Degenerate Diffusions. Initial Value Problems and Local Regularity Theory
This book is devoted to the study of nonnegative solutions of the Cauchy initial value problem or the Dirichlet boundary value problem for a class of nonlinear evolution differential equations ∂u/∂t = ∆ ϕ(u) on Rn x (0,T) or on a cylinder B x (t1,t2), where the nonlinearity ϕ is assumed to be continuous and increasing with ϕ (0) = 0, and is assumed to satisfy the growth condition a ≤ [u ϕ ´(u)]/[ ϕ (u)] ≤ 1/a for all u > 0 for a in (0,1) and the normalization condition ϕ (1) = 1. This class forms a natural generalization of the power case ϕ(u) = um and has a wide range of applications both in physics and in geometry. In 1940, D. Widder characterized the set of all nonnegative weak solutions of the Cauchy problem for the heat equation on the strip Rn x (0,T) by five fundamental properties.
The authors are interested in analogues of these properties for the nonlinearities ϕ described above. The results are divided into two groups according to the growth of ϕ. The first group concerns the case of “slow diffusion” and it is characterized in the power case by the condition m > 1. The second group includes the supercritical “fast diffusion” case, which generalizes the case ϕ (u) = um for (n-2)/n < m < 1. Besides the items mentioned above, the book collects together local regularity results in chapter 1 (a priori L∞ bounds, the Harnack inequality and equicontinuity of solutions to the slow diffusion case) and a proof of continuity of weak solutions to the porous medium equation in the final (fifth) chapter.
The book gives an up-to-date, clear and concise overview of results concerning degenerate diffusion together with powerful methods and useful techniques for studying existence and qualitative properties of solutions. A number of comments and discussions of various topics, a brief summary of further known results and some open problems are listed in the last section of each chapter and an up-to-date bibliography will be appreciated by both researchers and graduate students with a background in analysis and partial differential equations.