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In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L2-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the L2-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively. The distinctive feature of the paper is the monotonicity approach, which makes the analysis independent from the commonly used Strichartz estimates and allows us to work without artificial smoothing terms inserted into the main equation. We in addition implement a precise and efficient algorithm for studying the exponential decay established in the first part of the paper numerically. Our simulations illustrate the efficacy of the proposed control design. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence at the H 1 -level as well as the stability for the damped defocusing Schrödinger equation in R d . The considered damping coefficient is time-dependent and may vanish at infinity. To prove the existence, we employ the method devised by Özsarı, Kalantarov and Lasiecka [27] , which is based on monotone operators theory. In particular, when d = 1 or d = 2 , we obtain the uniqueness. Decay estimates for the L 2 -level and ( H 1 ∩ L p + 2 ) -level energies are established with the help of direct multipliers method, coupled with refined energy estimates and a lower semi-continuity argument. [ABSTRACT FROM AUTHOR]