We consider the critical norm blow-up problem for the nonlinear heat equation with power type nonlinearity |u|^{p-1}u in R^n.
In the Sobolev supercritical range p>(n+2)/(n−2), we show that if the maximal existence time T is finite, the scaling critical L^q norm of the solution becomes infinite at t=T. The range of p is optimal in view of known examples of blow-up solutions with the bounded critical norm for the Sobolev critical case. This is a joint work with Jin Takahashi (Tokyo institute of technology).