Uniform decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

The following coupled damped Klein-Gordon-Schrodinger equations are considered $$i\psi_{t} +\triangle\psi + i\alpha|\psi|^{2}\psi = \phi\psi\,\,{\rm in}\,\,\Omega \times (0,\infty), (\alpha > 0)$$ $$\phi_{tt} - \triangle\phi + a(x)\phi_{t} = |\psi|^{2}{\mathcal{X}}_{\omega}\,{\rm in}\,\Omega \times (0,\infty),$$ where Ω is a bounded domain of \({\mathbb{R}}^{n}, n \leq 3\), with smooth boundary Γ and ω is a neibourhood of \(\partial\Omega\). Here \({\mathcal{X}}_{\omega}\) represents the characteristic function of ω. Assuming that \(a \in W^{1,\infty}(\Omega)\) is a nonnegative function such that \(a(x) \geq a_{0} > 0\) a. e. in ω, polynomial decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous results given by the authors in the reference [CDC].