Mathematics > Analysis of PDEs
[Submitted on 28 Mar 2018 (v1), last revised 9 Jul 2018 (this version, v3)]
Title:Nonlocal Harnack inequalities for nonlocal heat equations
View PDFAbstract: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.
Submission history
From: Yong-Cheol Kim [view email][v1] Wed, 28 Mar 2018 23:16:05 UTC (31 KB)
[v2] Tue, 3 Apr 2018 06:40:32 UTC (32 KB)
[v3] Mon, 9 Jul 2018 10:38:31 UTC (34 KB)
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