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Stabilization of a transmission problem with past history and acoustic boundary conditions

Abstract

In this paper we consider a transmission problem for the wave equations with past history and acoustic boundary conditions, involving two distinct domains connected through a common interface. The frictional dampings are only distributed in a small neighbourhood of the interface. Under some geometric conditions, we obtain the energy decays at the rate quantified by the solution to a certain nonlinear ODE dependent on the damping terms. Our method is based on using appropriate weighted multipliers to establish the necessary observability inequality that allows to obtain the energy estimate.

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Acknowledgements

The authors are highly grateful to the anonymous referee for their valuable comments and suggestions for the improvement of the paper. This research work was partially supported by Natural Science Foundation of China (grant number 11871315, 61374089) and Natural Science Foundation of Shanxi Province of China (grant number 201901D111021).

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Correspondence to Jianghao Hao.

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Hao, J., Lv, M. Stabilization of a transmission problem with past history and acoustic boundary conditions. Z. Angew. Math. Phys. 73, 105 (2022). https://doi.org/10.1007/s00033-022-01751-7

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  • DOI: https://doi.org/10.1007/s00033-022-01751-7

Keywords

  • Transmission problem
  • Wave equation
  • Localized damping
  • Past history
  • Acoustic boundary conditions

Mathematics Subject Classification

  • 35B35
  • 35L70