Well-posedness and stability results for thermoelastic Bresse and Timoshenko type systems with Gurtin–Pipkin’s law through the vertical displacements

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 and strong stability of the systems for regular and weak solutions, respectively, where the polynomial decay rates depend on the smoothness of the 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 of the well-posedness result is based on the semigroups theory, whereas a combination of the energy method and the frequency domain approach is used for the proof of the stability results. © 2023, The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada.