Regularizing effect of homogeneous evolution equations with perturbation.

Since the pioneering work [C. R. Acad. Sci. Paris S\'er., 1979] by Aronson & B\'enilan and [Johns Hopkins Univ. Press, 1981] by B\'enilan & Crandall, it is well-known that first-order evolution problems governed by a nonlinear but homogeneous operator admit the smoothing effect that every corresponding mild solution is Lipschitz continuous for every positive time and if the underlying Banach space has the Radon-Nikodym property, then the mild solution is a.e. differentiable and the time-derivative satisfies global and point-wise bounds. In this paper, we show that these results remain true if the homogeneous operator is perturbed by a Lipschitz continuous mapping. More precisely, we establish "point-wise Aronson-B\'enilan type estimates" and global "$L^1$ B\'enilan-Crandall type estimates". We apply our theory to derive global $L^q$-$L^{\infty}$-estimates on the time-derivative of the evolution problem governed by the Dirichlet-to-Neumann operator associated with the $p$-Laplace-Beltrami operator on a compact Riemannian manifold with Lipschitz boundary perturbed by a Lipschitz nonlinearity.