# On the continuity of solutions to doubly singular parabolic equations

## TL;DR

This paper proves local continuity of bounded weak solutions to a class of doubly singular parabolic equations with a time derivative singularity, especially as the parameter p approaches 2, ensuring stability of solutions.

## Contribution

It establishes the local continuity of solutions for a range of p values close to 2, extending understanding of regularity in doubly singular parabolic equations.

## Key findings

- Solutions are locally continuous for 2−ε₀ ≤ p < 2.
- Continuity is stable as p approaches 2.
- Results apply to equations with singularities in the time derivative.

## 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 $p\to2$.