A mathematical characterization for patterns of a Keller-Segel model with a cubic source term
Summary: This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in a \(d\)-dimensional box (\(d = 1, 2, 3\)), which describes the movement of cells in response to the presence of a chemical signal substance. 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 differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.
scientific article; zbMATH DE number 6171650