PreprintRegularizing effect of homogeneous evolution equations with perturbationDaniel HauerAbstractSince the pioneering work [C. R. Acad. Sci. Paris Sér., 1979] by Aronson & Bénilan and [Johns Hopkins Univ. Press, 1981] by Bénilan & Crandall, it is wellknown that firstorder 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 RadonNikodým property, then the mild solution is a.e. differentiable and the timederivative satisfies global and pointwise 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 pointwise Aronson–Bénilan type estimates and global \(L^1\) BénilanCrandall type estimates. We apply our theory to derive global \(L^q\)\(L^{\infty}\)estimates on the timederivative of the evolution problem governed by the DirichlettoNeumann operator associated with the \(p\)LaplaceBeltrami operator on a compact Riemannian manifold with Lipschitz boundary perturbed by a Lipschitz nonlinearity. Keywords: Nonlinear semigroups, AronsonBénilan estimates, regularity of timederivative, homogenous operators, \(p\)Laplace Beltrami operator, DirichlettoNeumann operator on manifolds. AMS Subject Classification: Primary 47H20; secondary 47h06, 47H14, 47J35, 35B65. This paper is available as a pdf (324kB) file. It is also on the arXiv: arxiv.org/abs/2004.00483v2.
