a b s t r a c tWe consider vibrations modeled by the standard linear solid model of viscoelasticity which are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. We show that the exponential stability under the Fourier law of heat conduction holds. In order to obtain the asymptotic behaviour we use multiplier techniques.

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We consider the one-dimensional model of a thermoelastic mixture with second sound. We give a complete characterization of the asymptotic properties of the model in terms of the coefficients of the model. We establish the necessary and sufficient conditions for the model to be exponential or polynomial stable and also the conditions for which there exist initial data for where the energy is conserved.

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