Singularity formation in chemotaxis systems with volume-filling effect

A parabolic–elliptic model of chemotaxis which takes into account volume-filling effects is considered under the assumption that there is an a priori threshold for the cell density. For a wide range of nonlinear diffusion operators including singular and degenerate ones it is proved that if the taxis force is strong enough with respect to diffusion and the initial data are chosen properly then there exists a classical solution which reaches the threshold at the maximal time of its existence, no matter whether the latter is finite or infinite. Moreover, we prove that the threshold may even be reached in finite time provided the diffusion of cells is non-degenerate.