"The Porous Medium Equation: Mathematical Theory" by Juan Luis Vázquez

This volume is dedicated to the heat equation - 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. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME).

In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, author provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME).

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.

CONTENTS
PREFACE
1 INTRODUCTION
PART ONE:
2 MAIN APPLICATIONS
3 PRELIMINARIES AND BASIC ESTIMATES
4 BASIC EXAMPLES
5 THE DIRICHLET PROBLEM L WEAK SOLUTIONS
6 THE DIRICHLET PROBLEM IL LIMIT SOLUTIONS, VERY WEAK SOLUTIONS AND SOME OTHER VARIANTS
7 CONTINUITY OF LOCAL SOLUTIONS
8 THE DIRICHLET PROBLEM EL STRONG SOLUTIONS
9 THE CAUCHY PROBLEM. Ll-THEORY
10 THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH
11 THE NEUMANN PROBLEM AND PROBLEMS ON MANIFOLDS
PART TWO:
12 THE CAUCHY PROBLEM WITH GROWING INITIAL DATA
13 OPTIMAL EXISTENCE THEORY FOR NON-NEGATIVE SOLUTIONS
14 PROPAGATION PROPERTIES
15 ONE-DIMENSIONAL THEORY. REGULARITY AND INTERFACES
16 FULL ANALYSIS OF SELF SIMILARITY
17 TECHNIQUES OF SYMMETRIZATION AND CONCENTRATION
18 ASYMPTOTIC BEHAVIOUR L THE CAUCHY PROBLEM
19 REGULARITY AND FINER ASYMPTOTICS IN SEVERAL DIMENSIONS
20 ASYMPTOTIC BEHAVIOUR IL DIRICHLET AND NEUMANN PROBLEMS
COMPLEMENTS:
21 FURTHER APPLICATIONS
Appendix : BASIC FACTS
BIBLIOGRAPHY
INDEX