Partial regularity of solutions to a class of degenerate systems

We consider the system partial derivative u/partial derivative t - Delta u =sigma(u)\del phi\(2), div (sigma(u)del phi) = 0 in Q(T) = Omega x (0,T) coupled with suitable initial-boundary conditions, where Omega is a bounded domain in R-N with smooth boundary and sigma(u) is a continuous and positive function of u. Our main result is that under some conditions on sigma there exists a relatively open subset Q(0) of Q(T) such that u is locally Holder continuous on Q(0), the interior of Q(T)\Q(0) is empty, and u is essentially bounded on Q(T)\Q(0).