Blow-up rate for the subcritical semilinear heat equation in non-convex domains

We consider the semilinear heat equation $u_t=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in possibly non-convex and unbounded domains. Our main result shows the nonexistence of type II blow-up for possibly sign-changing solutions in the energy subcritical range $(n-2)p<n+2$. This resolves a long-standing open question dating back to the 1980s and also deduces the blow-up of the scaling critical norm.