Contents
Preface i
CHAPTER 1
Survey on Attractors in Time-Dependent Spaces 1
1.1 Time-Dependent Global Attractors 1
1.1.1 Oscillation Equations 1
1.1.2 Wave Equations 4
1.1.3 Reaction-Diffusion Equations 13
1.1.4 Berger Equations 18
1.1.5 Abstract Evolution Equations 19
1.2 Some Useful Definitions and Lemmas 29
CHAPTER 2
Time-Dependent Global Attractors for the Non-Classical Diffusion Equations with a Fading Memory 35
2.1 Introduction 35
2.2 Time-Dependent Global Attractors in A4t 38
2.2.1 Global Well-Posedness 38
2.2.2 Absorbing Sets 39
2.2.3 Time-Dependent Attractors 41
2.2.4 Regularity of Attractors 47
2.3 Bibliographic Comments 49
CHAPTER 3
Strong Attractors for the Non-Classical Diffusion Equation with a Fading Memory in Time-Dependent Spaces 51
3.1 Introduction 51
3.2 Existence and Uniqueness of Strong Solutions 54
3.3 Time-Dependent Global Attractors for Strong Solutions 58
3.3.1 Absorbing Sets in 58
3.3.2 Time-Dependent Attractors 59
3.4 Bibliographic Comments 61
CHAPTER 4
Long-Time Behavior of Solutions to the Non-Autonomous Non-Classical Diffusion Equations 63
4.1 Introduction 63
4.2 Global Well-Posedness of Solutions 65
4.3 Existence of Pullback Attractors 69
4.3.1 Pullback Dδ,Ht -Absorbing Set 69
4.3.2 Pullback Dδ,Ht -Asymptotical Compactness 71
4.4 Regularity of Attractors 74
4.5 Bibliographic Comments 76
CHAPTER 5
Existence and Upper Semicontinuity of Attractors for Non-Autonomous Nonlocal Diffusion Equations 79
5.1 Introduction 79
5.2 Existence and Uniqueness of Solutions 81
5.3 Existence of the Minimal Pullback P-Attractors 87
5.3.1 Pullback 2)^-Absorbing Family 88
5.3.2 Pullback Asymptotical Compactness 90
5.4 Existence of Pullback Attractors * and Upper Semicontinuity of * and Global Attractor A 93
5.5 Bibliographic Comments 101
CHAPTER 6
Pullback Attractors for Diffusion Equations with a Delay Function and a Nonlocal Diffusion Term in Time-Dependent Spaces 105
6.1 Introduction 105
6.2 Existence and Uniqueness of Solutions 107
6.3 Existence of Pullback Dη–Attractors 113
6.3.1 Pullback Dη-Absorbing Family 113
6.3.2 Pullback Dη- Asymptotical Compactness 119
6.4 Regularity of Pullback Attractors 121
6.5 Bibliographic Comments 123
CHAPTER 7
Existence and Regularity of Pullback Attractors for Non-Classical Diffusion Equations with a Delay Operator 127
7.1 Introduction 127
7.2 Existence and Uniqueness of Solutions 129
7.3 Existence and Priori Estimates of Regularity for Pullback Attractors 134
7.4 Regularity of Pullback Attractors 148
7.5 Bibliographic Comments 151
CHAPTER 8
Survey on Attractors for Kirchhoff Wave Equations with Strong Dampings 153
8.1 Attractors for Kirchhoff Wave Equations with Strong Dampings 153
CHAPTER 9
Existence, Regularity and Fractal Dimension of Global Attractors for a Kirchhoff Wave Equation with Strong Damping and Memory 219
9.1 Introduction 219
9.2 Existence of Global Attractor A 222
9.3 Regularity of Global Attractor A 235
9.4 Fractal Dimension of Global Attractor A of Problem (9.1.1) withδ= 0 238
9.5 Bibliographic Comments 243
Bibliography 247