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.
"Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions." Commun. Math. Sci. 6 (1) 1 - 28, March 2008.