Nonlinear Evolution Equations

Cover
......Page 1
Title: NONLINEAR EVOLUTION EQUATIONS......Page 4
Copyright
......Page 5
Contents......Page 8
Preface......Page 12
1.1 Motivations from Other Branches of Science......Page 16
1.2 Local Solutions and Global Solutions......Page 19
1.3.1 Sobolev Spaces W^{m, p}(Ω)......Page 23
1.3.2 The Gagliardo–Nirenberg and Poincar´e Inequalities......Page 25
1.3.3 Abstract Functions Valued in Banach Spaces......Page 26
1.3.4 Linear Elliptic Boundary Value Problems......Page 27
1.3.5 Interpolation Spaces......Page 30
1.3.6 Notation and Some Useful Inequalities......Page 32
1.4 Bibliographic Comments......Page 34
2.1.1 Semigroups......Page 36
2.1.2 The Infinitesimal Generator......Page 38
2.2 Hille–Yosida Theorem......Page 41
2.3 Regularities of Solutions......Page 50
2.4 Nonhomogeneous Equations......Page 56
2.5 Semilinear Equations......Page 61
2.6 Analytic Semigroups......Page 74
2.7 Applications......Page 85
2.8 Bibliographic Comments......Page 108
3.1 Compactness Method......Page 110
3.1.1 Some Results on Convergence and Compactness......Page 111
3.1.2 Method and Examples......Page 119
3.2 Monotone Operator Method......Page 129
3.2.1 General Framework......Page 130
3.2.2 Applications......Page 140
3.2.3 Generalizations and Supplements......Page 143
3.3 Bibliographic Comments......Page 145
4.1 Introduction......Page 148
4.2.1 Monotone Iterative Method for Nonlinear Parabolic Equations......Page 149
4.2.2 Monotone Iterative Method for Nonlinear Hyperbolic Equations......Page 164
4.3.1 Introduction......Page 169
4.3.2 Methods and Applications......Page 170
4.4 Bibliographic Comments......Page 189
5.1 Introduction......Page 190
5.2 IVP for Fully Nonlinear Parabolic Equations......Page 196
5.2.1 Decay Rate of the Solution to the Cauchy Problem for the Linear Heat Equation......Page 197
5.2.2 Global Solution to the Cauchy Problem for Fully Nonlinear Parabolic Equations......Page 199
5.3 IBVP for Fully Nonlinear Parabolic Equations......Page 215
5.3.1 Decay Rate of the Solution to IBVP for Linear Parabolic Equations......Page 217
5.3.2 Global Solution to IBVP for Fully Nonlinear Parabolic Equations......Page 219
5.4 Nonexistence of Global Solutions with Small Initial Data......Page 227
5.5 Bibliographic Comments......Page 234
6.1 Introduction......Page 236
6.2 A Lemma in Analysis and Its Applications......Page 240
6.3 Convergence to Stationary Solutions......Page 251
6.3.1 Some Basic Concepts......Page 252
6.3.2 Gradient Systems......Page 256
6.3.3 The Lojasiewicz–Simon Inequality......Page 260
6.3.4 Proof of Theorem 6.3.1......Page 265
6.4 Global Attractors......Page 270
6.5 Bibliographic Comments......Page 286
References......Page 288
Index......Page 302