g0 (x, τ, z, p)| ≤ |ζ(t) − ζ(τ )|N2 (x, z)(1 + |p|); |β(x, t, z) − β(x, τ, z)| ≤ |t − τ |N3 (x, z). Here again, it is to be noted that assuming linear growth of g0 (x, t, z, p) in p is stronger than assuming that of g0 (x, z, p)/α0 (x, p) in p in Chapter 4. The corresponding nonlinear evolution equation will be d u(t) = J(t)u(t), t ∈ (0, T ); dt (1.4) u(0) = u0 , in which the time-dependent, nonlinear operator J(t) : D(J(t)) ⊂ C(Ω) −→ C(Ω) is defined by J(t)v = α0 (x, t, Dv) v + g0 (x, t, v, Dv); v ∈ D(J(t)) ≡ {w ∈ C 2+μ (Ω) :

∂ w + β(x, t, w) = 0 on ∂Ω}. ∂n ˆ

ˆ The theory in Chapter 2 will be employed again to prove that, for u0 ∈ D(J(0)), the quantity n t t u(t) = lim [I − J(i )]−1 u0 n→∞ n n i=1 t

= lim

ν→0

[ν ]

[I − νJ(iν)]−1 u0

i=1

ˆ is a limit solution for equation (1.4) or, equivalently, equation (1.3). If u0 ∈ D(J(0)), this u(t) is even a strong solution. The rest of this chapter is organized as follows. Section 2 states the main results, and Sections 3 and 4 prove the main results. The material of this chapter is based on [25]. 2. Main Results Theorem 2.1. The time-dependent operator H(t) in (1.2) satisfies the four conditions, namely, the dissipativity condition (H1), the range condition (H2), the time-regulating condition (HA), and the embedding condition (HB) in Chapter 2. ˆ As a result, it follows from Theorem 2.1, Chapter 2 that, if u0 ∈ D(H(0)), then the equation (1.2) and then the equation (1.1) have a limit solution given by the quantity n t t [I − H(i )]−1 u0 u(t) = lim n→∞ n n i=1 t

= lim

ν→0

[ν] i=1

[I − νH(iν)]−1 u0 .

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ˆ If u0 ∈ D(H(0)), this u(t) is even a strong solution by Theorem 2.2. In that case, u(t) also satisfies the middle equation in (1.1). Remark. • The condition (z2 − z1 )[g(x, z2 , p) − g(x, z1 , p)] ≤ 0 can be replaced by the condition (z2 − z1 )[g(x, z2 , p) − g(x, z1 , p)] ≤ ω(z2 − z1 )2 for some positive constant ω. This is because, in that case, the operator (H − ωI) is instead considered. Theorem 2.2. The time-dependent operator J(t) in (1.4) satisfies the four conditions, namely, the dissipativity condition (H1), the weaker range condition (H2) , the time-regulating condition (HA), and the embedding condition (HB) in Chapter 2. As a result, it follows from Theorem 2.1, Chapter 2 that, if u0 ∈ ˆ D(J(0)), then equation (1.4) or, equivalently, equation (1.3) has a limit solution given by the quantity n t t [I − J(i )]−1 u0 u(t) = lim n→∞ n n i=1 t

= lim

ν→0

[ν ]

[I − νJ(iν)]−1 u0 .

i=1

ˆ this u(t) is even a strong solution by Theorem 2.2. In that If u0 ∈ D(J(0)), case, u(t) also satisfies the middle equation in (1.3). Remark. • The condition (z2 − z1 )[g0 (x, z2 , p) − g0 (x, z1 , p)] ≤ 0 can be weakened to (z2 − z1 )[g0 (x, z2 , p) − g0 (x, z1 , p)] ≤ ω(z2 − z1 )2 for some ω > 0, because, in that case, it suffices to consider the operator (J − ωI). 3. Proof of One Space Dimensional Case Proof of Theorem 2.1: Proof. We now begin the proof, which is composed of five steps. Step 1. (H(t) satisfies the dissipativity condition (H1).) This is done as in the proof of Theorem 2.1 of Chapter 4. Step 2. (H(t) satisfies the range condition (H2).) This is established by analogy with the proof of Theorem 2.1 in Chapter 4. Step 3. (H(t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C[0, 1],

i = 1, 2,

and let v1 = (I − λH(t))−1 g1 ;

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v2 = (I − λH(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ[α(x, t, v1 )(v1 − v2 ) + g(x, t, v1 , v1 ) − g(x, t, v2 , v2 )] = λ{

α(x, t, v1 ) − α(x, τ, v2 ) [H(τ )v2 − g(x, τ, v2 , v2 )] α(x, τ, v2 ) + [g(x, t, v2 , v2 ) − g(x, τ, v2 , v2 )]} + (g1 − g2 );

v2 ∞ ≤ δ0−1 [H(τ )v2 ∞ + g(x, τ, v2 , v2 )∞ ]; (v1 − v2 ) (0) ∈ [β0 (v1 (0)) − β0 (v2 (0))]; (v1 − v2 ) (1) ∈ −[β1 (v1 (1)) − β1 (v2 (1))]; so there holds, for some function L in the condition (HA), v1 − v2 ∞ ≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|L(v2 ∞ )[1 + H(τ )v2 ∞ ]. This proves the condition (HA), where v2 − g2 λ [I − λH(τ )]−1 g2 − g2 = . λ Here the linear growth in p at most of both |g(x, t, z, p)−g(x, τ, z, p)| and g(x, t, z, p) was used, together with the interpolation inequality [1], [13, page 135] H(τ )v2 =

v2 ∞ ≤ v2 ∞ + C()v2 ∞ for each > 0 and for some constant C(), where δ0−1

max

x∈[0,1];t∈[0,T ]

|M0 (x, t, v2 ∞ )| < 1

if = 1/[δ0−1

max

x∈[0,1];t∈[0,T ]

|M0 (x, t, v2 ∞ )| + 2], for example.

This is because, as in the proof of Theorem 2.1 of Chapter 4, the maximum principle applies; that is, there is an x0 ∈ (0, 1) such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0. Here x0 ∈ {0, 1} is impossible, due to the boundary conditions. Step 4. (H(t) satisfies the embedding condition (HB) of embeddedly quasidemi-closedness.) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(H(tn ) converge to v in C[0, 1], and let H(tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (0, 1) = (L2 (0, 1))∗ , η(Hv) exists and |η(Hvn ) − η(Hv)| −→ 0. Here (C[0, 1]; · ∞ ) is continuously embedded into L2 (0, 1); · ). Since vn ∞ and H(tn )vn ∞ are uniformly bounded, so is vn C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 [0, 1] to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (0, 1), whence, by Alaoglu theorem

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[36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (0, 1), 1 |η(H(tn )vn ) − η(H(t)v)| = | (H(tn )vn − H(t)v)η dx| 0

−→ 0 because

0

1

α(x, t, v )(v − +

−→ 0.

0

1

vn )η dx

1

+ 0

[α(x, t, v ) − α(x, tn , vn )]vn η dx

[g(x, t, v, v ) − g(x, tn , vn , vn )]η dx

Therefore H(t) satisfies the embedding condition (HB). ˆ Step 5. (u(t) for u0 ∈ D(H(0)) satisfies the middle equation in (1.1).) Consider the discretized equation ui −νH(ti )ui = ui−1 , (3.1) ui ∈ D(H(ti )), ˆ i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that where u0 ∈ D(H(0)), νω < 1 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νH(tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νH(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νH(tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t H(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4 of Chapter 2, we have that ui ∞ and H(ti )ui ∞ = (ui − ui−1 )/ν∞ are uniformly bounded. Those, in turn, result in a bound for ui C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1 [0, 1] to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.1), as ui does so. The proof is complete.

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4. Proof of Higher Space Dimensional Case Proof of Theorem 2.2: Proof. We now begin the proof, which consists of five steps. Step 1. That J(t) satisfies the dissipativity (H1) follows as in the proof of Theorem 2.2 of Chapter 4. Step 2. The proof of Theorem 2.2 in Chapter 4 also shows that J(t) satisfies the weaker range condition (H2) . Step 3. (J(t) satisfies the time-regulating condition (HA).) Let gi (x) ∈ C μ (Ω),

i = 1, 2,

and let v1 = (I − λJ(t))−1 g1 ; v2 = (I − λJ(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ[α0 (x, t, Dv1 ) (v1 − v2 ) + g0 (x, t, v1 , Dv1 ) − g0 (x, t, v2 , Dv2 )] = λ{

α0 (x, t, Dv1 ) − α0 (x, τ, Dv2 ) [J(τ )v2 − g0 (x, τ, v2 , Dv2 )] α0 (x, τ, Dv2 ) + [g0 (x, t, v2 , Dv2 ) − g0 (x, τ, v2 , Dv2 )]} + (g1 − g2 ), x ∈ Ω;

∂(v1 − v2 ) + [β(x, t, v1 ) − β(x, t, v2 )] = −[β(x, t, v2 ) − β(x, τ, v2 )], ∂n ˆ

x ∈ ∂Ω;

so, there holds, for some function L in the condition (HA), v1 − v2 ∞ ≤ g1 − g2 ∞ + λ|ζ(t) − ζ(τ )|L(v2 ∞)[1 + J(τ )v2 ∞ ]; 1 v1 − v2 ∞ ≤ N3 (x, v2 )∞ |t − τ | ≤ L(v2 ∞ )|t − τ |, δ1

or

where v2 − g2 λ [I − λJ(τ )]−1 g2 − g2 = . λ

J(τ )v2 =

Here the linear growth at most of |g0 (x, t, z, p) − g0 (x, τ, z, p)| in p was used. This is because, as in proving the dissipativity condition (H1) in Step 1, the maximum principle argument applies, that is, there is an x0 ∈ Ω such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ Ω, D(v1 − v2 )(x0 ) = 0; (v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0, and that, for x0 ∈ ∂Ω, ∂(v1 − v2 ) (x0 ) ≥ 0 ∂n ˆ

or ≤ 0 according as (v1 − v2 )(x0 ) > 0 or < 0.

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Here, to derive, for some numbers C1 (v2 ∞ ) and C2 (v2 ∞ ), depending on v2 ∞ , Dv2 ∞ ≤ v2 C 1+μ (Ω)

(4.1) v2 − g2 ∞ + C2 (v2 ∞ ), λ we used the integral representation of v2 , with the Green’s function Z(x, y) of the second kind [29], v2 = − Z(x, y)[−β2 (y, τ, v2 )] dσy ∂Ω v2 − g 2 + Z(x, y)α0 (y, τ, Dv2 )−1 [ − g0 (y, τ, v2 , Dv2 )] dy. λ Ω ≤ C1 (v2 ∞ )

Indeed, the result follows from differentiating the above v2 for (1 + μ) times, with respect to the variable x, to obtain, for some positive constants d1 and d2 and for some number C3 (v2 ∞ ) depending on v2 ∞ , Dv2 C μ (Ω) v2 − g2 ∞ ≤ d1 λ + d2 [ max

x∈Ω;t∈[0,T ]

|M1 (x, t, v2 ∞ )|Dv∞ + C3 (v2 ∞ )],

and from combining the interpolation inequality [1], [13, page 135] Dv2 ∞ ≤ Dv2 C μ (Ω) + C()v2 ∞ for each > 0 and for some constant C(). Here d2

max

x∈Ω;t∈[0,T ]

|M1 (x, t, v2 ∞)| < 1

if = 1/[d2

max

x∈Ω;t∈[0,T ]

|M1 (x, t, v2 ∞ )| + 2], for example.

Also to be noticed are the linear growth at most of g0 (x, t, z, p) in p and, for some constants b1 and b2 , |Di Z(x, y)| ≤ b1 |x − y|1−N ; |Dij Z(x, y)| ≤ b2 |x − y|−N ; |x − y|N (μ−1) dx is finite for 0 < μ < 1 [13, page 159]. Ω

Step 4. (J(t) satisfies the embedding condition (HB) of embeddedly quasidemi-closedness.) [20] Let tn ∈ [0, T ] converge to t, vn ∈ D(G(tn )) converge to v in C(Ω), and G(tn )vn ∞ be uniformly bounded. It will be shown that, for each η in the self-dual space L2 (Ω) = (L2 (Ω))∗ , η(G(t)v) exists and |η(J(tn )vn ) − η(J(t)v)| −→ 0. Here (C(Ω); · ∞ ) is continuously embedded into L2 (Ω); · ). Since vn ∞ and J(tn )vn ∞ are uniformly bounded, so is vn C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.5) in Chapter 4. (Alternatively, uniformly bounded are vn W 2,p (Ω) for any p ≥ 2 and then vn C 1+λ (Ω) for any 0 < λ < 1, on using the Lp elliptic estimates [37] and the Sobolev embedding theorem [1, 13].) Hence, by the Ascoli-Arzela theorem [33], a subsequence of vn and then itself converge in C 1 (Ω) to v. Also, vn is uniformly bounded in the Hilbert space W 2,2 (Ω), whence,

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by the Alaoglu theorem [36], a subsequence of vn and then itself converge weakly to v [36]. It follows that, for each η ∈ L2 (Ω), |η(G(tn )vn ) − η(G(t)v)| = | [G(tn )vn − G(t)v]η dx| Ω

−→ 0, because

Ω

α0 (x, t, Dv)( v − vn )η dx + [α0 (x, t, Dv) − α0 (x, tn , Dvn )] vn η dx Ω + [g0 (x, t, v, Dv) − g0 (x, tn , vn , Dvn )]η dx Ω

−→ 0.

Therefore J satisfies the embedding condition (A3). ˆ satisfies the middle equation in (1.3).) Consider Step 5. (u(t) for u0 ∈ D(J(0)) the discretized equation ui −νJ(ti )ui = ui−1 , (4.2) ui ∈ D(J(ti )), ˆ where u0 ∈ D(J(0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that νω < 1 and 0 ≤ ti = iν ≤ T . Here ui =

i

[I − νG(tk )]−1 u0

k=1

exists uniquely by the range condition (H2) and the dissipativity condition (H1). For convenience, we also define u−1 = u0 − νG(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νJ(tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t J(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Proposition 4.2 in Section 4 of Chapter 2, we have ui∞ and J(ti )ui ∞ = (ui − ui−1 )/ν∞ are uniformly bounded, whence so is uiC 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.1). (Alternatively, those, in turn, result in a bound for ui W 2,p (Ω) for any p ≥ 2, by the Lp elliptic estimates [37]. Hence, a bound exists for uiC 1+η (Ω) for any 0 < η < 1, as a result of the Sobolev embedding theorem [1, 13].) Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1+μ (Ω) to a limit,

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as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.3), as ui does so. The proof is complete.

