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Applications of Mathematics

, Volume 59, Issue 2, pp 205–215 | Cite as

Stability of vibrations for some Kirchhoff equation with dissipation

  • Prasanta Kumar NandiEmail author
  • Ganesh Chandra Gorain
  • Samarjit Kar
Article
  • 158 Downloads

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.

Keywords

Kirchhoff equation dissipation vibration stabilization energy decay estimate 

MSC 2010

35B35 35L70 45K05 37L15 

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References

  1. [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.)zbMATHMathSciNetGoogle Scholar
  2. [2]
    G. Autuori, P. Pucci: Asymptotic stability for Kirchhoff systems in variable exponent Sobolev spaces. Complex Var. Elliptic Equ. 56 (2011), 715–753.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    G. Autuori, P. Pucci: Local asymptotic stability for polyharmonic Kirchhoff systems. Appl. Anal. 90 (2011), 493–514.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    G. Autuori, P. Pucci, M. C. Salvatori: Asymptotic stability for anisotropic Kirchhoff systems. J. Math. Anal. Appl. 352 (2009), 149–165.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    G. Autuori, P. Pucci, M. C. Salvatori: Asymptotic stability for nonlinear Kirchhoff systems. Nonlinear Anal., Real World Appl. 10 (2009), 889–909.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    P. D’Ancona, S. Spagnolo: Nonlinear perturbations of the Kirchhoff equation. Commun. Pure Appl. Math. 47 (1994), 1005–1029.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [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.CrossRefMathSciNetGoogle Scholar
  9. [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.zbMATHMathSciNetGoogle Scholar
  10. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    G. P. Menzala: On classical solutions of a quasilinear hyperbolic equation. Nonlinear Anal., Theory Methods Appl. 3 (1979), 613–627.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [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.CrossRefzbMATHGoogle Scholar
  13. [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.CrossRefMathSciNetGoogle Scholar
  14. [14]
    R. Narasimha: Non-linear vibration of an elastic string. J. Sound Vib. 8 (1968), 134–146.CrossRefzbMATHGoogle Scholar
  15. [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. [16]
    W. G. Newman: Global solution of a nonlinear string equation. J. Math. Anal. Appl. 192 (1995), 689–704.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    K. Nishihara: On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 7 (1984), 437–459.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [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.zbMATHMathSciNetGoogle Scholar
  19. [19]
    K. Ono: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Differ. Equations 137 (1997), 273–301.CrossRefzbMATHGoogle Scholar
  20. [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.zbMATHMathSciNetGoogle Scholar
  21. [21]
    S. M. Shahruz: Bounded-input bounded-output stability of a damped nonlinear string. IEEE Trans. Autom. Control 41 (1996), 1179–1182.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    T. Yamazaki: Global solvability for the Kirchhoff equations in exterior domains of dimension three. J. Differ. Equations 210 (2005), 290–316.CrossRefzbMATHGoogle Scholar
  24. [24]
    Y. Ye: On the exponential decay of solutions for some Kirchhoff-type modelling equations with strong dissipation. Applied Mathematics 1 (2010), 529–533.CrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014

Authors and Affiliations

  • Prasanta Kumar Nandi
    • 1
    Email author
  • Ganesh Chandra Gorain
    • 2
  • Samarjit Kar
    • 1
  1. 1.Department of MathematicsNational Institute of TechnologyDurgapurIndia
  2. 2.Department of MathematicsJ.K.CollegePuruliaIndia

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