ASYMPTOTIC STABILITY OF THE WAVE EQUATION ON COMPACT MANIFOLDS AND LOCALLY DISTRIBUTED VISCOELASTIC DISSIPATION

We discuss the asymptotic stability of the wave equation on a compact Riemannian manifold (M, g) subject to locally distributed viscoelastic effects on a subset omega subset of M. Assuming that the well-known geometric control condition (omega, T-0) holds and supposing that the relaxation function is bounded by a function that decays exponentially to zero, we show that the solutions of the corresponding partial viscoelastic model decay exponentially to zero. We give a new geometric proof extending the prior results in the literature from the Euclidean setting to compact Riemannian manifolds (with or without boundary).