The model under consideration in this paper describes a vibrating structure of an interfacial slip and consists of three coupled hyperbolic equations in one-dimensional bounded interval under mixed homogeneous Dirichlet–Neumann boundary conditions. The first two equations are related to Timoshenko-type systems and the third one is subject to the dynamics of the slip. The main problem we discuss here is stabilizing the system by a viscoelastic damping generated by an infinite memory and acting only on one equation. First, we prove the existence, uniqueness and regularity of solutions using the semigroup theory. After that, we combine the energy method and the frequency domain approach to show that the infinite memory is capable alone to guarantee the strong and polynomial stability of the model, that is bringing it back to its equilibrium state with a decay rate of type |$t^{-d}$|⁠, where |$d$| is a positive constant depending on the regularity of initial data. Moreover, we prove that, when the infinite memory is effective on the first equation, the model is not exponentially stable independently of the values of the parameters. However, when the infinite memory is effective on the second or the third equation, we prove that the exponential stability is equivalent to the equality between the three speeds of wave propagations. An extension of our results to the frictional damping case is also given. Our results improve and extend some existing results in the literature subject to other types of controls.

中文翻译：

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具有界面滑动和无限记忆的叠层Timoshenko梁的适定性和稳定性结果

本文中考虑的模型描述了界面滑移的振动结构，并且在混合齐次Dirichlet-Neumann边界条件下，由一维有界区间中的三个耦合双曲方程组成。前两个方程与Timoshenko型系统有关，第三个方程受滑移动力学的影响。我们在这里讨论的主要问题是通过无限记忆产生的粘弹性阻尼来稳定系统，并且仅作用于一个方程。首先，我们使用半群理论证明了解的存在性，唯一性和正则性。之后，我们将能量方法和频域方法结合起来，证明了无限记忆本身就能够保证模型的强大和多项式稳定性，| $ t ^ {-d} $ |⁠，其中| $ d $ | 是一个正常数，取决于初始数据的规律性。此外，我们证明，当无限记忆对第一个方程有效时，该模型就不会独立于参数值而呈指数稳定。但是，当无限记忆对第二个或第三个方程有效时，我们证明了指数稳定性等于三种波传播速度之间的相等性。我们的结果也扩展到了摩擦阻尼情况。我们的结果改进并扩展了文献中现有的一些结果，并使其适用于其他类型的对照。