Skip to main content
SpringerLink
Log in

Energy decay of the wave equation with variable coefficients and a localized half-linear dissipation in an exterior domain

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

Abstract

The wave equation of variable coefficients in an exterior domain with a localized half-linear dissipation is considered. The energy decay estimate for the system is established by Riemannian geometry method.

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

Similar content being viewed by others

References

  1. Nakao M.: Energy decay for the linear and semilinear wave equation in exterior domains with some Localized dissipations. Math. Z. 238(4), 781–797 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Nakao M., Jung I.H.: Energy decay for the wave equations in exterior domains with some half-linear dissipation. Differ. Integral Equ. 16(8), 927–948 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Nakao M., Bae J.J.: Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations. Nonlinear Anal. Theory Methods Appl. 66, 301–323 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Nakao M.: Existence of global solutions for the Kirchhoff-type quasilinear wave equation in exterior domains with a half-linear dissipation. Kyushu J. Math. 58(2), 373–391 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Nakao M., Ono K.: Global existence to the Cauchy problem for the semilinear dissipative wave equation. Math. Z. 214(1), 325–342 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Nakao M.: Remarks on the energy decay for a nonlinear wave equation with a nonlinear localized dissipative term. Nonlinear Anal. Theory Methods Appl. 73, 2158–2169 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Nakao M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305(3), 403–417 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Todorova G.: Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Nonlinear Anal. Ser. A 41(7-8), 891–905 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Todorova G., Yordanov B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174(2), 464–489 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Yao P.F.: On the observability inequalities for the exact controllability of the wave equation with variable coefficients. SIAM. J. Control Optim. 37(5), 1568–1599 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chai S.G., Guo Y.X.: Boundary stabilization of wave equation with variable coefficients and memory. Differ. Integral Equ. 17, 669–680 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Feng S.J., Feng D.X.: Nonlinear internal damping of wave equations with variable coefficients. Acta Math. Sin. Engl. Ser. 20(6), 1057–1072 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cavalcanti M.M., Khemmoudj A., Medjden M.: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions. J. Math. Anal. Appl. 328, 900–930 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wu J.Q., Li S.J.: Stabilization of wave equation with variable coefficients by nonlinear boundary feedback. J. Syst. Sci. Compl. Engl. Ser. 24, 875–882 (2011)

    Article  MATH  Google Scholar 

  15. Wu J.Q., Li S.J., Chai S.G.: Exact controllability of wave equations with variable coefficients coupled in parallel. Asian J. Control 12(5), 650–655 (2010)

    Article  MathSciNet  Google Scholar 

  16. Wu J.Q., Li S.J., Chai S.G.: Uniform decay of the solution to a wave equation with memory conditions on the boundary. Nonlinear Anal. Theory Methods Appl. 73, 2213–2220 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yao, P.F.: Modeling and Control in Vibrational and Structural Dynamics, A Differential Geometric Approach. Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press (2011)

  18. Yao P.F.: Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. Chin. Ann. Math. Ser. B 31(1), 59–70 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wu H., Shen C.L., Yu Y.L.: An Introduction to Riemannian Geometry. Peking University Press, Beijing (1989)

    Google Scholar 

  20. Lions J.L., Strauss W.A.: Some nonlinear evolution equations. Bull. Soc. Math. France 93, 43–96 (1965)

    MATH  MathSciNet  Google Scholar 

  21. Tataru D.: The \({X_{\theta}^{s}}\) spaces and unique continuation for solutions to the semilinear wave equations. Commun. Partial Differ. Equ. 21, 841–887 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Jing Li, Shugen Chai & Jieqiong Wu

Authors
  1. Jing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shugen Chai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jieqiong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shugen Chai.

Additional information

This work is supported by the National Natural Science Foundation of China (No. 11171195), the Nature Science Foundation of Shanxi Province(2013011003-2), the National Nature Science Foundation of China for the Youth (No. 61104129) and the Youth Science Foundation of Shanxi Province(2011021002-1).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J., Chai, S. & Wu, J. Energy decay of the wave equation with variable coefficients and a localized half-linear dissipation in an exterior domain. Z. Angew. Math. Phys. 66, 95–112 (2015). https://doi.org/10.1007/s00033-013-0385-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-013-0385-2

Mathematics Subject Classification (2000)

Keywords

Search

Navigation