  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].
