Asymptotics near an extinction point for some semilinear heat equations

Abstract

We consider the Cauchy problem u t -u xx +u p =0,x∈ℝ,t>0,u(x,0)=u 0 (x),x∈ℝ, where 0<p<1, u 0 (x) is continuous, nonnegative, and bounded, with a single maximum at x=0 and such that u 0 (-x)=u 0 (x) for any x, lim x→∞ u 0 (x)=0. It is well known that the solution has some features which are absent in the superlinear case p≥1. For instance, there exists T>0 such that u(x,t)¬≡0 if t<T and u(x,t)≡0 for t≥T. Moreover, there exists a continuous curve ζ(t) such that lim t→T ζ(t)=0 and Ω + (t)={x: u(x,t)>0}={x: -ζ(t)<x<ζ(t)}. In this communication we describe some asymptotic results. Namely, lim t→T (T-t) 1 1-p u(ξ(T-t) 1/2 |ln(T-t)| 1/2 ,t)=(1-p) 1 1-p 1 - (1-p) 4p ξ 2 + 1 1-p uniformly on sets |ξ|≤C(T-t) 1/2 |ln(T-t)| 1/2 , and lim t→T ζ(t) (T-t) 1/2 |ln(T-t)|=4p 1-p 1/2 ·