This paper is devoted to a study of the asymptotic behaviour 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 to the one-dimensional case is the motion of these plateau-like solutions, respectively of the interfacial regions describing from zero to the maximal density. 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 mainly change its shape if cells diffuse along its boundary. The theoretical results on the asymptotic behaviour are supplemented by several numerical simulations on two- and three-dimensional domains.

@Article{BDS08,
  author       = {Burger, M. and Dolak-Struss, Y. and Schmeiser, C.},
  title        = {Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions},
  journal      = {Commun. Math. Sci.},
  volume       = {6},
  pages        = {1-28},
  year         = {2008},
  url          = \{/2008/BDS08},
}