Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation

Abstract

We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation u_t=Δu+u^p in R^N, where p>1+2/N, p(N-2)≤N+2. The initial data are of the form u(x, 0)=αϕ(x), where ϕ is a fixed function with suitable decay at |x|=∞ and α>0 is a parameter. There exists a threshold parameter α^* such that the solution exists globally if and only if α≤α^*. Our main results describe the asymptotic behavior of the solutions for α∈(0, α^*] and in particular exhibit the difference between the behavior of sub-threshold solutions (α<α^*) and the threshold solution (α=α^*).