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

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 $c>0$, $\theta \ge 0$ and that $f(r)\ge f(0)=0$ is Lipschitz continuous and monotonically increasing for $r\in [0,{\sup }_{x\in \Omega }{u}^0(x)]$. Throughout the paper we restrict ourselves to the model degenerate case $b(u,v):=\sigma uv$, where $\sigma >0$. 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.