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
