Well-posedness and exponential stability of a thermoelastic system with internal delay

TL;DR

This paper establishes the well-posedness and exponential stability of a thermoelastic system with internal delay by adding Kelvin-Voigt damping and employing semigroup theory and Lyapunov functionals.

Contribution

It introduces a damping mechanism to restore well-posedness and stability in delayed thermoelastic systems, providing rigorous proofs using semigroup theory and Lyapunov methods.

Findings

The system is well-posed under the proposed damping.

Exponential stability is achieved with appropriate assumptions.

The approach ensures stability despite internal delays.

Abstract

The presence of a delay in a thermoelastic system destroys the well-posedness and the stabilizing effect of the heat conduction [17]. To avoid this problem we add to the system, at the delayed equation, a Kelvin-Voigt damping. At first, we prove the well-posedness of the system by the semigroup theory. Next, under appropriate assumptions, we prove the exponential stability of the system by introducing a suitable Lyapunov functional.