In this paper, we investigate the decay properties of the Bresse–Cattaneo system in the whole space. We show that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and we prove that the decay rate of the solution is very slow. In fact, we show that the [Formula: see text]-norm of the solution decays with the rate of [Formula: see text]. The behavior of solutions depends on a certain number [Formula: see text] (which is the same stability number for the Timoshenko–Cattaneo system [Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. Differential Equations 255(4) (2013) 611–632; The stability number of the Timoshenko system with second sound, J. Differential Equations 253(9) (2012) 2715–2733]) which is a function of the parameters of the system. In addition, we show that we obtain the same decay rate as the one of the solution for the Bresse–Fourier model [The Bresse system in thermoelasticity, to appear in Math. Methods Appl. Sci.].
The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement
