Liouville rigidity and time-extrinsic Harnack estimates for an anisotropic slow diffusion

TL;DR

This paper establishes Liouville-type rigidity results and improves Harnack estimates for solutions to anisotropic slow diffusion equations, revealing conditions under which solutions are constant and refining time-extrinsic inequalities.

Contribution

It introduces new rigidity theorems for anisotropic diffusion and enhances Harnack estimates to include non-intrinsic times, broadening understanding of solution behavior.

Findings

Ancient solutions are constant under finite speed and boundedness conditions.

H"older estimates underpin the rigidity results.

Intrinsic Harnack estimates are extended to non-intrinsic times.

Abstract

We prove that ancient non-negative solutions to a fully anisotropic prototype evolution equation are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in space at a single time level. A similar statement is valid when the bound is given at a single space point. As a general paradigm, H\"older estimates provide the basics for rigidity. Finally, we show that recent intrinsic Harnack estimates can be improved to a Harnack inequality valid for non-intrinsic times. Locally, they are equivalent.