Parabolic Harnack estimates for anisotropic slow diffusion

TL;DR

This paper establishes a scale-invariant Harnack inequality for positive solutions of anisotropic slow diffusion equations, leading to regularity results like Hölder continuity and Liouville theorems.

Contribution

It introduces a novel anisotropic Harnack inequality for slow diffusion equations by leveraging self-similar solutions and translation invariance, extending classical results to anisotropic settings.

Findings

Proved a sharp anisotropic Harnack inequality.

Derived Hölder continuity and Liouville theorems.

Constructed self-similar Barenblatt solutions for the problem.

Abstract

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer H\"older continuity, an elliptic Harnack inequality and a Liouville theorem.