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摘要
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.., 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.
