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

Article type: Research Article

Authors: Poláčik, Peter | Quittner, Pavol^;

Affiliations: School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA | Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia

Note: [] Supported in part by VEGA Grant 1/3021/06.

Abstract: We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation ut=Δu+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*).

DOI: 10.3233/ASY-2008-0872

Journal: Asymptotic Analysis, vol. 57, no. 3-4, pp. 125-141, 2008