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CHAPTER 7

The Associated Elliptic Equations 1. Introduction In this chapter, the solutions for the associated elliptic equations with solving the parabolic problems in Chapters 3, 4, 5, and 6 will be further shown to be explicit functions of the solution φ to the elliptic equation − v(y) = h,

y ∈ Ω;

∂v + v = 0, y ∈ ∂Ω. ∂n ˆ Here for the dimension of the space variable y equal to 2 or 3, the φ can be computed numerically and efficiently by the boundary element methods. As a consequence, a solution for a parabolic, initial-boundary value problem in the previous chapters might be computed numerically. Thus we recall the following. In Chapter 6, the nonlinear, non-autonomous, parabolic initial-boundary value problem was considered ut (x, t) = α0 (x, t, Du) u(x, t) + g0 (x, t, u, Du), (x, t) ∈ Ω × (0, T ); ∂ u(x, t) + β(x, t, u) = 0, x ∈ ∂Ω; ∂n ˆ u(x, 0) = u0 (x).

(1.1)

It was written as the nonlinear evolution equation d u(t) = J(t)u(t), dt u(0) = u0 ,

t ∈ (0, T );

where the time-dependent, nonlinear operator J(t) : D(J(t)) ⊂ C(Ω) −→ C(Ω) was defined by J(t)v = α0 (x, t, Dv) v + g0 (x, t, v, Dv); v ∈ D(J(t)) ≡ {w ∈ C 2+μ (Ω) :

∂ w + β(x, t, w) = 0 on ∂Ω}. ∂n ˆ

ˆ It was then proved that, for u0 ∈ D(J(0)), the quantity u(t) = lim

n→∞

n

[I −

i=1 119

t t J(i )]−1 u0 n n

(1.2)

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7. THE ASSOCIATED ELLIPTIC EQUATIONS t

= lim

ν→0

[ν ]

[I − νJ(iν)]−1 u0

i=1

is a strong solution for equation (1.2) or, equivalently, equation (1.1). Here the quantity [I − nt J(i nt )]−1 u0 is the unique solution to the elliptic equation t t J(i )v = u0 , n n that is associated with the nonlinear parabolic equation (1.1). In this chapter, it will be further shown that the quantity, for h ∈ C μ (Ω) where 0 < μ < 1, t t [I − J(i )]−1 h, i = 1, 2, . . . , n n n is the limit of a sequence where each term in the sequence is an explicit function of the solution φ to the elliptic equation v−

− v(y) = h, y ∈ Ω; (1.3) ∂v + v = 0, y ∈ ∂Ω. ∂n ˆ Here for the dimension of the space variable y equal to 2 or 3, the φ can be computed numerically and efficiently by the boundary element methods [11, 34]. Thus, a solution for a parabolic, initial-boundary value problem in the previous chapters might be computed numerically. The same thing will be done to the linear, non-autonomous, parabolic initialboundary value problem in Chapter 5 N

ut (x, t) =

aij (x, t)Dij u(x, t) +

i.j=1

N

bi (x)Di u(x, t)

i=1

+ c(x)u(x, t) + f0 (x, t), ∂ u(x, t) + β2 (x, t)u(x, t) = 0, ∂n ˆ u(x, 0) = u0 (x),

(x, t) ∈ Ω × (0, T );

(1.4)

x ∈ ∂Ω;

N where i,j=1 aij (x, t)Dij u(x, t) = a0 (x, t) u(x, t). To (1.4), the corresponding evolution equation there in Chapter 5 was d u(t) = G(t)u(t), dt u(0) = u0 ,

0 < t < T;

(1.5)

in which the time-dependent operator G(t) : D(G(t)) ⊂ C(Ω) −→ C(Ω) was defined by G(t)v =

N i,j=1

aij (x, t)Dij v +

N

bi (x, t)Di v + c(x, t)v + f0 (x, t);

i=1

v ∈ D(G(t)) ≡ {w ∈ C 2+μ (Ω) :

∂ w + β2 (x, t)w = 0 ∂n ˆ

on ∂Ω}.

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ˆ It was shown there in Chapter 5 that, for u0 ∈ D(G(0)), the quantity u(t) = lim

n

n→∞

i=1

[ νt ]

= lim

ν→0

[I −

t t G(i )]−1 u0 n n

[I − νG(iν)]−1 u0

i=1

is a strong solution for equation (1.5) or, equivalently, equation (1.4). The rest of this chapter is organized as follows. Section 2 states the main results, and Sections 3 and 4 prove the main results. The material of this chapter is taken from [25]. 2. Main Results Here we first consider the linear, time-dependent operator G(t) in (1.5), where i,j=1 aij (x, t)Dij u(x, t) is not necessarily equal to a0 (x, t) u(x, t). In this case, we additionally assume that aij (x, t) = aji (x, t) is in C μ (Ω) for all t ∈ [0, T ], and satisfies, for ξ ∈ RN , x ∈ Ω, and t ∈ [0, t],

N

λmin |ξ|2 ≤

N

aij ξi ξj ≤ λmax |ξ|2

i,j=1

for some positive constants λmin and λmax . Theorem 2.1. For h ∈ C μ (Ω), the solution u to the equation [I − G(t)]u = h

(2.1)

where 0 ≤ t ≤ T and > 0, is the limit of a sequence where each term in the sequence is an explicit function of the solution φ to the elliptic equation (1.3). We next consider the case with the nonlinear, time-dependent operator J(t) in (1.2). Theorem 2.2. For h ∈ C μ (Ω), the solution u to the equation [I − J(t)]u = h

(2.2)

where 0 ≤ t ≤ T and > 0, is the limit of a sequence where each term in the sequence is an explicit function of the solution φ to the elliptic equation (1.3). Here β(x, t, 0) ≡ 0 is assumed additionally. 3. Proof of Linear Case Proof of Theorem 2.1: Proof. Solvability of the equation (2.1) follows from [13, Pages 128-130], where the method of continuity [13, Page 75] is used. By writing out fully how the method of continuity is used, it will be seen that the solution u is the limit of a sequence where each term in the sequence is an explicit function of the solution φ to the elliptic equation (1.3).

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7. THE ASSOCIATED ELLIPTIC EQUATIONS

To this end, set U1 = C 2+μ (Ω);

U2 = C μ (Ω) × C 1+μ (∂Ω);

Lτ u = τ [u − G(t)u] + (1 − τ )(− u) in Ω; ∂u ∂u + β(x, t)u] + (1 − τ )( + u) on ∂Ω; Nτ u = τ [ ∂n ˆ ∂n ˆ where 0 ≤ τ ≤ 1. Define the linear operator £τ : U1 −→ U2 by £τ u = (Lτ u, Nτ u) for u ∈ U1 , and assume that £s is onto for some s ∈ [0, 1]. It follows from [13, Pages 128-130] in which used are the maximum principle and the Schauder global estimate [13] that uU1 ≤ C£τ uU2 ,

(3.1)

where the constant C is independent of τ . This implies that £s is one to one, so −1 £−1 s exists. By making use of £s , the equation, for w0 ∈ U2 given, £τ u = w0 is equivalent to the equation −1 u = £−1 s w0 + (τ − s)£s (£0 − £1 )u,

from which a linear map S : U1 −→ U1 , −1 Su = Ss u ≡ £−1 s w0 + (τ − s)£s (£0 − £1 )u

is defined. The unique fixed point u of S = Ss will be related to the solution of (2.1). By choosing τ ∈ [0, 1] such that |s − τ | < δ ≡ [C(£0 U1 →U2 + £1 U1 →U2 )]−1 ,

(3.2)

it follows that S = Ss is a strict contraction map. Therefore S has a unique fixed point w, and the w can be represented by lim S n 0 = lim (Ss )n 0

n→∞

n→∞

because of 0 ∈ U1 . Thus £τ is onto for |τ − s| < δ. It follows that, by dividing [0, 1] into subintervals of length less than δ and repeating the above arguments in a finite number of times, £τ becomes onto for all τ ∈ [0, 1], provided that it is onto for some τ ∈ [0, 1]. Since £0 is onto by the potential theory [13, Page 130], we have that £1 is also onto. Therefore, for w0 = (h, 0), the equation £1 u = w0 has a unique solution u, and the u is the sought solution to (2.1). Here it is to −1 be observed that φ ≡ £−1 0 (h, 0) is the unique solution £0 (h, ϕ) to the elliptic equation (3.3) with ϕ ≡ 0: − v = h,

x ∈ Ω,

∂v + v(x) = ϕ on ∂Ω, ∂n ˆ

(3.3)

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and that S0 = S0 0 = £−1 0 (h, 0), −1 −1 S 2 0 = (S0 )2 0 = £−1 0 (h, 0) + £0 [|τ − 0|(£0 − £1 )£0 (h, 0)],

.. ..

The proof is complete. Remark. • The solution u is eventually represented by u(x) = £−1 0 H((h, 0)),

where H((h, 0)) is a convergent series in which each term is basically to obtained by, repeatedly, applying the linear operator (£0 − £1 )£−1 0 (h, 0) for a certain number of times. • The quantity £−1 0 (h, ϕ), for each (h, ϕ) ∈ U2 given, can be computed numerically and efficiently by the boundary element methods [11, 34], if the dimension of the space variable x equals 2 or 3. • The constant C above in (3.1) and (3.2) depends on n, μ, λmin , Ω, and on the coefficient functions aij (x, t), bi (x, t), c(x, t), β(x, t), and is not known explicitly [13]. Therefore, the corresponding δ cannot be determined in advance. Thus, when dealing with the elliptic equation (2.1) in Theorem 2.1 numerically, it is more possible, by choosing τ ∈ [0, 1] such that |s − τ | is smaller, that the sequence S n 0 will converge, for which |s − τ | < δ occurs. 4. Proof of the Nonlinear Case Proof of Theorem 2.2: Proof. The equation (2.2) has been solved in Chapter 6, but here the proof will be based on the contraction mapping theorem as in the proof of Theorem 2.1. To this end, set U1 = C 2+μ (Ω); U2 = C μ (Ω) × C 1+μ (∂Ω); Lτ u = τ [u − J(t)u] + (1 − τ )(u − u),

x ∈ Ω;

∂u ∂u + β(x, t, u)] + (1 − τ )( + u) on ∂Ω; ∂n ˆ ∂n ˆ where 0 ≤ τ ≤ 1. Define the nonlinear operator £τ : U1 −→ U2 by Nτ u = τ [

£τ u = (Lτ u, Nτ u) for u ∈ U1 , and assume that £s is onto for some s ∈ [0, 1]. As in proving that J(t) satisfies the dissipativity (H1) where the maximum principle was used, £s is one to one, so £−1 exists. By making use of £−1 s s , the equation, for w0 ∈ U2 given, £τ u = w0

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is equivalent to the equation u = £−1 s [w0 + (τ − s)(£0 − £1 )u], from which a nonlinear map S : U1 −→ U1 , Su = Ss u ≡ £−1 s [w0 + (τ − s)(£0 − £1 )u] for u ∈ U1 is defined. The unique fixed point of S = Ss will be related to the solution of (2.2). By restricting S = Ss to the closed ball of the Banach space U1 , Bs,r,w0 ≡ {u ∈ U1 : u − £−1 s w0 C 2+μ ≤ r > 0}, and choosing small enough |τ −s|, we will show that S = Ss leaves Bs,r,w0 invariant. This will be done by the following Steps 1 to 4. Step 1. It follows as in Chapter 6 that for £τ v = (f, χ), v∞ ≤ k{f ∞ ,χC(∂Ω) } ; DvC μ ≤ k{v∞ } Dv∞ + k{v∞ ,f ∞ ,χC(∂Ω) } ; vC 1+μ ≤ k{χC(∂Ω) ,f ∞ } ;

(4.1)

vC 2+μ ≤ K£τ vU2 = K£τ vC μ (Ω)×C 1+μ (∂Ω) . Here k{f ∞ } is a constant depending on f ∞ , and similar meaning is defined for other constants k’s; further, K is independent of τ , and while it depends on N, λmin , μ, Ω, and on the C 1+μ (Ω) norm of the coefficient functions α0 (x, t, Dv), g0 (x, t, v, Dv), β(x, t, v), the K has incorporated its dependence on vC 1+μ into £τ vU2 , by using the third equation in (4.1). Step 2. It is readily seen that, for v ∈ C 2+μ (Ω) with vC 2+μ ≤ R > 0, we have £τ vU2 ≤ k{R} vC 2+μ ,

(4.2)

where k{R} is independent of τ . Step 3. It will be shown that, if uC 2+μ ≤ R,

vC 2+μ ≤ R > 0,

then £τ u − £τ vU2 ≤ k{R} u − vC 2+μ .

(4.3)

It will be also shown that, if £τ u = (f, χ1 ),

£τ v = (w, χ2 ),

then u − vC 2+μ ≤ k{£τ uU2 ,£τ vU2 } [f − wC μ + χ1 − χ2 C 1+μ ] = k{£τ uU2 ,£τ vU2 } £τ u − £τ vU2 . Here K{R} and K{£τ uU2 ,£τ vU2 } are independent of τ .

