# Lower Bound and optimality for a nonlinearly damped Timoshenko system with thermoelasticity

## TL;DR

This paper establishes lower energy bounds and optimal decay rates for a nonlinear Timoshenko system with thermoelasticity and second sound, extending previous results and confirming their optimality through explicit examples.

## Contribution

It introduces new lower energy estimates and proves the optimality of decay rates for a nonlinear thermoelastic Timoshenko system, extending prior work with explicit damping examples.

## Key findings

- Strong stability of the system is established.
- Optimality of decay rates is proved using explicit damping examples.
- Lower energy bounds are derived using Alabau--Boussouira's method.

## Abstract

In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the strong stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau--Boussouira's energy comparison principle introduced in \cite{2} (see also \cite{alabau}). One of the main advantages of these results is that they allows us to prove the optimality of the asymptotic results (as $t\rightarrow \infty$) obtained in \cite{ali}. We also extend to our model the nice results achieved in \cite{alabau} for the case of nonlinearly damped Timoshenko system with thermoelasticity. The optimality of our results is also investigated through some explicit examples of the nonlinear damping term. The proof of our results relies on the approach in \cite{AB1, AB2}.