An Exponential Function Approach To Parabolic Equations

by Chin-yuan Lin

Overview

This volume is on initial-boundary value problems for parabolic partial differential equations of second order. It rewrites the problems as abstract Cauchy problems or evolution equations, and then solves them by the technique of elementary difference equations. Because of this, the volume assumes less background and provides an easy approach for readers to understand.

Product Details

ISBN-13:	9789814616386
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/13/2014
Series:	Series On Concrete And Applicable Mathematics , #15
Pages:	176
Product dimensions:	6.60(w) x 9.80(h) x 0.70(d)

Table of Contents

Preface vii

Chapter 1 Existence Theorems for Cauchy Problems 1

1 Introduction 1

2 Main Results 5

2.1 Linear Homogeneous Equations 5

2.2 Nonlinear Homogeneous Equations 6

2.3 Linear Nonhomogeneous Equations 7

2.4 Nonlinear Nonhomogeneous Equations 9

3 Examples 10

4 Some Preliminary Results 19

5 Difference Equations Theory 23

6 Proof of the Main Results 25

6.1 The Nonlinear Homogeneous Case 35

6.2 The Linear Homogeneous Case 30

6.3 The Linear Nonhomogeneous Case 34

6.4 The Nonlinear Nonhomogeneons Case 35

Chapter 2 Existence Theorems for Evolution Equations (I) 37

1 Introduction 37

2 Main Results 39

3 Examples 41

4 Some Preliminary Estimates 54

5 Proof of the Main Results 60

6 Difference Equations Theory 66

Chapter 3 Linear Autonomous Parabolic Equations 75

1 Introduction 75

2 Main Results 77

3 Proof of One Space Dimensional Case 77

4 Proof of Higher Space Dimensional Case 79

Chapter 4 Nonlinear Autonomous Parabolic Equations 83

1 Introduction 83

2 Main Results 85

3 Proof of One Space Dimensional Case 86

4 Proof of Higher Space Dimensional Case 89

Chapter 5 Linear Non-autonomous Parabolic Equations 95

1 Introduction 95

2 Main Results 97

3 Proof of One Space Dimensional Case 99

4 Proof of Higher Space Dimensional Case 103

Chapter 6 Nonlinear Non-autonomous Parabolic Equations (I) 109

1 Introduction 109

2 Main Results 111

3 Proof of One Space Dimensional Case 112

4 Proof of Higher Space Dimensional Case 115

Chapter 7 The Associated Elliptic Equations 119

1 Introduction 119

2 Main Results 121

3 Proof of Linear Case 121

4 Proof of the Nonlinear Case 123

Chapter 8 Existence Theorems for Evolution Equations (II) 129

1 Introduction 129

2 Main Results 132

3 Some Preliminary Estimates 134

4 Proof of the Main Results 141

Chapter 9 Nonlinear Non-autonomous Parabolic Equations (II) 145

1 Introduction 145

2 Main Results 148

3 Proof of One Space Dimensional Case 149

4 Proof of Higher Space Dimensional Case 151

Appendix 155

1 Existence of a Solution 155

2 Apriori Estimates 158

3 Hopf Boundary Point Lemma 159

4 Interpolation Inequality 160

5 Sobolev Embedding Theorem 160

Bibliography 161

Index 163