preprint

Inserted: 4 nov 2025
Last Updated: 4 nov 2025

Year: 2025
Doi: https://doi.org/10.48550/arXiv.2511.01480

Abstract:

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega\times I, \] where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interval, $p\geq2$, $\delta_{1},\ldots,\delta_{n}$ are non-negative numbers and $\left(\,\cdot\,\right)_{+}$ denotes the positive part. We prove that the local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. We emphasize that our result can be considered, on the one hand, as the parabolic counterpart of the elliptic result established in 12, and on the other hand as an extension to a significantly more degenerate framework of the findings contained in 13.

Keywords: lipschitz continuity, Degenerate parabolic equations, Moser iteration, anisotropic equations