Boundary symmetry breaking via logistic damping in a chemotaxis-growth system

TL;DR

This paper proves global stability for a chemotaxis-growth model with dynamic boundary conditions by introducing dynamic reference profiles and an entropy functional, addressing boundary data challenges in 1D systems.

Contribution

It introduces a novel approach using dynamic reference profiles and entropy functionals to establish stability in chemotaxis-growth models with boundary dynamics.

Findings

Proved global stability of the chemotaxis-growth system.

Developed a method to handle time-dependent boundary data.

Established energy estimates ensuring solution boundedness.

Abstract

We establish global stability for a chemotaxis-growth model with logarithmic sensitivity under dynamic Dirichlet boundary conditions on a 1D domain. We analyze both parabolic-parabolic and parabolic-hyperbolic systems. The key challenge is handling time-dependent boundary data for the unknown functions. We overcome this by introducing dynamic reference profiles which suitably interpolate boundary values. Using an expanded entropy functional measuring deviation from these profiles, we prove energy estimates the uniform boundedness of solutions and global asymptotic stability of perturbations.