Study on the stability of thermoelastic Bresse and Timoshenko type systems with Gurtin-Pipkin's law via the vertical displacements

TL;DR

This paper investigates the stability of thermoelastic Bresse and Timoshenko systems with Gurtin-Pipkin heat conduction, establishing conditions for exponential and polynomial stability depending on coupling order and system parameters.

Contribution

It introduces new stability criteria for coupled thermoelastic systems with Gurtin-Pipkin law, including conditions for exponential and polynomial decay based on coupling order.

Findings

Polynomial stability depends on initial data smoothness.

Exponential stability occurs under specific parameter conditions for order one coupling.

Both couplings ensure polynomial stability regardless of parameters.

Abstract

The objective of this paper is to study the stability of a linear one-dimensional thermoelastic Bresse system in a bounded domain, where the coupling is given through the first component of the Bresse model with the heat conduction of Gurtin-Pipkin type. Two kinds of coupling are considered; the first coupling is of order one with respect to space variable, and the second one is of order zero. We state the well-posedness and show the polynomial stability of the systems, where the decay rates depend on the smoothness of initial data. Moreover, in case of coupling of order one, we prove the equivalence between the exponential stability and some new conditions on the parameters of the system. However, when the coupling is of order zero, we prove the non-exponential stability independently of the parameters of the system. Applications to the corresponding particular Timoshenko models are also given, where we prove that both couplings lead to the exponential stability if and only if some conditions on the parameters of the systems are satisfied, and both couplings guarantee the polynomial stability independently of the parameters of the systems. The proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.