Exponential stability for the \(2\)-D defocusing Schrödinger equation with locally distributed damping. (English) Zbl 1240.35509

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Natali, F.

Differ. Integral Equ. 22, No. 7-8, 617-636 (2009).

Summary: This paper is concerned with the study of the unique continuation property associated with the defocusing Schrödinger equation \(iu_{t}+\Delta u-| u| ^{2}u=0\) in \(\Omega \times (0,\infty )\), subject to Dirichlet boundary conditions, where \(\Omega \subset \mathbb {R}^2\) is a bounded domain with smooth boundary \(\partial \Omega \). In addition, we prove exponential decay rates of the energy for the damped problem \(iu_{t}+\Delta u-| u| ^{2}u+ia(x)u=0\) in \(\mathbb {R}^2\times (0,\infty )\), provided that \(a(x)\geq a_0>0\) almost everywhere in \(\{x\in \mathbb {R}^2:| x| \geq R\}\), where \(R>0\).

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B40	Asymptotic behavior of solutions to PDEs
35B60	Continuation and prolongation of solutions to PDEs

Keywords:

\(2\)-D defocusing Schrödinger equation; locally distributed damping; exponential stability; unique continuation property