EXISTENCE ANALYSIS FOR A REACTION-DIFFUSION CAHN-HILLIARD-TYPE SYSTEM WITH DEGENERATE MOBILITY AND SINGULAR POTENTIAL MODELING BIOFILM GROWTH

. The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reactiondiffusion 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.