Higher regularity for weak solutions to degenerate parabolic problems

Abstract

In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-÷\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in }Ω_T, \end{equation*} where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ for $n\ge 2$, $p\ge 2$ 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 $f\in L^{2}_{\loc}\left(Ω_T\right)$. This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum $f$.