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 I, consisting of the first nine 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 actually lose mass over 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 paper. 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 equations offer difficult borderline cases in the presence of some critical parameters that we identify. New results are proved to complete the picture; we establish mass conservation or its failure in those pending cases. We also explain the occasional disappearance of the fundamental solutions in a very graphical way, as “escape to infinity”. In this part of the paper, we aim to provide a comprehensive overview of the types of behaviour to be expected in these basic models. Additionally, Section 10 on the doubly nonlinear equation complements this foundational block.

The next sections extend the detailed study to other models that occupy a relevant niche in the current literature. Thus, the three sections of Part II 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 III, with surprising similarities. We conclude with a long review of related equations and topics in Part IV. The survey contains an extensive bibliography.

Summing up, the paper aims at surveying important topics in nonlinear diffusion related to mass conservation. At the same time, we solve a number of open problems on key issues and point out interesting connections and new directions.