Exponential stability for the coupled Klein-Gordon-Schrodinger equations with locally distributed damping in unbounded domains

摘要

The following coupled damped Klein-Gordon-Schrodinger equations are consideredi psi(t) + Delta psi + i alpha b(x)(vertical bar psi vertical bar(2) + 1)psi = phi psi chi(Omega R) in R-n x (0, infinity) (alpha 0),phi(tt) - Delta phi + phi + a(x)phi(t) = vertical bar psi vertical bar(2)chi(Omega R) in R-n x (0, infinity),where Omega(R) = R-n B-R = {vertical bar x vertical bar = R} and chi(Omega R) represents the characteristic function of Omega(R). Assuming that a, b is an element of W-1,W-infinity (R-n) are nonnegative functions such that a(x) = a(0) 0 in Omega(R) and b(x) = b(0) 0 in Omega(R), 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 references (NoDEA, Nonlinear Differ. Equ. Appl. 15 (2008) 91-113), (NoDEA, Nonlinear Differ Equ. Appl. 7 (2000) 285-307), (Communications on Pure and Applied Analysis 17 (2018) 2039-2061) and (Evolution Equations and Control Theory 8 (2019) 847-865).