On Similarity Solutions of a Heat Equation with a Nonhomogeneous Nonlinearity

AbstractWe are interested in positive radially symmetric solutions of the semilinear equationΔw−y·∇w2−Aw+|y|lwp=0,inRn,n⩾3, where p>1, l>−2 and A≡l+22(p−1). 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