Oxford Mathematical Monographs; Oxford Science Publications. Oxford: Oxford University Press (ISBN 0-19-856903-3/hbk). xxii, 624 p. £ 65.00 (2007).

This monograph presents and develops mathematical tools needed for a systematic studying the nonlinear heat equation $\partial_t = \Delta(u^m)$, $m > 1$ posed in $d$-dimensional Euclidean space. This equation have a number of applications in practice such as the description of the flow of an isentropic gas through a porous medium, the study of groundwater infiltration, heat radiation in plasmas, spread of viscous fluids, and in other fields. The author concentrates on such mathematical questions for this equation as existence, uniqueness, stability and practical construction of suitable defined solutions of the equation together with well-posed setting of initial-boundary value problems and studying qualitative properties. The book consists of twenty one chapters which are combined in two parts. The first part of the book presents an introductory course on the porous medium equation and the generalized porous medium equation. The author reviews the main facts, introduces the basic a priori estimates and exposes some of the main topics of the theory, like the property of finite propagation of disturbances, the appearance of free boundaries, the need for generalized solutions, the question of limited regularity, blow-up phenomenon and the evolution of signed solutions. For interested readers, the classical problems of existence, uniqueness and regularity of a (generalized) solution for the three main problems to the equations under consideration are studied in details. This completes the first half of the book.
With this foundation, the second part of the book enters into more peculiar aspects of the theory for the porous medium equation; existence with optimal data (solutions with the so-called growing data and solutions whose initial value is a Radon measure), free boundaries and evolution of the support of a solution, self-similar solutions, higher regularity, symmetrization and asymptotic behavior as $t$ goes to infinity both for the Cauchy problem and Dirichlet, Neumann problems. The question of regularity of the solutions of the Cauchy problem is concentrated on describing two of the main results for the non-negative and compactly supported solutions: the Lipschitz continuity of the so-called pressure function and the free boundary for large times and the lesser regularity for small times of the so-called focusing solutions (or hole-filling solutions). Partial $C^{\infty}$ regularity of solutions and the concavity properties are also studied. The final two chapters gather complements on the previous material and show further applications to the physical sciences.