Global regularity versus infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion

A system of quasi-linear parabolic and elliptic-parabolic equations describing chemotaxis is studied. Due to the assumed presence of a volume-filling effect it is assumed that there is an impassable threshold for the density of cells. This assumption leads to singular or degenerate operators in both the diffusive and the chemotactic components of the flux of cells. We improve results from earlier works and find critical conditions which reflect the interplay between diffusion and chemotaxis and warrant that classical solutions are global in time and separated uniformly from the threshold. In the case of degenerate diffusion for the elliptic-parabolic version of the model we prove the existence of radially symmetric solutions which exhibit a phenomenon of infinite-time singularity formation in that they are global and smooth but attain the threshold in the large time limit.