本書研究的內容為非經典擴散方程在時間依賴空間中的吸引子，受到時間依賴整體吸引子的一些研究成果的啟發，我們首先研究了時間依賴整體吸引子和強吸引子的存在性，之后通過調整對時間依賴函數的假設，如重新設置其下界和單調性，得到了一些在時間依賴空間中關于拉回吸引子的存在性和正則性、以及拉回吸引子和整體吸引子的上半連續性的成果，它們都是新的嘗試，并且通過這些模型的研究為在時間依賴空間中研究吸引子提供了一些新的思路和方法。此外，注意到時間依賴空間的范數中包含了時間依賴函數，因此很容易知道在此類空間中研究吸引子的存在性或其吸引子的其他性質要比在Sobolev空間中更為復雜和困難，例如在證明吸收集和漸近緊性時計算量會大大增加等。雖然計算和分析較為困難，但相空間范數中時間相關項的存在拓寬了以往的研究框架，使人們能夠在更接近物理現實的模型中對解的長時間行為進行討論，促進了對動力系統解的適定性的研究進程，具有重要意義。
. 非線性發展方程整體適定性和吸引子的研究，上海市自然科學二等獎，2015年，排名第一
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
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