Radial Symmetry of Self-Similar Solutions for Semilinear Heat Equations

The symmetry properties of positive solutions of the equation 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 supx→∞ x2/(p−1)u(x)>0.