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

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 $R>0$.