This paper is devoted to existence, uniqueness and asymptotic behavior, as time tends to infinity, of the solutions of an integro-partial differential equation arising from the theory of heat conduction with memory, in presence of a temperature-dependent heat supply. A linearized heat flux law involving positive instantaneous conductivity is matched with the energy balance, to generate an autonomous semilinear system subject to initial history and Dirichlet boundary conditions. Existence and uniqueness of solution is provided. Moreover, under proper assumptions on the heat flux memory kernel, the existence of absorbing sets in suitable function spaces is achieved.

Cauchy fluxes induced by locally summable tensor fields with divergence measure are characterized. The equivalence between integral formulations involving subsets of finite perimeter and much more restricted classes of subsets is proved.

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SUMMARYWe 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 ÿrst, second and third order energy associated with the system, we show that there exist positive constants C and satisfyingMoreover, the existence of absorbing sets is achieved in the non-homogeneous case.

Abstract.In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of a non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of uniform absorbing sets and of a global uniform attractor is achieved. In the case of quasiperiodic dependence of time of the external heat supply, the above attractor is shown to have finite Hausdorff dimension. Introduction.Let S] C M3 be a fixed bounded domain occupied by a rigid, isotropic, homogeneous heat conductor with linear memory. We consider the following integro-partial differential equation, which is derived in the framework of the wellestablished theory of heat flow with memory due to Coleman and Gurtin [8]:Received June, 1998.