Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type

In the present paper, we deal with the degenerate parabolic system of porous medium type with non-linear diffusion m and interaction q in in the critical case of . We establish the -regularity theorem for weak solutions. As an application, the structure of asymptotics of blow-up solution is clarified. In particular, we show that the solution behaves like the delta-function at the blow-up points. Moreover, we prove that the number of blow-up points is finite, which can be controlled in terms of the mass of initial data. We also give a sharp constant for the -regularity theorem.