# Revista Matemática Iberoamericana

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**Volume 28, Issue 2, 2012, pp. 535–576**

**DOI: 10.4171/rmi/684**

Published online: 2012-04-22

Potential estimates and gradient boundedness for nonlinear parabolic systems

Tuomo Kuusi^{[1]}and Giuseppe Mingione

^{[2]}(1) Aalto University, Finland

(2) Università di Parma, Italy

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary $p$-Laplacean system $$ u_t - \operatorname{div} (|Du|^{p-2}Du)=V(x,t) , $$ 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 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 $V \equiv 0$.

*Keywords: *Nonlinear potentials, $p$-Laplacean, parabolic equations

Kuusi Tuomo, Mingione Giuseppe: Potential estimates and gradient boundedness for nonlinear parabolic systems. *Rev. Mat. Iberoamericana* 28 (2012), 535-576. doi: 10.4171/rmi/684