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

Abstract

In the present paper, we deal with the degenerate parabolic system of porous medium type with non-linear diffusion m and interaction q in \({\rm{I\!R}}^N\) in the critical case of \({q =m+\frac{2}{N}}\). We establish the \({\varepsilon}\) -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 δ-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 \({\varepsilon}\) -regularity theorem.