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Lectures on Nonlinear Evolution Equations

Initial Value Problems

  • Reinhard Racke

Table of contents

  1. Front Matter
    Pages i-viii
  2. Reinhard Racke
    Pages 1-6
  3. Reinhard Racke
    Pages 21-33
  4. Reinhard Racke
    Pages 34-56
  5. Reinhard Racke
    Pages 78-81
  6. Reinhard Racke
    Pages 82-88
  7. Reinhard Racke
    Pages 89-103
  8. Reinhard Racke
    Pages 104-107
  9. Reinhard Racke
    Pages 108-111
  10. Reinhard Racke
    Pages 112-206
  11. Reinhard Racke
    Pages 207-220
  12. Reinhard Racke
    Pages 221-270
  13. Back Matter
    Pages 271-306

About this book

Introduction

This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other
examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value problems and for open questions are provided.


In this second edition, initial-boundary value problems in waveguides are additionally considered.

Keywords

Cauchy problem Klein-Gordon equation Schrödinger equation heat equation plate equation thermoelasticity wave equation waveguide

Authors and affiliations

  • Reinhard Racke
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany

Bibliographic information