Coupled nonlinear degenerate systems arise in various applications of societal relevance, such as biofilm growth, wildfires, and reactive transport in porous media. One of the equations in the system, describing the evolution of a biomass, exhibits degenerate and singular diffusion behaviour. The other equations are either of advection-reaction-diffusion type or ordinary differential equations modelling nutrient dispersion. In our analysis, we first propose a backward Euler time-discretization of the problem where the reactive terms coupling the equations are estimated semi-implicitly, and thus, the equations can be solved sequentially. Properties such as well-posedness, boundedness, and positivity of the time-discrete solutions are proven, and the existence of the time-continuous solutions is shown by passing the time-step size to zero. Global-in-time well-posedness is established for Dirichlet and mixed boundary conditions, whereas only local well-posedness can be shown for homogeneous Neumann boundary conditions. Using a suitable barrier function and comparison theorems we formulate sufficient conditions for finite-time blow-up or uniform boundedness of solutions. Assuming additional structural assumptions we also prove the uniqueness of solutions. The time-discretization method serves as an efficient way to solve such problems and numerical experiments are provided that support this.
Then, we show the existence of traveling wave (TW) solutions for a special case of PDE-ODE coupled models arising in the growth of cellulolytic biofilms. TW solutions for such systems have previously been observed numerically as well as in experiments. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. We present numerical results that exhibit the existence and stability of the TWs, along with corroborating predictions on parametric dependence.
This is joint work with J.D. Dockery (Montana State University), H.J. Eberl (University of Guelph), J.M. Hughes (University of British Columbia), I.S. Pop (Hasselt University), R.K.H. Smeets (University of Amsterdam) and S. Sonner (Radboud University).