In this paper, we consider the energy decay of a damped hyperbolic system ofwave-wave type which is coupled through the velocities. We are interested inthe asymptotic properties of the solutions of this system in the case ofindirect nonlinear damping, i.e. when only one equation is directly damped by anonlinear damping. We prove that the total energy of the whole system decays asfast as the damped single equation. Moreover, we give a one-step generalexplicit decay formula for arbitrary nonlinearity. Our results shows that thedamping properties are fully transferred from the damped equation to theundamped one by the coupling in velocities, different from the case ofcouplings through displacements as shown in \cite{AB01, ACK01, AB02, AL12} forthe linear damping case, and in \cite{AB07} for the nonlinear damping case. Theproofs of our results are based on multiplier techniques, weighted nonlinearintegral inequalities and the optimal-weight convexity method of \cite{AB05,AB10}.
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