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

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• 作者：Marcelo M. Cavalcanti^a ; ¹ ; ^{mmcavalcanti@uem.br} ; Wellington J. Corrê ; a^b ; Valé ; ria N. Domingos Cavalcanti^a ; ² ; Louis Tebou^c
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2521-2539
• 全文大小：340 K
• 卷排序：262

文摘

In this paper, we study the existence at the H1H1-level as well as the stability for the damped defocusing Schrödinger equation in RdRd. 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=1d=1 or d=2d=2, we obtain the uniqueness. Decay estimates for the L2L2-level and (H1∩Lp+2)(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.