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 discontinuous 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 non-negative solutions - the latter in the case of small (in the L1-norm) initial data - as well as their uniqueness.
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