  Motivated by applications to gas filtration problems, we study the regularity
of weak solutions to the strongly degenerate parabolic PDE
  $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert
Du\vert}\right)=f$ in $\Omega_{T}=\Omega\times(0,T)$,
  where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$,
$p\geq2$, $\nu$ is a positive constant and $\left(\,\cdot\,\right)_{+}$ stands
for the positive part. Assuming that the datum $f$ belongs to a suitable
Lebesgue-Sobolev parabolic space, we establish the Sobolev spatial regularity
of a nonlinear function of the spatial gradient of the weak solutions, which in
turn implies the existence of the weak time derivative $u_{t}$. The main
novelty here is that the structure function of the above equation satisfies
standard growth and ellipticity conditions only outside a ball with radius
$\nu$ centered at the origin. We would like to point out that the first result
obtained here can be considered, on the one hand, as the parabolic counterpart
of an elliptic result established in [5], and on the other hand as the
extension to a strongly degenerate context of some known results for less
degenerate parabolic equations.
