发表机构
University of Wuppertal, School of Mathematics and Natural Sciences（伍珀塔尔大学数学与自然科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究含空间依赖热系数的耦合波动-热系统，采用闭包关系框架与C₀半群等方法，建立其强渐近稳定性，将非均匀衰减率扩展至空间依赖热系数情形，得到对数与多项式衰减结果。
AI 中文摘要
我们研究耦合波动-热系统的长期行为，该系统由波动方程与热方程构成，定义在两个相邻的Lipschitz域上，通过公共界面耦合，且热方程可包含空间依赖的系数。首先，我们建立了与域和系数无关的强渐近稳定性，为此采用闭包关系框架，将耦合系统的谱分析简化为波动方程的谱分析。其次，我们分析耦合系统经典解的非均匀衰减率，运用C₀半群理论的方法及状态空间的非正交分解，将问题简化为与热域和热系数无关、仅单调依赖于波动域和界面的残差估计。据此，我们将已知的非均匀衰减率扩展至空间依赖热系数的情形，在无额外假设下得到对数衰减，在一维情形或满足几何控制条件时得到多项式衰减。
英文摘要
We study of the long-term behavior of a coupled wave-heat system. The system consists of a wave equation and a heat equation on two adjacent Lipschitz domains coupled by a common interface, with the heat equation being allowed to incorporate spatially dependent coefficients. We first establish strong asymptotic stability independent of the domains and coefficients. To this end, we employ the framework of closure relations, which reduces the spectral analysis of the coupled system to that of a wave equation. Secondly, we analyze non-uniform decay rates for classical solutions to the coupled system. Using a non-orthogonal decomposition of the state space, we reduce the problem to a resolvent-type estimate which is independent of the heat domain and heat coefficients. With this, we extend the known non-uniform decay rates to spatially dependent heat coefficients, yielding logarithmic decay under no assumptions and polynomial decay under the Geometric Control Condition.
Comments38 pages
