Interior H\"older continuity for singular-degenerate porous medium type equations with an application to a biofilm model

TL;DR

This paper proves interior H"older continuity for a class of singular-degenerate reaction-diffusion equations with porous medium type degeneracy, motivated by biofilm growth models, using a novel approach that does not rely on approximation sequences.

Contribution

It establishes interior regularity results for a broad class of degenerate equations without the common approximation assumptions, extending the theory to more general biofilm-related models.

Findings

Proves interior H"older continuity for the equations.

Handles equations with singular and degenerate features.

Provides a new method avoiding approximation sequences.

Abstract

We show interior H\"older continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyse is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior H\"older continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solutions of regularized problems, which is a common assumption.