On the continuity of solutions to doubly singular parabolic equations

Qifan Li

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arXiv · 1812.04864

On the continuity of solutions to doubly singular parabolic equations

Qifan Li

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Abstract

This paper considers a certain doubly singular parabolic equations with one singularity occurs in the time derivative, whose model is \begin{equation*} \partial_t\beta(u)-\operatorname{div}|Du|^{p-2}Du\ni0,\qquad \text{in}\quad \Omega\times(0,T)\end{equation*} where $\Omega\subset\mathbb{R}^N$ and $N\geq3$. We show that the bounded weak solutions are locally continuous in the range $$2-\epsilon_0\leq p<2,$$ provided $\epsilon_0>0$ is small enough, and the continuity is stable as $p\to2$.

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Qifan Li. 2018-12-12. On the continuity of solutions to doubly singular parabolic equations. https://arxiv.org/abs/1812.04864

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