Regularity results for a class of widely degenerate parabolic equations

Abstract

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in $\Omega_{T}=\Omega\times(0,T)$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$, $\nu$ is a positive constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative $u_{t}$. The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius $\nu$ centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [5], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.