Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

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.