Stabilization of a wave-wave transmission problem with generalized acoustic boundary conditions

We investigate the energy decay of a hyperbolic system of wave-wave type with generalized acoustic boundary conditions in $ d $-dimensional space, with the equations being coupled through a boundary connection. First, by spectrum approach combined with a general criteria of Arendt-Batty, we prove that our model is strongly stable. Further, we prove the lack of exponential stability of our system. Then, under appropriate geometric conditions we establish different types of polynomial energy decay rates provided that the coefficients of the acoustic boundary condition satisfy some assumptions. After that, we present some proper examples and show that our assumptions have been made correctly. Finally, we prove that the obtained energy decay rate is optimal in a particular case.