Abstract
In this paper, we study the exponential stability for the semilinear defocusing Schrödinger equation with locally distributed damping on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$. The proofs are based on a result of unique continuation property due to Cavalcanti et al. [15] and on a forced smoothing effect due to Aloui [2] combined with ideas from Cavalcanti et. al. [15], [16] adapted to the present context.
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