Higher regularity for weak solutions to degenerate parabolic problems

TL;DR

This paper proves higher regularity and integrability of weak solutions to a class of degenerate parabolic equations with minimal assumptions on the data, advancing understanding of their smoothness properties.

Contribution

It establishes higher differentiability and integrability of the spatial gradient for weak solutions to a degenerate parabolic PDE, with only $L^2$ data assumptions.

Findings

Higher differentiability of nonlinear functions of the gradient.

Higher integrability of the spatial gradient.

Results hold under minimal $L^2$ assumptions on the source term.

Abstract

In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-\div\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in }\Omega_T, \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$ 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 $f\in L^{2}_{\loc}\left(\Omega_T\right)$. This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum $f$.