Asymptotic behavior of a flexible structure with Cattaneo type of thermal effect

We consider vibrations of a non uniform flexible structure modeled by a 11D viscoelastic equation with Kelvin–Voigt, coupled with an expected dissipative effect : heat conduction governed by Cattaneo’s law (second sound). We establish the well-posedness of the system and we prove the stabilization to be exponential for one set of boundary conditions, and at least polynomial for another set of boundary conditions. Two different methods are used: the energy method and another more original, using the semigroup approach and studying the Resolvent of the system.