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 ut = ∆u + up in RN , 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∗). ∗Supported in part by VEGA Grant 1/3021/06