Partial regularity of solutions to a class of degenerate systems

We consider the system $\frac{\partial u }{\partial t}-\Delta u=\sigma \left ( u\right ) \left | \nabla \varphi \right | ^2$, $\mathrm {div}\left ( \sigma \left ( u\right ) \nabla \varphi \right ) =0$ in $Q_T\equiv \Omega \times \left ( 0,T\right ]$ coupled with suitable initial-boundary conditions, where $\Omega$ is a bounded domain in $\mathbf {R}^N$ with smooth boundary and $\sigma \left ( u\right )$ 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 Hölder continuous on $Q_0$, the interior of $Q_T\backslash Q_0$ is empty, and $u$ is essentially bounded on $Q_T\backslash Q_0$.