Asymptotic behavior of a delayed wave equation without displacement term

Abstract : This paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any displacement term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter, LaSalle's invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. More importantly, without any geometric condition such as BLR condition (Bardos et al. in SIAM J Control Optim 30 1024-1064, 1992; Lebeau and Robbiano in Duke Math J 86 465-491, 1997) in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method.
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Submitted on : Monday, July 1, 2019 - 11:37:04 AM
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Kaïs Ammari, Boumediene Chentouf. Asymptotic behavior of a delayed wave equation without displacement term. Zeitschrift für Angewandte Mathematik und Physik, Springer Verlag, 2017, 68 (5), ⟨10.1007/s00033-017-0865-x⟩. ⟨hal-02169458⟩

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