Skip to main content
SpringerLink
Log in

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

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Aliev, A.B., Isayeva, S.E.: Exponential stability of the nonlinear transmission acoustic problem. Math. Methods Appl. Sci. 41(16), 7055–7073 (2018)

    Article  MathSciNet  Google Scholar 

  2. Ammari, K., Nicaise, S.: Stabilization of a transmission wave/plate equation. J. Differ. Equ. 249(3), 707–727 (2010)

    Article  MathSciNet  Google Scholar 

  3. Andrade, D., Fatori, L.H., Muñoz Rivera, J.E.: Nonlinear transmission problem with a dissipative boundary condition of memory type. Electron. J. Differ. Equ. 2006(53), 1–16 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Bae, J.J.: Nonlinear transmission problem for wave equation with boundary condition of memory type. Acta Appl. Math. 110(2), 907–919 (2010)

    Article  MathSciNet  Google Scholar 

  5. Beale, J.T., Rosencrans, S.I.: Acoustic boundary conditions. Bull. Am. Math. Soc. 80, 1276–1278 (1974)

    Article  MathSciNet  Google Scholar 

  6. Beale, J.T.: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25(9), 895–917 (1976)

    Article  MathSciNet  Google Scholar 

  7. Beale, J.T.: Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26(2), 199–222 (1977)

    Article  MathSciNet  Google Scholar 

  8. Boukhatem, Y., Benabderrahmane, B.: Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. Nonlinear Anal. 97, 191–209 (2014)

    Article  MathSciNet  Google Scholar 

  9. Boukhatem, Y., Benabderrahmane, B.: Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions. Acta Math. Sin. (Engl. Ser.) 32(2), 153–174 (2016)

    Article  MathSciNet  Google Scholar 

  10. Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)

    Article  MathSciNet  Google Scholar 

  11. Duyckaerts, T.: Optimal decay rates of the energy of a hyperbolic–parabolic system coupled by an interface. Asymptot. Anal. 51(1), 17–45 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Frota, C.L., Medeiros, L.A., Vicente, A.: Wave equation in domains with non-locally reacting boundary. Differ. Integral Equ. 24, 1001–1020 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Hao, J.H., Rao, B.P.: Influence of the hidden regularity on the stability of partially damped systems of wave equations. J. Math. Pures Appl. 143, 257–286 (2020)

    Article  MathSciNet  Google Scholar 

  14. Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507–533 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Liu, W.J.: Stabilization and controllability for the transmission wave equation. IEEE Trans. Automat. Control 46(12), 1900–1907 (2001)

    Article  MathSciNet  Google Scholar 

  16. Liu, W.J.: Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions. Appl. Math. Lett. 38, 155–161 (2014)

    Article  MathSciNet  Google Scholar 

  17. Liu, W.J., Sun, Y.: General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions. Z. Angew. Math. Phys. 65(1), 125–134 (2014)

    Article  MathSciNet  Google Scholar 

  18. Liu, W.J., Williams, G.H.: The exponential stability of the problem of transmission of the wave equation. Bull. Austral. Math. Soc. 57(2), 305–327 (1998)

    Article  MathSciNet  Google Scholar 

  19. Liu, Z.Y., Rao, B.P.: Frequency domain approach for the polynomial stability of partially damped wave equations. J. Math. Anal. Appl. 335, 860–881 (2007)

    Article  MathSciNet  Google Scholar 

  20. Messaoudi, S.A.: General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl. 341(2), 1457–1467 (2008)

    Article  MathSciNet  Google Scholar 

  21. Morse, P.M., Ingard, K.U.: Theoretical Acoustics. McGraw-Hill, New York (1968)

    Google Scholar 

  22. Muñoz Rivera, J.E., Oquendo, H.P.: The transmission problem of viscoelastic waves. Acta Appl. Math. 62(1), 1–21 (2000)

    Article  MathSciNet  Google Scholar 

  23. Oquendo, H.P.: Nonlinear boundary stabilization for a transmission problem in elasticity. Nonlinear Anal. 52(4), 1331–1345 (2003)

    Article  MathSciNet  Google Scholar 

  24. Park, S.H.: General decay of a transmission problem for Kirchhoff type wave equations with boundary memory condition. Acta Math. Sci. Ser. B (Engl. Ed.) 34(5), 1395–1403 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Rauch, J., Zhang, X., Zuazua, E.: Polynomial decay for a hyperbolic–parabolic coupled system. J. Math. Pures Appl. 84(4), 407–470 (2005)

    Article  MathSciNet  Google Scholar 

  26. Vicente, A.: Wave equation with acoustic/memory boundary conditions. Bol. Soc. Parana. Mat. 27(1), 29–39 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Vicente, A., Frota, C.L.: General decay of solutions of a wave equation with memory term and acoustic boundary condition. Math. Methods Appl. Sci. 40(6), 2140–2152 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Vila Bravo, J.C., Muñoz Rivera, J.E.: The transmission problem to thermoelastic plate of hyperbolic type. IMA J. Appl. Math. 74(6), 950–962 (2009)

    Article  MathSciNet  Google Scholar 

  29. Zhang, X., Zuazua, E.: Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal. 184(1), 49–120 (2007)

    Article  MathSciNet  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, China

    Jianghao Hao & Mengxian Lv

Authors
  1. Jianghao Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mengxian Lv
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianghao Hao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

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

Keywords

Mathematics Subject Classification

Search

Navigation