Convergence rate of solutions towards spiky steady state for the Keller-Segel system with logarithmic sensitivity

We study the large time behaviors of solutions to the Keller-Segel system with logarithmic singular sensitivity in the half space, where biological mixed boundary conditions are prescribed. The existence and asymptotic stability of spiky steady states of this system were proved by Carrillo et al. (2021). In this paper we obtain convergence rate of solutions towards the steady state under appropriate initial perturbations. The proofs are based on a Cole-Hopf type transformation and a weighted energy method, where the weights are artfully constructed. (c) 2023 Elsevier Ltd. All rights reserved.