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