In this study, we consider a one-dimensional Timoshenko system with two damping terms in the context of the second frequency spectrum. One damping is viscoelastic with infinite memory, while the other is a non-linear frictional damping of variable exponent type. These damping terms are simultaneously and complementary acting on the shear force in the domain. We establish, for the first time to the best of our knowledge, explicit and general energy decay rates for this system with infinite memory. We use Sobolev embedding and the multiplier approach to get our decay results. These results generalize and improve some earlier related results in the literature.

中文翻译：

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基于无限记忆和变指数型非线性阻尼的第二频率谱的Timoshenko系统新的衰减结果

在本研究中，我们考虑在第二频谱背景下具有两个阻尼项的一维 Timoshenko 系统。一种阻尼是具有无限记忆的粘弹性阻尼，而另一种阻尼是变指数型非线性摩擦阻尼。这些阻尼项同时且互补地作用于域中的剪切力。据我们所知，我们首次为这个具有无限记忆的系统建立了明确和一般的能量衰减率。我们使用 Sobolev 嵌入和乘数方法来获得衰减结果。这些结果概括并改进了文献中一些早期的相关结果。