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

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