Smoothing and Decay Estimates for Nonlinear Diffusion Equations - Equations of Porous Medium Type
The central object of this book is the nonlinear partial differential equation, ut – div (|u|m-1grad u); x Rn, t > 0, equipped with the initial value condition u = u0; x Rn, t = 0. The author is concerned with the smoothing effect of the equation and the time decay of positive solutions, i.e. whether the fact that u0 belongs to some function space X implies that the solution u(t) in time t > 0 is a member of some "better" function space Y and if it is possible to get estimates of the form |u(t)|Y < C(t, X, n, m,|u0|X). Well-posedness of the problem and some other substantial results such as the comparison theorem are mentioned in the preliminary part of the book and references are given for the proofs. Smoothing is carefully studied for all n N, m R (if m = 1 the classical results for the heat equation are reconstructed), X and Y being Lebesgue or weak Lebesgue spaces, which naturally appear as the correct spaces for studies of smoothing. It is very interesting that depending on m, n, X, Y, the solutions of the equation exhibit qualitatively very different properties, which are sometimes very surprising. The last chapter is devoted to the question of whether the results for the equation introduced at the beginning of the review also remain valid for the p-Laplacian equation. The book is very nicely written, well ordered and gives a rather complete overview of known results in the chosen field. At the beginning of each chapter there is a summary of the whole chapter with remarks of which sections of the chapter are essential for the following sections. The text is equipped with historical notes, remarks and a number of exercises. These properties make the book useful as a graduate textbook or a source of information for graduate students and researchers.