On Similarity Solutions of a Heat Equation with a Nonhomogeneous Nonlinearity

Abstract

We are interested in positive radially symmetric solutions of the semilinear equation Δw - y·∇w/2 - Aw + |y|l wp = 0, in ℝn, n ≥ 3, where p > 1, l > -2 and A ≡ l+2/2(p-2) This equation is satisfied by self-similar solutions of a semilinear heat equation. We prove existence and non existence of solutions for various values of the parameters l and p. When solutions exist we study their asymptotic behavior and discuss their uniqueness. Our proofs are based on various continuity, comparison and Pohozaev type arguments. © 2000 Academic Press.