Regularity results for a class of widely degenerate parabolic equations

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ 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 establish the higher differentiability and the higher integrability of the spatial gradient of the weak solutions, as well as the existence of a weak time derivative $u_{t}$, by initially assuming that $f\in L^{\frac{p}{p-1}}\left(0,T;W^{1,\frac{p}{p-1}}\left(\Omega\right)\right)$. Later, we also prove that it is possible to recover most of the previous results by suitably weakening the assumption on the datum $f$. The main novelty here is that the structure function of the above equation satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that some of the results obtained here can be considered, on the one hand, as the parabolic counterpart of some elliptic results established, for example, in [2, 4] and [5], and on the other hand as the extension to a strongly degenerate setting of the known results for parabolic equations