Asymptotic behaviour of solutions of some semilinear parabolic problems

We consider the Cauchy problem: u(t) - Delta u + u(p) = 0 for x is an element of IRN, t > 0, (0.1) u(x, 0) = u(o)(x) for x is an element of IRN. (0.2) Here p > 1, N greater than or equal to 1 and u(o)(x) is a continuous, nonnegative and bounded function such that: u(o)(x) similar to A\x\(-alpha), as \x\ --> infinity, (0.3) for some A > 0 and alpha > 0. In this paper we discuss the asymptotic behaviour of solutions to (0.1)-(0.3) in terms of the various values of the parameters p, N, alpha and A. A common pattern that emerges from our analysis is the existence of an external zone where u(x, t) similar to u(o)(x) and one (or several) internal regions, where the influence of diffusion and absorption is most strongly felt. We present a complete classification of the size of these regions, as well as that of the stabilization profiles that unfold therein, in terms of the aforementioned parameters, (C) Elsevier, Paris.