阻尼波动方程解的整体存在性和衰减估计

本书内容涉及正则性赢得型和正则性损失型两类耗散波动方程的渐近性态。其一是广义双色散方程等正则性赢得型耗散波动方程的整体存在性、衰减估计、渐近性态、逐点衰减估计及加权衰减估计等等。通过对解算子的详尽分析，建立由波动方程、热方程的基本解以及非线性项表示的渐近性态。其二是结构耗散的广义双色散方程、具有阻尼项六阶Boussinesq型方程等正则性损失型耗散波动方程的整体存在性、衰减估计、渐近性态等等等。结合解算子的衰减性质，引进时间加权的能量克服正则性损失型造成的衰减性质差的困难，进而证明整体解的存在性和渐近性态。

Preface
Chapter 1 Preliminary
1.1 Lp space
1.2 Sobolev spaces
1.3 Abstract functions valued in Banach spaces
1.4 Fourier transform
1.4.1 Fourier transform of L1 space
1.4.2 Fourier transform of Schwartz function space
1.4.3 Fourier transform of Lp(1≤p≤2) space
1.5 Decay rate of solutions to the initial value problem for linear heat equation
1.6 Calculus of inequalities
1.7 Some useful inequalities
Chapter 2 Asymptotic Profile of Solutions to the Double Dispersion Equation
2.1 Introduction
2.2 Analysis of Solution operators
2.3 Decay estimate of solutions to (2.2.1), (2.1.2)
2.4 Asymptotic profile of solutions to (2.2.1), (2.1.2)
2.5 Applications of asymptotic profile
2.5.1 Application Ⅰ
2.5.2 Application Ⅱ
2.6 Decay estimate of solutions to (2.1.1), (2.1.2)
Chapter 3 Asymptotic Profile of Solutions to the Linearized Double Dispersion Equation on the Half Space Rn+
3.1 Introduction
3.2 Solution operators
3.3 Asymptotic profile of solutions to (3.1.7), (3.1.8)
3.4 Asymptotic profile of solutions to the initial boundary value problem (3.1.4)–(3.1.6)
Chapter 4 Asymptotic Profile of Global Solutions to the Generalized Double Dispersion Equation via the Nonlinear Term
4.1 Introduction
4.2 Analysis of solution operators
4.3 Weighted decay estimate of solutions
4.4 Asymptotic profile
Chapter 5 Pointwise Estimate of Global Small Solutions to the Generalized Double Dispersion Equation
5.1 Introduction
5.2 Analysis of solution operators
5.3 Global classical solutions
5.4 Pointwise estimate of solution operators
5.4.1 Pointwise estimate for GL and HL
5.4.2 Pointwise estimate for GH and HH
5.4.3 Pointwise estimate of G and H
5.5 Pointwise estimate of global classical solutions
Chapter 6 Global Existence and Asymptotic Behavior of Solutions for a Semi-linear Wave Equation
6.1 Introduction
6.2 Solution formula
6.3 Decay property
6.4 Decay estimate of solutions to (6.1.3), (6.1.2) with L1 data
6.5 Global existence and asymptotic behavior with L1 data
6.6 Decay of solutions to (6.1.3), (6.1.2) with L2 data
6.7 Global existence and asymptotic behavior with L2 data
Chapter 7 Asymptotic Behavior of Solutions to a Class of Nonlinear Wave Equations of Sixth Order with Damping
7.1 Introduction
7.2 Solution operators
7.3 Decay rate of solutions to linear problem
7.4 Asymptotic profile of solutions to linear problem
7.5 Decay estimate of solutions to nonlinear problem
Chapter 8 Asymptotic Profile of Solutions to the Generalized Double Dispersion Equation with Structural Damping
8.1 Introduction
8.2 Mild formulation and solution operators
8.2.1 Mild formulation
8.2.2 Decay property of solution operators
8.3 Global existence and time-decay
8.4 Asymptotic profile
8.4.1 The case for ν∈(0, 1/2)
8.4.2 The case ν∈(1/2, 1)
8.4.3 The case for ν=1/2
Bibliography
Index