We consider the Bresse system with temperature and we show that there exist exponential stability if and only if the wave propagation is equal. We show that, in general, the system is not exponentially stable but that there exists polynomial stability with rates that depend on the wave propagations and the regularity of the initial data. Moreover, we introduce a necessary condition to dissipative semigroup decay polynomially. This result allows us to show some optimality to the polynomial rate of decay.

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This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.

Abstract.In this paper we study the hyperbolic thermoelastic system, which is obtained when, instead of Fourier's law for the heat flux relation, we follow the linearized model proposed by Gurtin and Pipkin concerning the memory theory of heat conduction. In this case the thermoelastic model is fully hyperbolic. We show that the linear system is well posed and that the solution decays exponentially to zero as time goes to infinity.

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