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

### Robert Nürnberg

Imperial College London, United Kingdom### John W. Barrett

Imperial College London, United Kingdom

## Abstract

We consider a fully practical finite-element approximation of the following nonlinear degenerate parabolic system [part]u[divide][part]t ? c [Dgr]u = ?f (u) v in [OHgr]T := [OHgr] [times] (0 T), [OHgr] [sub] Rd, d [le] 2; [part]v[divide][part]t ? [nabla].(b(u, v) [nabla]v) = [thgr]f (u) v in [OHgr]T subject to no flux boundary conditions, and non-negative initial data u0 and v0 on u and v. Here we assume that c > 0, [thgr] [ge] 0 and that f (r) [ge] f (0) = 0 is Lipschitz continuous and monotonically increasing for r [isin] [0 supx[isin][OHgr]u0(x)]. Throughout the paper we restrict ourselves to the model degenerate case b(u, v) := [sgr] u v, where [sgr] > 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.