IMA Journal of Applied Mathematics (2003) 68, 23–46 Transmission problem in thermoelasticity with symmetry ALFREDO MARZOCCHI† Dipartimento di Matematica, Universit` a degli Studi di Brescia, Via Valotti 9, I - 25133 Brescia, Italia JAIME E. MU ˜ NOZ RIVERA‡ National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitadinha-Petr´ opolis 25651-070, Rio de Janeiro, RJ, Brazil AND MARIA GRAZIA NASO§ Dipartimento di Matematica, Universit` a degli Studi di Brescia, Via Valotti 9, I - 25133 Brescia, Italia [Received on 26 June 2001; revised on 5 February 2002] In this paper we show the existence, uniqueness and regularity of the solutions to the thermoelastic transmission problem. Moreover, when the solutions are symmetrical we show that the energy decays exponentially as time goes to infinity, no matter how small is the size of the thermoelastic part. Keywords: transmission problem; n-d-thermoelasticity; radial symmetry; exponential decay; simultaneous stabilization. 1. Introduction We consider a model describing oscillations of a body which is composed of two different materials, one of them is a thermoelastic material while the other is indifferent to thermal effects. We assume that the density as well as the elastic coefficients are different constants in each component. Therefore, we have a transmission problem where the damping effect given by the difference of temperature is effective only in a part of the material. Concerning thermoelastic systems we have the work of Dafermos (1968), where it is proved that the solution of the n-dimensional anisotropic thermoelastic material is asymptotically stable as time tends to infinity, and the decay of the displacement is not to zero but to an undamped oscillation. On the other hand the difference of temperature as well as the divergence of the displacement always tend to zero as time tends to infinity. For one-dimensional models it was proved that the dissipation given by difference of temperature is strong enough to produce uniform rate of decay of the solution; see (Kim, 1992; Marzocchi et al., 2002; Mu˜ noz Rivera, 1992). The situation is different in two and three dimensions because the displacement vector field has two or three degrees of freedom † Email: amarzocc@ing.unibs.it ‡ Email: rivera@lncc.br § Email: naso@ing.unibs.it c  The Institute of Mathematics and its Applications 2003