Optimal decay rate for the local energy of a unbounded network

We consider the wave equation on a unbounded network with Dirichlet and Kirchhoff conditions. We study the local energy decay of the solution. We prove that the energy is exponentially decaying and we give an exact formula for exponential decay rate. The limit energy is also given in terms of the initial conditions. The results are obtained using two approaches. A direct one uses type τ operators in the case of equal edge lengths. The other one is based on a spectral investigation of an associated linear operator leading to the correspondence between the resonances width and the local energy decay rate.