Contents Preface Chapter 1 Preliminary 1 1.1 Lp space 1 1.2 Sobolev spaces 2 1.3 Abstract functions valued in Banach spaces 4 1.4 Fourier transform 5 1.4.1 Fourier transform of L1 space 5 1.4.2 Fourier transform of Schwartz function space 6 1.4.3 Fourier transform of Lp(1≤p≤2) space 7 1.5 Decay rate of solutions to the initial value problem for linear heat equation 9 1.6 Calculus of inequalities 10 1.7 Some useful inequalities 11 Chapter 2 Asymptotic Profile of Solutions to the Double Dispersion Equation 13 2.1 Introduction 13 2.2 Analysis of Solution operators 16 2.3 Decay estimate of solutions to (2.2.1), (2.1.2) 21 2.4 Asymptotic profile of solutions to (2.2.1), (2.1.2) 25 2.5 Applications of asymptotic profile 33 2.5.1 Application I 34 2.5.2 Application II 34 2.6 Decay estimate of solutions to (2.1.1), (2.1.2) 35 Chapter 3 Asymptotic Profile of Solutions to the Linearized Double Dispersion Equation on the Half Space Rn+ 39 3.1 Introduction 39 3.2 Solution operators 41 3.3 Asymptotic profile of solutions to (3.1.7), (3.1.8) 47 3.4 Asymptotic profile of solutions to the initial boundary value problem (3.1.4)–(3.1.6) 55 Chapter 4 Asymptotic Profile of Global Solutions to the Generalized Double Dispersion Equation via the Nonlinear Term. 57 4.1 Introduction 57 4.2 Analysis of solution operators 59 4.3 Weighted decay estimate of solutions 64 4.4 Asymptotic profile 69 Chapter 5 Pointwise Estimate of Global Small Solutions to the Generalized Double Dispersion Equation 76 5.1 Introduction 76 5.2 Analysis of solution operators 78 5.3 Global classical solutions 81 5.4 Pointwise estimate of solution operators 86 5.4.1 Pointwise estimate for GL and HL 86 5.4.2 Pointwise estimate for GH and HH 91 5.4.3 Pointwise estimate of G and H 92 5.5 Pointwise estimate of global classical solutions 92 Chapter 6 Global Existence and Asymptotic Behavior of Solutions for a Semi-linear Wave Equation 95 6.1 Introduction 95 6.2 Solution formula 97 6.3 Decay property 99 6.4 Decay estimate of solutions to (6.1.3), (6.1.2) with L1 data 105 6.5 Global existence and asymptotic behavior with L1 data 110 6.6 Decay of solutions to (6.1.3), (6.1.2) with L2 data 121 6.7 Global existence and asymptotic behavior with L2 data 124 Chapter 7 Asymptotic Behavior of Solutions to a Class of Nonlinear Wave Equations of Sixth Order with Damping 133 7.1 Introduction 133 7.2 Solution operators 135 7.3 Decay rate of solutions to linear problem 138 7.4 Asymptotic profile of solutions to linear problem 143 7.5 Decay estimate of solutions to nonlinear problem 152 Chapter 8 Asymptotic Profile of Solutions to the Generalized Double Dispersion Equation with Structural Damping 157 8.1 Introduction 157 8.2 Mild formulation and solution operators 160 8.2.1 Mild formulation 160 8.2.2 Decay property of solution operators 161 8.3 Global existence and time-decay 167 8.4 Asymptotic profile 173 8.4.1 The case for ν ∈ (0, 1/2) 173 8.4.2 The case ν ∈ (1/2, 1) 180 8.4.3 The case for ν = 1/2 189 Bibliography 196 Index 203
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