We consider the scaling critical Lebesgue norm of blow-up solutions to a semilinear heat equation (the Fujita equation).In the Sobolev supercritical range, we show that the critical norm also blows up at the blow-up time.The range is optimal in view of the existence of blow-up solutions with bounded critical norm for the Sobolev critical case.

This talk is based on a joint work with Hideyuki Miura (Tokyo Institute of Technology).