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Higher regularity for weak solutions to degenerate parabolic problems (2301.11795v1)

Published 27 Jan 2023 in math.AP

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 $f$.

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