Nonlocal Harnack inequalities for nonlocal heat equations

TL;DR

This paper establishes nonlocal Harnack inequalities and weak Harnack inequalities for solutions of nonlocal parabolic equations involving integro-differential operators, extending classical results to nonlocal settings using the De Giorgi method.

Contribution

It introduces nonlocal Harnack inequalities for nonlocal heat equations with integro-differential operators, a novel extension of classical parabolic regularity results.

Findings

Proved nonlocal Harnack inequalities for weak solutions.

Established nonlocal weak Harnack inequalities.

Applied De Giorgi method to nonlocal parabolic equations.

Abstract

In this paper, applying the De Giorgi method, we obtain nonlocal Harnack inequalities for weak solutions of nonlocal parabolic equations given by an integro-differential operator $\rL_K$ as follows; \begin{equation*}\begin{cases} \rL_K u+\pa_t u=0 &\text{ in $\Om\times(-T,0]$ } u=g &\text{ in $\bigl((\BR^n\s\Om)\times (-T,0]\bigr)\cup\bigl(\Om\times\{t=-T\}\bigr)$ } \end{cases}\end{equation*} where $g\in C(\BR^n\times [-T,0])\cap L^{\iy}(\BR^n\times(-T,0])$ and $\,\Om\,$ is a bounded domain in $\BR^n$ with Lipschitz boundary. Moreover, we get nonlocal parabolic weak Harnack inequalities of the weak solutions.