Self-similar solutions for a semilinear heat equation with critical Sobolev exponent

Abstract

The Cauchy problem for a semilinear heat equation with singular initial data (w t =Δw+w p in R N x(0,∞) w(x,0) =λa(x/|x|)|x| -2/(p-1) in R N \{0} is studied, where N > 2, p = (N+2)/(N-2), A > 0 is a parameter, and a > 0, a ≢ 0. We show that there exists a constant λ* > 0 such that the problem has at least two positive self-similar solutions for λ∈ (0,λ*) when N = 3,4,5, and that, when N ≥ 6 and a ≡ 1, the problem has a unique positive radially symmetric self-similar solution for λ∈ (0, λ * ) with some λ * ∈ (0, λ*). Our proofs are based on the variational methods and Pohozaev type arguments to the elliptic problem related to the profiles of self-similar solutions.