Superlinear Parabolic Problems

1. Front Matter

   Pages i-xi

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2. Preliminaries

   Pages 1-5

3. Model Elliptic Problems

   Pages 7-74

4. Model Parabolic Problems

   Pages 75-250

5. Systems

   Pages 251-312

6. Equations with Gradient Terms

   Pages 313-376

7. Nonlocal Problems

   Pages 377-427

8. Back Matter

   Pages 429-584

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This book is devoted to the qualitative study of solutions of superlinear elliptic and parabolic partial differential equations and systems. This class of problems contains, in particular, a number of reaction-diffusion systems which arise in various mathematical models, especially in chemistry, physics and biology.

The first two chapters introduce to the field and enable the reader to get acquainted with the main ideas by studying simple model problems, respectively of elliptic and parabolic type. The subsequent three chapters are devoted to problems with more complex structure; namely, elliptic and parabolic systems, equations with gradient depending nonlinearities, and nonlocal equations. They include many developments which reflect several aspects of current research. Although the techniques introduced in the first two chapters provide efficient tools to attack some aspects of these problems, they often display new phenomena and specifically different behaviors, whose study requires new ideas. Many open problems are mentioned and commented.

The book is self-contained and up-to-date, it has a high didactic quality. It is devoted to problems that are intensively studied but have not been treated so far in depth in the book literature. The intended audience includes graduate and postgraduate students and researchers working in the field of partial differential equations and applied mathematics.

• A-priori bounds
• Cauchy problem
• Dirichlet problem
• blow-up
• parabolic problem
• partial differential equation
• partial differential equations

From the reviews:

“This book was written by two authors who in the past have made numerous substantial contributions to this field … . A list of applicable tricks, tools and techniques proves very helpful. Quittner and Souplet have opened their fat bag and let the reader profit from their large collection of methods described in this appendix. …The authors have put a lot of hard work into this book, and their efforts have certainly paid off. … recommended to anyone interested in this important and extensive field.” (R. Sperb, Zeitschrift für angewandte Mathematik und Physik, Vol. 60, 2009)

• Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia

  Pavol Quittner

• Laboratoire Analyse Géométrie et Applications UMR CNRS 7539 -Institut Galilée, Université Paris-Nord, Villetaneuse, France

  Philippe Souplet

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