Nonlinear instability for a volume-filling chemotaxis model with logistic growth (Q1722431)

Summary: This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a \(d\)-dimensional box \(\mathbb{T}^d = (0, \pi)^d (d = 1,2, 3)\). It is proved that given any general perturbation of magnitude \(\delta\), its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order \(\ln(1 / \delta)\). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.