Stability analysis of hyperbolic-parabolic free boundary problems modelling biofilms
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.
Keywords
Cite
@article{arxiv.2411.16977,
title = {Stability analysis of hyperbolic-parabolic free boundary problems modelling biofilms},
author = {Dieudonné Zirhumanana Balike and Maria Rosaria Mattei and Luigi Frunzo},
journal= {arXiv preprint arXiv:2411.16977},
year = {2025}
}
Comments
We kindly request the withdrawal of our submission due to a misunderstanding among the co-authors regarding the decision to upload the work to arXiv. Not all authors had agreed to its public release, and the submission was made based on an internal miscommunication. We regret this oversight and appreciate your assistance in addressing this matter