Partial regularity for parabolic systems of double phase type

We study partial regularity for nondegenerate parabolic systems of double phase type, where the growth function is given by $H(z,s)=s^p+a(z)s^q$, $z=(x,t)\in\Omega_T$, with $\tfrac{2n}{n+2}<p\le q$ and $a(z)$ a nonnegative $C^{0,\alpha,\frac{\alpha}{2}}$-continuous function for some $\alpha\in(0,1]$. As the main result we prove that if $q< \min \{p+\tfrac{\alpha p }{n+2}, p+1 \}$ the spatial gradient of any weak solution is locally H\"older continuous, except on a set of measure zero.