Abstract In this note, we investigate stability issues for a mixture of two elastic solids. The governing equations consist of two coupled systems of elastodynamic equations with a weak damping. The damping involves the difference of velocities, so that the damping operator is degenerate. First, we show that the underlying semigroup is strongly stable under a weaker parameters constraint, thereby improving an earlier strong stability result of Dafermos. Next, we show that under extra appropriate constraints on the parameters of the coupled system, the semigroup is exponentially stable. Our proof relies on the frequency domain method, the multipliers technique, and a seemingly new vector calculus identity (Lemma B). Our exponential stability result is new in the framework of mixtures involving elastodynamic equations with weak damping. We also show that for some sets of parameters, the system is asymptotically unstable; in particular, we use a concrete example to show that strong stability fails for a certain parameters constraint.

中文翻译：

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具有简并弱阻尼机制的两种弹性固体混合物的指数稳定性

摘要 在本说明中，我们研究了两种弹性固体混合物的稳定性问题。控制方程由两个具有弱阻尼的弹性动力学方程耦合系统组成。阻尼涉及速度的差异，因此阻尼算子是退化的。首先，我们证明了底层半群在较弱的参数约束下是强稳定的，从而改进了早期的 Dafermos 强稳定性结果。接下来，我们证明在耦合系统参数的额外适当约束下，半群是指数稳定的。我们的证明依赖于频域方法、乘法器技术和一个看似新的向量微积分恒等式（引理 B）。我们的指数稳定性结果在涉及具有弱阻尼的弹性动力学方程的混合物框架中是新的。我们还表明，对于某些参数集，系统是渐近不稳定的；特别是，我们用一个具体的例子来说明对于某个参数约束强稳定性失效。