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

This paper is concerned with the study of the unique continuation property associated with the defocusing Schrödinger equation {eqnarray*} iuₜ +Δ u - |u|^2u =0 ~ in Ω × (0,∞), {eqnarray*} subject to Dirichlet boundary conditions, where Ω ⊂ R² is a bounded domain with smooth boundary ∂ Ω=Γ. In addition, we prove exponential decay rates of the energy for the damped problem {eqnarray*} iuₜ +Δ u - |u|^2u +i a(x) u =0 in R^2 × (0,∞), {eqnarray*} provided that a(x) ≥ a₀ >0 almost everywhere in ΩR:=∈ R² : |x| ≥ R\, where $R>0$.