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

We consider the following Cauchy problem with a singular nonlinearity(P)(P)ut=Δu−u−ν,x∈Rn,t>0,ν>0,u|t=0=ϕ∈CLB(Rn) with n⩾3n⩾3 (and ϕ having a positive lower bound). We find some conditions on the initial value ϕ   such that the local solutions of (P)(P) vanish in finite time. Meanwhile, we obtain optimal conditions on ϕ   for global existence and study the large time behavior of those global solutions. In particular, we prove that if ν>0ν>0 and n⩾3n⩾3,ϕ(x)⩾γus(x)=γ[2ν+1(n−2+2ν+1)]−1/(ν+1)|x|2/(ν+1), where usus is a singular equilibrium of (P)(P) and γ>1γ>1, then (P)(P) has a (unique) global classical solution u   with u⩾γusu⩾γus andu(x,t)⩾(ν+1)1/(ν+1)(γν+1−1)1/(ν+1)t1/(ν+1).u(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)(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)(P) are also discussed.