上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1086-1099.

上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性

马晶晶¹(),魏娜^2,^*()

¹陕西师范大学数学与统计学院西安 710119
²中南财经政法大学统计与数学学院武汉 430073

收稿日期:2024-11-27 修回日期:2025-02-28 出版日期:2025-08-26 发布日期:2025-08-01
通讯作者: *E-mail: weina@zuel.edu.cn
作者简介:E-mail: mjjwnm@126.com
基金资助:
国家自然科学基金(12071482);陕西省高校青年创新团队, 陕西省数学和物理基础科学研究基金(22JSZ012);中央高校基本科研业务费专项资金(GK202202007);中央高校基本科研业务费专项资金(GK202307001);中央高校基本科研业务费专项资金(GK202402004)

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)

摘要/Abstract

摘要：

该文考虑上半空间分数阶 $p$-Laplace 算子相关的非线性抛物方程

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

首先, 证明有界域和无界域上反对称函数狭窄区域原理和极大值原理; 然后建立反对称函数 Hopf 引理; 最后, 利用移动平面法证明上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性.

关键词: 分数阶 $p$-Laplace 算子, 抛物方程, 单调性, 反对称函数 Hopf 引理, 移动平面法

Abstract:

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

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

中图分类号:

马晶晶, 魏娜. 上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性[J]. 数学物理学报, 2025, 45(4): 1086-1099.

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.

参考文献 20

[1]	Bidaut-Véron M F. Initial blow-up for the solutions of a semilinear parabolic equation with source term//Gauthier-Villars. Équations aux Dérivées Partielles et Applications, Paris: Elsevier, 1998: 189-198
[2]	Brändle C, Colorado E, de Pablo A, Sànchez U. A concave-convex elliptic problem involving the fractional Laplacian. Proc Roy Soc Edinb Sec A, 2013, 1.3(1): 39-71
[3]	Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Commun Partial Differ Equ, 2007, 32(8): 1245-1260
[4]	Chen W, Fang Y, Yang R. Liouville theorems involving the fractional Laplacian on a half space. Adv Math, 2015, 2.4(9): 167-198
[5]	Chen W, Li C. Maximum principles for the fractional $p$-Laplacian and symmetry of solutions. Adv Math, 2018, 3.5: 735-758
[6]	Chen W, Li C, Li Y. A direct method of moving planes for the fractional Laplacian. Adv Math, 2017, 3.8: 404-437
[7]	Chen W, Li C, Ou B. Classification of solutions for an integral equation. Comm Pure Appl Math, 2006, 59(3): 330-343
[8]	Chen W, Li Y, Ma P. The Fractional Laplacian. Hackensack: World Scientific, 2020
[9]	Chen W, Wang P, Niu Y, Hu Y. Asymptotic method of moving planes for fractional parabolic equations. Adv Math, 2021, 3.7: 107463
[10]	Chen W, Wu L. Liouville theorems for fractional parabolic equations. Adv Nonlinear Stud, 2021, 21(4): 939-958
[11]	Chen W, Wu L, Wang P. Nonexistence of solutions for inde nite fractional parabolic equations. Adv Math, 2021, 3.2: 108018
[12]	Chen W, Zhu J. Indefinite fractional elliptic problem and Liouville theorems. J Differ Equ, 2016, 2.0(5): 4758-4785
[13]	Ding M, Zhang C, Zhou S. Local boundedness and Hölder continuity for the parabolic fractional $p$-Laplace equations. Calc Var Partial Differ Equ, 2021, 60(1): 1-45
[14]	Li C, Chen W. A Hopf type lemma for fractional equations. Proc Amer Math Soc, 2019, 1.7(4): 1565-1575
[15]	Merle F, Zaag H. Optimal estimates for blow-up rate and behavior for nonlinear heat equations. Comm Pure Appl Math, 1998, 51(2): 139-196
[16]	Polàčik P, Quittner P, Souplet P. Singularity and decay estimates in superlinear problems via Liouville-type theorems. II: Parabolic equations. Indiana Univ Math J, 2007, 56(2): 879-908
[17]	Wang P. Symmetry and monotonicity of solutions to elliptic and parabolic fractional $p$-equations. arXiv: 2502.16158
[18]	Wang P, Chen W. Hopf's lemmas for parabolic fractional $p$-Laplacians. Commun Pure Appl Anal, 2022, 21(9): 3055-3069
[19]	Wu L, Chen W. The sliding methods for the fractional $p$-Laplacian. Adv Math, 2020, 3.1: 106933
[20]	Xing R. The blow-up rate for positive solutions of indefinite parabolic problems and related Liouville type theorems. Acta Math Sin (Engl Ser), 2009, 25(3): 503-518