Transmission problem for hyperbolic thermoelastic systems Luci Harue Fatori, Edson Lueders Department of Mathematics, Universidade Estadual de Londrina, 86.051-990 Londrina-PR,Brazil. Jaime E. Mu˜ noz Rivera National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitandinha 25651-070, Petr´opolis RJ, Brazil, IM, Federal University of Rio de Janeiro. Abstract In this paper we study a transmission problem in thermoelasticity. We show that the linear system is well posed and that the solution decays exponentially to zero as time goes to infinity. That is, denoting by E(t) the first order energy associated to the thermoelastic system, then there exists positive constants c and γ such that E(t) ≤ cE(0)e -γt Introduction In this paper we consider the asymptotic behaviour of one dimensional bodies which are com- posed of two different types of materials, one of them is of thermoelastic type, while the other has no thermal effect. That is, we have a material with a localized thermal effect. Since the body is composed of two different types of material the density is not necessarily a continuous function and since the stress strain relation changes from the thermoelastic part to the elastic part, the corresponding model is not a continuous one. The mathematical model which deals with the above situation is called a transmission problem. Concerning the thermal effect, in the classical linear theory of thermoelasticity the Fourier’s law is used to describe the heat conduction in the body. This theory has two principal short- comings. First, it is unable to account for memory effect which may prevail in some materials, particularly at low temperatures. Secondly, the correspondig parabolic part of the system pre- dicts an unrealistic result, that a thermal disturbance at one point of the body is instantly felt 1