In this paper we study the asymptotic dynamics for semilinear defocusing Schrödinger equation subject to a damping locally distributed on a n-dimentional compact Riemannian manifold $ M^n $ without boundary. The proofs are based on a result of unique continuation property, in the construction of a function $ f $ whose Hessian is positive definite and $ \Delta f = C_0 $ in some region contained in $ M $ and about the smoothing effect due to Aloui adapted to the present context.
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