Transition layers to chemotaxis-consumption models with volume-filling effect

TL;DR

This paper studies a chemotaxis-consumption model with volume-filling effects, analyzing steady states, their transition layer behavior as chemotactic sensitivity increases, and proving nonlinear stability without parameter restrictions.

Contribution

It introduces a new analysis of transition layers and stability for chemotaxis models with volume-filling effects, using Helly's theorem and energy methods.

Findings

Existence of a unique nonconstant steady state.

Asymptotic transition layer profile as chemotactic coefficient increases.

Nonlinear stability of steady state without parameter restrictions.

Abstract

We are interested in the dynamical behaviors of solutions to a parabolic-parabolic chemotaxis-consumption model with a volume-filling effect on a bounded interval, where the physical no-flux boundary condition for the bacteria and mixed Dirichlet-Neumann boundary condition for the oxygen are prescribed. By taking a continuity argument, we first show that the model admits a unique nonconstant steady state. Then we use Helly's compactness theorem to show that the asymptotic profile of steady state is a transition layer as the chemotactic coefficient goes to infinity. Finally, based on the energy method along with a cancellation structure of the model, we show that the steady state is nonlinearly stable under appropriate perturbations. Moreover, we do not need any assumption on the parameters in showing the stability of steady state.