A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term. (27th February 2013) - Fu, Shengmao; Liu, Ji
Abstract : 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 δ, 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 / δ ). 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.