Existence and general stabilization of the Timoshenko system with a thermo-viscoelastic damping and a delay term in the internal feedback

TL;DR

This paper investigates the existence and stability of solutions for a Timoshenko system incorporating thermo-viscoelastic damping and a delay term, establishing conditions for global existence and various energy decay rates.

Contribution

It introduces new conditions linking delay and damping weights, proving global solutions and general energy decay, including exponential and polynomial rates.

Findings

Proved global existence of solutions using Faedo-Galerkin approximations.

Established general energy decay results under specific weight constraints.

Identified conditions for exponential and polynomial decay types.

Abstract

In this paper, we consider a Timoshenko system with a thermo-viscoelastic damping and a delay term in the internal feedback together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and {\mu}1, {\mu}2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the the weight of the friction damping term, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the weights of the two feedbacks and the coefficients, we establish the general energy decay result from which the exponential and polynomial types of decay are only special cases.