MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2002; 25:955–980 (DOI: 10.1002/mma.323) MOS subject classication: 35 B 40; 35 L 05; 35 L 70; 73 B 30; 73 K 03 Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity Alfredo Marzocchi 1; † , Jaime E. Mu˜ noz Rivera 2; ∗; ‡ and Maria Grazia Naso 3; § 1 Dipartimento di Matematica e Fisica; Universit a Cattolica del Sacro Cuore; Via Musei 41; I-25121 Brescia; Italia 2 National Laboratory for Scientic Computation; Rua Getulio Vargas 333; Quitadinha-Petr opolis 25651-070; Rio de Janeiro; RJ; Brazil 3 Dipartimento di Matematica; Universit a degli Studi di Brescia; Via Valotti 9; I-25133 Brescia; Italia Communicated by Y. Shibata SUMMARY We show that the solution of a semilinear transmission problem between an elastic and a thermoelastic material, decays exponentially to zero. That is, denoting by E(t ) the sum of the rst, second and third order energy associated with the system, we show that there exist positive constants C and satisfying E(t )6CE(0)e -t Moreover, the existence of absorbing sets is achieved in the non-homogeneous case. Copyright ? 2002 John Wiley & Sons, Ltd. KEY WORDS: transmission problem; thermoelasticity; exponential decay; simultaneous stabilization; asymptotic behaviour; absorbing set 1. INTRODUCTION The wave equation without any dissipation is a conservative system, that is, its total energy is constant for any time. Several authors introduced dierent types of dissipative mechanisms to stabilize the oscillations. For example, the frictional damping u t (see Reference [1]) where the dissipation works in the whole domain, or frictional boundary conditions (see References [2; 3]) where the dissipation is working in a part of the boundary or also localized frictional damping, that is when the frictional damping is of the form (x)u t where vanishes in ∗ Correspondence to: J. E. Mu˜ noz Rivera, National Laboratory for Scientic Computation, Rua Getulio Vargas 333, Quitadinha-Petropolis 25651-070, Rio de Janeiro, RJ, Brazil † E-mail: a.marzocchi@dmf.bs.unicatt.it ‡ E-mail: rivera@lncc.br § E-mail: naso@bsing.ing.unibs.it Contract=grant=sponsor: CNPq-BRASIL; contract=grant number: 305406=88-4 Contract=grant=sponsor: Italian MURST Copyright ? 2002 John Wiley & Sons, Ltd. Received 19 September 2001