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 Schrodinger equation iut + δu - |u|² u = 0 in Ω X (0, ∞), subject to Dirichlet boundary conditions, where Ω ⊆ ℝ² is a bounded domain with smooth boundary ρΩ = Γ. In addition, we prove exponential decay rates of the energy for the damped problem iut + δu - |u| ² u + ia(x)u = 0 in ℝ² X (0, ∞), provided that a(x) ≥ a0 > 0 almost everywhere in ΩR:= {x ∈ ℝ²: |x| ≥ R}, where R ≥ 0.