Attractors for Singular-Degenerate Porous Medium Type Equations Arising in Models for Biofilm Growth

TL;DR

This paper studies the long-term behavior of solutions to singular-degenerate porous medium equations, proving the existence of global and exponential attractors, and extending results to biofilm growth models.

Contribution

It establishes the existence of global and exponential attractors for a broad class of singular-degenerate equations, including coupled systems relevant to biofilm growth.

Findings

Existence of global attractors under general conditions

Existence of exponential attractors with finite fractal dimension

Extension of results to coupled reaction-diffusion systems

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 H\"older 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.