Decay for a Keller–Segel Chemotaxis Model

We study a model due to E.F. Keller and L.A. Segel, which describes the movement of a microbiological population under the influence of chemical substances produced by the biological organisms themselves. There has been an enormous amount of recent activity on Keller–Segel models, especially with regard to blow-up or spike formation of the solution. In this paper, we concentrate on deriving conditions from a fully nonlinear analysis which ensure the solution will decay to a constant steady state. These thresholds are useful in understanding when organisms will not aggregate to form growths.