Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation

Abstract. 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 ω ⊂ M . Assuming that the well-known geometric control condition (ω, T0) 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).

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