Asymptotics for Wave Equations with Damping Only on the Dynamical Boundary

In this work, we use a semigroup approach to study the asymptotics of the linear wave equation with frictional damping only on the dynamic boundary. We reformulate the model into an abstract Cauchy problem and show that the spectrum of the differential operator corresponding to the Cauchy problem has no purely imaginary values. Moreover, by controlling the trace of the first derivative, we establish the estimate for the order of unboundedness of the resolvent on the imaginary axis and obtain the asymptotics for the system.