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

∂v ∂t −∇ .(b(u ,v) ∇v) = θ f (u )v in ΩT 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, supx∈Ω u 0 (x)]. Throughout the paper we restrict ourselves to the model degenerate case b(u ,v) := σ u v, 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.