Vázquez J.L. The Porous Medium Equation: Mathematical Theory

NY: Oxford University Press Inc., 2007. — 647 p. — ISBN: 0198569033.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.Main applications
Preliminaries and basic estimates
Basic examples
The Dirichlet problem I. Weak solutions
The Dirichlet problem II. Limit solutions, very weak solutions and some other variants
Continuity of local solutions
The Dirichlet problem III. Strong solutions
The Cauchy problem. L1 -theory
The PME as an abstract evolution equation. Semigroup approach
The Neumann problem and problems on manifoldsThe Cauchy problem with growing initial data
Optimal existence theory for non-negative solutions
Propagation properties
One-dimensional theory. Regularity and interfaces
Full analysis of self-similarity
Techniques of symmetrization and concentration
Asymptotic behaviour I. The Cauchy problem
Regularity and finer asymptotics in several dimensions
Asymptotic behaviour II. Dirichlet and Neumann problems
Complements
Further applications
Basic facts