Exponential decay for the cubic Schrödinger equation on a dissipative waveguide

We derive an exponential decay rate of energy for the cubic Schrödinger equation on a dissipative waveguide with a damping term that is effective locally on a neighborhood at the boundary and at infinity. In this paper, we give a new example for which the geometric control condition is sufficient to obtain an exponential decay of energy Schrödinger equation. The proof is based on Strichartz's estimates and the smoothing effects argument associated with the linear Schrödinger equation(LS).