## Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain.(English)Zbl 1449.35045

Summary: In this paper, we study the exponential stability for the semilinear defocusing Schrödinger equation with locally distributed damping on a bounded domain $$\Omega\subset\mathbb{R}^n$$ with smooth boundary $$\partial\Omega$$. The proofs are based on a result of unique continuation property due to M. M. Cavalcanti et al. [Differ. Integral Equ. 22, No. 7–8, 617–636 (2009; Zbl 1240.35509)] and on a forced smoothing effect due to L. Aloui [Asymptotic Anal. 59, No. 3–4, 179–193 (2008; Zbl 1173.35392)] combined with ideas from M. M. Cavalcanti et. al. [loc. cit.; J. Differ. Equations 248, No. 12, 2955–2971 (2010; Zbl 1190.35206)] adapted to the present context.

### MSC:

 35B35 Stability in context of PDEs 35Q55 NLS equations (nonlinear Schrödinger equations)

### Keywords:

decay rate estimate; undamped semilinear problem

### Citations:

Zbl 1240.35509; Zbl 1173.35392; Zbl 1190.35206