Sharp Sobolev regularity for widely degenerate parabolic equations

Abstract

We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ \Omega_{T}=\Omega\times(0,T), \] where $p\geq2$, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $\lambda$ 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 [6], 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 [5], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations.