In this paper, we consider a new Timoshenko beam model with thermal and mass diffusion effects where heat and mass diffusion flux are governed by Cattaneo’s law. Necessary and sufficient conditions for exponential stability are provided in terms of the physical parameters of the model. Firstly, by the \(C_0\)-semigroup theory, we prove the well-posedness of the considered problem. Then, we prove the lack of exponential stability of the system when one of these conditions is not valid. Finally, we prove in this case that the semigroup decays to zero polynomially as \(1/\sqrt{t}\). Moreover, we show that the rate is optimal.

中文翻译：

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具有第二声音的热扩散 Timoshenko 系统的稳定性条件

在本文中，我们考虑了具有热和质量扩散效应的新 Timoshenko 梁模型，其中热和质量扩散通量受 Cattaneo 定律支配。在模型的物理参数方面提供了指数稳定性的充分必要条件。首先，通过\(C_0\)-半群理论，我们证明了所考虑问题的适定性。然后，当这些条件之一不成立时，我们证明系统缺乏指数稳定性。最后，我们在这种情况下证明半群以多项式衰减为零为\(1/\sqrt{t}\)。此外，我们表明该速率是最佳的。