Exponential stability for the nonlinear Schrodinger equation on a star-shaped network

In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrodinger equation on a star-shaped network R, where the damping is localized on one branch at the infinity and the initial data are assumed to be in L-2(R). We use the fixed point argument and Strichartz estimates on a star-shaped network to obtain results of local and global well-posedness. The proof of the exponential decay is based on smoothing properties for Schrodinger equation, on the unique continuation and on the semigroup properties.