Stability of traveling waves of the Keller–Segel system with logarithmic sensitivity

Abstract

Proceeding with a series of works (Refs. 12, 23–25) by the authors, this paper establishes the nonlinear asymptotic stability of traveling wave solutions of the Keller–Segel system with nonzero chemical diffusion and linear consumption rate, where the right asymptotic state of cell density is vacuum (zero) and the initial value is a perturbation with zero integral from the spatially shifted traveling wave. The main challenge of the problem is various singularities caused by the logarithmic sensitivity and the vacuum asymptotic state, which are overcome by a Hopf–Cole type transformation and the weighted energy estimates with an unbounded weight function introduced in the paper.