The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement

In this paper, we consider the thermoelastic Bresse system in one-dimensional bounded interval under mixed homogeneous Dirichlet–Neumann boundary conditions and two different kinds of dissipation working only on the longitudinal displacement and given by heat conduction of types I and III. We prove that the exponential stability of the two systems is equivalent to the equality of the three speeds of the wave propagations. Moreover, when at least two speeds of the wave propagations are different, we show the polynomial stability for each system with a decay rate depending on the smoothness of the initial data. The results of this paper complete the ones of Afilal et al. [On the uniform stability for a linear thermoelastic Bresse system with second sound (submitted), 2018], where the dissipation is given by a linear frictional damping or by the heat conduction of second sound. The proof of our results is based on the semigroup theory and a combination of the energy method and the frequency domain approach.[Figure not available: see fulltext.].