Communications in Mathematical Sciences

Volume 6 (2008)

Number 1

Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions

Pages: 1 – 28

DOI: http://dx.doi.org/10.4310/CMS.2008.v6.n1.a1

Authors

Martin Burger

Yasmin Dolak-Struss

Christian Schmeiser

Abstract

This paper is devoted to a study of the asymptotic behavior of solutions of a chemotaxis model with logistic terms in multiple spatial dimensions. Of particular interest is the practically relevant case of small diffusivity, where (as in the one-dimensional case) the cell densities form plateau-like solutions for large time.

The major difference from the one-dimensional case is the motion of these plateau-like solutions with respect to the plateau boundaries separating zero density regions from maximum density regions. This interface motion appears on a non-logarithmic time scale and can be interpreted as a surface diffusion law. The biological interpretation of the surface diffusion is that a packed region of cells can change its shape mainly if cells diffuse along its boundary.

The theoretical results on the asymptotic behavior are supplemented by several numerical simulations on two- and three-dimensional domains.

Keywords

chemotaxis models; asymptotic behavior; interface motion; surface diffusion

2010 Mathematics Subject Classification

35R35, 92C17

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