Attractors for Singular-Degenerate Porous Medium Type Equations Arising in Models for Biofilm Growth
Zehra Şen, Stefanie Sonner
TL;DR
This work studies the long-time dynamics of singular-degenerate porous-medium type equations and their biofilm-inspired coupled systems in bounded domains with Dirichlet boundaries. It develops a two-tier approach: first establishing global attractors via smooth nondegenerate approximations and asymptotic compactness, then constructing exponential attractors under global Hölder continuity and a sign condition near degeneracy, which yields finite fractal dimension for the global attractor. The results cover scalar equations and extend to coupled reaction-diffusion systems, providing a rigorous framework for finite-dimensional long-time behavior in models with sharp interfaces and diffusion degeneracies. This offers theoretical grounding for biofilm-growth models and related processes, with potential implications for understanding pattern formation and interface propagation in bounded domains.
Abstract
We investigate the long-time behaviour of solutions of a class of singular-degenerate porous medium type equations in bounded domains with homogeneous Dirichlet boundary conditions. The existence of global attractors is shown under very general assumptions. Assuming, in addition, that solutions are globally Hölder continuous and the reaction terms satisfy a suitable sign condition in the vicinity of the degeneracy, we also prove the existence of an exponential attractor, which, in turn, yields the finite fractal dimension of the global attractor. Moreover, we extend the results for scalar equations to systems where the degenerate equation is coupled to a semilinear reaction-diffusion equation. The study of such systems is motivated by models for biofilm growth.