Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients

Abstract

We consider vector valued weak solutions $u:{\Omega }_T\to R^N$ with $N\in N$ of degenerate or singular parabolic systems of type ${\partial }_{t}u-\mathrm{div}a(z,u,Du)=0\text{in}{\Omega }_T=\Omega \times (0,T),$where $\Omega$ denotes an open set in $R^n$ for $n\ge 1$ and $T>0$ a finite time. Assuming that the vector field $a$ is not of Uhlenbeck-type structure, satisfies $p$-growth assumptions and $(z,u)\mapsto a(z,u,\xi )$ is Hölder continuous for every $\xi \in R^{Nn}$, we show that the gradient $Du$ is partially Hölder continuous, provided the vector field degenerates like that of the $p$-Laplacian for small gradients.