QUARTERLY OF APPLIED MATHEMATICS VOLUME LII, NUMBER 4 DECEMBER 1994, PAGES 629-648 ASYMPTOTIC BEHAVIOUR IN LINEAR VISCOELASTICITY By JAIME E. MUNOZ RIVERA National Laboratory for Scientific Computation (LNCC-CNPQ), and IMUFRJ, Rio de Janeiro, Brasil Abstract. We study the asymptotic behaviour of the solution of the viscoelastic equation, and we prove for a bounded domain that the energy associated to this system approaches zero exponentially as time goes to infinity. Moreover, for the whole space R" we will prove that the displacement vector field can be decomposed into two parts, solenoidal and irrotational, whose corresponding energies decay to zero uniformly as time goes to infinity with rates that depend on the regularity of the initial data. 1. Introduction. In this paper we will discuss the asymptotic behaviour of the total energy associated to the model whose solution describes the evolution of elastic waves in a homogeneous, isotropic, and viscoelastic medium. The corresponding mathematical scheme involves a system of integro-differential equations, with the integrals reflecting the memory effect.. Through this work we will suppose that the specific relaxation function E is of the form E(/, r) = e(t - t) ; therefore, the constitutive relation for viscoelasticity assumes the expression a = e(0)s + [ g(t - t)e(t) ch: Jo where a and e stand for the stress and strain tensors, respectively, and g is the derivative of the specific relaxation function e (see [1]). Denoting the mass density by p, the dynamic equation is written as p\ilt = V - a where by uO, t) = (ux{x,t),..., un(x, t)) =y(x, t) -X we denote the displacement vector field associated with the motion and by y the positive vector at time t of the particle located at the point x. Expressing a in terms of the displacement vector field we finally obtain putt - pAu - (A + /i)V{divu} + pg * Au + (A + p)h * V{divu} = 0 in Q (1.1) subject to the initial conditions u(x, 0) = u0(x), ut(x, 0) = u,(x) (1.2) Received January 16, 1992. 1991Mathematics SubjectClassification. Primary 35B40,35L15, 73F15. Key words and phrases. Linear viscoelasticity, energy decay rates, asymptotic behaviour. ©1994 Brown University 629