Blow-up of the critical norm for a supercritical semilinear heat equation

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in an arbitrary smooth domain of ${\mathbf{R}}^n$. In the range $p>p_S:=(n+2)\mathrm{/}(n-2)$, we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range $p>p_S$ is optimal in view of the existence of type II blow-up solutions with bounded critical norm for $p=p_S$.