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

We consider the following nonlinear parabolic system delta u/delta t - c Delta u = -f(u)v in Omega(T) := Omega x (0, T), Omega subset of R-d, delta v/delta t - del center dot (b(u) del [psi (v)]) = theta f(u)v in Omega(T) 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 > theta >= 0 and that f is an element of C-loc(0,1) ([0,infinity)) is increasing with f(0) = 0. The system is possibly doubly-degenerate in that b is an element of C-loc(1,1)([0,infinity)) is only non-negative, and psi is an element of C-1([0,infinity)) boolean AND C-2((0,infinity)) is convex, strictly increasing with psi(0) = 0 and possibly psi'(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., 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) := r(q) and psi( r) := r(p) with q >= 2 and p >= 1, for example.