A Nonlinear Parabolic Problem in an Exterior Domain

We consider the problem:[formula]Herep>1,N 2, 0 is a finite union of disjoint open sets, anduo(x) is a continuous, nonnegative, and bounded function such thatuo(x) A x−α as x→∞, ((0.4))for someA>0 andα>0. In this paper we discuss the asymptotic behaviour of solutions to (0.1)–(0.4) in terms of the various values of the parametersp,A,N,Ω, andα. A common pattern that emerges from our analysis is the existence of an external zone whereu(x, t) uo(x), and one or several internal regions, where the influence of the setΩ, as well as that of diffusion and absorption, is most strongly felt. We present a complete classification of the stabilization profiles in terms of these parameters.