(4.4)

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Using the mean value theorem, we have that f − w = Lτ u − L τ v = (u − v) − (1 − τ ) (u − v) − τ [α (u − v) + αp (x, t, p1 )(Du − Dv) v + gp (x, t, u, p2 )(Du − Dv) + gz (x, t, z1 , Dv)(u − v)],

x ∈ Ω;

∂(u − v) + [β(x, t, u) − β(x, t, v)] = χ1 − χ2 on ∂Ω; ∂n ˆ where p1 , p2 are some functions between Du and Dv, and z1 is some function between u and v. It follows as in (4.2) that £τ u − £τ vU2 ≤ k{R} u − vC 2+μ , which is one desired estimate. On the other hand, the maximum principle yields u − v∞ ≤ k{f −w∞ ,χ1 −χ2 ∞ } , and the fourth equation in (4.1) delivers uC 2+μ ≤ K£τ uU2 ; v||C 2+μ ≤ K£τ vU2 . Thus, it follows from the Schauder global estimate [13] that u − vC 2+μ ≤ k{£τ uU2 ,£τ U2 } £τ u − £τ vU2 , the other desired estimate. Step 4. Consequently, for u ∈ Bs,r,w0 , we deduce that, by the fourth equation in (4.1), uC 2+μ ≤ r + £−1 s w0 C 2+μ ≤ r + Kw0 U2 (4.5) ≡ R{r,w0 U2 } , and that Su − £−1 s w0 C 2+μ ≤ k{w0 U2 ,w0 +(τ −s)(£0 −£1 )uU2 } (τ − s)(£0 − £1 )uU2 ≤ |τ − s|k{w0 U2 ,R{r,w0 U

2

}}

by (4.4)

by (4.2) and (4.5).

Here the constant k{w0 U2 ,R{r,w0 U } } when w0 given and r chosen, is independent 2 of τ and s. Hence, by choosing some sufficiently small δ1 > 0, there results S = Ss : Bs,r,w0 ⊂ U1 −→ Bs,r,w0 ⊂ U1 for |τ − s| < δ1 ; that is, Bs,r,w0 is left invariant by S = Ss . Next, it will be shown that, for small |τ − s|, S = Ss is a strict contraction on Bs,r,w0 , from which S = Ss has a unique fixed point. Because, for u, v ∈ Bs,r,w0 , uC 2+μ ≤ R{r,w0 U2 } ,

vC 2+μ ≤ R{r,w0 U2 }

by (4.5),

it follows that, by (4.2), w0 + (τ − s)(£0 − £1 )uU2 ≤ k{w0 U2 ,R{r,w0 U

}}

w0 + (τ − s)(£0 − £1 )vU2 ≤ k{w0 U2 ,R{r,w0U

}}

2

2

;

;

(4.6)

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and that, by (4.3), (τ − s)[(£0 − £1 )u − (£0 − £1 )v]U2 ≤ |τ − s|k{R{r,w0 U

2

}}

u − vC 2+μ .

(4.7)

Therefore, on account of (4.4), (4.6), and (4.7), we obtain Su − SvC 2+μ ≤ |τ − s|k{R{r,w0 U

2

} ,w0 U2 }

k{R{r,w0 U

2

}}

u − vC 2+μ .

Here the constant k{R{r,w0 U } ,w0 U2 } k{R{r,w0 U } } when w0 given and r chosen, 2 2 is independent of τ and s. Hence, by choosing some sufficiently small δ2 > 0, it follows that S = Ss : Bs,r,w0 −→ Bs,r,w0 is a strict contraction for |τ − s| < δ2 ≤ δ1 . Furthermore, the unique fixed point w of S = Ss can be represented by lim S n 0 = lim (Ss )n 0

n→∞

n→∞

if β(x, t, 0) ≡ 0 and if r = r{Kw0 U2 } is chosen such that r = r{Kw0 U2 } ≥ Kw0 U2 ≥ £−1 s w0 C 2+μ

by (4.5);

(4.8)

this is because 0 belongs to Bs,r,w0 in this case. Thus £τ is onto for |τ − s| < δ2 . It follows that, by dividing [0, 1] into subintervals of length less than δ2 and repeating the above arguments in a finite number of times, £τ becomes onto for all τ ∈ [0, 1], provided that it is onto for some τ ∈ [0, 1]. Since £0 is onto by linear elliptic equations theory [13], we have that £1 is also onto. Therefore, the equation, for w0 = (h, 0), £1 u = w0 has a unique solution u, and the u is the sought solution to (2.2). Here it is to be observed that ψ ≡ £−1 0 (h, 0) is the unique solution to the elliptic equation v − v = h, x ∈ Ω, ∂v + v(x) = 0 on ∂Ω, ∂n ˆ and that, by Theorem 2.1, the ψ is the limit of a sequence where each term in the sequence is an explicit function of the solution φ to the elliptic equation (1.3). It is also to be observed that S0 = S0 0 = £−1 0 (h, 0), −1 S 2 0 = (S0 )2 0 = £−1 0 [(h, 0) + |τ − 0|(£0 − £1 )£0 (h, 0)],

.. ., where (£0 − £1 )£−1 0 is a nonlinear operator. The proof is complete.

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Remark. • The constants k{R{r,w0 U } } and k{R{r,w0 U } ,w0 U2 } k{R{r,w0 U } } , when 2 2 2 w0 is given and when r is chosen and conditioned by (4.8), is not known explicitly, so the corresponding δ2 cannot be determined in advance. Hence, when dealing with the elliptic equation (2.2) in Theorem 2.2 numerically, it is more possible, by choosing τ ∈ [0, 1] such that |τ − s| is smaller, that the sequence S n 0 will converge, for which |τ − s| < δ2 ≤ δ1 occurs.

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CHAPTER 8

Existence Theorems for Evolution Equations (II) 1. Introduction In Chapter 6, the two initial-boundary value problems for parabolic, partial differential equations with the Robin boundary conditions were solved by applying the results in Chapter 2: ut (x, t) = α(x, t, ux )uxx (x, t) + g(x, t, u, ux ), (x, t) ∈ (0, 1) × (0, T ); ux (0, t) ∈ β0 (u(0, t)),

ux (1, t) ∈ −β1 (u(1, t));

u(x, 0) = u0 (x), and ut (x, t) = α0 (x, t, Du) u(x, t) + g0 (x, t, u, Du), (x, t) ∈ Ω × (0, T ); ∂ u(x, t) + β(x, t, u) = 0, x ∈ ∂Ω; ∂n ˆ u(x, 0) = u0 (x). However, the results in Chapter 2 become inapplicable, if the coefficient functions in the leading terms, α = α(x, t, u, ux ) and α0 = α0 (x, t, u, Du), have the u dependence. This is due to the failure of the dissipativity ccondition (H1) in Chapter 2, as is checked by the maximum principle arguments. But this problem will be resolved here, as the following describes it. In this chapter, we will continue the study in Chapter 2 by changing its dissipativity condition (H1), range condition (H2), and time-regulating condition (HA) or (HA) . The results obtained will be applied to solve more general, nonlinear parabolic partial differential equations in Chapter 9, where the coefficient function in the leading term has the u dependence additionally. Because of this u dependence, the dissipativity condition (H1) will fail and the results obtained in Chapter 2 cannot be applied. The failure of (H1) is readily checked by the familiar maximum principle arguments in Sections 3 and 4 in Chapter 9 (or in Sections 3 and 4 in Chapter 6). As is the case with Chapter 2, let (X, · ) be a real Banach space with the norm · , and let T > 0 be a real constant. Consider the nonlinear evolution equation du(t) ∈ A(t)u(t), dt u(s) = u0 , 129

0 ≤ s < t < T,

(1.1)

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8. EXISTENCE THEOREMS FOR EVOLUTION EQUATIONS (II)

where A(t) : D(A(t)) ⊂ X −→ X is a nonlinear, time-dependent, and multi-valued operator. Equation (1.1) will be solved under the set of hypotheses, namely, the nondissipativity condition (H3), the weaker range condition (H2) , and the timeregulating condition (HC) or (HC) . (H2) The range of (I − λA(t)), denoted by E, is independent of t and contains D(A(t)) for all t ∈ [0, T ] and for small 0 < λ < λ0 , where λ0 is some positive number. (H3) For all 0 ≤ t ≤ T and all 0 < λ < λ0 , the λ0 in (H2) , A(t) satisfies the non-dissipativity condition. Namely, there are some number δ0 with λ0 δ0 < 1 and some nonnegative numbers a and a ˜, and there are some function L0 and some nonnegative function L1 on [0, ∞) × [0, ∞), such that, for all u, v ∈ E, the following hold: ˜ u ≤ [(1 − λδ0 )−1 (u + λa) or a ˜], where u ˜ ∈ Jλ (t)u ≡ (I − λA(t))−1 u; u, ˜ v)˜ u − v˜ ˜ u − v˜ ≤ [u − v + λL0 (˜ ˜ v − v ˜ u − v˜], λ where u ˜ ∈ Jλ (t)u and v˜ ∈ Jλ (t)v.

+ λL1 (˜ u, ˜ v)

Here Lk , k = 0, 1 are bounded on bounded subsets of [0, ∞) × [0, ∞), and, for τ, s1 , s2 ≥ 0, the expression τ ≤ [s1 or s2 ] means τ ≤ s1 or τ ≤ s2 . In fact, it will be defined for later use that τ ≤ [s1 or s2 or . . . or sn ] means τ ≤ s1 or τ ≤ s2 or . . . or τ ≤ sn , and that τ ≤ [s1 and s2 and . . . and sn ] means τ ≤ s1 and τ ≤ s2 and . . . and τ ≤ sn . Here τ, si ≥ 0, i = 1, 2, . . . , n ∈ N. To be noticed is that the dissipativity condition (H1) implies the nondissipativity condition (H3) here, if 0 ∈ D(A(t)) for all t and ζ≡

sup v0 ∈A(t)0;t∈[0,T ]

v0

is finite. This follows because then a = ζ, L1 ≡ 0, and L0 ≡ δ0 = ω, where the constant ω is from (H1).

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(HC) There are two continuous functions f, g : [0, T ] −→ R, of bounded variation, and there are one function M1 and four nonnegative functions Mk , k = 0, 2, 3, 4, on [0, ∞) × [0, ∞) with Mk (s1 , s2 ) bounded for bounded s1 , s2 , such that, for each 0 < λ < λ0 , the λ0 in (H2) , we have S1 (λ) ∪ S2 (λ) ≡ {˜ x − y˜ : x ˜ ∈ Jλ (t)x, y˜ ∈ Jλ (τ )y; 0 ≤ t, τ ≤ T ; x, y ∈ E}. Here S1 (λ) denotes the set: {˜ x − y˜ : x˜ ∈ Jλ (t)x, y˜ ∈ Jλ (τ )y; 0 ≤ t, τ ≤ T ; x, y ∈ E; x, ˜ y)|t − τ |}, ˜ x − y˜ ≤ M0 (˜ while S2 (λ) denotes the set: {˜ x − y˜ : x ˜ ∈ Jλ (t)x, y˜ ∈ Jλ (τ )y; 0 ≤ t, τ ≤ T ; x, y ∈ E; x, ˜ y)˜ x − y˜ ˜ x − y˜ ≤ [x − y + λM1 (˜ ˜ y − y ˜ x − y˜ λ ˜ y − y x, ˜ y) + λ|f (t) − f (τ )|M3 (˜ λ x, ˜ y)]}. + λ|g(t) − g(τ )|M4 (˜

x, ˜ y) + λM2 (˜

Observed is that (HC) is reduced to the second condition in (H3) when S1 (λ) = ∅ and t = τ . (HC) There are two continuous functions f, g : [0, T ] −→ R, of bounded vari˜1 and four nonnegative functions ation, and there are one function M 2 ˜ Mk , k = 0, 2, , 3, 4, on [0, ∞) × [0, ∞)2 with M˜k (s1 , s2 , s3 , s4 ) bounded for bounded si , i = 1, 2, 3, 4, such that, for each 0 < λ < λ0 , the λ0 in (H2) , we have S1 (λ) ∪ S2 (λ) ≡ {˜ x − y˜ : x ˜ ∈ Jλ (t)x, y˜ ∈ Jλ (τ )y; 0 ≤ t, τ ≤ T ; x, y ∈ E}. Here S1 (λ) denotes the set: {˜ x − y˜ : x˜ ∈ Jλ (t)x, y˜ ∈ Jλ (τ )y; 0 ≤ t, τ ≤ T ; x, y ∈ E; ˜0 (˜ ˜ x − y˜ ≤ M x, x, ˜ y, y)|t − τ |}, while S2 (λ) denotes the set: {˜ x − y˜ : x ˜ ∈ Jλ (t)x, y˜ ∈ Jλ (τ )y; 0 ≤ t, τ ≤ T ; x, y ∈ E; ˜1 (˜ ˜ x − y˜ ≤ [x − y + λM x, x, ˜ y, y)˜ x − y˜ ˜ y − y ˜2 (˜ ˜ x − y˜ + λM x, x, ˜ y, y) λ ˜ y − y ˜3 (˜ x, x, ˜ y, y) + λ|f (t) − f (τ )|M λ ˜4 (˜ x, x, ˜ y, y)]}. + λ|g(t) − g(τ )|M Again, (HC) becomes essentially the second condition in (H3), as is the case with (HC), when S1 (λ) = ∅ and t = τ .

