Abstract:
We consider blow-up solutions for the nonlinear heat equation with power type nonlinearity |u|^p-1u in Rⁿ. 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 (Institute of Science Tokyo).

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