# Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation

### John W. Barrett

Imperial College London, United Kingdom### Robert Nürnberg

Imperial College London, United Kingdom

## Abstract

We consider a fully practical finite-element approximation of the following nonlinear degenerate parabolic system

subject to no flux boundary conditions, and non-negative initial data $u_{0}$ and $v_{0}$ on $u$ and $v$. Here we assume that $c>0$, $θ≥0$ and that $f(r)≥f(0)=0$ is Lipschitz continuous and monotonically increasing for $r∈[0,sup_{x∈Ω}u_{0}(x)]$. Throughout the paper we restrict ourselves to the model degenerate case $b(u,v):=σuv$, where $σ>0$. The above models the spatiotemporal evolution of a bacterium on a thin film of nutrient, where $u$ is the nutrient concentration and $v$ is the bacterial cell density. In addition to showing stability bounds for our approximation, we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system. Finally, some numerical experiments in one and two space dimensions are presented.

## Cite this article

John W. Barrett, Robert Nürnberg, Finite-element approximation of a nonlinear degenerate parabolic system describing bacterial pattern formation. Interfaces Free Bound. 4 (2002), no. 3, pp. 277–307

DOI 10.4171/IFB/62