  In this paper, we study the regularity of weak solutions to the following
strongly degenerate parabolic equation
  \begin{equation*}
  u_t-\div\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{
in }\Omega_T,
  \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for
$n\geq2$, $p\geq2$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive
part. We prove the higher differentiability of a nonlinear function of the
spatial gradient of the weak solutions, assuming only that $f\in
L^{2}_{\loc}\left(\Omega_T\right)$. This allows us to establish the higher
integrability of the spatial gradient under the same minimal requirement on the
datum $f$.
