Submitted Paper

Inserted: 15 sep 2026
Last Updated: 15 sep 2026

Year: 2026
Doi: https://doi.org/10.48550/arXiv.2609.15360

Abstract:

We study the regularity of weak solutions to doubly nonlinear orthotropic evolution equations of the form \[ \partial_{t}(\vert u\vert^{α-1}u)-\sum_{i=1}^{N}\partial_{i}\left[a_{i}(x,t)\,(
\partial_{i}u
-δ_{i})_{+}^{p-1}\frac{\partial_{i}u}{\vert\partial_{i}u\vert}\right]=f\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded open subset of $\mathbb{R}^{N}$ for $N\geq2$, the coefficients $a_{i}$ are measurable and bounded, $α>0$, $p\in(1,\infty)$ and $δ_{1},\ldots,δ_{N}$ are non-negative numbers. We show that weak solutions are locally bounded due to their membership in a suitable De Giorgi-type energy class. We also obtain contractive estimates and global boundedness in space for solutions to a Cauchy problem associated with the above PDE. Our analysis extends analogous results available in the literature for diffusion equations that either do not exhibit double nonlinearity or are less degenerate than those considered here. Another main novelty of this paper is the presence of a source term $f$ on the right-hand side of the equation, for which we impose suitable integrability assumptions in the space-time variables.

Keywords: Degenerate parabolic equations, boundedness, anisotropic equations, Doubly nonlinear parabolic equations, contractive bounds