Optimal stabilization rate for the wave equation with hyperbolic boundary condition

We consider the wave equation on a bounded domain with a mixed boundary: a dynamic part governed by a coupled lower-dimensional wave equation (dynamic Wentzell boundary condition) subject to viscous damping, and a complementary part (possibly empty) left at rest. When the damped dynamic boundary satisfies the Geometric Control Condition, we prove that the energy of strong solutions decays like1/t. The proof is based on resolvent estimates obtained from high-frequency quasimodes, microlocal boundary trace inequalities in several regimes, and a special decoupling argument. Optimality is assessed via an appropriate quasimode construction in the particular case of the disk. This is joint work with Nicolas Vanspranghe