Accepted Paper

Inserted: 9 jul 2024
Last Updated: 4 nov 2024

Journal: Calculus of Variations and Partial Differential Equations
Year: 2024

Abstract:

We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega_{T}=\Omega\times(0,T), \] 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.

Keywords: Sobolev regularity, Besov spaces, Degenerate parabolic equations, higher differentiability