Stability of coupled wave equations with variable coefficients, localised Kelvin–Voigt damping and time delay

We consider the stabilization of two weakly dissipative wave equations with variable coefficients, localized Kelvin–Voigt damping, mixed boundary condition and time delay. The well posedness is obtained by the semigroup method. Using a unique continuation result, we show that the system is strongly stable. Then, we show that the system is not exponentially stable. Finally, using a frequency domain approach combined with multiplier method, we establish a polynomial energy decay rate for the undelayed system. Then, we prove that the system with delay has the same decay rate.