Potential estimates and gradient boundedness for nonlinear parabolic systems

Abstract

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary $p$-Laplacean system

and provide $L^{\infty }$-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 $V\in L(n+2,1)$ then $Du\in L_{\mathrm{loc}}^{\infty }$, 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 $V\in L(n+2,1)$ is independent of $p$. Moreover, we find explicit forms of local a priori estimates extending those from [5] valid for the homogeneous case $V\equiv 0$.