Analysis of a degenerate and singular volume-filling cross-diffusion system modeling biofilm growth

TL;DR

This paper investigates a complex multi-species biofilm model with volume-filling cross-diffusion, addressing mathematical challenges like degeneracy and singularity, and establishes existence, uniqueness, and asymptotic behavior of solutions with supporting numerical simulations.

Contribution

The paper provides the first rigorous analysis of a volume-filling cross-diffusion system with degeneracy and singularity, including existence, uniqueness, and asymptotic results.

Findings

Existence of global weak solutions

Asymptotic behavior characterized

Numerical simulations support theoretical results

Abstract

We analyze the mathematical properties of a multi-species biofilm cross-diffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volume-filling type cross-diffusion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdiffusion-type singularity when the biomass reaches its maximum cell capacity, which make the analysis extremely challenging. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.