The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function has a smoothing effect, discovered by Haïm Brezis, 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 are not only closed, but even compact. We apply our results to the -Laplacian and also to the Dirichlet-to-Neumann operator with respect to -harmonic functions.
"Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth." Pure Appl. Anal. 2 (1) 23 - 34, 2020. https://doi.org/10.2140/paa.2020.2.23