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arXiv · 2305.08539

H\"older Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations

Verena Bögelein·Frank Duzaar·Ugo Gianazza·Naian Liao·Christoph Scheven
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Abstract

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time cylinder. H\"older estimates are established for the gradient of its weak solutions in the super-critical fast diffusion regime $0<p-1< q<\frac{N(p-1)}{(N-p)_+}$ where $N$ is the space dimension. Moreover, decay estimates are obtained for weak solutions and their gradient in the vicinity of possible extinction time. Two main components towards these regularity estimates are a time-insensitive Harnack inequality that is particular about this regime, and Schauder estimates for the parabolic $p$-Laplace equation.

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Verena Bögelein, Frank Duzaar, Ugo Gianazza, Naian Liao, Christoph Scheven. 2023-05-15. H\"older Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations. https://arxiv.org/abs/2305.08539

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