# Potential estimates and gradient boundedness for nonlinear parabolic systems

### Tuomo Kuusi

Aalto University, Finland### Giuseppe Mingione

Università di Parma, Italy

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

## Cite this article

Tuomo Kuusi, Giuseppe Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 535–576

DOI 10.4171/RMI/684