Regularizing effect of homogeneous evolution equations with perturbation

Abstract

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.