Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks

In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.