Existence analysis for a reaction-diffusion Cahn-Hilliard-type system with degenerate mobility and singular potential modeling biofilm growth

TL;DR

This paper proves the global existence of bounded weak solutions for a biofilm growth model involving a reaction-diffusion equation and a Cahn-Hilliard-type equation with degenerate mobility and singular potential, using advanced mathematical techniques.

Contribution

It introduces a novel analytical approach to handle degenerate diffusivity, singular potentials, and nonlinear reactions in a biofilm growth model, establishing existence results.

Findings

Existence of bounded weak solutions is proven.

Numerical experiments illustrate solution behavior.

Handling of degenerate and singular terms is demonstrated.

Abstract

The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reaction-diffusion equation for the substrate concentration and a fourth-order Cahn-Hilliard-type equation for the volume fraction of the biomass, considered in a bounded domain with no-flux boundary conditions. The main difficulties are coming from the degenerate diffusivity and mobility, the singular potential arising from a logarithmic free energy, and the nonlinear reaction rates. These issues are overcome by a truncation technique and a Browder-Minty trick to identify the weak limits of the reaction terms. The qualitative behavior of the solutions is illustrated by numerical experiments in one space dimension, using a BDF2 (second-order backward Differentiation Formula) finite-volume scheme.