On the dynamics of a semilinear heat equation with strong absorption

Abstract

This paper deals with the following initial value problem: (1) ∂u/∂t−Δu+u p=0 on RN, N≥1, 0<p<1, u(x,0)=u0(x) in RN and u0 is nonnegative, continuous and such that (2) u0(x)≤Aeα|x|2 for some A>0 and α≥0. For any x∈RN, the extinction time of x is defined as follows: TE(x)=sup{t>0,u(x,t)>0}, u(x,t) being the solution of (1). The following results are established. (i) If u0(x)≤A(|x|)+B|x−a|2/(1−p) for some a∈RN where A(|x|)=o(|x| 2/(1−p)) as |x|→+∞ and
0≤B<[(1−p)2/2(N(1−p)+2p)]1/(1−p), then for any y∈RN there exists TE(y)<+∞ such that u(y,t)=0 for any t>TE(y). (ii) It is assumed that N=1. Let u(x,t) be the solution of (1). For a convenient behaviour of u0(x) as |x|→+∞, then u(x,t) tends to a limit as t→+∞, uniformly in compact sets on R. (iii) Let u0(r) be a nonnegative radial function satisfying (2). Then the solution of (1) with initial value u0(r) is radially symmetric and for a convenient behaviour of u0(r) as r→+∞, u(r,t) tends to a limit as r→+∞, uniformly on compact sets in R. For N≥2, the authors restrict themselves to the radial case.