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

We consider the following Cauchy problem with a singular nonlinearity (P) with n 3 (and having a positive lower bound). We find some conditions on the initial value such that the local solutions of (P) vanish in finite time. Meanwhile, we obtain optimal conditions on for global existence and study the large time behavior of those global solutions. In particular, we prove that if ν>0 and n 3, where us is a singular equilibrium of (P) and γ>1, then (P) has a (unique) global classical solution u with u γus andu(x,t) (ν+1)1/(ν+1)(γν+1−1)1/(ν+1)t1/(ν+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.