Nonlinear Schrödinger equations on a finite interval with point dissipation

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doi:10.3934/mcrf.2019017

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2019, Volume 9, Issue 2: 351-384. Doi: 10.3934/mcrf.2019017

This issue Previous Article The generalised singular perturbation approximation for bounded real and positive real control systems Next Article Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I

Nonlinear Schrödinger equations on a finite interval with point dissipation

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Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA USA

^* Corresponding author: Shu-Ming Sun
^* Corresponding author: Shu-Ming Sun

Received: June 30, 2017

Revised: June 30, 2018

Published: May 2019

The research was partially supported by the National Science Foundation under grant No. DMS-1210979

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Abstract

The paper considers the initial value problem of a general type of nonlinear Schrödinger equations

$ iu_t+u_{xx}+f(u) = 0 , \;\;\;\; u ( x, 0 ) = w_0 (x) $

posed on a finite domain $ x\in [0, L] $ with an $ L^2 $-stabilizing feedback control law $ u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t), $ where $ L>0 $, $ \alpha, \beta $ are real constants with $ \alpha\beta<0 $ and $ \beta\neq \pm 1 $, and $ f(u) $ is a smooth function from $ \mathbb{C} $ to $ \mathbb{C} $ satisfying some growth conditions. It is shown that for $ s \in \left ( \frac12, 1\right ] $ and $ w_0 (x) \in H^s(0, L ) $ with the boundary conditions described above, the problem is locally well-posed for $ u \in C([0, T]; H^s (0, L )) $. Moreover, the solution with small initial condition exists globally and approaches to 0 as $ t \rightarrow + \infty $.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q93.

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