Asymptotic behavior of an inhomogeneous flexible structure with Cattaneo type of thermal effect

TL;DR

This paper analyzes the long-term behavior of vibrations in an inhomogeneous flexible structure with thermal effects modeled by Cattaneo's law, establishing conditions for exponential and polynomial stabilization.

Contribution

It introduces a comprehensive analysis of a 1D viscoelastic system with Cattaneo heat conduction, proving well-posedness and stabilization rates using novel and classical methods.

Findings

Exponential stabilization under certain boundary conditions

Polynomial decay under alternative boundary conditions

Use of semigroup and resolvent methods for analysis

Abstract

We consider vibrations of an inhomogeneous flexible structure modeled by a $1$D 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.