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The purpose of this chapter is to show, in a way similar to Chapter 2, that, with (H2) , (H3), and (HC) or (HC) assumed, the quantity, for x ∈ D(A(s)) and for large enough n ∈ N, n t−s )x J t−s (s + i n n i=1 is single-valued and its limit, as n −→ ∞, U (t, s)x ≡ lim

n

n→∞

i=1

J t−s (s + i n

t−s )x n

exists, provided that certain condition of smallness is satisfied. Again, this limit U (t, s)x for x = u0 ∈ D(A(s)) will be only intepreted as a limit solution to equation (1.1), but it will be a strong solution if A(t) satisfies additionally the embedding property (HB) of embeddedly quasi-demi-closedness in Chapter 2 (see Section 2). In addition to this section, there are three more sections in this chapter. Section 2 states the main results, and Section 3 obtains some preliminary estimates. Finally, Section 4 proves the main results. The material of this chapter is based on our article [26]. 2. Main Results There are two theorems in relation to the evolution equation (1.1). Theorem 2.1 (Existence of a limit). Let the nonlinear operator A(t) satisfy the non-dissipativity condition (H3), the weaker range condition (H2) , and the timeregulating condition (HC) or (HC) . Then, for u0 ∈ D(A(s)) and for large enough n ∈ N, the quantity n t J nt (s + i )u0 n i=1 is single-valued and its limit, as n −→ ∞, n

t J nt (s + i )u0 n→∞ n i=1

U (s + t, s)u0 ≡ lim

[t]

= lim

μ→0

μ

Jμ (s + iμ)u0

i=1

exists, if T or α−1 is sufficiently small, or if so are both K and α0 where both KD and KE remain finite. Here s, t ≥ 0 and 0 ≤ (s + t) ≤ T , and moreover, α−1 , K, α0 , KD , and KE are defined in Proposition 3.5 in Section 3 where how small the mentioned quantities are is also conditioned there. This limit U (s + t, s)u0 is Lipschitz continuous in t ≥ 0 for u0 ∈ D(A(s)). Remark 2.2. In loose terms, the smallness of α−1 means that of both M2 (and ˜ 2 , respectively) and L1 , but the smallness of both K and α0 means that of both M ˜ 0 , respectively) and v0 , a less or equal order of which is possessed by M0 (and M the total variation of g on [0, T ], multiplied by α2 (defined in Proposition 3.5 in Section 3). Here v0 is any element in A(s)u0 , and the smallness of α2 means that ˜ 4 , respectively). of M4 (and M

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The next theorem, Theorem 2.3, concerns a limit solution and a strong solution, whose concepts are stated in Chapter 2 and will be recalled now. We make two preparations, the first of which is for a limit solution. Discretize (1.1) on [0, T ] as ui − A(ti )ui ui−1 , ui ∈ D(A(ti )),

(2.1)

where n ∈ N is large, and 0 < < λ0 is such that s ≤ ti = s + i ≤ T for each i = 1, 2, . . . , n. Here to be noticed is that, for u0 ∈ D(A(s)) and for small enough , ui will exist uniquely (see Proposition 3.6 in Section 3) by hypotheses (H2) , (H3), and (HC) or (HC) , if the condition of smallness in Theorem 2.1 is satisfied (see also Remark 2.2). Next, let u0 ∈ D(A(s)), and construct the Rothe functions [12, 32] by defining χn (s) = u0 ,

C n (s) = A(s);

χn (t) = ui ,

C n (t) = A(ti )

(2.2)

for t ∈ (ti−1 , ti ], and

un (s) = u0 ; t − ti−1 for t ∈ (ti−1 , ti ] ⊂ [s, T ].

un (t) = ui−1 + (ui − ui−1 )

(2.3)

Then it follows that, for some constant K−1 , lim

sup un (t) − χn (t) = 0;

n→∞ t∈[0,T ]

un (t) − un (τ ) ≤ K−1 |t − τ |,

(2.4)

where t, τ ∈ (ti−1 , ti ], and that dun (t) ∈ C n (t)χn (t); dt un (s) = u0 ,

(2.5)

where t ∈ (ti−1 , ti ]. Here the last equation has values in B([s, T ]; X), the real Banach space of all bounded functions from [s, T ] to X. The second preparation is for a strong solution. Let (Y, .Y ) be a real Banach space, into which the real Banach space (X, .) is continuously embedded. Assume additionally that A(t) satisfies the embedding condition of embeddedly quasi-demiclosedness: (HB) If tn ∈ [0, T ] −→ t, if xn ∈ D(A(tn )) −→ x, and if yn ≤ M0 for some yn ∈ A(tn )xn and for some positive constant M0 , then η(A(t)x) exists and |η(ynl ) − z| −→ 0 for some subsequence ynl of yn , for some z ∈ η(A(t)x), and for each η ∈ Y ∗ ⊂ X ∗ , the real dual space of Y . Theorem 2.3 (A limit or a strong solution [25]). Following Theorem 2.1, if u0 ∈ D(A(s)), then the function u(t) ≡ U (t, s)u0

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= lim

n

n→∞

i=1

J t−s (s + i n

t−s )u0 n

[ t−s μ ]

= lim

μ→0

Jμ (s + iμ)u0

i=1

is a limit solution of the evolution equation (1.1) on [0, T ], in the sense that it is also the uniform limit of un (t) on [0, T ], where un (t) satisfies (2.5). Furthermore, if A(t) satisfies the embedding property (HB), then u(t) is a strong solution in Y , in the sense that d u(t) ∈ A(t)u(t) in Y dt for almost every t ∈ (0, T ); u(s) = u0 is true. The strong solution is unique if Y ≡ X. 3. Some Preliminary Estimates In order to prove the main results, Theorems 2.1 and 2.3 in Section 2, we need to prepare some preliminary estimates, which are in this section. Lemma 3.1. Let xi , i = 0, 1, 2, . . . , be a sequence of nonnegative numbers that satisfies the difference inequality xi ≤ λγαxi xi−1 + γ[1 + |F (ti ) − F (ti−1 )|]xi−1 , (3.1) i = 1, 2, . . . . Here the numbers λ, ω, γ > 0, α ≥ 0, are such that γ = (1 − λω)−1 > 1 and λω < 1; the function F (t) is continuous and of bounded variation on [0, T ]; 0 ≤ ti−1 < ti ≤ T, and iλ ≤ T . Then either the following (P1 ) or (P2 ) holds: (P1 ) There is an i0 ∈ {0} ∪ N, such that xi0 = 0 and such that, for all i ≥ 0, xi ≤ max{x0 , x1 , . . . , xi0 }. (P2 ) xi = 0 for all i ≥ 0, and, on setting yi = 1/xi , we have yi ≥ e−ωT e−KC y0 − T α. This yi is strictly greater than zero, if x0 or T or α is small enough. Here KC is the total variation of F on [0, T ]. Hence, xi is uniformly bounded for all i ≥ 0, if T or α or x0 is small enough. Proof. We divide the proof into two cases. Case 1, where xi = 0 for all i. On dividing (3.1) by xi xi−1 and setting yi = 1/xi , we have yi ≥ ci yi−1 − di , where ci = γ −1e−|F (ti )−F (ti−1 )| ≤ 1;

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di = λαe−|F (ti )−F (ti−1 )| ≤ λα. Here we have used [1 + |F (ti ) − F (ti−1 )|] ≤ e|F (ti )−F (ti−1 )| . Solving this linear difference inequality yields yi ≥ (ci ci−1 · · · c1 )y0 − [(ci ci−1 · · · c2 )d1 + · · · + ci di−1 + di ] ≥ γ −i e−KC y0 − (iλ)α ≥e

−ωT −KC

e

(3.2)

y0 − T α,

which is strictly greater than zero, if x0 or T or α is sufficiently small. Here iλ ≤ T , and KC is the total variation of F (t) on [0, T ]. Case 2, where xi0 = 0 for some i0 ∈ {0} ∪ N. Then clearly, (3.1) implies xi = 0 for i ≥ i0 , so for all i ≥ 0, xi ≤ max{x0 , x1 , . . . , xi0 }.


Lemma 3.2. Let α0 be a positive number, and let the xi in Lemma 3.1 satisfy instead the difference inequality xi ≤ λγαxi xi−1 + γ[1 + |F (ti ) − F (ti−1 )|]xi−1 (3.3) + γ|G(ti ) − G(ti−1 )|, i = 1, 2, . . . . Here λ, ω, α, γ and F (t) are as in Lemma 3.1, and the function G(t) is continuous and of bounded variation on [0, T ]. Then the following (P3 ) holds: (P3 ) If x0 ≤ α0 , then, for all i ≥ 1, xi ≤ eωT α0 eKD /[1 − T αα0 eωT eKD ], if T or α is small enough. This is also true for small enough α0 , where KD should be finite however small α0 is. Here iλ ≤ T , and KD is the sum of the total variation of F (t) and that of G(t)/α0 , respectively, on [0, T ]. Thus, a condition for KD to be finite, regardless of the small value of α0 , is to reguire that the total variation KG of G(t) on [0, T ] be small enough, so that KG /α0 is finite. For instance, assume that KG is proportional to α0 . Proof. Let x0 ≤ α0 . Then it follows from (3.3) that, for |h(t) − h(τ )| ≡ |F (t) − F (τ )| + |G(t) − G(τ )|/α0 , γα0 [1 + |h(t1 ) − h(t0 )|] 1 − λαα0 γ |h(t1 )−h(t0 )| γα0 e , ≤ 1 − λαα0 γ if λαα0 is small enough. Since the right hand side Rd of the above is greater than or equal to α0 , we have x1 ≤

γ|G(t2 ) − G(t1 )| ≤ γ[|G(t2 ) − G(t1 )|/α0 ]Rd . Hence (3.3) delivers x2 ≤ ≤

γ 2 α0 [1 + |h(t2 ) − h(t1 )|][1 + |h(t1 ) − h(t0 )|] 1 − λαα0 γ − λαα0 γ 2 [1 + |h(t1 ) − h(t0 )|] γ 2 α0 e[|h(t2 )−h(t1 )|+|h(t1 )−h(t0 )|] , 1 − λαα0 γ − λαα0 γ 2 e|h(t1 )−h(t0 )|

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if λαα0 is small enough. Again, the right hand side of the above is greater than or equal to α0 , so, as in estimating x2 , we derive, on using [1 + |h(t) − h(τ )|] ≤ e|h(t)−h(τ )|, x3 ≤

γ 3 α0 e[|h(t3 )−h(t2 )|+|h(t2 )−h(t1 )|+|h(t1 )−h(t0 )|] , 1 − λαα0 γ{1 + γe|h(t1 )−h(t0 )| + γ 2 e[|h(t2 )−h(t1 )|+|h(t1 )−h(t0 )|] }

if λαα0 is small enough. Clearly, the right side of the above is greater than or equal to α0 . Continued in this way, the following is obtained i−1

γ i α0 e j=0 |h(tj+1 )−h(tj )| xi ≤ j−1 i−1 1 − λαα0 γ[1 + j=1 γ j e k=0 |h(tk+1 )−h(tk )| ] ≤ γ i α0 eKD /[1 − λαα0 (iγ i eKD )]

(3.4)

≤ eωT α0 eKD /[1 − T αα0 eωT eKD ], if small enough is T or α, or if so is α0 where KD remains finite. Here iλ ≤ T , and KD is the sum of the total variation of F (t) and that of G(t)/α0 , respectively, on [0, T ]. The proof is complete. Lemma 3.3. Let u0 ∈ D(A(s)), and let the weaker range condition (H2) hold. Let 0 < λ < λ0 and 0 ≤ ti = s+iλ ≤ T . Then there is a sequence ui ∈ D(A(ti )), i = 1, 2, . . . , such that ui − λA(ti )ui ui−1 . Proof. By (H2) , there is a u1 ∈ D(A(t1 )) that satisfies u1 − λA(t1 )u1 u0 .

For this u1 , we have, by (H2) again, a u2 ∈ D(A(t2 )) that satisfies u2 − λA(t2 )u2 u1 . Continuing in this way, there are ui ∈ D(A(ti )), i = 1, 2, . . ., such that ui − λA(ti )ui ui−1 . This completes the proof.

Lemma 3.4. Let the weaker range condition (H2) and the non-dissipativity condition (H3) hold. Then the ui in Lemma 3.3 satisfies, for all i ≥ 0 and η = 1/(1 − λδ0 ), ˜ + ηi u0 + (iλ)η i a, if δ0 > 0; ηi a (3.5) ui ≤ a ˜ + u0 + (iλ)a, if δ0 ≤ 0. Hence, ui is uniformly bounded for all i, λ. Proof. The proof will be made with the case δ0 > 0 for which η > 1, as the othe case is similar. The method of induction will complete the proof, as the arguments below show. If i = 1, we have, by the nondissipativity condition (H1) , ˜, u1 ≤ (ηu0 + ληa) or a

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which is ≤ the right side of (3.5) for i = 1. Now we show (3.5) is true for i = i when it is so for i = i − 1. By the non-dissipativity condition (H1) , we have ˜, ui ≤ (ηui−1 + ηλa) or a which, combined with the induction assumption, concludes the proof. Here used was 1 ≤ η ≤ η i for η > 1. In view of (3.4), we are led to the claim: Proposition 3.5. Let the weaker range condition (H2) , the non-dissipativity condition (H3), and the time-regulating condtion (HC) or (HC) hold. Then, under (HC), the ui in Lemma 3.3 satisfies, for small enough λ > 0, for i = 0, 1, 2, . . ., and for u−1 = u0 − λv0 , where v0 is any element in A(s)u0 , the following (Q1), (Q2), (Q3), and (Q4): (Q1) There is a positive number ω0 , such that L0 (ui , ui−1 ),

M1 (ui , ui−1 ) ≤ ω0 .

(Q2) There are one positive number K > 0 and three nonnegative numbers α−1 , α1 , α2 ≥ 0, such that M0 (ui , ui−1 ) ≤ K;

M2 (ui , ui−1 ) ≤ α−1 ;

L1 (ui , ui−1 ) ≤ α−1 ; M3 (ui , ui−1 ) ≤ α1 ;

M4 (ui , ui−1 ) ≤ α2 .

