Gradient estimates for a class of parabolic systems

We establish Calder\'on \& Zygmund type estimates for a class of parabolic problems whose model is the non-homogeneous, degenerate/singular parabolic $p$-Laplacan system $$u_t - div \ (|Du|^{p-2}Du) = div \ (|F|^{p-2}F)\;, $$ proving that $$ F \in L_{\loc} ^{ q} \Longrightarrow Du\in L^{q}_{\loc} \qquad \forall \ q\geq p\;.$$ We also treat systems with discontinuous coefficients of vanishing mean oscillation (VMO) type.