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

Linlin Fan, Linfen Cao, Peibiao Zhao

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)+(-Δ-λ_{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..

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

Abstract

In this paper, we study a nonlinear system involving a generalized tempered fractional -Laplacian in : \begin{equation*} \left\{ \begin{array}{ll} \partial_tu(x,t)+(-Δ-λ_{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 , . We establish Hopf's lemma for parabolic equations involving a generalized tempered fractional -Laplacian. Hopf's lemma will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations..

Paper Structure

This paper contains 3 sections, 6 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Some lemmas
  3. Proof of the Main Result

Key Result

Theorem 1.1

(Asymptotic Hopf's lemma for a generalized tempered fractional p-Laplacian) Assume that $u(x,t)\in(C_{loc}^{1,1}(B_{1}(0))\cap\mathcal{L}_{sp})\times C^{1}(0,\infty)$ is a positive solution to where $0<s<1$, $p>2,\ n\geq2$. Assume that Then there exists a positive constant $c$, such that for any $t\rightarrow\infty$ and for all $x$ near the boundary of $B_{1}(0)$, we have where $d(x)=dist(x,\pa

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 2 more