  Since the pioneering works by Aronson & B\'enilan [C. R. Acad. Sci. Paris
S\'er., 1979] and B\'enilan & Crandall [Johns Hopkins Univ. Press, 1981], 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 at every positive time. Moreover, if the
underlying Banach space has the Radon-Nikod\'ym property, then these 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 global $L^1$ Aronson-B\'enilan type estimates and point-wise
Aronson-B\'enilan type estimates. We apply our theory to derive global
$L^q$-$L^{\infty}$-estimates on the time-derivative of the perturbed diffusion
problem governed by the Dirichlet-to-Neumann operator associated with the
$p$-Laplace-Beltrami operator and lower-order terms on a compact Riemannian
manifold with a Lipschitz boundary.
