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 L 2 -norm of the solution decays with the rate of (1 + t) −1/12 . The behavior of solutions depends on a certain number α (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.].

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In this paper, we are concerned with asymptotic stability of a class of Bresse-type system with three boundary dissipations. The beam has a rigid body attached to its free end. We show that exponential stabilization can be achieved by applying force and moment feedback boundary controls on the shear, longitudinal, and transverse displacement velocities at the point of contact between the mass and the beam. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method.