  The global existence of bounded weak solutions to a diffusion system modeling
biofilm growth is proven. The equations consist of a reaction-diffusion
equation for the substrate concentration and a fourth-order Cahn-Hilliard-type
equation for the volume fraction of the biomass, considered in a bounded domain
with no-flux boundary conditions. The main difficulties are coming from the
degenerate diffusivity and mobility, the singular potential arising from a
logarithmic free energy, and the nonlinear reaction rates. These issues are
overcome by a truncation technique and a Browder-Minty trick to identify the
weak limits of the reaction terms. The qualitative behavior of the solutions is
illustrated by numerical experiments in one space dimension, using a BDF2
(second-order backward Differentiation Formula) finite-volume scheme.
