Radial Symmetry of Self-Similar Solutions for Semilinear Heat Equations

Abstract

The symmetry properties of positive solutions of the equation Δu+1/x · ∇u+ 1/p-1 u + up = 0 in Rn, where n ≥ 2, p > (n + 2)/n, was studied. It was proved that u must be radially symmetric about the origin provided u(x) = o(|x|-2/(p-1)) as \x\ → ∞, and that there exist non-radial solutions u satisfying lim sup|x| → ∞ |x\2/(p-1)u(x) > 0. © 2000 Academic Press.