On the continuity of solutions to doubly singular parabolic equations

Abstract

This paper considers a certain doubly singular parabolic equations with one singularity occurs in the time derivative, whose model is \begin{equation*} \partial_tβ(u)-\operatorname{div}|Du|^{p-2}Du\ni0,\qquad \text{in}\quad Ω\times(0,T)\end{equation*} where $\Omega \subset {\mathbb{R}}^N$ and $N\ge 3$. We show that the bounded weak solutions are locally continuous in the range $$2-{\epsilon }_0\le p<2,$$ provided ${\epsilon }_0>0$ is small enough, and the continuity is stable as $p\to 2$.