Stability analysis of hyperbolic-parabolic free boundary problems modelling biofilms

TL;DR

This paper analyzes the stability of two complex free boundary problems in biofilm modeling, demonstrating existence, uniqueness, and convergence of solutions, with extensions to substrate precipitation models.

Contribution

It provides the first rigorous stability analysis of hyperbolic-parabolic free boundary biofilm models, including convergence to stationary solutions and extensions to variable diffusion systems.

Findings

Existence and uniqueness of stationary solutions established.

Classical solutions converge to stationary states.

Extended analysis to models with variable diffusion coefficients.

Abstract

We present the stability analysis of two free boundary problems arising in biofilm modelling. The first, introduced in the 1980s by Wanner and Gujer, is related to the competition between autotrophic and heterotrophic bacteria in a biofilm bioreactor. It is a free boundary problem consisting of a system of hyperbolic PDEs modelling biofilm growth and the competition between species, a parabolic system accounting for substrate consumption, and an ODE for biofilm thickness. The second, also based on the former, arises from the modelling of trace-metal precipitation in biofilms, with a special focus on the role of sulfate-reducing bacteria in the methane production process. The analysis is made into two steps, the first one being the existence and uniqueness of the stationary solutions. The second one allows to show that the calssical solutions converge to the stationary solutions by using a semigroup approach and the energy method. We also extend the study to the precipitation model, in which the substrates are modelled by a parabolic system with variable diffusion coefficients.