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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]

Inspired by the highly cited work due to Lasiecka and Tataru (1993) [20] , a semilinear model of the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks is considered. While for a homogeneous medium Ω and for certain geometries nonlinear feedback is strong enough to derive uniform decay rates of the energy associated with the model, in contrast, in an inhomogeneous medium, and for general geometry of Ω, the problem requires a new approach. In the first case, the rays of the geometric optics are straight lines and the geometry is favorable. However, in the second one, more damping is required to well-known obtain a similar result. For this purpose, arguments of microlocal analysis are used instead of the multipliers technique. [ABSTRACT FROM AUTHOR]