Global smooth solutions for a hyperbolic chemotaxis model on a network
  In this paper we study a semilinear hyperbolic-parabolic system modeling biological phenomena evolving on a network composed by oriented arcs. We prove the existence of global (in time) smooth solutions to this problem. The result is obtained by using energy estimates with suitable transmission conditions at nodes. 
Francesca Romana Guarguaglini and Roberto Natalini
math.AP