Blow‐up profile for radial solutions of the nonlinear heat equation

Abstract

Assuming that the maximal classical solution $u$ of $u_t-\Delta u=u^p$ on $B(0,R)$ has, at time $T_{\max}$ , a blow‐up set $\mathbf{B}(u)$ that is not the singleton $\left\{0\right\}$ , we prove that ${\|u\|}_{L^q (\varOmega)}$ blows up for all $q>(p-1)/2$ , the energy blows up and $u$ blows up completely after $T_{\max}$ in the sense of Baras–Cohen (equivalent to the fact that there is no weak solution extending $u$ beyond $T_{\max}$ ). We also study this type of blow‐up through numerical experiments as well as degenerate blow‐up.