The Porous Medium Equation

Aimed at research students and academics in mathematics and engineering, as well as engineering specialists, this book provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation. It contains a detailed introduction and is supplied with a section of notes.

The author of this monograph skillfully guides the reader, whether mathematician or physicist, through the background needed to understand and use the modern techniques developed...This book is a pleasure to read. It will be an excellent source, allowing the reader to build a proper intuition and of understand the basic facts of the theory...This book is also strongly recommended to researchers in Applied Mathematics who study various linear and nonlinear phenomena, but it may also be useful and comprehensive for a broader community od mathmaticians, physicists and engineers. I deeply believe that the present book is one of the most important works in its fieldthat have appeared until now...beautifully writtien and well organized, and I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of nonlinear partial differential equations.

Klappentext
The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.


Inhalt
Preface; 1. Introduction; PART 1; 2. Main applications; 3. Preliminaries and basic estimates; 4. Basic examples; 5. The Dirichlet problem I. Weak solutions; 6. The Dirichlet problem II. Limit solutions, very weak solutions and some other variants; 7. Continuity of local solutions; 8. The Dirichlet problem III. Strong solutions; 9. The Cauchy problem. L' theory; 10. The PME as an abstract evolution equation. Semigroup approach; 11. The Neumann problem and problems on manifolds; PART 2; 12. The Cauchy problem with growing initial data; 13. Optimal existence theory for nonnegative solutions; 14. Propagation properties; 15. One-dimensional theory. Regularity and interfaces; 16. Full analysis of selfsimilarity; 17. Techniques of symmetrization and concentration; 18. Asymptotic behaviour I. The Cauchy problem; 19. Regularity and finer asymptotics in several dimensions; 20. Asymptotic behaviour II. Dirichlet and Neumann problems; COMPLEMENTS; 21. Further applications; 22. Basic facts and appendices; Bibliography; Index

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