A survey on mass conservation 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. 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, and posed in the whole Euclidean space, the solutions actually lose mass in time, they even disappear in finite time. This is a surprising fact. In Part 1 we pay attention to the connection between the validity of mass conservation and the existence of finite-mass self-similar solutions, as well as the role in the asymptotic behaviour of more general classes of solutions. We examine the situations when mass conservation is replaced by its extreme alternative, extinction in finite time. The conservation laws offer difficult borderline cases in the presence of critical parameters that we identify. New results are proved. We establish mass conservation in those pending cases. We also explain the disappearance of the fundamental solutions in a very graphical way. The 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 pending theorems are proved. We conclude with a long review of related equations and topics in Part 3. Summing up, the paper aims at surveying an important topic in nonlinear diffusion; at the time we solve a number of open problems on key issues and point out new directions.