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

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  • Shengmao Fu
  • Ji Liu

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.

Suggested Citation

  • Shengmao Fu & Ji Liu, 2013. "A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-11, February.
    • Handle: RePEc:hin:jnlamp:934745
    DOI: 10.1155/2013/934745
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