Traveling wave solutions of a singular Keller-Segel system with logistic source

This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.