Lectures on Nonlinear Evolution Equations

1. Front Matter

   Pages I-VIII

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2. Introduction

   • Reinhard Racke

   Pages 1-6

3. Global solutions to wave equations — existence theorems

   • Reinhard Racke

   Pages 7-14

4. L ^p-L ^q-decay estimates for the linear wave equation

   • Reinhard Racke

   Pages 15-20

5. Linear symmetric hyperbolic systems

   • Reinhard Racke

   Pages 21-33

6. Some inequalities

   • Reinhard Racke

   Pages 34-57

7. Local existence for quasilinear symmetric hyperbolic systems

   • Reinhard Racke

   Pages 58-78

8. High energy estimates

   • Reinhard Racke

   Pages 79-83

9. Weighted a priori estimates for small data

   • Reinhard Racke

   Pages 84-90

10. Global solutions to wave equations — proofs

    • Reinhard Racke

    Pages 91-105

11. Other methods

    • Reinhard Racke

    Pages 106-109

12. Development of singularities

    • Reinhard Racke

    Pages 110-113

13. More evolution equations

    • Reinhard Racke

    Pages 114-211

14. Further aspects and questions

    • Reinhard Racke

    Pages 212-225

15. Back Matter

    Pages 226-260

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The book in hand is based on lectures which were given at the University of Bonn in the winter semesters of 1989/90 and 1990/91. The aim of the lectures was to present an elementary, self-contained introduction into some important aspects of the theory of global, small, smooth solutions to initial value problems for non linear evolution equa­ tions. The addressed audience included graduate students of both mathematics and physics who were only assumed to have abasie knowledge of linear partial differential equations. Thus, in the spirit of the underlying series, this book is intended to serve as a detailed basis for lectures on the subject as weIl as for self-studies for students or for other newcomers to this field. The presentation of the theory is made using the classical method of continuation of local solutions with the help of apriori estimates obtained for small data. The corre­ sponding global existence theorems have been proved mainly in the last decade, focussing on fully nonlinear systems; Related questions concerning large data problems, the ex­ istence of weak solutions or the analysis of &.hock waves are not discussed. Also the question of optimal regularity assumptions on the coefficients is beyond the scope of the book and is touched only in part and exemplarily.

• a priori estimates
• equation
• evolution
• global solutions
• hyperbolic system
• initial value problems
• interpolation
• local solutions
• nonlinear evolution equations
• online
• presentation
• proof
• theorem

• Institute of Applied Mathematics, University of Bonn, Germany

  Reinhard Racke

Dr. Reinhard Racke, Institut für Angewandte Mathematik,Universität Bonn/ Prof. Dr. Klas Diederich, Mathematisches Institut der Universität Wuppertal.

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