Preventing blow-up in a chemotaxis model

This paper deals with a chemotactic model of Keller-Segel type. The main feature of the Keller-Segel model is the possibility of blow-up of solutions in a finite time. To eliminate the possibility of blow-up a modified version of Keller-Segel model is introduced. The blow-up control relies on the presence of a pressure function, which increases faster than a logarithm for high enough cells densities: for such a pressure function the solutions cannot blow up in a finite time. Some other conditions are introduced to ensure the global boundedness.