Asymptotic models for the evolution of a circular biofilm

We study a class of lubrication-type equations modeling the spread of thin poroelastic biofilms at air/agar interfaces. Starting from a biofilm slab model, we perform a formal multi-scale asymptotic expansion to derive a reduced nonlinear evolution equation with time-dependent coefficients. First, we establish a local-in-time existence result as well as a continuation criteria. Moreover, under suitable assumptions on the radius of the biofilm, we show the existence of global in time solutions. We conclude with some numerical simulations illustrating the emergence of instabilities and potential singularities in finite time.