Asymptotic properties of a semilinear heat equation with strong absorption and small diffusion

Abstract

In this paper the authors study the asymptotic behaviour of solutions uε(x,t) of the Cauchy problems as ε goes to zero: ut−εΔu+up=0, x∈RN, t>0; u(x,0)=u0(x), x∈RN, 0<p<1. Compared with the explicit solution u¯(x,t) and the extinction time T0E(x) of the corresponding spatially independent initial value problem: ut+up=0, x∈RN, t>0; u(x,0)=u0(x), x∈RN, it is proved under certain assumptions that uε(x,t)→u¯(x,t) as ε↓0 uniformly on compact subsets of RN ×[0,∞) and, moreover, a precise estimate is given. Local and global estimates for extinction time are also given. The proofs are somewhat technical