Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

The following coupled damped Klein-Gordon-Schrödinger equations are considered {document}$ {eqnarray*} iψ_t + Δ ψ + i α b(x)(|ψ|² + 1)ψ & = & φ ψ χω \; in\; Ω × (0, ∞), \; (α >0)\ \\φₜₜ - Δ φ + a(x) φ_t & = & |ψ|^2 χω\; in\; Ω × (0, ∞), {eqnarray*} document where Ω $ is a bounded domain of $ R^2 $, with smooth boundary $ Γ $ and $ ω $ is a neighbourhood of $ ∂ Ω $ satisfying the geometric control condition. Here $ χω $ represents the characteristic function of $ ω $. Assuming that $ a, b∈ L∞(Ω) $ are nonnegative functions such that $ a(x) ≥ a_0 >0 $ in $ ω $ and $ b(x) ≥ b₀ > 0 $ in $ ω $, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous ones given by Cavalcanti et. al in the reference [9] and [1].