Skip to main content

Advertisement

SpringerLink
Log in

Stability of vibrations for some Kirchhoff equation with dissipation

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we consider the boundary value problem of some nonlinear Kirchhoff-type equation with dissipation. We also estimate the total energy of the system over any time interval [0, T] with a tolerance level γ. The amplitude of such vibrations is bounded subject to some restrictions on the uncertain disturbing force f. After constructing suitable Lyapunov functional, uniform decay of solutions is established by means of an exponential energy decay estimate when the uncertain disturbances are insignificant.

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. M. Aassila, A. Benaissa: Global existence and asymptotic behavior of solutions of mildly degenerate Kirchhoff equations with nonlinear dissipative term. Funkc. Ekvacioj, Ser. Int. 44 (2001), 309–333. (In French.)

    MATH  MathSciNet  Google Scholar 

  2. G. Autuori, P. Pucci: Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces. Complex Var. Elliptic Equ. 56 (2011), 715–753.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Autuori, P. Pucci: Local asymptotic stability for polyharmonic Kirchhoff systems. Appl. Anal. 90 (2011), 493–514.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Autuori, P. Pucci, M. C. Salvatori: Asymptotic stability for anisotropic Kirchhoff systems. J. Math. Anal. Appl. 352 (2009), 149–165.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Autuori, P. Pucci, M. C. Salvatori: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal., Real World Appl. 10 (2009), 889–909.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. D’Ancona, S. Spagnolo: Nonlinear perturbations of the Kirchhoff equation. Commun. Pure Appl. Math. 47 (1994), 1005–1029.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. C. Gorain: Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in R n. J. Math. Anal. Appl. 319 (2006), 635–650.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. C. Gorain: Exponential energy decay estimates for the solutions of n-dimensional Kirchhoff type wave equation. Appl. Math. Comput. 177 (2006), 235–242.

    Article  MathSciNet  Google Scholar 

  9. V. Komornik, E. Zuazua: A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl., IX. Sér. 69 (1990), 33–54.

    MATH  MathSciNet  Google Scholar 

  10. I. Lasiecka, J. Ong: Global solvability and uniform decays of solutions to quasilinear equation with nonlinear boundary dissipation. Commun. Partial Differ. Equations 24 (1999), 2069–2107.

    Article  MATH  MathSciNet  Google Scholar 

  11. G. P. Menzala: On classical solutions of a quasilinear hyperbolic equation. Nonlinear Anal., Theory Methods Appl. 3 (1979), 613–627.

    Article  MATH  MathSciNet  Google Scholar 

  12. D. S. Mitrinović, J. E. Pečarić, A. M. Fink: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications: East European Series 53, Kluwer Academic Publishers, Dordrecht, 1991.

    Book  MATH  Google Scholar 

  13. P. K. Nandi, G. C. Gorain, S. Kar: Uniform exponential stabilization for flexural vibrations of a solar panel. Appl. Math. (Irvine) 2 (2011), 661–665.

    Article  MathSciNet  Google Scholar 

  14. R. Narasimha: Non-linear vibration of an elastic string. J. Sound Vib. 8 (1968), 134–146.

    Article  MATH  Google Scholar 

  15. A. H. Nayfeh, D. T. Mook: Nonlinear Oscillations. Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, New York, 1979.

    Google Scholar 

  16. W. G. Newman: Global solution of a nonlinear string equation. J. Math. Anal. Appl. 192 (1995), 689–704.

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Nishihara: On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 7 (1984), 437–459.

    Article  MATH  MathSciNet  Google Scholar 

  18. K. Nishihara, Y. Yamada: On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms. Funkc. Ekvacioj, Ser. Int. 33 (1990), 151–159.

    MATH  MathSciNet  Google Scholar 

  19. K. Ono: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differ. Equations 137 (1997), 273–301.

    Article  MATH  Google Scholar 

  20. K. Ono, K. Nishihara: On a nonlinear degenerate integro-differential equation of hyperbolic type with a strong dissipation. Adv. Math. Sci. Appl. 5 (1995), 457–476.

    MATH  MathSciNet  Google Scholar 

  21. S. M. Shahruz: Bounded-input bounded-output stability of a damped nonlinear string. IEEE Trans. Autom. Control 41 (1996), 1179–1182.

    Article  MATH  MathSciNet  Google Scholar 

  22. T. Yamazaki: Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three. Math. Methods Appl. Sci. 27 (2004), 1893–1916.

    Article  MATH  MathSciNet  Google Scholar 

  23. T. Yamazaki: Global solvability for the Kirchhoff equations in exterior domains of dimension three. J. Differ. Equations 210 (2005), 290–316.

    Article  MATH  Google Scholar 

  24. Y. Ye: On the exponential decay of solutions for some Kirchhoff-type modelling equations with strong dissipation. Applied Mathematics 1 (2010), 529–533.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Durgapur, 713209, West Bengal, India

    Prasanta Kumar Nandi & Samarjit Kar

  2. Department of Mathematics, J.K.College, Purulia, 723101, West Bengal, India

    Ganesh Chandra Gorain

Authors
  1. Prasanta Kumar Nandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ganesh Chandra Gorain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samarjit Kar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Prasanta Kumar Nandi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nandi, P.K., Gorain, G.C. & Kar, S. Stability of vibrations for some Kirchhoff equation with dissipation. Appl Math 59, 205–215 (2014). https://doi.org/10.1007/s10492-014-0050-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-014-0050-x

Keywords

MSC 2010

Search

Navigation