上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性

Abstract

Abstract:

In this paper, we consider the nonlinear parabolic equation associated with the fractional $p$-Laplace operator on the upper half space

$\begin{cases} \frac{\partial u}{\partial t}(x,t)+(-\triangle)^s_pu(x,t)=f(u(x,t)), &\quad(x,t)\in\mathbb{R}^{n}_+\times(0,\infty),\\ u(x,t)>0, &\quad(x,t)\in\mathbb{R}^{n}_+\times(0,\infty),\\ u(x,t)=0, &\quad(x,t)\notin\mathbb{R}^{n}_+\times(0,\infty). \end{cases}$

First, the narrow region principle and the maximum principle for antisymmetric functions are proved in both bounded and unbounded domains. Then, the Hopf lemma for antisymmetric functions is established. Finally, using the method of moving planes, the monotonicity of solutions to the parabolic equation associated with the fractional $p$-Laplace operator on the upper half space is demonstrated.

Key words: fractional $p$-Laplace operator, parabolic equations, monotonicity, Hopf lemma for antisymmetric functions, moving plane method

CLC Number:

Ma Jingjing, Wei Na. Monotonicity of Solutions for Parabolic Equations Related to Fractional Order $p$-Laplace Operators on the Upper Half Space[J].Acta mathematica scientia,Series A, 2025, 45(4): 1086-1099.

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Monotonicity of Solutions for Parabolic Equations Related to Fractional Order $p$-Laplace Operators on the Upper Half Space

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