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• W1978224972 abstract "We discuss the asymptotic stability of the wave equation on a compact Riemannian manifold <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper M comma bold g bold right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant=bold>g</mml:mi> <mml:mo mathvariant=bold stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(M, bf g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> subject to locally distributed viscoelastic effects on a subset <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=omega subset-of upper M> <mml:semantics> <mml:mrow> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>omega subset M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assuming that the well-known geometric control condition <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis omega comma upper T 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ω<!-- ω --></mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(omega , T_0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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)." @default.
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• W1978224972 date "2013-05-29" @default.
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• W1978224972 title "Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation" @default.
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• W1978224972 doi "https://doi.org/10.1090/s0002-9939-2013-11869-1" @default.
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