Analysis of a one-dimensional biofilm model

TL;DR

This work rigorously analyzes a reduced one-dimensional moving-boundary biofilm model driven by a substrate-limited reaction. By transforming the parabolic free-boundary problem to a fixed-domain formulation and employing quasilinear evolution theory, the authors establish global well-posedness and Sobolev-regularity for the full evolution problem, along with qualitative behaviors such as extinction under negative growth. The quasi-steady approximation ($\varepsilon=0$) is analyzed in depth, yielding existence and smooth dependence of the substrate profile on the biofilm height, and a reduced ODE for the height; equilibria are shown to exist in general, with a unique, globally attracting equilibrium in the affine-growth case. Overall, the paper provides a solid mathematical foundation for the dynamics of a simplified biofilm–substrate system and informs stability and long-time behavior under various growth and consumption laws.

Abstract

In this paper a reduced one-dimensional moving boundary model is studied that describes the evolution of a biofilm driven by the presence of a reaction limiting substrate. Global well-posedness is established for the resulting parabolic free boundary value problem in strong form in Sobolev spaces and for a quasi-stationary approximation in spaces of classical regularity. The general existence results are complemented by results about the qualitative properties of solutions including the existence, in general, and, additionally, the uniqueness and stability of non-trivial equilibria, in a special case.

Figures (2)

• Figure 1: The "long" time evolution of the moving boundary (film depth) and that of substrate concentration at small times.
• Figure 2: The behavior of substrate concentration at larger times.

Theorems & Definitions (26)

• Theorem 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['T1']}