The existence and stability of spikes in the one-dimensional Keller-Segel model with logistic growth

It is well known that Keller-Segel models serve as a paradigm to describe the self aggregation phenomenon, which exists in a variety of biological processes such as wound healing, tumor growth, etc. In this paper, we study the existence of monotone decreasing spiky steady state and its linear stability property in the Keller-Segel model with logistic growth over one-dimensional bounded domain subject to homogeneous Neumann boundary conditions. Under the assumption that chemo-attractive coefficient is asymptotically large, we construct the single boundary spike and next show this non-constant steady state is locally linear stable via Lyapunov-Schmidt reduction method. As a consequence, the multi-symmetric spikes are obtained by reflection and periodic extension. In particular, we present the formal analysis to illustrate our rigorous theoretical results.