  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.
