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.
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.
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
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 in this work. mathematical reviews This book is a pleasure to read. It will be an excellent source, allowing the reader to build a proper intuition and to understand the basic facts of the theory.
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