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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 934745, 11 pages
http://dx.doi.org/10.1155/2013/934745
Research Article

A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

1College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China

Received 4 October 2012; Revised 25 December 2012; Accepted 9 January 2013

Academic Editor: M. Lakshmanan

Copyright © 2013 Shengmao Fu and Ji Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , 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( ). 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.