Regularizing effect of homogeneous evolution equations with perturbation. (May 2021)

• Record Type: Journal Article
• Authors: Hauer, Daniel
• Is Part Of: Nonlinear analysis. Volume 206(2021)
• Publication Date: 2021-05
• Subjects: 47H20 -- 47h06 -- 47H14 -- 47J35 -- 35B65; Nonlinear semigroups -- Aronson–Bénilan estimates -- Regularity of time-derivative -- Homogeneous operators -- p-Laplace Beltrami operator -- Dirichlet-to-Neumann operator on manifolds; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112245
• Languages: English
• ISSNs: 0362-546X

Abstract

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 L 1 Aronson–Bénilan type estimates and point-wise Aronson–Bénilan type estimates . We apply our theory to derive global L q - 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.