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 Aronson-B\'enilan type estimates and point-wise Aronson-B\'enilan type estimates. We apply our theory to derive global --estimates on the time-derivative of the perturbed diffusion problem governed by the Dirichlet-to-Neumann operator associated with the -Laplace-Beltrami operator and lower-order terms on a compact Riemannian manifold with a Lipschitz boundary.
Cite
@article{arxiv.2004.00483,
title = {Regularizing effect of homogeneous evolution equations with perturbation},
author = {Daniel Hauer},
journal= {arXiv preprint arXiv:2004.00483},
year = {2021}
}
Comments
Among some minor corrections of typos, the direction of the inequality in the point-wise Aronson-B\'enilan type estimates are corrected in this version. The proof was correct, but the final statement had previously the inequality in the wrong direction! arXiv admin note: text overlap with arXiv:1901.08691