## Talk by Dr. Hermann Eberl (Dept. of Mathematics and Statistics, University of Guelph, Canada)

### 11/5/2020  3:10-4:00pm  Webex Meeting

#### Abstract:

Starting point will be a quasilinear system of diffusion-reaction equations, one of which (i) degenerates if the dependent variable vanishes, and (ii) attains a super-diffusion
singularity if the dependent variable reaches an a priori known critical value.  We introduced this type of equations many years ago to model bacterial biofilms. Both effects (i) and (ii) introduce numerical (and analytical) challenges: (i)  like the porous medium equation, leads to the formation of sharp  interfaces and gradient blow-up. (ii) is characterized by blow-up of the density-dependent diffusion coefficient,
which prevents the application of standard arguments to show boundedness of solutions.

After semi-discretization in space one obtains a very stiff system of ordinary differential equations that, owing to (ii), {\it a priori} does not satisfy a Lipschitz condition. We explore this system using regulariztaion techniques. We are able to prove that (for practically reasonable assumptions on initial and boundary conditions) the numerical
solutions remain bounded by unity (as they ought to) and that traditional ODE integrators can be safely applied.  We present a small application of this method that is concerned with quorum sensing cross-talk in a biofilm community. This is joint work with Maryam Ghasemi (now at the University of Waterloo).