## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 7/8 (2009), 617-636.

### Exponential stability for the $2$-D defocusing Schrödinger equation with locally distributed damping

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

#### Abstract

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

#### Article information

**Source**

Differential Integral Equations Volume 22, Number 7/8 (2009), 617-636.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019541

**Mathematical Reviews number (MathSciNet)**

MR2532114

**Zentralblatt MATH identifier**

1240.35509

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Secondary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx]

#### Citation

Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Natali, F. Exponential stability for the $2$-D defocusing Schrödinger equation with locally distributed damping. Differential Integral Equations 22 (2009), no. 7/8, 617--636.https://projecteuclid.org/euclid.die/1356019541