內容大鋼
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，之後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於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，並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空間中更為複雜和困難，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。
目錄
Preface
CHAPTER 1 Survey on Attractors in Time-Dependent Spaces
1.1 Time-Dependent Global Attractors
1.1.1 Oscillation Equations
1.1.2 Wave Equations
1.1.3 Reaction-Diffusion Equations
1.1.4 Berger Equations
1.1.5 Abstract Evolution Equations
1.2 Some Useful Definitions and Lemmas
CHAPTER 2 Time-Dependent Global Attractors for the Non-Classical Diffusion Equations with a Fading Memory
2.1 Introduction
2.2 Time-Dependent Global Attractors in A4t
2.2.1 Global Well-Posedness
2.2.2 Absorbing Sets
2.2.3 Time-Dependent Attractors
2.2.4 Regularity of Attractors
2.3 Bibliographic Comments
CHAPTER 3 Strong Attractors for the Non-Classical Diffusion Equation with a Fading Memory in Time-Dependent Spaces
3.1 Introduction
3.2 Existence and Uniqueness of Strong Solutions
3.3 Time-Dependent Global Attractors for Strong Solutions
3.3.1 Absorbing Sets in
3.3.2 Time-Dependent Attractors
3.4 Bibliographic Comments
CHAPTER 4 Long-Time Behavior of Solutions to the Non-Autonomous Non-Classical Diffusion Equations
4.1 Introduction
4.2 Global Well-Posedness of Solutions
4.3 Existence of Pullback Attractors
4.3.1 Pullback Dδ,Ht -Absorbing Set
4.3.2 Pullback Dδ,Ht -Asymptotical Compactness
4.4 Regularity of Attractors
4.5 Bibliographic Comments
CHAPTER 5 Existence and Upper Semicontinuity of Attractors for Non-Autonomous Nonlocal Diffusion Equations
5.1 Introduction
5.2 Existence and Uniqueness of Solutions
5.3 Existence of the Minimal Pullback P-Attractors
5.3.1 Pullback 2)^-Absorbing Family
5.3.2 Pullback Asymptotical Compactness
5.4 Existence of Pullback Attractors * and Upper Semicontinuity of * and Global Attractor A
5.5 Bibliographic Comments
CHAPTER 6 Pullback Attractors for Diffusion Equations with a Delay Function and a Nonlocal Diffusion Term in Time-Dependent
Spaces
6.1 Introduction
6.2 Existence and Uniqueness of Solutions
6.3 Existence of Pullback Dη–Attractors
6.3.1 Pullback Dη-Absorbing Family
6.3.2 Pullback Dη- Asymptotical Compactness
6.4 Regularity of Pullback Attractors
6.5 Bibliographic Comments
CHAPTER 7 Existence and Regularity of Pullback Attractors for Non-Classical Diffusion Equations with a Delay Operator
7.1 Introduction
7.2 Existence and Uniqueness of Solutions
7.3 Existence and Priori Estimates of Regularity for Pullback Attractors
7.4 Regularity of Pullback Attractors
7.5 Bibliographic Comments
CHAPTER 8 Survey on Attractors for Kirchhoff Wave Equations with Strong Dampings
8.1 Attractors for Kirchhoff Wave Equations with Strong Dampings
CHAPTER 9 Existence, Regularity and Fractal Dimension of Global Attractors for a Kirchhoff Wave Equation with Strong Damping
and Memory
9.1 Introduction
9.2 Existence of Global Attractor A
9.3 Regularity of Global Attractor A
9.4 Fractal Dimension of Global Attractor A of Problem (9.1.1 ) withδ=
9.5 Bibliographic Comments
Bibliography
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