非局部 Hamilton-Jacobi 方程解的渐近对称性与单调性——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1961-1976.

非局部 Hamilton-Jacobi 方程解的渐近对称性与单调性——献给李工宝教授 70 寿辰

牛亚慧()

郑州大学数学与统计学院郑州 450001

收稿日期:2025-05-28 修回日期:2025-08-27 出版日期:2025-12-26 发布日期:2025-11-18
作者简介:牛亚慧，E-mail：yhniu@zzu.edu.cn
基金资助:
国家自然科学基金(12301264);河南省自然科学基金项目(232300421347)

The Asymptotic Symmetry and Monotonicity of Solutions to Nonlocal Hamilton-Jacobi Equations

Yahui Niu()

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001

Received:2025-05-28 Revised:2025-08-27 Online:2025-12-26 Published:2025-11-18
Supported by:
NSFC(12301264);Henan Natural Science Foundation Project(232300421347)

摘要/Abstract

摘要：

该文研究一类非局部一阶 Hamilton-Jacobi 方程解的渐近对称性与单调性问题. 通过将文献 [Adv Math, 2021, 377: Art 107463] 中关于非线性项 $H(t,u)$ 的经典结果推广至更一般的 $H(t,x,u,\nabla u)$ 情形, 我们突破了原有理论框架的限制. 研究采用文献 [Adv Math, 2021, 377: Art 107463] 提出的的渐近移动平面法作为核心工具, 但针对哈密顿量中梯度项 $\nabla u$ 带来的新挑战, 作者对构造下解方法进行了关键性改进. 这拓展了方法的适用范围, 使其能够处理更广泛的非线性项类型.

关键词: 非局部 Hamilton-Jacobi 方程, 渐近对称性, 渐近移动平面法.

Abstract:

This paper investigates the asymptotic symmetry and monotonicity of solutions to a class of nonlocal first-order Hamilton-Jacobi equations. By extending the classical results on the nonlinear term $H(t,u)$ from the literature [Adv Math, 2021, 377: Art 107463] to the more general case of $H(t,x,u,\nabla u)$, we overcome the limitations of the original theoretical framework. The study employs the asymptotic moving plane method proposed in [Adv Math, 2021, 377: Art 107463] as the core tool. However, to address the new challenges posed by the gradient term $\nabla$u in the Hamiltonian, we make critical improvements to the construction of lower solution methods. This expands the applicability of the approach, enabling it to handle a broader range of nonlinear term types.

Key words: nonlocal Hamilton-Jacobi equation, asymptotic symmetry, asymptotic moving plane method.

中图分类号:

牛亚慧. 非局部 Hamilton-Jacobi 方程解的渐近对称性与单调性——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1961-1976.

Yahui Niu. The Asymptotic Symmetry and Monotonicity of Solutions to Nonlocal Hamilton-Jacobi Equations[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1961-1976.

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