Pattern Dynamics Of A Keller-Segel Model

In this paper, nonlinear dynamics is considered near an unstable constant equi-librium in the Keller-Segel model with the source term of up(1u). This paper isdivided into five sections. In Section1, we prove that the positive constant steadystate of the model without chemotaxis is locally asymptotically stable. In Section2,growing modes of the chemotaxis model are investigated. In Section3, the Bootstraplemma is proved. In Section4, the main result for nonlinear instability and patternformation is given. In Section5, we show that given any general initial perturbation,its nonlinear evolution is dominated by the corresponding linear dynamics along afinite number of fixed fastest growing modes.