Self-similar solutions of semilinear heat equations with positive speed

Abstract

We classify the smooth self-similar solutions of the semilinear heat equation $u_t=\Delta u+|u{|}^{p-1}u$ in ${\mathbb{R}}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with $u(\cdot ,0)\ge 0$ and $u_t(\cdot ,0)\ge 0$ converges to a positive constant after rescaling at the blow-up point for all $p>1$.

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