Book contents
- Frontmatter
- Dedication
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- 1 Introduction and Main Results
- 2 Preliminaries from Functional Analysis
- 3 Theory of Analytic Semigroups
- 4 Sobolev Imbedding Theorems
- 5 Lp Theory of Pseudo-Differential Operators
- 6 Lp Approach to Elliptic Boundary Value Problems
- 7 Proof of Theorem 1.1
- 8 Proof of Theorem 1.2
- 9 Proof of Theorems 1.3 and 1.4
- Appendix A The Laplace Transform
- Appendix B The Maximum Principle
- Appendix C Vector Bundles
- References
- Index
9 - Proof of Theorems 1.3 and 1.4
Published online by Cambridge University Press: 05 April 2016
Book contents
- Frontmatter
- Dedication
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- 1 Introduction and Main Results
- 2 Preliminaries from Functional Analysis
- 3 Theory of Analytic Semigroups
- 4 Sobolev Imbedding Theorems
- 5 Lp Theory of Pseudo-Differential Operators
- 6 Lp Approach to Elliptic Boundary Value Problems
- 7 Proof of Theorem 1.1
- 8 Proof of Theorem 1.2
- 9 Proof of Theorems 1.3 and 1.4
- Appendix A The Laplace Transform
- Appendix B The Maximum Principle
- Appendix C Vector Bundles
- References
- Index
Summary
In this last chapter we study the imbedding properties of the domains of fractional powers associated with analytic semigroups. This allows us to solve by successive approximations the semilinear initial boundary value problem, proving Theorems 1.3 and 1.4. Our proof is based on the local existence and uniqueness theorem of the abstract semilinear Cauchy problem in a Banach space. Finally, we prove global existence and uniqueness theorems for the semilinear Cauchy problem if an a priori bound for the nonlinear term can be found (Theorems 1.5 and 1.6).
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- Publisher: Cambridge University PressPrint publication year: 2016