Asymptotic behaviour of solutions and free boundaries of the anisotropic slow diffusion equation

TL;DR

This paper investigates the asymptotic behavior and free boundary dynamics of solutions to the anisotropic porous medium equation in the slow diffusion regime, establishing existence, uniqueness, and convergence properties of self-similar solutions.

Contribution

It introduces the existence and uniqueness of self-similar fundamental solutions for the anisotropic porous medium equation in the slow diffusion range, including analysis of free boundary behavior.

Findings

Existence of self-similar fundamental solutions with compact support.

Asymptotic convergence of solutions to these fundamental solutions.

Analysis of free boundary support and anisotropic boundary behavior.

Abstract

In this paper we explore the theory of the anisotropic porous medium equation in the slow diffusion range. After revising the basic theory, we prove the existence of self-similar fundamental solutions (SSFS) of the equation posed in the whole Euclidean space. Each of such solutions is uniquely determined by its mass. This solution has compact support w.r.t. the space variables. We also obtain the sharp asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Special attention is paid to the convergence of supports and free boundaries in relative size, i.e., measured in the appropriate anisotropic way. The fast diffusion case has been studied in a previous paper by us, there no free boundaries appear.