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 $H(z,s)=s^p+a(z)s^q$, $z=(x,t)\in {\mathrm{\Omega }}_T$, with $\frac{2n}{n+2}<p\le q$ 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 $q<\min \{p+\frac{\alpha p}{n+2},p+1\}$ the spatial gradient of any weak solution is locally Hölder continuous, except on a set of measure zero.