Hopf's lemma for parabolic equations involving a generalized tempered fractional $p$-Laplacian

TL;DR

This paper establishes a Hopf's lemma for parabolic equations involving a generalized tempered fractional p-Laplacian, providing a key tool for analyzing qualitative properties of solutions to nonlocal parabolic equations.

Contribution

It introduces a Hopf's lemma for a class of nonlocal parabolic equations with a generalized tempered fractional p-Laplacian, advancing theoretical understanding.

Findings

Proved Hopf's lemma for the specified nonlocal operator.

Demonstrated the lemma's utility in qualitative analysis.

Extended classical results to nonlocal fractional operators.

Abstract

In this paper, we study a nonlinear system involving a generalized tempered fractional $p$-Laplacian in $B_{1}(0)$: \begin{equation*} \left\{ \begin{array}{ll} \partial_tu(x,t)+(-\Delta-\lambda_{f})_{p}^{s}u(x,t)=g(t,u(x,t)), &(x,t)\in B_{1}(0)\times[0,+\infty),\\ u(x)=0,&(x,t)\in B_{1}^{c}(0)\times[0,+\infty), \end{array} \right. \end{equation*} where $0<s<1$, $p>2,\ n\geq2$. We establish Hopf's lemma for parabolic equations involving a generalized tempered fractional $p$-Laplacian. Hopf's lemma will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations..