Analysis of a one-dimensional biofilm model

TL;DR

This paper develops and analyzes a one-dimensional free boundary model for biofilm growth driven by substrate availability, establishing well-posedness, qualitative properties, and equilibrium stability.

Contribution

It introduces a reduced one-dimensional moving boundary model for biofilm evolution and proves its well-posedness and qualitative solution properties.

Findings

Global well-posedness in Sobolev spaces

Existence and uniqueness of equilibria

Stability analysis of non-trivial equilibria

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.