Exponential Stabilization for the Nonlinear Schrödinger Equation with Localized Damping

Results of exponential stability related to the nonlinear Schrödinger equation posed on the whole real line will be discussed in this paper. We shall use local smoothing effect associated with the semigroup of the Schrödinger equation, Strichartz estimates, and compactness tools to determine the exponential decay of the energy, by assuming the initial data in convenient bounded sets of L2.