Well-posedness and energy decay estimates in the Cauchy problem for the damped defocusing Schrödinger equation

In this paper, we study the existence at the H1-level as well as the stability for the damped defocusing Schrödinger equation in Rd. 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 L2-level and (H1∩Lp+2)-level energies are established with the help of direct multipliers method, coupled with refined energy estimates and a lower semi-continuity argument.