*Published Paper*

**Inserted:** 28 sep 2012

**Last Updated:** 29 sep 2012

**Journal:** Revista Matematica Iberoamericana

**Volume:** 28

**Pages:** 535-576

**Year:** 2012

**Abstract:**

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary $p$-Laplacean system \[ u_t -div\ (|Du|^{p-2}Du)=V(x,t)\,, \] 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 Kilpel\"ainen \& Mal\'y \cite{KM} 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 $Du \in L^\infty$ locally, where $n$ is the space dimension, and this gives the borderline case of a result of DiBenedetto \cite{DB2}; 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 \cite{DB2} valid for the homogeneous case $V \equiv 0$.

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