Mathematical treatment of PDE model describing chemotactic E. coli colonies

We consider an initial-boundary value problem describing the formation of colony patterns of bacteria Escherichia coli. This model consists of reaction-diffusion equations coupled with the Keller-Segel system from the chemotaxis theory in a bounded domain, supplemented with zero-flux boundary conditions and with non-negative initial data. We answer questions on the global in time existence of solutions as well as on their large time behaviour. Moreover, we show that solutions of a related model may blow up in a finite time.