Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation

TL;DR

This paper investigates the fractional Sobolev regularity of solutions to a strongly degenerate parabolic equation, extending previous results by weakening assumptions on the right-hand side and establishing higher differentiability and summability of solutions.

Contribution

It introduces new fractional regularity results for solutions under weaker conditions on the source term, advancing understanding of degenerate parabolic equations.

Findings

Higher fractional differentiability of the spatial gradient Du.

Enhanced summability properties of Du.

Extension of elliptic regularity results to the parabolic setting.

Abstract

We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right]=f\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\Omega_{T}=\Omega\times(0,T), \] where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that $f\in L_{loc}^{p'}\left(0,T;B_{p',\infty,loc}^{\alpha}\left(\Omega\right)\right)$, with $\alpha\in(0,1)$ and $p'=p/(p-1)$. This leads us to obtain higher fractional differentiability results for a function of the spatial gradient $Du$ of the solutions. Moreover, we establish the higher summability of $Du$ with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [1], and on the other hand as the fractional version of some results established in [2].