Regularizing effect of homogeneous evolution equations with perturbation

Abstract

Since the pioneering works by Aronson and Bénilan (1979), and Bénilan and Crandall (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–Nikodým property, then this 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 L1Aronson–Bénilan type estimates and point-wise Aronson–Bénilan type estimates. We apply our theory to derive global Lq-L∞-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.