Uniform Attractors for a Non-autonomous Semilinear Heat Equation with Memory

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. 0. 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]: d r* cottO — k0A9 — / k(t — s)A6(s) ds + g(6) = h on SI x (r, +oo), 'di (x, 9(x,t) — Oq(x), x E fl. (0.1) 8 t) = 0, x G dfl, t > t, Received June, 1998. 2000 Mathematics Subject Classification. Primary 35B40, 35K05, 45K05, 47H20.