CAO系统的长期行为：退化平衡态的最优多项式$H^1$收敛
Long-term behaviour of the CAO-system: Optimal polynomial $H^1$-convergence to degenerate equilibria
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- Technical University of Darmstadt（达姆施塔特工业大学）
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中文总结 AI 辅助
本文证明CAO系统在非微扰机制下全局强解以最优多项式速率收敛到退化平衡态，揭示了由非线性耦合引发的慢流形导致的代数减速，首次为有界域非梯度非局部耗散系统建立了全局多项式衰减结果。
中文摘要 AI 辅助
本文对CAO系统的长期动力学进行了确定性分析。我们在非微扰机制下，建立了全局强解在完整$H^1$拓扑中收敛到常数平衡态的结论。值得注意的是，我们证明了CAO系统表现出多项式衰减速率，并确定了最优速率。这一现象与有界域上一致抛物系统通常预期的指数收敛形成鲜明对比。这种代数减速被证明是一种纯粹的非线性效应，由非线性耦合条件引发的慢流形（中心流形）所驱动。据我们所知，这为有界域上的非梯度、非局部耗散系统提供了全局多项式衰减结果的第一个实例。
英文摘要
This article is concerned with a definitive analysis of its long-term dynamics of the CAO-system. We establish the convergence of global strong solutions to constant equilibria in the full $H^1$-topology in the non-perturbative regime. Remarkably, we demonstrate that the CAO-system exhibits a polynomial rate of decay and we determine the optimal rate. This phenomenon stands in stark contrast to the exponential convergence typically expected for uniformly parabolic systems on bounded domains. This algebraic slowing is shown to be a purely nonlinear effect, driven by the emergence of a slow manifold (center manifold) induced by nonlinear coupling conditions. To the best of our knowledge, this provides the first instance of a global polynomial decay result for a non-gradient, non-local dissipative system on a bounded domain.
