Gradient bounds for strongly singular or degenerate parabolic systems

We consider weak solutions u:ΩT→RN to parabolic systems of the typeut−divA(x,t,Du)=finΩT=Ω×(0,T), where Ω is a bounded open subset of Rn 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|2nn+2 we establish that any weak solutionu∈C0((0,T);L2(Ω,RN))∩Lp(0,T;W1,p(Ω,RN)) admits a locally bounded spatial gradient Du. Moreover, assuming that u is essentially bounded, we recover the same result in the case 1<p≤2nn+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.