# A sufficient condition for the continuity of solutions to a logarithmic diffusion equation

## TL;DR

This paper establishes a sufficient condition ensuring the continuity of solutions to a logarithmic diffusion equation, providing estimates for the modulus of continuity and the Hausdorff measure of discontinuities.

## Contribution

It introduces a new sufficient condition for the continuity of solutions to a logarithmic diffusion equation and offers quantitative estimates for continuity and discontinuity sets.

## Key findings

- Solutions are continuous under the new condition.
- An explicit estimate of the modulus of continuity is provided.
- The Hausdorff measure of the discontinuity set is bounded.

## Abstract

This note gives a first sufficient condition that insures a non-negative, locally bounded, local solution to a logarithmically singular parabolic equation is continuous at a vanishing point and an estimate of the modulus of continuity is given. Moreover, an estimate of the Hausdorff measure of the set of discontinuity is established.