Stabilization through viscoelastic boundary damping: a semigroup approach

The undamped linear wave equation on a bounded domain in ℝn with C2 boundary is considered. The interaction of the interior waves and the viscoelastic boundary material is modeled by convolution boundary conditions. It is assumed that the convolution kernel is integrable and completely monotonic. The main result is that the derivatives of all solutions tend to zero. The proof is given by an application of the Arendt-Batty-Lyubic-Vu Theorem. To this end, the model is reformulated as an abstract first order Cauchy problem in an appropriate Hilbert space, including the memory of the boundary as a state component. It is shown that the differential operator of the Cauchy problem is the generator of a contraction semigroup on the state space by establishing the range condition for the Lumer-Phillips Theorem using a generalized Lax-Milgram argument and Fredholm’s alternative. Furthermore, it is shown that neither the generator nor its adjoint have purely imaginary eigenvalues.