Blowup and global solutions in a chemotaxis–growth system

We study a Keller–Segel type of system, which includes growth and death of the chemotactic species and an elliptic equation for the chemo-attractant. The problem is considered in bounded domains with smooth boundary as well as in the whole space. In case the random motion of the chemotactic species is neglected, a hyperbolic–elliptic problem results, for which we characterize blow-up of solutions in finite time and existence of regular solutions globally in time, in dependence on the systems parameters. In this case, convexity of the domain is needed. For the parabolic–elliptic problem in dimensions three and higher, we establish global existence of regular solutions in a limiting case, which is an extension of the results given by Tello and Winkler (2007).