半空间中满足 Cattaneo 定律的双曲系统的稳定性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1444-1462.

半空间中满足 Cattaneo 定律的双曲系统的稳定性

邓钧元^1,^*(),张峻皓^1,²()

¹武汉大学数学与统计学院武汉 430072
²香港中文大学数学系香港沙田

收稿日期:2024-05-22 修回日期:2025-06-10 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 邓钧元,E-mail:jy_deng@whu.edu.cn
作者简介:张峻皓,E-mail: jhzhang@math.cuhk.edu.hk

Stability to a Hyperbolic System with Cattaneo's Law in the Half Space

Junyuan Deng^1,^*(),Junhao Zhang^1,²()

¹School of Mathematics and Statistics, Wuhan University, Wuhan 430072
²Department of Mathematics, the Chinese University of Hong Kong, Hong Kong Shatin

Received:2024-05-22 Revised:2025-06-10 Online:2025-10-26 Published:2025-10-14

摘要/Abstract

摘要：

该文研究一维半空间中满足 Cattaneo 定律双曲方程的初边值问题稳态解的渐近非线性稳定性. 构造了该初边值问题的稳态解并得到了其正则性. 此外, 通过引入一个修正函数, 利用半空间中的 $ L^2 $-能量方法和庞加莱型不等式, 证明了上述稳态解在小初值扰动下的渐近稳定性.

关键词: Cattaneo 定律, 初边值问题, 稳态解, 渐近稳定性, 庞加莱型不等式

Abstract:

This paper is concerned with the time-asymptotically nonlinear stability of stationary solutions to the initial boundary value problem of hyperbolic equations with Cattaneo's law in one-dimensional half space. We construct the stationary solutions to such an initial boundary value problem and show their regularities. Moreover, by introducing a correction function, the asymptotic stability of the above stationary solutions under small initial perturbations is proved by using the $ L^2 $-energy method and Poincaré-type inequalities in half space.

Key words: Cattaneo's Law, initial-boundary value problem, stationary solutions, asymptotic stability, Poincaré-type inequalities

中图分类号:

O175.2

邓钧元, 张峻皓. 半空间中满足 Cattaneo 定律的双曲系统的稳定性[J]. 数学物理学报, 2025, 45(5): 1444-1462.

Junyuan Deng, Junhao Zhang. Stability to a Hyperbolic System with Cattaneo's Law in the Half Space[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1444-1462.

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