Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space

TL;DR

This paper proves that solutions to a singular chemotaxis PDE-ODE system in a half space converge to a uniquely determined traveling wavefront over time, using a Cole-Hopf transformation and energy estimates.

Contribution

It establishes the asymptotic stability and convergence to traveling waves for a singular chemotaxis system with boundary flux conditions in a half space.

Findings

Solutions converge to a traveling wavefront as time approaches infinity.

The wave profile and speed are uniquely determined by boundary flux data.

The proof employs a Cole-Hopf transformation and weighted energy estimates.

Abstract

This paper is concerned with the asymptotic stability of the initial-boundary value problem of a singular PDE-ODE hybrid chemotaxis system in the half space $\R_+=[0, \infty)$. We show that when the non-zero flux boundary condition at $x=0$ is prescribed and the initial data are suitably chosen, the solution of the initial-boundary value problem converges, as time tend to infinity, to a shifted traveling wavefront restricted in the half space $[0,\infty)$ where the wave profile and speed are uniquely selected by the boundary flux data. The results are proved by a Cole-Hopf type transformation and weighted energy estimates along with the technique of taking {\color{black} the} anti-derivative.