General decay of Bresse system by modified thermoelasticity of type III

Abstract

In this paper we have studied the model for arched beams problem

$$\begin{aligned}&\rho _{1}\varphi _{tt}-k(\varphi _{x}+\psi +l\omega )_{x}-k_{0}l(\omega _{x}-l\varphi )=0,\\&\rho _{2}\psi _{tt}-b\psi _{xx}+k(\varphi _{x}+\psi +l\omega )+\gamma \theta _{tx}=0,\\&\rho _{1}\omega _{tt}-k_{0}(\omega _{x}-l\varphi )_{x}+kl(\varphi _{x} +\psi +l\omega )=0,\\&\rho _{3}\theta _{tt}-\kappa \theta _{xx}+\beta (g*\theta _{xx})+\gamma \psi _{tx}=0, \end{aligned}$$

which reduces to the classical Timoshenko system when the arch curvature $l=0$, the asymptotic stability of one-dimensional Timoshenko system by thermoelasticity of type III was proved by Djebabla and Tatar in Djebabla (J. Dyn. Control Syst. 16(2):189-210, 2010). The subject of this paper is to supplement these previous results by proving that the Bresse system which is considered a generalization of the Timoshenko system, is also subjected to the same sufficient condition that controls the stability of the Timoshenko system and we have shown that (exponential / polynomial) energy stability is achieved in an (exponential and polynomial) kernel function state.