A survey on mass conservation, self-similarity and related topics in nonlinear diffusion

Abstract

We examine the validity of the principle of mass conservation for solutions of some typical equations in the theory of nonlinear diffusion, including equations in standard differential form and also their fractional counterparts. In Part 1, consisting of the first 9 sections, we use as main examples the heat equation, the porous medium equation and the $p$-Laplacian equation. Though these equations have the form of conservation laws, it happens that in some ranges of exponents the solutions posed in the whole Euclidean space lose mass in time. From the start we pay attention to the close connection between the validity of mass conservation and the existence of finite-mass self-similar solutions, as well as their role in the asymptotic behaviour of more general classes of solutions. Describing the extent of this connection is the common thread throughout the manuscript. When mass conservation does not hold, we are led to examine the situation when it is replaced by its extreme alternative, extinction in finite time, a very surprising fact. The next sections extend the detailed study to other models that occupy a relevant role in the current literature. Thus, the 3 sections of Part 2 are devoted to the discussion of mass conservation for some fractional nonlinear diffusion equations, where the situation is surveyed and a number of new theorems are proved. This is followed by the analogous study of Scalar Conservation Laws in Part 3, with surprising similarities. We conclude with a long review of related equations and topics in Part 4. The survey contains an extensive bibliography.