(Q3) Assume v0 ≤ α0 for some number α0 > 0. Then the inequality is true ui − ui−1 /λ ≤

˜ i ], [K or K1 (i) or K2 (i) or · · · or Ki−1 (i) or K

˜ 1 when ˜ 0 when i = 0, and as K or K where the right side is interpreted as K or K −1 i = 1. Here, for j = 1, 2, . . . , i − 1, and for small λ so that γ = (1 − λω0 ) exists, Kj (i) =

γ j Ke 1 − λα−1 Kγ[1 +

≤ γ Ke i

≤e

ω0 T

KD

Ke

i−1

k=i−j

j−1

k=1

|h(tk+1 )−h(tk )|

γke

i−2

l=i−k−1

/[1 − λα−1 K(iγ e

i KD

KD

/[1 − T α−1 Ke

|h(tl+1 )−h(tl )|

]

)]

ω0 T KD

e

],

provided that T or α−1 is small enough, or that K is so where KD remains finite. Here iλ ≤ T − s, |h(t) − h(τ )| = |f (t) − f (τ )|α1 + |g(t) − g(τ )|α2 /K, and KD is the sum of the total variation of f (t)α1 and that of g(t)α2 /K, respectively, on [0, T ]. ˜ 0 = α0 , and Furthermore, K ˜i = K

γ i α0 e

i−1

1 − λα−1 α0 γ[1 +

≤ γ α0 e i

≤e

ω0 T

KE

α0 e

j=0

|H(tj+1 )−H(tj )|

i−1 j=1

γje

/[1 − λα−1 α0 (iγ e

KE

j−1

k=0

i KE

/[1 − T α−1 α0 e

|H(tk+1 )−H(tk )|

]

)]

ω0 T KE

e

],

if T or α−1 is small enough, or if so is α0 where KE remains finite. Here iλ ≤ T −s, |H(t) − H(τ )| = |f (t) − f (τ )|α1 + |g(t) − g(τ )|α2 /α0 , and KE is the sum of the total variation of f (t)α1 and that of g(t)α2 /α0 , respectively, on [0, T ]. Thus, a condition for both KD and KE to be finite, regardless of the small value of K and α0 , is to require that either α2 or the total variation KG of g(t) on

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[0, T ] be small enough, so that both KG α2 /K and KG α2 /α0 are finite. For example, assume that either α2 or KG is proportional to Kα0 . ˜ i , the convention was used that m al = 0 for In the above Kj (i) and K l=k m < k. Hence, there is a constant N1 > K, α0 , such that ui − ui−1 /λ ≤ N1 ,

(3.6)

uniformly bounded for all i, λ, if T or α−1 is sufficiently small, or if so are both K and α0 where both KD and KE remain finite. Here the constant N1 depends on K, ω0 , T, α−1 , α0 , KD , and KE . (Q4) ui − u0 ≤ η i K(iλ) + [η i−1 b1 + η i−2 b2 + · · · + ηbi−1 + bi ], where η = 1/[1 − λ(ω0 + α−1 v0 )] exists for small λ, and bi = ηλv0 + ηλ[|f (ti ) − f (t0 )|α1 v0 + |g(ti ) − g(t0 )|α2 ]. Hence, there is a constant N2 such that ui − u0 ≤ N2 [1 − λ(ω0 + α−1 v0 )]−i (2i + 1)λ ≤ N2 [1 − λ(ω0 + α−1 N1 )]−i (2i + 1)λ

(3.7)

because v0 ≤ α0 < N1 , where the constant N2 depends on K, v0 , and the sum KF of the total variation of f (t)α1 v0 and that of g(t)α2 , respectively, on [0, T ]. On the other hand, the same results are obtained under (HC) , if the above Mk , k = 0, . . . , 4, are replaced by M˜k ’s. Proof. The proof will be made with (HC) assumed, as it is similar under (HC) . We divide the proof into four steps, where the simple inequality, for c > 0, (1 + c) ≤ ec will be frequently utilized. Step 1. The assertions (Q1) and (Q2) follow immediately from Lemma 3.4. Step 2. To prove (Q3), the method of mathematical induction will be used. Since u1 − λA(t1 )u1 u0 ; u0 − λv0 ≡ u−1 , (Q3) is true for i = 1 by the time-regulating condition (HC). This is because, for x1 ≡ u1 − u0 /λ, we have x1 ≤ K

if (u1 − u0 ) ∈ S1 (λ);

but, if (u1 − u0 ) ∈ S2 (λ), then x1 ≤ λγα−1 α0 x1 + γ[1 + |f (t1 ) − f (t0 )|α1 ]α0 + γ|g(t1 ) − g(t0 )|(α2 /α0 )α0 , ˜ 1 . Here γ ≡ 1/(1 − λω0 ) for small λ. so x1 ≤ K Step 3. Assume that (Q3) is true for i = i − 1, and we prove it is also true for i = i. This will follow from (HC), combined with the induction assumption, as the following arguments show. If (ui − ui−1 ) ∈ S1 (λ), then ui − ui−1 ≤ K(ti − ti−1 ) = Kλ.

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But if (ui − ui−1 ) ∈ S2 (λ), then, for xi ≡ ui − ui−1 /λ, xi ≤ λγα−1 xi xi−1 + γ[1 + |f (ti ) − f (ti−1 )|α1 ]xi−1 + γ|g(ti ) − g(ti−1 )|α2 , which, in conjuction with the induction assumption, yields (Q3) for i = i. This is because ˜ i−1 ]; xi−1 ≤ [K or K1 (i − 1) or K2 (i − 1) or · · · or Ki−1 (i − 1) or K 0 < K ≤ [K and Kj (i − 1), j = 1, 2, . . . , i − 1]; ˜ i−1 , α0 ≤ K so that γ|g(ti ) − g(ti−1 )|α2 ≤ γ|g(ti ) − g(ti−1 )|(α2 /K) [K and Kj (i − 1), j = 1, 2, . . . , i − 1]; ˜ i−1 . γ|g(ti ) − g(ti−1 )|α2 ≤ γ|g(ti ) − g(ti−1 )|(α2 /α0 )K Step 4. Again, (Q4) will be proved by induction. (Q4) is correct by (HC) if i = 1, because, for u−1 = u0 − λv0 , u1 − u0 ≤ Kλ ≤ ηKλ if (u1 − u0 ) ∈ S1 (λ); u1 − u0 ≤ b1

if (u1 − u0 ) ∈ S2 (λ).

Next, by assuming that (Q4) is correct for i = i − 1, we shall show that it is also correct for i = i. This follows from (HC), together with induction assumption. For ui − u0 ≤ K(iλ) ≤ η i K(iλ) if (ui − u0 ) ∈ S1 (λ); but, if (ui − u0 ) ∈ S2 (λ), then we have, for the η and bi in (Q4), ui − u0 ≤ ηui−1 − u0 + bi ≤ η{η i−1 K[(i − 1)λ] + [η i−2 b1 + ηi−3 b2 + · · · + ηbi−2 + bi−1 ]} + bi , which is less than or equal to the right side of (Q4) with i = i. The proof is complete.

Proposition 3.6. Let A(t) satisfy the weaker range condition (H2) , the nondissipativity condition (H3), and the time-regulating condition (HC) or (HC) . Then the ui in Lemma 3.3 is unique, if, as is conditioned in Proposition 3.5, T or α−1 is small enough, or if so are both K and α0 where both KD and KE remain finite. Thus ui = Jλ (ti )u0 . Proof. We will show that ui = vi , if vi ∈ D(A(ti )), i = 1, 2, . . . , is another sequence such that vi − λA(ti )vi vi−1 ,

i = 1, 2, . . . ,

v 0 = u0 . But this follows from the non-dissipativity condition (H3) because, then, ui − vi ≤ [ui−1 − vi−1 + λL0 (ui , vi )ui − vi + λL1 (ui , vi )(vi − vi−1 /λ)ui − vi ],

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whence, for some constant ξ with λξ small enough, ui − vi ≤ [1/(1 − λξ)]ui−1 − vi−1 . This inequality concludes the proof, where u0 − v0 = 0. Here used was boundedness of ui , vi , and vi − vi−1 /λ under smallness of T or α−1 or both K and α0 , by Lemma 3.4 and Proposition 3.5. It follows immediately from the proof of Propositions 3.5 that Corollary 3.7. Let u0 ∈ D(A(s)) and let ui , i = 1, 2, . . . , satisfy the difference relation, where 0 < λ < λ0 , ui − λA(ti )ui ui−1 ,

i = 1, 2, . . . .

Then the (Q3) and (Q4) in Propositions 3.5 are, respectively, still true, if we do not assume that A(t) satisfies the non-dissipativity condition (H3), the weaker range condition (H2) , and the time-regulating condition (HC) or (HC) , but assume that ui satisfies, respectively, both the conditions (D1) and (D2), and both the conditions (D3) and (D4): (D1) For x1 ≡ u1 − u0 /λ, ⎧ ⎪ ⎨K, or x1 ≤ λγα−1 α0 x1 + γ[1 + |f (t1 ) − f (t0 )|α1 ]α0 ⎪ ⎩ + γ|g(t1 ) − g(t0 )|(α2 /α0 )α0 . (D2) For xi ≡ ui − ui−1 /λ, ⎧ ⎪ ⎨K, or xi ≤ λγα−1 xi xi−1 + γ[1 + |f (ti ) − f (ti−1 )|α1 ]xi−1 ⎪ ⎩ + γ|g(ti ) − g(ti−1 )|α2 . (D3)

u1 − u0 ≤

(D4)

ui − u0 ≤

Kλ, b1 .

or

K(iλ), or ηui−1 − u0 + bi .

Here v0 is any element in A(s)u0 ; u0 − v0 = u−1 ; η = 1/[1 − λ(ω0 + α−1 v0 )]; bi = ηλv0 + ηλ[|f (ti ) − f (t0 )|α1 v0 + |g(ti ) − g(t0 )|α2 ; the functions f and g are as in (HC) or (HC) .

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4. Proof of the Main Results Proof of Theorem 2.1 will be done after those of Propositions 4.1 and 4.2 below, where the formulation of these two propositions resemble that of Propositions 5.1 and 5.4 in Chapter 2. We are enabled to obtain Proposition 4.1 below, as in the proof of Proposition 5.1 in Section 6 of Chapter 2, by the preliminary estimates in Section 3, together with the difference equation theory in Chapter 2. Proposition 4.1. Under the assumptions of Proposition 4.2, the inequality is true

am,n ≤

K|nμ − mλ|, cm,n + sm,n + dm,n + fm,n + gm,n ,

if S2 (μ) = ∅; if S1 (μ) = ∅;

where am,n, cm,n , sm,n , fm,n , gm,n and K are defined in Proposition 4.2. In view of this and Proposition 3.5, we are led to the claim: Proposition 4.2. Let x ∈ D(A(s)) where 0 ≤ s ≤ T , and let small enough λ, μ > 0 and let n, m ∈ N, be such that 0 ≤ (s + mλ), (s + nμ) ≤ T , and such that λ0 > λ ≥ μ > 0. Let A(t) satisfy the non-dissipativity condition (H3), the weaker range condition (H2) , and the time-regulating condition (HC) or (HC) . Then the inequality is true if, as is conditioned in Proposition 3.5, T or α−1 is sufficiently small, or if so are both K and α0 where KD and KE remain finite: am,n ≤ cm,n + sm,n + dm,n + em,n + fm,n + gm,n. Here α−1 ,K, and α0 are defined in Proposition 3.5; am,n ≡

n

Jμ (s + iμ)x −

i=1

m

Jλ (s + iλ)x

i=1

exists by Proposition 3.6; μ γ ≡ [1 − μ(ω0 + α−1 N1 ]−1 > 1; α ≡ ; β ≡ 1 − α; λ n 2 cm,n = 2N2 γ [(nμ − mλ) + (nμ − mλ) + (nμ)(λ − μ)]; sm,n = 2N2 γ n (1 − λω)−m (nμ − mλ)2 + (nμ)(λ − μ); ρ(T ) n γ [(mλ)(nμ − mλ)2 δ2 m(m + 1) 2 + (λ − μ) λ ]}; 2 = Kγ n (nμ − mλ)2 + (nμ)(λ − μ);

dm,n = [N3 ρ(δ)γ n (mλ)] + {N3

em,n

fm,n = N2 [γ n μ + γ n (1 − λω)−m λ]; gm,n = N3 ρ(|λ − μ|)γ n (mλ); N3 = γ(α1 N1 + α2 );

δ>0

is arbitrary;

ρ(r) ≡ ρ1 (r) + ρ2 (r); ρ1 (r) ≡ sup{|f (t) − f (τ )| : 0 ≤ t, τ ≤ T, |t − τ | ≤ r}; ρ2 (r) ≡ sup{|g(t) − g(τ )| : 0 ≤ t, τ ≤ T, |t − τ | ≤ r};

(4.1)

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ρ1 and ρ2 are the modulus of continuity of f and g, respectively, on [0, T ]; ω0 , α−1 , N1 , and N2 are defined in Proposition 3.5. Proof. We will use the method of mathematical induction and divide the proof into two steps. Step 1. (4.1) is clearly true by (3.7) in Proposition 3.5, if (m, n) = (0, n) or (m, n) = (m, 0). Step 2. By assuming that (4.1) is true for (m, n) = (m − 1, n − 1) or (m, n) = (m, n − 1), we will show that it is also true for (m, n) = (m, n). This is done by the arguments below. Using the nonlinear resolvent identity in Lemma 4.3 in Chapter 1, we have am,n = Ju (s + nμ)

n−1

Jμ (s + iμ)x − Jμ (s + mλ)

i=1

[α

m−1

Jλ (s + iλ)x + β

i=1

m

Jλ (s + iλ)x)].

i=1

. Here α = μλ and β = λ−μ λ Under the time-regulating condition (HC) or (HC) , it follows that, if the element inside the norm of the right side of the above equality is in S1 (μ), then, by (Q2) in Proposition 3.5, am,n ≤ K|mλ − nμ|, which is less than or equal to the right side of (4.1) with (m, n) = (m, n), where γ n = [1 − μ(ω0 + α−1 N1 )]−n > 1. If that element instead lies in S2 (μ), then, by (Q1), (Q2), and (3.6) in Proposition 3.5, am,n ≤ γ(αam−1,n−1 + βam,n−1 )+ γμ[|f (s + mλ) − f (s + nμ)|α1 N1 + |g(s + mλ) − g(s + nμ)|α2 ] ≤ [γαam−1,n−1 + γβam,n−1] + N3 μρ(|nμ − mλ|), where N3 = γ(α1 N1 + α2 ), ρ(r) = ρ1 (r) + ρ2 (r), and ρ1 (r) and ρ2 (r) are the modulus of continuity of f and g, respectively, on [0, T ]. From here on, the rest of the proof can be completed by being patterned after that of Proposition 5.2 in Chapter 2. We are ready for Proof of Theorem 2.1: Proof. Let the quantities T or α−1 or both K and α0 in Proposition 3.5 be small where KD and KE remain finite. Then, for x ∈ D(A(s)),√ it follows from t Proposition 4.2, by setting μ = nt ≤ λ = m < λ0 , and δ 2 = λ − μ, that, as n, m −→ ∞, am,n converges to 0 uniformly for 0 ≤ (s + t) ≤ T . Thus n

t J nt (s + i )x n→∞ n i=1 lim

exists for x ∈ D(A(s)).

