Fully parabolic Keller-Segel model for chemotaxis with prevention of overcrowding

In this paper we study a fully parabolic version of the Keller-Segel system in the presence of a volume filling effect which prevents blow-up of the L norm. This effect is sometimes referred to as prevention of overcrowding. As in the parabolic-elliptic version of this model (previously studied in (Burger et al 2006 SIAM J. Math. Anal. 38 1288-315)), the results in this paper basically infer that the combination of the prevention of the overcrowding effect with a linear diffusion for the density of cells implies domination of the diffusion effect for large times. In particular, first we show that both the density of cells and the concentration of the chemical vanish uniformly for large times, then we prove that the density of cells converges in L1 towards the Gaussian profile of the heat equation as time goes to infinity, at a rate which differs from the rate of convergence to self-similarity for the heat equation by an arbitrarily small constant ('quasi-sharp rate').