We consider a multilayer hyperbolic-parabolic PDE system which constitutes a coupling of 3D thermal – 2D elastic – 3D elastic dynamics, in which the boundary interface coupling between 3D fluid and 3D structure is realized via a 2D elastic equation. Our main result here is one of strong decay for the given multilayered – heat system. That is, the solution to this composite PDE system is stabilized asymptotically to the zero state. Our proof of strong stability takes place in the ‘frequency domain’ and ultimately appeals to the pointwise resolvent condition introduced by Tomilov [23]. This very useful result, however, requires that the semigroup associated with our multilayered FSI system be completely non-unitary (c.n.u). Accordingly, we firstly establish that the semigroup $\{e^{\mathcal{A}t}{\}}_{t\ge 0}$ is indeed c.n.u., in part by invoking relatively recent results of global uniqueness for overdetermined Lamé systems on non-smooth domains. Although the entire proof also requires higher regularity results for some trace terms, this ‘resolvent criterion approach’ allows us to establish a ‘classially soft’ proof of strong decay. In particular, it avoids the sort of technical PDE multipliers invoked in [Avalos G, Geredeli PG, Muha B. Wellposedness, spectral analysis and asymptotic stability of a multilayered heat-wavewave system. J Differ Equ. 2020;269:7129–7156].

我们考虑一个多层双曲抛物 PDE 系统，它构成了 3D 热 - 2D 弹性 - 3D 弹性动力学的耦合，其中 3D 流体和 3D 结构之间的边界界面耦合通过 2D 弹性方程实现。我们这里的主要结果是给定多层热系统的强烈衰减之一。也就是说，这个复合 PDE 系统的解逐渐稳定到零状态。我们对强稳定性的证明发生在“频域”中，并最终诉诸于 Tomilov [23] 引入的逐点分解条件。然而，这个非常有用的结果要求与我们的多层 FSI 系统相关的半群是完全非单一的 (cnu)。因此，我们首先建立半群$\{e^{\mathcal{一种}吨}{\}}_{吨\ge 0}$确实是 cnu，部分是通过在非光滑域上为超定 Lamé 系统调用相对最近的全局唯一性结果。尽管整个证明还要求某些迹项具有更高的规律性结果，但这种“解决标准方法”使我们能够建立强衰减的“经典软”证明。特别是，它避免了在 [Avalos G, Geredeli PG, Muha B. Wellposedness、光谱分析和多层热波系统的渐近稳定性中调用的那种技术 PDE 乘数。J 差分方程。2020；269:7129–7156]。