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 $L^q$-$L^{\infty }$-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.

自从Aronson和Bénilan（1979）以及Bénilan和Crandall（1981）的开创性工作以来，众​​所周知，由非线性但齐次的算子控制的一阶演化问题承认平滑效果，即每个相应的温和解在Lipschitz连续于每个积极的时间。此外，如果基础的Banach空间具有Radon–Nikodým属性，则该温和解是可微的，并且时间导数满足全局和逐点边界。

在本文中，我们证明，如果Lipschitz连续映射扰动齐次算子，则这些结果仍然成立。更确切地说，我们建立全球${\mathrm{大号}}^{\mathrm{1个}}$ Aronson-Bénilan类型估计和逐点Aronson-Bénilan类型估计。我们运用我们的理论来推导全球${\mathrm{大号}}^q$--${\mathrm{大号}}^{\infty }$估计由Dirichlet-to-Neumann算子控制的扰动扩散问题的时间导数，该算子与 $p$-Laplace–Beltrami算子和具有Lipschitz边界的紧致黎曼流形上的低阶项。