EXISTENCE, UNIQUENESS AND APPROXIMATION OF A DOUBLY-DEGENERATE NONLINEAR PARABOLIC SYSTEM MODELLING BACTERIAL EVOLUTION

Abstract

We consider the following nonlinear parabolic system [Formula: see text] subject to no flux boundary conditions, and non-negative initial data u0 and v0 on u and v. Here we assume that c > 0, θ ≥ 0 and that [Formula: see text] is increasing with f(0) = 0. The system is possibly doubly-degenerate in that [Formula: see text] is only non-negative, and ψ ∈ C1([0,∞)) ∩ C2((0,∞)) is convex, strictly increasing with ψ(0) = 0 and possibly ψ'(0) = 0. The above models the spatiotemporal evolution of a bacterium species on a thin film of nutrient, where u is the nutrient concentration and v is the bacterial cell density. Under some further mild technical assumptions on b and ψ, we prove the existence and uniqueness of a weak solution to the above system. Moreover, we prove error bounds for a fully practical finite element approximation of this system. All of our results apply to the choices b(r) ≔ rq and ψ(r) ≔ rp with q ≥ 2 and p ≥ 1, for example.