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On the other hand, setting μ = λ = it follows that lim

n→∞

n i=1

t n

< λ0 , m = [ μt ] and setting δ 2 =

143

√ λ − μ,

t [μ ]

t J nt (s + i )u0 = lim Jμ (s + iμ)u0 . μ→0 n

(4.2)

i=1

Now, to show the Lipschitz property, (4.2) and Crandall-Pazy [8, Page 71] will be used. From (3.6) in Proposition 3.5, it is derived that un − um ≤ un − un−1 + un−1 − un−2 + ... + um+1 − um ≤ N1 μ(n − m) for x ∈ D(A(s)); un =

n

Jμ (s + iμ)x;

um =

i=1

m

Jμ (s + iμ)x.

i=1

Here n = [ μt ], m = [ μτ ], t > τ and 0 < μ < λ0 . The proof is completed by making μ −→ 0 and using (4.2). Proof of Theorem 2.3 will be done after those of Propositions 4.3 and 4.4 below. Use will be made of the setup in (2.1), (2.2), and (2.3). Due to ui − ui−1 / ≤ N1 by Proposition 3.5, we have the results in Propositions 4.3 and 4.4 below, as in the proof of Propositions 5.3 and 5.4 in Chapter 2. Proposition 4.3. For u0 ∈ D(A(s)), we have that lim

sup un (t) − χn (t) = 0;

n→∞ t∈[0,T ]

un (t) − un (τ ) ≤ N1 |t − τ |, where t, τ ∈ (ti−1 , ti ], and that dun (t) ∈ C n (t)χn (t); dt un (s) = u0 , where t ∈ (ti−1 , ti ]. Here the last equation has values in B([s, T ]; X), the real Banach space of all bounded functions from [s, T ] to X. Proposition 4.4. If A(t) satisfies the assumptions in Theorem 2.1, then n t−s n )u0 J t−s (s + i lim u (t) = lim n n→∞ n→∞ n i=1 [ t−s μ ]

= lim

μ→0

Jμ (s + iμ)u0

i=1

uniformly for finite 0 ≤ (s + t) ≤ T and for u0 ∈ D(A(s)). Here is Proof of Theorem 2.3: Proof. That u(t) is a limit solution follows from Propositions 4.3 and 4.4. That u(t) is a strong solution under the embedding property (HB) follows as in the Step 5 for the proof of Theorem 2.5 in Chapter 1.

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CHAPTER 9

Nonlinear Non-autonomous Parabolic Equations (II) 1. Introduction In this chapter, nonlinear non-autonomous, parabolic initial-boundary value problems with the u dependence in the coefficient function of the leading term will be solved by using the results in Chapter 8. The obtained solutions will be strong ones under suitable assumptions. Thus, by allowing the u dependence in the coefficient function α(x, u, ux ), we shall extend the problem in (1.1) of Chapter 6 to this nonlinear, non-autonomous equation with the nonlinear Robin boundary condition, as well as to its higher space dimensional analogue ut (x, t) = α(x, t, u, ux )uxx (x, t) + g˜(x, t, u, ux ), (x, t) ∈ (0, 1) × (0, T ); ux (0, t) ∈ β0 (u(0, t)),

ux (1, t) ∈ −β1 (u(1, t));

(1.1)

u(x, 0) = u0 (x). Here, as before, made are the similar assumptions: • β0 , β1 : R −→ R, are multi-valued, maximal monotone functions with 0 ∈ β0 (0) ∩ β1 (0). • α(x, t, z, p) and g˜(x, t, z, p) are real-valued, continuous functions of their arguments x ∈ [0, 1], t ∈ [0, T ], z ∈ R, and p ∈ R. Here T > 0. • α(x, t, z, p) is greater than or equal to some positive constant δ0 for all its arguments x, t, and p. • g˜(x, t, z, p) satisfies, for some real constant δ00 and some positive constant a > 0, z˜ g(x, t, z, p) ≤ δ00 |z|2 + |z|a. • g˜(x, t, z, p) is of at most linear growth in p, that is, for some nonnegative, continuous function M00 (x, t, z), |˜ g (x, t, z, p)| ≤ M00 (x, t, z)(1 + |p|). • The following are true for some continuous, nonnegative functions N00 and N01 and for some continuous function ζ of bounded variation: |α(x, t, z, p) − α(x, τ, z, p)|/α(x, τ, z, p) ≤ |ζ(t) − ζ(τ )|N00 (x, t, τ, z); |˜ g (x, t, z, p) − g˜(x, τ, z, p)| ≤ |ζ(t) − ζ(τ )|N01 (x, t, τ, z)(1 + |p|). 145

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9. NONLINEAR NON-AUTONOMOUS PARABOLIC EQUATIONS (II)

• The following are true for some continuous, nonnegative functions N02 and N03 : |α(x, t, z1 , p) − α(x, t, z2 , p)|/α(x, τ, z2 , p) ≤ N02 (x, t, τ, z1 , z2 )|z1 − z2 |; |˜ g (x, t, z1 , p) − g˜(x, t, z2 , p)| ≤ N03 (x, t, z1 , z2 )(1 + |p|)|z1 − z2 |. To be observed is that (1.1) is different from (1.1) of Chapter 6, in that here α(x, t, z, p) has the z dependence, and g˜(x, t, z, p) is not necessarily monotone nonincreasing in z. (1.1) will be written as the nonlinear evoution d u(t) = H(t)u(t), dt u(0) = u0 ,

t ∈ (0, T );

(1.2)

where the nonlinear, multi-valued operator H(t) : D(H(t)) ⊂ C[0, 1] −→ C[0, 1] is defined by H(t)v = α(x, t, v, v )v + g˜(x, t, v, v ); v ∈ D(H(t)) ≡ {w ∈ C 2 [0, 1] : w (j) ∈ (−1)j βj w(j), j = 0, 1}. It will be shown by using the theory in Chapter 8 that the equation (1.2) and then the equation (1.1), for u0 ∈ D(H(0)), have a strong solution given by u(t) = lim

n

n→∞

i=1

[ νt ]

= lim

ν→0

I−

t t H(i )]−1 u0 n n

I − νH(iν)]−1 u0 ,

i=1

provided that certain condition of smallness is fulfilled. The higher space dimensional analogue of (1.1) will be of the form ut (x, t) = α0 (x, u, t, Du) u(x, t) + g˜0 (x, t, u, Du), (x, t) ∈ Ω × (0, T ); ∂ u(x, t) + β(x, t, u) = 0, ∂n ˆ u(x, 0) = u0 (x);

x ∈ ∂Ω;

(1.3)

in which eight assumptions are made. • Ω is a bounded smooth domain in RN , N ≥ 2, and ∂Ω is the boundary of Ω. • n ˆ (x) is the unit outer normal to x ∈ ∂Ω, and μ is a real number such that 0 < μ < 1. • α0 (x, t, z, p) ∈ C 1+μ (Ω × R × RN ) is true for each t ∈ [0, T ] where T > 0, and is continuous in all its arguments. Furthermore, α0 (x, t, z, p) ≥ δ1 > 0 is true for all x, z, p, and all t ∈ [0, T ], and for some constant δ1 > 0.

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• g˜0 (x, t, z, p) ∈ C 1+μ (Ω × R × RN ) is true for each t ∈ [0, T ], is continuous in all its arguments, and satisfies z˜ g0 (x, t, z, p) ≤ δ11 |z|2 + |z|b for some real number δ11 and some positive number b > 0. • g˜0 (x, t, z, p) is of at most linear growth in p, that is, |˜ g0 (x, t, z, p)| ≤ M10 (x, t, z)(1 + |p|) for some nonnegative, continuous function M10 and for all t ∈ [0, T ]. • β(x, t, z) ∈ C 2+μ (Ω × R) is true for each t ∈ [0, T ], is continuous in all its arguments, and is strictly monotone increasing in z so that βz ≥ δ11 > 0 for the constant δ11 > 0. • The following are true for some continuous, nonnegative functions N10 , N20 , N30 and for some continuous function ζ of bounded variation: |α0 (x, t, z, p) − α0 (x, τ, z, p)|/α0 (x, τ, z, p) ≤ |ζ(t) − ζ(τ )|N10 (x, t, τ, z); |˜ g0 (x, t, z, p) − g˜0 (x, τ, z, p)| ≤ |ζ(t) − ζ(τ )|N20 (x, t, τ, z)(1 + |p|); |β(x, t, z) − β(x, τ, z)| ≤ |t − τ |N30 (x, t, τ, z). • The following are true for continuous, nonnegative functions N11 and N21 : |α0 (x, t, z1 , p) − α0 (x, t, z2 , p)|/α0 (x, τ, z2 , p) ≤ N11 (x, t, τ, z1 , z2 )|z1 − z2 |; |˜ g0 (x, t, z1 , p) − g˜0 (x, t, z2 , p)| ≤ N21 (x, t, z1 , z2 )(1 + |p|)|z1 − z2 |. Again, it is to be noted that, relative to those in (1.3) of Chapter 6, here α0 (x, t, z, p) has the z dependence, and g˜0 (x, t, z, p) need not be monotone non-increasing in z. The corresponding nonlinear evolution equation will be d u(t) = J(t)u(t), t ∈ (0, T ); dt (1.4) u(0) = u0 , in which the time-dependent, nonlinear operator J(t) : D(J(t)) ⊂ C(Ω) −→ C(Ω) is defined by J(t)v = α0 (x, t, v, Dv) v + g˜0 (x, t, v, Dv); ∂ w + β(x, t, w) = 0 on ∂Ω}. ∂n ˆ The theory in Chapter 8 will be employed again to prove that, for u0 ∈ D(J(0)), the quantity n t t [I − J(i )]−1 u0 u(t) = lim n→∞ n n i=1 v ∈ D(J(t)) ≡ {w ∈ C 2+μ (Ω) :

t

= lim

ν→0

[ν ]

[I − νJ(iν)]−1 u0

i=1

is a strong solution for equation (1.4) or, equivalently, equation (1.3), as long as some condition of smallness is met.

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The rest of this chapter is organized as follows. Section 2 states the main results, and Sections 3 and 4 prove the main results. The material of this chapter is taken from our article [26].

2. Main Results Theorem 2.1. The time-dependent operator H(t) in (1.2) satisfies the four conditions, namely, the non-dissipativity condition (H3), the weaker range condition (H2) , the time-regulating condition (HC), and the embedding condition (HB) in Chapter 8. As a result, it follows from Theorems 2.1 and 2.3 of Chapter 8 that, for u0 ∈ D(H(0)), the equation (1.2) and then the equation (1.1) have a strong solution given by the quantity n

u(t) = lim

n→∞

i=1

[ νt ]

= lim

ν→0

[I −

t t H(i )]−1 u0 n n

[I − νH(iν)]−1 u0 ,

i=1

providing that the condition of smallness in Theorem 2.1 of Chapter 8 is satisfied, where involved are the Remark 2.2 in Chapter 8 and the (3.1) in Section 3. This condition of smallness includes the cases where sufficiently small is T or both N03 (x, t, z1 , z2 ) and N02 (x, t, τ, z1 , z2 ) for finite |z1 | and |z2 |, or both H(0)u0 and the total variation of ζ(t) on [0, T ]. In that case, u(t) also satisfies the middle equation in (1.1). Theorem 2.2. The time-dependent operator J(t) in (1.4) satisfies the four conditions, namely, the non-dissipativity condition (H3), the weaker range condition (H2) , the time-regulating condition (HC), and the embedding condition (HB) in Chapter 8. As a result, it follows from Theorems 2.1 and 2.3 of Chapter 8 that, for u0 ∈ D(J(0)), the equation (1.4) and then the equation (1.3) have a strong solution given by the quantity u(t) = lim

n

n→∞

i=1

[ νt ]

= lim

ν→0

[I −

t t J(i )]−1 u0 n n

[I − νJ(iν)]−1 u0 ,

i=1

providing that the condition of smallness in Theorem 2.1 of Chapter 8 is satisfied, where involved are the Remark 2.2 in Chapter 8 and the (4.1) in Section 4. This condition of smallness includes the cases where small enough is T , or both N21 (x, t, z1 , z2 ) and N11 (x, t, τ, z1 , z2 ) for finite |z1 | and |z2 |, or, in addition to N30 (x, t, τ, z) for finite |z|, both J(0)u0 and the total variation of ζ(t) on [0, T ]. In that case, u(t) also satisfies the middle equation in (1.3).

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3. Proof of One Space Dimensional Case Proof of Theorem 2.1: Proof. We now begin the proof, which is composed of five steps. Step 1. (H(t) satisfies the non-dissipativity condition (H3).) The second condition in (H3) follows from Step 3 below, while the first condition in (H3) is done as in the proof of Theorem 2.1 of Chapter 4, where the maximum principle arguments were used. Step 2. (H(t) satisfies the weaker range condition (H2) .) This is established by analogy with the proof of Theorem 2.1 in Chapter 4. Step 3. (H(t) satisfies the time-regulating condition (HC).) Let gi (x) ∈ C[0, 1],

i = 1, 2,

and let v1 = (I − λH(t))−1 g1 ; v2 = (I − λH(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ[α(x, t, v1 , v1 )(v1 − v2 ) + g˜(x, t, v1 , v1 ) − g˜(x, t, v2 , v2 )] = λ{

[α(x, t, v1 , v1 ) − α(x, t, v2 , v1 )] + [α(x, t, v2 , v1 ) − α(x, τ, v2 , v2 )] α(x, τ, v2 , v2 ) g (x, t, v2 , v2 ) − g˜(x, τ, v2 , v2 )]} [H(τ )v2 − g˜(x, τ, v2 , v2 )] + [˜ + (g1 − g2 );

|˜ g(x, t, v1 , v2 ) − g˜(x, t, v2 , v2 )| ≤ N03 (x, t, v1 , v2 )(1 + |v2 |)|v1 − v2 |; |˜ g(x, t, v2 , v2 ) − g˜(x, τ, v2 , v2 )| ≤ |ζ(t) − ζ(τ )|N01 (x, t, τ, v2 )(1 + |v2 |); |α(x, t, v1 , v2 ) − α(x, t, v2 , v2 )|/α(x, τ, v2 , v2 ) ≤ N02 (x, t, τ, v1 , v2 )|v1 − v2 |; |α(x, t, v2 , v2 )

− α(x, τ, v2 , v2 )|/α(x, τ, v2 , v2 ) ≤ |ζ(t) − ζ(τ )|N00 (x, t, τ, v2 );

v2 ∞

≤

δ0−1 [H(τ )v2 ∞

+ ˜ g(x, τ, v2 , v2 )∞ ];

H(τ )v2 = (v2 − g2 )/λ = [(I − λH(τ ))−1 g2 − g2 ]/λ; |˜ g(x, τ, v2 , v2 )| ≤ M00 (x, τ, v2 )(1 + |v2 |); (v1 − v2 ) (0) ∈ [β0 (v1 (0)) − β0 (v2 (0))]; (v1 − v2 ) (1) ∈ −[β1 (v1 (1)) − β1 (v2 (1))]; so there holds, for some functions Mi , i = 1, 2, 3, 4, in the condition (HC), v1 − v2 ∞ ≤ [g1 − g2 ∞ + λM1 (v1 ∞ , v2 ∞ )v1 − v2 ∞ + λM2 (v1 ∞ , v2 ∞ )(v2 − g2 /λ)v1 − v2 ∞ + λ|ζ(t) − ζ(τ )|M3 (v1 ∞ , v2 ∞)(v2 − g2 /λ) + λ|ζ(t) − ζ(τ )|M4 (v1 ∞ , v2 ∞ )].

