Gradient bounds for strongly singular or degenerate parabolic systems

We consider weak solutions u : \Omega_T → R^N to parabolic systems of the type u t − div A(x, t, Du) = f in \Omega_T = \Omega × (0, T), where \Omega is a bounded open subset of R^n for n ≥ 2, T > 0 and the datum f belongs to a suitable Orlicz space. The main novelty here is that the partial map ξ → A(x, t, ξ) satisfies standard p-growth and ellipticity conditions for p > 1 only outside the unit ball {|ξ | < 1}. For p > 2n/(n+2) we establish that any weak solution u ∈ C^0 ((0, T); L^2 (\Omega, R^N)) ∩ L^p (0, T ; W^1,p (\Omega, R^N)) admits a locally bounded spatial gradient Du. Moreover, assuming that u is essentially bounded, we recover the same result in the case 1 < p ≤ 2n/(n+2) and f = 0. Finally, we also prove the uniqueness of weak solutions to a Cauchy-Dirichlet problem associated with the parabolic system above. We emphasize that our results include both the degenerate case p ≥ 2 and the singular case 1 < p < 2.