Local boundedness for weak solutions to strongly degenerate orthotropic parabolic equations
Pasquale Ambrosio, Simone Ciani
TL;DR
The paper addresses local boundedness of local weak solutions to the strongly degenerate, anisotropic parabolic equation $\partial_t u = \sum_{i=1}^n \partial_{x_i}[(|u_{x_i}|-\delta_i)_+^{p-1} \frac{u_{x_i}}{|u_{x_i}|}]$ on $\Omega_T$, introducing an orthotropic degeneracy that vanishes on $\bigcup_i\{|u_{x_i}|\le\delta_i\}$. It adapts De Giorgi iteration through a local energy estimate and a sequence of shrinking cylinders, controlling superlevel sets via carefully chosen cut-offs and Steklov-average techniques. The authors prove local $L^{\infty}$ bounds for all $p\ge2$, with explicit cylinder-dependent estimates for $p>2$ and $p=2$, and also derive a corresponding isotropic corollary. Overall, the work extends classical boundedness results from the parabolic $p$-Laplacian to a wide class of degenerate anisotropic problems, providing foundational steps toward further regularity analysis in this setting.
Abstract
We prove the local boundedness of local weak solutions to the parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-δ_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}=Ω\times(0,T]\,, \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, $p\geq2$, $δ_{1},\ldots,δ_{n}$ 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.