Local boundedness for weak solutions to strongly degenerate orthotropic parabolic equations

Abstract

We prove the local boundedness of local weak solutions to the parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega_{T}=\Omega\times(0,T]\,, \] where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, $p\geq2$, $\delta_{1},\ldots,\delta_{n}$ are non-negative numbers and $\left(\,\cdot\,\right)_{+}$ denotes the positive part. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. The core result of this paper thus extends a classical boundedness theorem, originally proved for the parabolic $p$-Laplacian, to a widely degenerate anisotropic setting. As a byproduct, we also obtain the local boundedness of local weak solutions to the isotropic counterpart of the above equation.