On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity u(t) = Delta u-u(-v), x is an element of R-n, t > 0, v > 0, u|(t=0) = phi is an element of C-LB(R-n) with n >= 3 (and phi having a positive lower bound). We find some conditions on the initial value phi such that the local solutions of (P) vanish in finite time. Meanwhile, we obtain optimal conditions on phi for global existence and study the large time behavior of those global solutions. In particular, we prove that if v > 0 and n > 3, phi(x) >= gamma u(s)(x) = gamma[2/v+1 (n-2 + 2/v+1)](-1/(v+1)) |x|(2/(v+1)), where u(s) is a singular equilibrium of (P) and gamma > 1, then (P) has a (unique) global classical solution u with u > gamma u(s) and u(x,t) >= (v+1)(1/(v+1))(gamma(v+1)-1)(1/(v+1))t(1/(v+1)) On the other hand, the structure of positive radial solutions of the steady-state of (P) is studied and some interesting properties of the positive solutions are obtained. Moreover, the stability and weakly asymptotic stability of the positive radial solutions of the steady-state of (P) are also discussed. (c) 2007 Elsevier Inc. All rights reserved.