Gradient bounds for a widely degenerate orthotropic parabolic equation

Abstract

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-δ_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω\times I, \] where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interval, $p\geq2$, $δ_{1},\ldots,δ_{n}$ 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. 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].