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

We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded L n ( p − 1 ) / 2 , ∞ ( R n ) -norm up to the blow-up time. As a consequence of this, we obtain optimal blow-up rates for certain radial solutions undergoing type II blow-up. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative ε -regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek 31 and Tao 38 involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.