  In this paper we study a semilinear hyperbolic-parabolic system as a model
for some chemotaxis phenomena evolving on networks; we consider transmission
conditions at the inner nodes which preserve the fluxes and non- homogeneous
boundary conditions having in mind phenomena with inflow of cells and food
providing at the network exits. We give some conditions on the boundary data
which ensure the existence of stationary solutions and we prove that these ones
are asymptotic profiles for a class of global solutions.
