Well-posedness and asymptotic limits for a degenerate Keller-Segel system with volume filling

TL;DR

This paper investigates a degenerate Keller-Segel system with volume filling, establishing global solutions, pattern formation, and asymptotic limits through analytical and numerical methods.

Contribution

It introduces new analytical results for existence, uniqueness, and limits of solutions in a complex Keller-Segel model with volume filling effects.

Findings

Existence of global weak solutions

Exponential convergence to steady state

Pattern formation in one dimension

Abstract

A class of parabolic-parabolic Keller-Segel systems with degenerate diffusion and volume filling is studied in a bounded domain subject to no-flux boundary conditions. The equations are derived from a multiphase fluid model. The interplay between nonlinear diffusion and density saturation leads to a rich variety of behaviors across different parameter regimes. We establish the existence of global weak solutions, a weak-strong uniqueness result, the exponential convergence to the homogeneous steady state, pattern formation in one spatial dimension, as well as the parabolic-elliptic and vanishing diffusion limits. The analysis relies on a priori estimates derived from suitable entropy functionals. Pattern formation is demonstrated by reducing the system to a first-order equation and conducting a detailed analysis of the resulting nonlinearity. Numerical simulations from a one-dimensional finite-volume scheme illustrate the asymptotic regimes.