Critical norm blow-up for the energy supercritical nonlinear heat equation

We address the critical norm blow-up problem for the nonlinear heat equation $u_t-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in ${\mathbf{R}}^n\times (0,T)$. In the supercritical range $p>(n+2)\mathrm{/}(n-2)$, we prove that if the maximal existence time $T$ is finite, then ${\lim }_{t\to T}\mathrm{\parallel }u(\cdot ,t){\mathrm{\parallel }}_{L^{n(p-1)\mathrm{/}2}({\mathbf{R}}^n)}=\mathrm{\infty }$ without assuming extra conditions such as radial symmetry or the type of blow-up.