上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性

Acta mathematica scientia,Series A ›› 2025, Vol. 45 ›› Issue (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

Ma Jingjing¹(),Wei Na^2,^*()

¹School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119
²School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073

Received:2024-11-27 Revised:2025-02-28 Online:2025-08-26 Published:2025-08-01
Supported by:
NSFC(12071482);Youth Innovation Team of Shaanxi Universities, Shaanxi Fundamental Science Research Project for Mathematics and Physics(22JSZ012);Fundamental Research Funds for the Central Universities(GK202202007);Fundamental Research Funds for the Central Universities(GK202307001);Fundamental Research Funds for the Central Universities(GK202402004)

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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