We consider local weak solutions to the widely degenerate parabolic PDE \(\begin{aligned} \partial _{t}u-\textrm{div}\left( (\vert Du\vert -\lambda )_{+}^{p-1}\frac{Du}{\vert Du\vert }\right) =f\,\,\,\,\,\,\,\,\textrm{in}\,\,\,\Omega _{T}=\Omega \times (0,T), \end{aligned}\) where \(p\ge 2\) , \(\Omega \) is a bounded domain in \(\mathbb {R}^{n}\) for \(n\ge 2\) , \(\lambda \) is a non-negative constant and \(\left( \,\cdot \,\right) _{+}\) stands for the positive part. Assuming that the datum f belongs to a suitable Lebesgue–Besov parabolic space when \(p>2\) and that \(f\in L_{loc}^{2}(\Omega _{T})\) if \(p=2\) , we prove the Sobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary p-Poisson equation. The main novelty here is that f only has a Besov or Lebesgue spatial regularity, unlike the previous work [7], where f was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [6], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations.
