We consider a linear system of two coupled plate equations. One of these equations is conservative and the other has dissipative properties. Alabau et al. (J. Evol. Equ. 2:127–150, 2002) studied this kind of systems. In particular, they considered a system with frictional damping and have shown that, if both plates have the same elasticity/density ratio, the associated semigroup decays polynomially with the rate \(t^{-1/2}\). When the equations of this system have different elasticity/density ratio, the semigroup decays slower. Oquendo and Raya (Z. Angew. Math. Phys. 68:77, 2017) proved that the semigroup decays polynomially with the optimal rate \(t^{-1/4}\). When one of the equations of the coupled plates has the rotational inertia and a viscous damping the situation is completely different. In this work, we prove that no matter the elasticity/density relation, the rotational inertia forces the semigroup to decay polynomially with the optimal rate \(t^{-1/5}\).

中文翻译：

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具有转动惯量的部分耗散板的最佳衰减率

我们考虑两个耦合板方程的线性系统。这些方程之一是保守的，另一个方程具有耗散特性。Alabau等。（J. Evol。Equ。2：127–150，2002）研究了这种系统。特别是，他们考虑了具有摩擦阻尼的系统，并表明，如果两个板都具有相同的弹性/密度比，则关联的半群将以\（t ^ {-1/2} \）的比率多项式衰减。当该系统的方程具有不同的弹性/密度比时，半群的衰减变慢。Oquendo和Raya（Z.Angew.Math.Phys.68：77，2017）证明半群以最优速率\（t ^ {-1/4} \）多项式衰减。当耦合板的方程之一具有转动惯量和粘性阻尼时，情况就完全不同了。在这项工作中，我们证明了，无论弹性/密度关系如何，旋转惯性都迫使半群以最佳速率\（t ^ {-1/5} \）多项式衰减。