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The nonlinear semigroup generated by the subdifferential of a convex lowersemicontinuous function $\varphi$ has a smoothing effect, discovered by H.Br\'ezis, which implies maximal regularity for the evolution equation. We usethis and Schaefer's fixed point theorem to solve the evolution equationperturbed by a Nemytskii-operator of sublinear growth. For this, we need thatthe sublevel sets of $\varphi$ are not only closed but even compact. We applyour results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operatorwith respect to $p$-harmonic functions.

Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth