Abstract
In this paper, we consider the wave equation on a bounded domain with mixed Dirichlet-impedance type boundary conditions coupled with oscillators on the Neumann boundary. The system has either a delay in the pressure term of the wave component or the velocity of the oscillator component. Using the velocity as a boundary feedback it is shown that if the delay factor is less than that of the damping factor then the energy of the solutions decays to zero exponentially. The results are based on the energy method, a compactness-uniqueness argument and an appropriate weighted trace estimate. In the critical case where the damping and delay factors are equal, it is shown using variational methods that the energy decays to zero asymptotically.
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Acknowledgements
The author would like to thank Georg Propst for his helpful comments and suggestions. This research is partially supported by the Austrian Science Fund (FWF) under SFB grant Mathematical Optimization and Applications in Biomedical Sciences.
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Communicated by Abdelaziz Rhandi.
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Peralta, G.R. Stabilization of the wave equation with acoustic and delay boundary conditions. Semigroup Forum 96, 357–376 (2018). https://doi.org/10.1007/s00233-018-9930-9
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DOI: https://doi.org/10.1007/s00233-018-9930-9