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.

中文翻译：

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半空间中奇异 PDE-ODE 混合趋化系统的行波收敛

本文研究半空间$\R_+=[0, \infty)$中奇异PDE-ODE混合趋化系统初边值问题的渐近稳定性。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在半空间 $[0,\infty)$ 中，波浪剖面和速度由边界通量数据唯一选择。结果由 Cole-Hopf 类型变换和加权能量估计以及采用 {\color{black}} 反导数的技术证明。