The Keller-Segel model with small diffusivity

We study the classical model for chemotaxis, the so-called Keller-Segel model, which is a drift-diffusion equation for the cell density coupled with an elliptic equation describing the evolution of the chemoattractant. We investigate the case of small cell diffusivity and, in particular, the hyperbolic limit of the system as the diffusion coefficient goes to zero. Considering a model where the drift term vanishes at high cell densities leads to a nonlinear equation which allows the formation of shocks in the limit. Moreover, we investigate the long term behaviour of solutions and derive a system of ordinary differential equations describing the slow motion of internal layers.