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

We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation u(t) =Delta u + 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) = alpha phi(x), where phi is a fixed function with suitable decay at vertical bar x vertical bar = infinity and alpha > 0 is a parameter. There exists a threshold parameter alpha* such that the solution exists globally if and only if alpha <= alpha*. Our main results describe the asymptotic behavior of the solutions for alpha is an element of (0, alpha*] and in particular exhibit the difference between the behavior of sub- threshold solutions (alpha < alpha*) and the threshold solution (alpha= alpha*).