(3.1)

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This is the condition (HC). Here used was the interpolation inequality [1], [13, page 135] v2 ∞ ≤ v2 ∞ + C()v2 ∞ for each > 0 and for some constant C(), where δ0−1

max

x∈[0,1];t∈[0,T ]

|M00 (x, t, v2 ∞ )| < 1

if = 1/[δ0−1

max

x∈[0,1];t∈[0,T ]

|M00 (x, t, v2 ∞ )| + 2], for example.

This is because, as in the proof of Theorem 2.1 of Chapter 4, the maximum principle applies; that is, there is an x0 ∈ (0, 1) such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|; (v1 − v2 ) (x0 ) = 0; (v1 − v2 )(x0 )(v1 − v2 ) (x0 ) ≤ 0. Here x0 ∈ {0, 1} is impossible, due to the boundary conditions. Step 4. (H(t) satisfies the embedding condition (HB) of embeddedly quasidemi-closedness.) This follows as in the proof of Theorem 2.1 of Chapter 6. Step 5. (u(t) for u0 ∈ D(H(0)) satisfies the middle equation in (1.1).) Consider the discretized equation ui −νH(ti )ui = ui−1 , (3.2) ui ∈ D(H(ti )), where u0 ∈ D(H(0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that νω < 1 and 0 ≤ ti = iν ≤ T . Here, for small enough ν, ui =

i

[I − νH(tk )]−1 u0

k=1

exists uniquely by Proposition 3.6 of Chapter 8. For convenience, we also define u−1 = u0 − νH(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 of Chapter 8 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νH(tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t H(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Lemma 3.4 and Proposition 3.5 in Chapter 8, we have that ui ∞ and H(ti )ui ∞ = (ui − ui−1 )/ν∞ are uniformly bounded. Those, in turn, result in a bound for ui C 2 [0,1] by the interpolation inequality [1], [13, page 135]. Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1 [0, 1] to a limit, as ν −→ 0. This

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limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.1), as ui does so. The proof is complete. 4. Proof of Higher Space Dimensional Case Proof of Theorem 2.2: Proof. We now begin the proof, which consists of five steps. Step 1. As in the proof of Theorem 2.2 of Chapter 4, the maximum principle arguments prove the first condition in (H3), whereas the second condition in (H3) is a result of Step 3 below. Here notice β(x, t, z) = [β(x, t, z) − β(x, t, 0)] + β(x, t, 0) = βz (x, t, θz)(z − 0) + β(x, t, 0),

0≤θ≤1

by the mean value theorem, where βz ≥ δ11 > 0. Step 2. The proof of Theorem 2.2 in Chapter 4 also shows that J(t) satisfies the weaker range condition (H2) . Step 3. (J(t) satisfies the time-regulating condition (HC).) Let gi (x) ∈ C μ (Ω),

i = 1, 2,

and let v1 = (I − λJ(t))−1 g1 ; v2 = (I − λJ(τ ))−1 g2 ; where λ > 0 and 0 ≤ t, τ ≤ T . Then (v1 − v2 ) − λ[α0 (x, t, v1 , Dv1 ) (v1 − v2 ) + g˜0 (x, t, v1 , Dv1 ) − g˜0 (x, t, v2 , Dv2 )] = λ{

[α0 (x, t, v1 , Dv1 ) − α0 (x, t, v2 , Dv1 )] + [α0 (x, t, v2 , Dv1 ) − α0 (x, τ, v2 , Dv2 )] α0 (x, τ, v2 , Dv2 ) [J(τ )v2 − g˜0 (x, τ, v2 , Dv2 )] + [˜ g0 (x, t, v2 , Dv2 ) − g˜0 (x, τ, v2 , Dv2 )]} + (g1 − g2 ),

x ∈ Ω;

|˜ g0 (x, t, v1 , Dv2 ) − g˜0 (x, t, v2 , Dv2 )| ≤ N21 (x, t, v1 , v2 )(1 + |Dv2 |)|v1 − v2 |; |˜ g0 (x, t, v2 , Dv2 ) − g˜0 (x, τ, v2 , Dv2 )| ≤ |ζ(t) − ζ(τ )|N20 (x, t, τ, v2 )(1 + |Dv2 |); |α0 (x, t, v1 , Dv2 ) − α0 (x, t, v2 , Dv2 )|/α0 (x, τ, v2 , Dv2 ) ≤ N11 (x, t, τ, v1 , v2 )|v1 − v2 |; |α0 (x, t, v2 , Dv2 ) − α0 (x, τ, v2 , Dv2 )|/α0 (x, τ, v2 , Dv2 ) ≤ |ζ(t) − ζ(τ )|N10 (x, t, τ, v2 ); J(τ )v2 = (v2 − g2 )/λ = [(I − λJ(τ ))−1 g2 − g2 ]/λ; |˜ g0 (x, τ, v2 , Dv2 )| ≤ M10 (x, τ, v2 )(1 + |Dv2 |); ∂(v1 − v2 ) + [β(x, t, v1 ) − β(x, t, v2 )] ∂n ˆ = −[β(x, t, v2 ) − β(x, τ, v2 )], |β(x, t, v2 ) − β(x, τ, v2 )| ≤ |t − τ |N30 (x, t, τ, v2 );

x ∈ ∂Ω;

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so there holds, for some functions Mi , i = 0, 1, 2, 3, 4, in the condition (HC), v1 − v2 ∞ ≤ [g1 − g2 ∞ + λM1 (v1 ∞ , v2 ∞ )v1 − v2 ∞ + λM2 (v1 ∞ , v2 ∞ )(v2 − g2 /λ)v1 − v2 ∞ + λ|ζ(t) − ζ(τ )|M3 (v1 ∞ , v2 ∞)(v2 − g2 /λ) + λ|ζ(t) − ζ(τ )|M4 (v1 ∞ , v2 ∞ )];

(4.1)

or

v1 − v2 ∞ ≤ M0 (v1 ∞ , v2 ∞ )|t − τ |; proving the condition (HC). This is because, as in proving the non-dissipativity condition (H3) in Step 1, the maximum principle argument applies, that is, there is an x0 ∈ Ω such that v1 − v2 ∞ = |(v1 − v2 )(x0 )|, that, for x0 ∈ Ω, D(v1 − v2 )(x0 ) = 0; (v1 − v2 )(x0 ) (v1 − v2 )(x0 ) ≤ 0, and that, for x0 ∈ ∂Ω, ∂(v1 − v2 ) (x0 ) ≥ 0 or ≤ 0 according as (v1 − v2 )(x0 ) > 0 or < 0. ∂n ˆ Here, to derive, for some numbers C1 (v2 ∞ ) and C2 (v2 ∞ ), depending on v2 ∞ , Dv2 ∞ ≤ v2 C 1+μ (Ω)

(4.2) v2 − g2 ∞ + C2 (v2 ∞ ), λ we used the integral representation of v2 , with the Green’s function Z(x, y) of the second kind [29], Z(x, y)[−β2 (y, τ, v2 )] dσy v2 = − ∂Ω v2 − g 2 − g0 (y, τ, v2 , Dv2 )] dy. + Z(x, y)α0 (y, τ, Dv2 )−1 [ λ Ω ≤ C1 (v2 ∞ )

Indeed, the result follows from differentiating the above v2 for (1 + μ) times, with respect to the variable x, to obtain, for some constants d1 and d2 and for some number C3 (v2 ∞) depending on v2 ∞, v2 − g2 ∞ λ + d2 [ max

Dv2 C μ (Ω) ≤ d1

x∈Ω;t∈[0,T ]

|M10 (x, t, v2 ∞ )|Dv2 ∞ + C3 (v2 ∞ )],

and from combining the interpolation inequality [1], [13, page 135] Dv2 ∞ ≤ Dv2 C μ (Ω) + C()v2 ∞ for each > 0 and for some constant C(). Here d2

max

x∈Ω;t∈[0,T ]

|M10 (x, t, v2 ∞ )| < 1

if = 1/[d2

max

x∈Ω;t∈[0,T ]

|M10 (x, t, v2 ∞ )| + 2], for example,

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and, for some constants b1 and b2 , |Di Z(x, y)| ≤ b1 |x − y|1−N ; |Dij Z(x, y)| ≤ b2 |x − y|−N ; |x − y|N (μ−1) dx is finite for 0 < μ < 1 [13, page 159]. Ω

Step 4. (J(t) satisfies the embedding condition (HB) of embeddedly quasidemi-closedness.) This follows as in the proof of Theorem 2.2 of Chapter 6. Step 5. (u(t) for u0 ∈ D(J(0)) satisfies the middle equation in (1.3).) Consider the discretized equation ui −νJ(ti )ui = ui−1 , (4.3) ui ∈ D(J(ti )), where u0 ∈ D(J(0)), i = 1, 2, . . . , n, n ∈ N is large, and ν > 0 is such that νω < 1 and 0 ≤ ti = iν ≤ T . Here, for small enough ν, ui =

i

[I − νJ(tk )]−1 u0

k=1

exists uniquely by Proposition 3.6 of Chapter 8. For convenience, we also define u−1 = u0 − νJ(0)u0 . Now, for each t ∈ [0, T ), we have t ∈ [ti , ti+1 ) for some i, so i = [ νt ]. It follows from Theorem 2.1 of Chapter 8 that, for each above t with the corresponding i, t

lim ui = lim

ν→0

ν→0

= lim

[ν]

[I − νJ(tk )]−1 u0

k=1 n

n→∞

k=1

[I −

t t J(k )]−1 u0 n n

≡ u(t) exists. On the other hand, by utilizing Lemma 3.4 and Proposition 3.5 in Chapter 8, we have ui ∞ and J(ti )ui ∞ = (ui −ui−1 )/ν∞ are uniformly bounded, whence so is ui C 1+λ (Ω) for any 0 < λ < 1, using the proof of (4.2). (Alternatively, those, in turn, result in a bound for ui W 2,p (Ω) for any p ≥ 2, by the Lp elliptic estimates [37]. Hence, a bound exists for uiC 1+η (Ω) for any 0 < η < 1, as a result of the Sobolev embedding theorem [1, 13].) Therefore it follows from Ascoli-Arzela theorem [33] that a subsequence of ui and then itself converge in C 1+μ (Ω) to a limit, as ν −→ 0. This limit equals u(t) as shown above. Consequently, u(t) satisfies the middle equation in (1.3), as ui does so. The proof is complete.

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Appendix In this Appendix, some essential background results from elliptic partial differential equations of second order will be collected for the convenience of the reader. There are five small sections in this Appendix. 1. Existence of a Solution Theorem 1.1 ([5, 24]). The nonhomogeneous boundary value problem for the ordinary differential equations of second order p(x)y + q(x)y + r(x)y = f (x),

a < x < b; 2

B y˜ = γ ∈ R , has a unique solution, if the homogeneous boundary value problem p(x)y + q(x)y + r(x)y = 0,

a < x < b,

B y˜ = 0 ∈ R2 , only has the trivial zero solution. Here p(x), q(x), r(x),

and f (x) are continuous, real-valued

functions on [a, b]; p(x) ≥ δ > 0 for some constant δ; B is a real 2 × 4 matrix of rank 2; ⎞ ⎛ y(a) ⎜y (a)⎟ 4 ⎟ y˜ = ⎜ ⎝ y(b) ⎠ ∈ R is a four dimensional, real vector; y (b) γ ∈ R2

is a given two dimensional, real vector.

