Pattern formation (I): The Keller–Segel model

We investigate nonlinear dynamics near an unstable constant equilibrium in the classical Keller–Segel model. Given any general perturbation of magnitude δ, 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 . Our result can be interpreted as a rigorous mathematical characterization for early pattern formation in the Keller–Segel model.