  We are interested in evolution phenomena on star-like networks composed of
several branches which vary considerably in physical properties. The initial
boundary value problem for singularly perturbed hyperbolic differential
equation on a metric graph is studied. The hyperbolic equation becomes
degenerate on a part of the graph as a small parameter goes to zero. In
addition, the rates of degeneration may differ in different edges of the graph.
Using the boundary layer method the complete asymptotic expansions of solutions
are constructed and justified.
