Potential estimates and gradient boundedness for nonlinear parabolic systems

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary p-Laplacean system ut−div(|Du|p−2Du)=V(x,t), and provide L∞-bounds for the spatial gradient of solutions Du via nonlinear potentials of the right hand side datum V. Such estimates are related to those obtained by Kilpeläinen and Malý [22] in the elliptic case. In turn, the potential estimates found imply optimal conditions for the boundedness of Du in terms of borderline rearrangement invariant function spaces of Lorentz type. In particular, we prove that if V∈L(n+2,1) then Du∈L∞loc, where n is the space dimension, and this gives the borderline case of a result of DiBenedetto [5]; a significant point is that the condition V∈L(n+2,1) is independent of p. Moreover, we find explicit forms of local a priori estimates extending those from [5] valid for the homogeneous case V≡0.