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Stability of the weakly coupled wave system with hybrid boundary conditions

We investigate the stability of a weakly coupled wave system with hybrid boundary conditions. Between the two coupled equations, only one is directly damped. In particular, the coupling coefficients allow to be different. Without any geometric conditions on the damping region, we show that the strong solutions of the system will decay logarithmically. The proof is essentially based on an interpolation type inequality which can be obtained by the method of Carleman estimates.