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

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