Pattern formation (I): The Keller-Segel Model

Abstract

We investigate nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Given any general perturbation of magnitude delta, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of In 1/delta. Our result can be interpreted as a rigorous mathematical characterization for early pattern formation in the Keller-Segel model. (C) 2010 Elsevier Inc. All rights reserved.