Local and Global Solutions for a Hyperbolic-Elliptic Model of Chemotaxis on a Network

In this paper we study a hyperbolic-elliptic system on a network which arises in biological models involving chemotaxis. We also consider suitable transmission conditions at internal points of the graph which on one hand allow discontinuities density functions at nodes, and on the other guarantee the continuity of the fluxes at each node. Finally, we prove local and global existence of nonnegative solutions – the latter in the case of small (in the L1-norm) initial data – as well as their uniqueness.