  We classify the smooth linearly stable self-similar solutions of the
semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^n\times
(0,T)$ 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 $u(\cdot,0)\geq 0$ and $u_t(\cdot,0)\geq 0$ converges to a constant after
rescaling at the blow-up point for all $p>1$.
