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 presence of a volume filling effect which prevents blow up of the L1 norm. This effect is sometimes referred to as prevention of overcrowding. As in the parabolic elliptic version of this model (previously studied in [BDFDS06]), the results in this paper basically infer that the combination of the prevention of overcrowding effect with a linear diusion for the density of cells implies domination of the diusion 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 toward the Gaussian profile of the heat equation as time goes to infinity, with a rate which differs from the rate of convergence to self similarity for the heat equation by an arbitrarily small constant (`quasi sharp rate').