We consider the Timoshenko beam with localized Kelvin–Voigt dissipation distributed over two components: one of them with constitutive law of the type \(C^1\), and the other with discontinuous law. The third component is simply elastic, where the viscosity is not effective. Our main result is that the decay depends on the position of the components. We will show that the system is exponentially stable if and only if the component with discontinuous constitutive law is not in the center of the beam. When the discontinuous component is in the middle, the solution decays polynomially.

中文翻译：

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具有局部开尔文-沃格特耗散的Timoshenko系统的稳定性

我们考虑具有局部Kelvin-Voigt耗散的Timoshenko光束分布在两个分量上：一个分量具有本构律\（C ^ 1 \），另一个分量具有不连续律。第三组分是简单的弹性，其中粘度无效。我们的主要结果是衰减取决于组件的位置。我们将证明，当且仅当具有不连续本构律的分量不在光束中心时，系统才是指数稳定的。当不连续分量位于中间时，解多项式衰减。