Anisotropic Fast Diffusion Equations

TL;DR

This paper establishes the existence and uniqueness of self-similar solutions for anisotropic fast diffusion equations, analyzes their asymptotic behavior, and introduces novel methods to handle the mathematical challenges posed by anisotropy.

Contribution

It proves the existence and uniqueness of self-similar fundamental solutions for anisotropic porous medium equations in the fast diffusion range, addressing complex anisotropic effects.

Findings

Existence of self-similar fundamental solutions (SSF) for anisotropic equations.

Asymptotic behavior of solutions in terms of SSF.

Derived decay rates and regularity properties.

Abstract

We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite-mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.