EXPONENTIAL DECAY FOR THE COUPLED KLEIN-GORDON-SCHRODINGER EQUATIONS WITH LOCALLY DISTRIBUTED DAMPING

The following coupled damped Klein-Gordon-Schrodinger equations are considered i psi(t) + Delta psi + i alpha b(x)(-Delta)(1/2) b(x)psi = phi psi chi(omega) in Omega x (0, infinity), (alpha > 0) phi(tt) - Delta phi + a(x)phi(t) = vertical bar psi vertical bar(2)chi(omega) in Omega x (0, infinity), where Omega is a bounded domain of R-n, n = 2, with smooth boundary Gamma and omega is a neighbourhood of partial derivative Omega satisfying the geometric control condition. Here chi(omega) represents the characteristic function of omega. Assuming that a, b is an element of W-1,W-infinity (Omega) boolean AND C-infinity (Omega) are nonnegative functions such that a(x) >= a(0) > 0 in omega and b(x) >= b(0) > 0 in omega, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous results given by Cavalcanti et. al in the reference [7].