Notice that the boundary conditions of the Robin, Neumann, Dirichlet or periodic type meet the requirements of Theorem 1.1. Theorem 1.2 ([13]). The nonhomogeneous boundary value problem for the elliptic partial differential equation of second order with the Robin boundary condition Lu =

N

aij (x)Dij u(x) +

i,j=1

N

bi (x)Di u(x)

i=1

+ c(x)u = f (x), γ(x)u(x) + β(x)

∂u(x) = ϕ(x), ∂ν 155

x ∈ Ω ⊂ RN ; x ∈ ∂Ω,

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APPENDIX

has a unique solution in C 2+α (Ω) for all f ∈ C α (Ω) and ϕ ∈ C 1+α (Ω). Here and f (x) are in C α (Ω);

aij (x), bi (x), c(x), Ω

is a bounded smooth domain in RN , N = 2, 3, . . .;

Ω is the closure of Ω; ∂Ω is the boundary of Ω; c(x) ≤ 0 for x ∈ Ω; and L is strictly elliptic, that is, the matrix function (aij (x))N ×N N

is strictly positive definite:

aij (x)ξi ξj ≥ λ|ξ|

for some λ > 0,

i,j=1

⎛

⎞ ξ1 ⎜ ⎟ for all x ∈ Ω and for all ξ = ⎝ ... ⎠ ∈ RN ; ξN

γ(x), β(x)

are in C

1+α

(Ω) with γ(x) and β(x)

positive on ∂Ω and β(x) ≥ κ > 0 on ∂Ω; and ∂u(x) is the outer normal derivative of u(x) on ∂Ω; ∂ν Dij u(x) are the second partial derivatives of u(x); Di u(x)

are the first partial derivatives of u(x);

and C k (Ω), k = 0, 1, . . . ,

is the real vector space of,

k times, continuously differentiable, real-valued function on Ω; uC k (Ω) =

k

sup |D η u(x)|

j=1 |β|=k Ω

for u ∈ C k (Ω);

∂ |η| u(x) , x = (x1 , . . . , xN ) ∈ RN ; · · · ∂xηNN η = (η1 , . . . , ηN ), ηi ∈ {0} ∪ N, is a multi-index

D η u(x) =

∂xη11

with |η| =

N

ηi ;

i=1

and C α (Ω), 0 < α < 1,

is the real vector space of all

α-Holder continuous, real-valued functions on Ω;

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1. EXISTENCE OF A SOLUTION

uC α(Ω) =

|u(x) − u(y)| |x − y|α x =y;x,y∈Ω sup

i=1

157

for u ∈ C α (Ω); ⎛

⎞ x1 ⎜ ⎟ for x = ⎝ ... ⎠,

N |x − y| = [ |xi − yi |2 ]1/2

⎛

main4

xN

⎞

y1 ⎜ ⎟ y = ⎝ ... ⎠ ∈ RN ; yN and C k+α (Ω) is the subspace of C k (Ω) with functions whose k-th order, partial derivatives are are α-Holder continuous on Ω; uC k+α(Ω) = uC k (Ω) + sup D η uC α (Ω) . |η|=k

Theorem 1.3 ([13]). Following Theorem 1.2, the nonhomogeneous boundary value problem for the elliptic partial differential equation of second order with the Dirichlet boundary condition Lu =

N

aij (x)Dij u(x) +

i,j=1

N

bi (x)Di u(x)

i=1

+ c(x)u = f (x), u(x) = ϕ(x),

x ∈ Ω ⊂ RN ;

x ∈ ∂Ω,

has a unique solution in C 2+α (Ω) for all f ∈ C α (Ω) and ϕ ∈ C 2+α (Ω). Theorem 1.4 ([13], Contraction Mapping Principle). If T : X −→ X is a strict contraction, then T has a unique fixed point, that is, Tx = x for some unique x ∈ X. Here X

is a real Banach space with the norm · ;

T

is a strict contraction, if there is a number 0 < θ < 1, such that T x1 − T x2 ≤ θx1 − x2 holds for x1 , x2 ∈ X;

Theorem 1.5 ([13], The Method of Continuity). Let X and Y be two real Banach spaces with the norms · X and · Y , respectively. Let L0 , L1 : X −→ Y

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158

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APPENDIX

be two linear bounded operators. Let, for each t ∈ [0, 1], Lt = (1 − t)L0 + tL1 . Supppose that there is a constant C, such that, for x ∈ X, xX ≤ CLt xY holds for all t ∈ [0, 1]. Then L1 is onto if and only if so is L0 . Proof. Assume for a moment that Ls is onto for some s ∈ [0, 1]. It will be shown that Ls is also one to one, from which L−1 s exists. But this follows from the assumption that there is a constant C, such that, for x ∈ X, xX ≤ CLt xY holds for all t ∈ [0, 1]. By making use of £−1 s , the equation, for y ∈ Y given, Lτ x = y is equivalent to the equation −1 x = L−1 s y + (τ − s)Ls (L0 − L1 )x

from which a linear map S : X −→ X, −1 Sx = Ss x ≡ L−1 s y + (τ − s)Ls (L0 − L1 )u

is defined. A fixed point x of S = Ss will be a solution of the equation Lτ x = y. By choosing τ ∈ [0, 1] such that |s − τ | < δ ≡ [C(L0 X→Y + L1 X→Y )]−1 , it follows that S = Ss is a strict contraction map. Therefore S has a unique fixed point by the Contraction Mapping Principle, Theorem 1.4. Hence Lτ is onto for τ satisfying |τ − s| < δ. It follows that, by dividing [0, 1] into subintervals of length less than δ and repeating the above arguments in a finite number of times, Lτ becomes onto for all τ ∈ [0, 1], provided that it is onto for some τ ∈ [0, 1]. In particular, L1 is onto if and only if so is L0 . 2. Apriori Estimates Theorem 2.1 ([13]). Following Theorem 1.2, let u ∈ C 2+α (Ω) be a solution to the nonhomogeneous boundary value problem for the elliptic partial differential equation of second order with the Robin boundary condition Lu =

N

aij (x)Dij u(x) +

i,j=1

N

bi (x)Di u(x)

i=1

+ c(x)u = f (x), γ(x)u(x) + β(x)

∂u(x) = ϕ(x), ∂ν

x ∈ Ω ⊂ RN ; x ∈ ∂Ω.

Let aij , bi , cC α (Ω) ,

γ, βC 1+α(Ω) ≤ Λ

holds for some constant Λ, where i, j = 1, . . . , N .

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3. HOPF BOUNDARY POINT LEMMA

main4

159

Then the apriori estimate uC 2+α(Ω) ≤ C(uC 0 (Ω) + f C α (Ω) + ϕC 1+α (Ω) ) holds for some constant C, where C = C(N, α, λ, Λ, κ, Ω) depends on N, α, . . . , Ω. Theorem 2.2 ([13]). Following Theorem 1.2, let u ∈ C 2+α (Ω) be a solution to the nonhomogeneous boundary value problem for the elliptic partial differential equation of second order with the Dirichlet boundary condition Lu =

N

aij (x)Dij u(x) +

i,j=1

N

bi (x)Di u(x)

i=1

x ∈ Ω ⊂ RN ;

+ c(x)u = f (x), x ∈ ∂Ω.

u(x) = ϕ(x), Let

aij , bi , cC α (Ω) ≤ Λ holds for some constant Λ, where i, j = 1, . . . , N . Then the apriori estimate uC 2+α(Ω) ≤ C(uC 0 (Ω) + f C α (Ω) + ϕC 2+α (Ω) ) holds for some constant C, where C = C(N, α, λ, Λ, Ω) depends on N, α, . . . , Ω. 3. Hopf Boundary Point Lemma Lemma 3.1 ([13]). Let u be a function defined on Ω, such that Lu =

N

aij (x)Dij u(x) +

i,j=1

N

bi (x)u(x)

i=1

+ c(x)u(x) ≥ 0

for x ∈ Ω.

Suppose that, at a point x0 ∈ ∂Ω, u(x0 ) is a non-negative maximum. Then the 0) of u at x0 , if it exists, satisfies outer, normal derivative ∂u(x ∂ν ∂u(x0 ) > 0. ∂ν Here Ω is a bounded smooth domain in RN , N = 2, 3, . . .; u is continuous at x0 ; L is uniformly elliptic, that is, the matrix function (aij (x))N ×N

satisfies the estimate:

0 < λ(x)|ξ|2 ≤

N i,j=1

aij (x)ξi ξj ≤ Λ(x)|ξ|2

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160

main4

APPENDIX

for some functions λ(x) and Λ(x) with

λ(x) Λ(x)

bounded in Ω; c(x) ≤ 0 for x ∈ Ω; bi (x) c(x) , λ(x) λ(x)

are bounded in Ω. 4. Interpolation Inequality

Lemma 4.1 ([13]). Let Ω be a bounded smooth domain in RN , N ≥ 2. Let j + β < k + α, where 0 < α, β < 1;

j = 0, 1, . . . ;

k = 1, 2, . . . .

Let u ∈ C (Ω). Then for any > 0, there is a constant C = C(, j, k, Ω), such that k+α

uC j+β (Ω) ≤ CuC 0 (Ω) + uC k+α(Ω) holds. 5. Sobolev Embedding Theorem Theorem 5.1 ([1]). Let Ω be a bounded smooth domain in RN , N ≥ 2. Let j and m be non-negative integers, and let p satisfy 1 ≤ p < ∞. Suppose that mp > N > (m − p). Then the real Sobolev space W j+m,p (Ω) is imbedded into the real vector space C j+λ (Ω) for 0 < λ < m − Np . That is, for u ∈ W j+m,p (Ω), there is a constant K, such that uC j+λ (Ω) ≤ KuW j+m,p (Ω) holds. Here the real Sobolev space W k,p (Ω) for k = 0, 1, 2, . . ., is defined as follows. W k,p (Ω) = {u ∈ W k (Ω) : the weak derivatives Dη u ∈ Lp (Ω) for all multi-indices η with |η| ≤ k}; W k (Ω) = {u : u is a k times, weakly differentiable, real-valued function on Ω}; L (Ω) = {u : u is a measurable, real-valued p

function on Ω that is p-integrable}.

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main4

Bibliography [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] T. M. Apostol, Mathematical Analysis, second edition, Addison-Wesley Publishing Company, Inc., 1974. [3] V. Barbu, Semigroups and Differential Equations in Banach Spaces, Leyden: Noordhoff, 1976. [4] C.-C. Chen and K.-M. Koh, Principles and Techniques in Combinatorics, World Scientific, Singapore, 1992. [5] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company Inc., New York, 1955. [6] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 256-298. [7] M. G. Crandall, A generalized domain for semigroup Generators, Proceedings of the AMS, 2, (1973), 435-440. [8] M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math., 11 (1972), 57-94. [9] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, SpringerVerlag, New York, 1999. [10] K. Engel and R. Nagel, A Short Course on Operator semigroups, Springer-Verlag, New York, 2006. [11] L. Gaul, M. Kogl, and M. Wagner, Boundary Element Methods for Engineers And Scientists: An Introductory Course with Advanced Topics, Springer-Verlag, New York, 2003. [12] J. Kacur, Method of Rothe in Evolution Equations, Teubner Texte Zur Mathematik, Band bf80, BSB B. G. Teubner Verlagsgessellschaft, Leipzig, 1985. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer-Verlag, New York, 1983. [14] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985. [15] J. A. Goldsetin and C.-Y. Lin, Singular nonlinear parabolic boundary value problems in one space dimension, J. Diff. Eqns., 68 (1987), 429-443. [16] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Coll. Publ., Vol. 31, Providence, R. I., 1957. [17] T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966. [18] C.-Y. Lin, Degenerate nonlinear parabolic boundary value problems, Nonlinear Analysis, T. M. A., 13 (1989), 1303-1315. [19] C.-Y. Lin, Quasilinear parabolic equations with nonlinear boundary conditions, J. Computational and Applied Math., 126 (2000), 339-349. [20] C.-Y. Lin, Cauchy problems and applications, Topological Methods in Nonlinear Analysis, 15 (2000), 359-368. [21] C.-Y. Lin, On generation of C0 semigroups and nonlinear operator semigroups, Semigroup Forum, 66 (2003), 110-120. [22] C.-Y. Lin, On generation of nonlinear operator semigroups and nonlinear evolution operators, Semigroup Forum, 67 (2003), 226-246. [23] C.-Y. Lin, Nonlinear evolution equations, Electronic Journal of Differential Equations, Vol. 2005 (2005), No. 42, pp. 1-42. [24] C.-Y. Lin, Theory and Examples of Ordinary Differential Equations, World Scientific, Singapore, 2011.

161

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162


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BIBLIOGRAPHY

[25] C.-Y. Lin, Time-dependent domains for nonlinear evolution operators and partial differential equations, Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 92, pp. 1-30. [26] C.-Y. Lin, Some non-dissipativity condition for evolution equations, Int. J. Math., Vol. 24, No. 2 (2013), 1350002 (28 pages). [27] C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. [28] R. E. Mickens, Difference Equations, Theory and Applications, Second Edition, Van Mostrand Reinhold, New York, (1990). [29] C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York, 1970. [30] I. Miyadera, Nonlinear Semigroups, Translations of Mathematical Monographs, Vol. 109, American Mathematical Society, 1992. [31] A. Pazy, Semigroups of Linear Operators and Applications in Partial Differential Equations, Springer-Verlag, New York, 1983. [32] E. Rothe, Zweidimensionale parabolische Randvertaufgaben als Grenfall eindimensionale Renvertaufgaben, Math. Ann., 102 (1930), 650-670. [33] H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, 1989. [34] A. Schatz, V. Thomee, and W. Wendland, Mathematical Theory of Finite and Boundary Element Methods, Birkhauser, Basel, Boston, 1990. [35] H. Serizawa, M-Browder-accretiveness of a quasi-linear differential operator, Houston J. Math., 10 (1984), 147-152. [36] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, Wiley, New York, (1980). [37] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987. [38] U. Westphal, Sur la saturation pour des semi-groups ono lineaires, C. R. Acad. Sc. Paris 274 (1972), 1351-1353. [39] K. Yosida, Functional Analysis, Springer, New York, 1980.

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main4

Index

apriori estimate, 159

Robin boundary condition, 75, 83, 95, 109, 145, 155 Rothe functions, 6, 40, 64, 133

Cauchy problem, 1, 3 closed operator, 2 Contraction Mapping Principle, 157 Crandall Liggett theorem, 6 Crandall-Pazy theorem, 40

Sobolev Embedding Theorem, 160 stricly positive definite, 156 strict contraction, 157 strictly elliptic, 156 strong solution, 7, 41, 134

Difference Equations Theory, 23, 66 Dirichlet boundary condition, 157 Dirichlet condition, 75, 76, 96, 97 dissipativity condition, 2, 3, 38

The Method of Continuity, 157 time-regulating condition, 38, 130 weaker range condition, 7, 9, 41, 130

embeddedly quasi-demi-closed, 7 embeddedly quasi-demi-closedness, 41, 133 embedding condition, 7, 41, 133 evolution equation, 37, 129 exponential function, 4 fixed point, 157 generalized domain, 39 Green’s function, 91, 104, 116, 152 Hille-Yosida theorem, 5 Holder continuous, 156 Hopf Boundary Point Lemma, 159 Interpolation Inequality, 160 interpolation inequality, 87, 100, 104, 113, 116, 150, 152 limit solution, 7, 41, 134 maximal monotone graph, 16, 50 maximum principle, 43, 47, 51, 100, 104, 113, 115, 150, 152 mixture condition, 2 multi-index, 156 Neumann condition, 75, 76, 96, 97 non-dissipativity condition, 130 Nonlinear resolvent identity, 21 Periodic condition, 76, 96 range condition, 2, 3, 38 163

page 163

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