Energy decay for damped Timoshenko-Ehrenfest systems under influence of the second spectrum of frequency

Abstract

In this paper we consider the Timoshenko-Ehrenfest beam models (2019, {\it Who developed the so-called Timoshenko beam theory? Mathematics and Mechanics of Solids)} and we stablished exponential decay results based on influence of the second spectrum of frequency and it’s damaging consequences for wave propagation speeds. For the classical case, having two wave speeds governing the stress waves and shear waves, we prove that the corresponding semigroup associated to the system decays exponentially under equal wave speeds assumption. On the contrary, there is a lack of exponential stability and we prove it’s optimality based on Borichev-Tomilov approach. For the truncated case, we assure the well-posedness by using the Faedo-Galerkin method and we prove that the total energy decays exponentially regardless any relationship between coefficients of the system by using the energy method.