Resolvent Estimate and Energy Decay for Partially Damped Hyperbolic Systems in Exterior Domain

In this paper, we study the resolvent associated with a system of two coupled hyperbolic equations (wave–wave or wave‐Klein–Gordon) in exterior domains, where only one equation is directly damped. Under the assumptions, that the damping and coupling terms are effective at infinity and satisfy the geometric control condition near the boundary, we prove that the resolvent norm in the energy space grows as , while its norm in is of order . These estimates allow us to establish a decay rate of at least for the total energy of the system with initial data in . Moreover, we improve upon existing results in two key aspects: For the wave–wave system, we obtain the same energy decay rate as in previous works, but under less restrictive assumptions. For the wave‐Klein–Gordon case, we show that the energy of the second component decays at the rate of , improving the previously known decay.