H\"older Estimates for Singular Non-local Parabolic Equations

TL;DR

This paper proves local H"older continuity for solutions of certain singular non-local parabolic equations using extension techniques, and describes solution behavior near extinction time.

Contribution

It introduces a novel approach to establish regularity for non-local equations via extension methods, overcoming difficulties with direct energy estimates.

Findings

Established local H"older estimates for solutions

Applied extension method to non-local operators

Described solution behavior near extinction time

Abstract

In this paper, we establish local H\"older estimate for non-negative solutions of the singular equation \eqref{eq-nlocal-PME-1} below, for $m$ in the range of exponents $(\frac{n-2\sigma}{n+2\sigma},1)$. Since we have trouble in finding the local energy inequality of $v$ directly. we use the fact that the operator $(-\La)^{\sigma}$ can be thought as the normal derivative of some extension $v^{\ast}$ of $v$ to the upper half space, \cite{CS}, i.e., $v$ is regarded as boundary value of $v^{\ast}$ the solution of some local extension problem. Therefore, the local H\"older estimate of $v$ can be obtained by the same regularity of $v^{\ast}$. In addition, it enables us to describe the behaviour of solution of non-local fast diffusion equation near their extinction time.