A NONLINEAR PARABOLIC PROBLEM IN AN EXTERIOR DOMAIN

We consider the problem: u(t) - Delta u + u(p) = 0 for x epsilon R-n\O mega, t > 0, (0.1) u(x, t) = 0 for x epsilon partial derivative Omega, t > 0, (0.2) u(x, 0) = u(o)(x) for x is an element of R-n\<(Omega)ove r bar>. (0.3) Here p > 1, N greater than or equal to 2 Omega is a fini te union of disjoint open sets, and u(o)(x) is a continuous, nonnegati ve, and bounded function such that u(o)(x) similar to A\x\(-alpha) as \x\ --> infinity, (0.4) for some A > 0 and alpha > 0. In this paper we discuss the asymptotic behaviour of solutions to (0.1)-(0.4) in terms of the various values of the parameters p, A, N, Omega, and alpha. A common pattern that emerges from our analysis is the existence of an e xternal zone where u(x, t) similar to u(o)(x), and one or several inte rnal regions, where the influence of the set Omega, as well as that of diffusion and adsorption, is most strongly felt. We present a complet e classification of the stabilization profiles in terms of these param eters. (C) 1998 Academic Press.