Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type.

*(English)*Zbl 1113.35004
Oxford Lecture Series in Mathematics and its Applications 33. Oxford: Oxford University Press (ISBN 0-19-920297-4/hbk). xiii, 234 p. (2006).

This book is concerned with quantitative aspects of the theory of nonlinear diffusion equations with nonlinearities of power type that lead to degenerate or singular parabolicity. Its aim is to obtain sharp a priori estimates and decay rates for solutions of general classes of equations in terms of estimates of particular problems, and a deeper knowledge of the evolution semigroup. The estimates are building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Basic tools are results of symmetrization and mass concentration comparison, combined with scaling properties. The functional setting consists of Lebesgue and Marcinkiewicz spaces.

Being based on estimates, it is a book about mathematical inequalities and their impact. These inequalities determine or control the behavior of nonlinear diffusion semigroups in terms of data and parameters.

The book provides an introduction to the subject of smoothing estimates of nonlinear diffusion equations, centered in the study of some model equations and on the Cauchy problem. It gives a rather comprehensive account of PME/FDE.

The book consists of three parts. The analysis of the results based on comparison with source-type solutions occupies Part I, and concerns the porous medium equation (PME), the heat equation (HE), and the supercritical range of the fast diffusion equation (FDE). Part II deals with the critical and subcritical range of the FDE. Part III is devoted to extensions (such as \(p\)-Laplacian equation and the doubly nonlinear equation), and appendices containing the technical material needed in the book, and parallel topics. At the end of each chapter, additional information on the contents, main references and possible developments are given.

The book may serve as an advanced graduate text, and as a source of information for graduate students and researchers. It contains a fair amount of new results and open problems. Although the basic part of the text is self-contained, materials from other sources are used in the advanced sections introducing current problems. In such cases, pertinent references are given.

Being based on estimates, it is a book about mathematical inequalities and their impact. These inequalities determine or control the behavior of nonlinear diffusion semigroups in terms of data and parameters.

The book provides an introduction to the subject of smoothing estimates of nonlinear diffusion equations, centered in the study of some model equations and on the Cauchy problem. It gives a rather comprehensive account of PME/FDE.

The book consists of three parts. The analysis of the results based on comparison with source-type solutions occupies Part I, and concerns the porous medium equation (PME), the heat equation (HE), and the supercritical range of the fast diffusion equation (FDE). Part II deals with the critical and subcritical range of the FDE. Part III is devoted to extensions (such as \(p\)-Laplacian equation and the doubly nonlinear equation), and appendices containing the technical material needed in the book, and parallel topics. At the end of each chapter, additional information on the contents, main references and possible developments are given.

The book may serve as an advanced graduate text, and as a source of information for graduate students and researchers. It contains a fair amount of new results and open problems. Although the basic part of the text is self-contained, materials from other sources are used in the advanced sections introducing current problems. In such cases, pertinent references are given.

Reviewer: Chiu Yeung Chan (Lafayette)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35K55 | Nonlinear parabolic equations |

35K65 | Degenerate parabolic equations |

47H20 | Semigroups of nonlinear operators |