Linearly stable self-similar solutions of semilinear heat equations

We classify the smooth linearly stable self-similar solutions of the semilinear heat equation uₜ=Δ u+|u|ᵖ⁻¹u in Rⁿ× (0,T) under an integral condition for all $p>1$. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with u(·,0)≥ 0 and uₜ(·,0)≥ 0 converges to a constant after rescaling at the blow-up point for all $p>1$.