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Chapter 1 Generation and Representation 1

1.1 Uniformly Continuous Semigroups of Bounded Linear Operators 1

1.2 Strongly Continuous Semigroups of Bounded Linear Operators 4

1.3 The Hille-Yosida Theorem 8

1.4 The Lumer Phillips Theorem 13

1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups 17

1.6 Groups of Bounded Operators 22

1.7 The Inversion of the Laplace Transform 25

1.8 Two Exponential Formulas 32

1.9 Pseudo Resolvents 36

1.10 The Dual Semigroup 38

Chapter 2 Spectral Properties and Regularity 42

2.1 Weak Equals Strong 42

2.2 Spectral Mapping Theorems 44

2.3 Semigroups of Compact Operators 48

2.4 Differentiability 51

2.5 Analytic Semigroups 60

2.6 Fractional Powers of Closed Operators 69

Chapter 3 Perturbations and Approximations 76

3.1 Perturbations by Bounded Linear Operators 76

3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups 80

3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups 81

3.4 The Trotter Approximation Theorem 84

3.5 A General Representation Theorem 89

3.6 Approximation by Discrete Semigroups 94

Chapter 4 The Abstract Cauchy Problem 100

4.1 The Homogeneous Initial Value Problem 100

4.2 The Inhomogeneous Initial Value Problem 105

4.3 Regularity of Mild Solutions for Analytic Semigroups 110

4.4 Asymptotic Behavior of Solutions 115

4.5 Invariant and Admissible Subspaces 121

Chapter 5 Evolution Equations 126

5.1 Evolution Systems 126

5.2 Stable Families of Generators 130

5.3 An Evolution System in the Hyperbolic Case 134

5.4 Regular Solutions in the Hyperbolic Case 139

5.5 The Inhomogeneous Equation in the Hyperbolic Case 146

5.6 An Evolution System for the Parabolic Initial Value Problem 149

5.7 The Inhomogeneous Equation in the Parabolic Case 167

5.8 Asymptotic Behavior of Solutions in the Parabolic Case 172

Chapter 6 Some Nonlinear Evolution Equations 183

6.1 Lipschitz Perturbations of Linear Evolution Equations 183

6.2 Semilinear Equations with Compact Semigroups 191

6.3 Semilinear Equations with Analytic Semigroups 195

6.4 A Quasilinear Equation of Evolution 200

Chapter 7 Applications to Partial Differential Equations—Linear Equations 206

7.1 Introduction 206

7.2 Parabolic Equations—L2 Theory 208

7.3 Parabolic Equations—Lp Theory 212

7.4 The Wave Equation 219

7.5 A Schr？dinger Equation 223

7.6 A Parabolic Evolution Equation 225

Chapter 8 Applications to Partial Differential Equations—Nonlinear Equations 230

8.1 A Nonlinear Schr？dinger Equation 230

8.2 A Nonlinear Heat Equation in R1 234

8.3 A Semilinear Evolution Equation in R3 238

8.4 A General Class of Semilinear Initial Value Problems 241

8.5 The Korteweg-de Vries Equation 247

Bibliographical Notes and Remarks 252

Bibliography 264

Index 277