Pergamon Nonlinear Analysis, 7kmy, Methods & Application, Vol. 31, No. l/2, pp. 149-162, 1998 0 1998 Elsevia science zyxwvutsrqpon Ltd AU rights reserved. Printed in Great Britain 0362-%x/98 s19.oo+o.oo PII: SO362-546X(%)OO3OOg zyxwvutsrqponmlkjihgfedcbaZYXWV EXISTENCE AND EXPONENTIAL DECAY IN NONLINEAR THERMOELASTICITY JAIME E. MUROZ RIVERAQ and RIOCO KAMEI BARRET TNational Laboratory for Scientific Computation, Department of Research and Development, Rua Lauro Milller 455, Botafogo Cep. 22290, Rio de Janeiro, RJ, Brasil and IM. Federal University of Rio de Janeiro; and ODepartment of Mathematics of Fluminense University, Rio de Janeiro, Brasil (Received I1 October 1995; received for publication 27 November 1996) Key words andphrases: Thermoelasticity, thermoelastic rod, exponential decay, global solution, initial boundary value problems. 1. INTRODUCTION It is well known that in the absence of dissipation, smooth solution of nonlinear elastic materials develop singularities in finite time, while for thermoelastic materials the conduction of the heat equation provides dissipation that competes with the destabilizing effect of nonlinearity in the elastic response. The level of subtlety of this dissipation depends on the boundary condition that the displacement and the thermal difference are supporting. Slemrod [I] showed the global existence of smooth solution for small data when the boundary is either traction-free and at a constant temperature or rigidly clamped and thermally insulated. A similar result was obtained by Zheng [2]. These boundary conditions get a simpler damping mechanism because they imply additional boundary conditions for u and the thermal difference 8, that is, if an end is clamped then the displacement u and the thermal difference 0 satisfy u,, = 0 and 0,, = 0 there respectively. So we can make additional partial integrations which led to the desire a priori L2-estimate. In case of Dirichlet boundary condition for which the boundary is rigidly clamped and held at a constant temperature we lost the value of u,, in that point and instead of it we get u, + of& = 0. So this case leads ill behaved boundary terms and it is not possible to apply directly the multiplicative techniques to secure global estimate. Recently Racke and Shibata [3] proved Global existence of a smooth solution for these boundary conditions. To do this the authors showed the algebraic decay of the energy for the linear equation by studying the spectral properties of the stationary linearized problem. The rate of decay depends on higher regularity of the initial data and therefore the global existence result depends on the initial data to be small in Hm(O, L) with zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ m large. One of the authors of this paper proved in [4] (see also the work of Kim [5]) that the solution of the linearized thermoelastic system decays exponentially as time goes to infinity. This fact allows us to get simpler existence result for the corresponding nonlinear equation as was shown in [6] for small data (uO, ul) in H3(0, L) x H2(0, L). The system in question is written as follows Utt - [W,, a, = 09 in IO, L[ X IO, a[ zyxwvutsrqponmlkjihgfedcbaZYXWV (e + t dw o.4 , e)i, - ~ 2(4,e,, a, = 0, in IO, L[ x IO, a[ (1.1) (1.2) $ Partially supported by a grant of CNPq-BRASIL. 1 Partially supported by a grant of CNPq-BRASIL. 149