Quasilinear Degenerate Evolution Systems Modelling Biofilm Growth: Well‐Posedness and Qualitative Properties
Koondanibha Mitra, Stefanie Sonner · Mathematical Methods in the Applied Sciences · 2025
ABSTRACT We analyze nonlinear degenerate coupled partial differential equation (PDE)‐PDE and PDE‐ordinary differential equation (ODE) systems that arise, for example, in the modelling of biofilm growth. One of the equations, describing the evolution of a biomass density, exhibits degenerate and singular diffusion. The other equations are either of advection‐reaction‐diffusion type or ODEs. Under very general assumptions, the existence of weak solutions is proven by considering regularized systems, deriving uniform bounds, and using fixed point arguments. Assuming additional structural assumptions we also prove the uniqueness of solutions. 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. Finally, we show that solutions of the degenerate parabolic equation inherit additional global spatial regularity if the diffusion coefficient has a power‐law growth.