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

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,∞). 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,∞) 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 the anti-derivative.