发表机构
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对具有 past history 的耦合悬索桥系统，结合Gearhart--Prüss定理等证明其线性半群指数稳定性，填补相关文献渐近光滑性证明的空白，得到自然能量空间中的紧全局吸引子。
AI 中文摘要
本文研究具有 past history 的耦合非线性悬索桥系统的长时间动力学，其中桥面与主缆通过非线性恢复力 \(k[u-v]^{+}\) 相互作用。基于文献[Mukiawa2025]建立的适定性和解半群，我们首先借助Gearhart--Prüss定理结合详细的预解估计，证明相关线性半群的指数稳定性。随后，我们将该指数稳定性与非线性项的局部Lipschitz性质、非线性项仅作用于弹性分量的事实、弹性分量的光滑性以及记忆核的指数衰减相结合，得出该解半群在Chueshov--Lasiecka意义下是拟稳定的，因此具有渐近光滑性。结合文献[Mukiawa2025]中提及的耗散性，这在自然能量空间中得到了紧全局吸引子。特别地，我们的方法可视为填补了文献[Mukiawa2025]中渐近光滑性证明的一处空白，该文献中一个非衰减的初差项被无依据地吸收到任意小的常数中。
英文摘要
This paper is concerned with the long-time dynamics of a coupled nonlinear suspension bridge system with past history, in which the deck and the main cable interact through the nonlinear restoring force \(k[u-v]^{+}\). Building on the well-posedness and the solution semigroup established in \cite{Mukiawa2025}, we first show the exponential stability of the associated linear semigroup by means of the Gearhart--Prüss theorem together with a detailed resolvent estimate. We then combine this exponential stability with the local Lipschitz property of the nonlinearity, the fact that the nonlinearity acts only on the elastic component, the smoothing of the elastic component, and the exponential decay of the memory kernels to conclude that the solution semigroup is quasi-stable in the sense of Chueshov--Lasiecka, and therefore asymptotically smooth. Together with the dissipativity recalled from \cite{Mukiawa2025}, this leads to a compact global attractor in the natural energy space. In particular, our approach may be viewed as closing a gap in the proof of asymptotic smoothness in \cite{Mukiawa2025}, where a non-decaying initial-difference term was absorbed into an arbitrarily small constant without justification.
