Stability of spiky solution of Keller–Segel's minimal chemotaxis model

Abstract

A huge volume of research has been done for the simplest chemotaxis model (Keller–Segel's minimal model) and its variants, yet, some of the basic issues remain unresolved until now. For example, it is known that the minimal model has spiky steady states that can be used to model the important cell aggregation phenomenon, but the stability of monotone spiky steady states was not shown. In this paper, we derive, first formally and then rigorously, the asymptotic expansion of these monotone steady states, and then we use this fine information on the spike to prove its local asymptotic stability. Moreover, we obtain the uniqueness of such steady states. We expect that the new ideas and techniques for rigorous asymptotic expansion and spectrum analysis presented in this paper will be useful in attacking and hence stimulating research on other more sophisticated chemotaxis models.