Nonlocal Harnack inequalities for nonlocal heat 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.