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

34 pages