Skip to main content
SpringerLink
Log in

Existence and exponential decay of the Dirichlet problem for a nonlinear wave equation with the Balakrishnan–Taylor term*

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

In this paper, we consider the Dirichlet problem for a nonlinear wave equation with Balakrishnan–Taylor term. We establish the local existence is established by the linearization method together with the Faedo–Galerkin method. Next, we prove an exponential energy decay result by suitable Lyapunov functionals.

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. A.V. Balakrishnan and L.W. Taylor, Distributed parameter nonlinear damping models for flight structures, Report No. WRDC-TR-89-3116 Vol. II, Air Force Wright Aeronautical Laboratories, 1989.

  2. R.W. Bass and D. Zes, Spillover nonlinearity and flexible structures, in L.W. Taylor (Ed.), The Fourth NASA Workshop on Computational Control of Fexible Aerospace Systems, NASA Conference Publication 10065, Elsevier, 1991, pp. 1–14.

  3. M.M. Cavalcanti, U.V.N. Domingos Cavalcanti, and J.S. Prates Filho, Existence and exponential decay for a Kirchhoff–Carrier model with viscosity, J. Math. Anal. Appl., 226:40–60, 1998.

    MathSciNet  MATH  Google Scholar 

  4. I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differ. Equations, 252:1229–1262, 2012.

    MathSciNet  MATH  Google Scholar 

  5. E.L.A. Coddington, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  6. M. Ghisi and M. Gobbino, Hyperbolic–parabolic singular perturbation for mildly degenerate Kirchhoff equations: Decay error estimates, J. Differ. Equations, 252:6099–6132, 2012.

    MathSciNet  MATH  Google Scholar 

  7. T.G. Ha, General decay rate estimates for viscoelastic wave equation with Balakrishnan–Taylor damping, Z. Angew. Math. Phys., 67(32):1–17, 2016.

    MathSciNet  MATH  Google Scholar 

  8. T.G. Ha, Stabilization for the wave equation with variable coefficients and Balakrishnan–Taylor damping, Taiwanese J. Math., 21(4):807–817, 2017.

    MathSciNet  MATH  Google Scholar 

  9. T.G. Ha, On the viscoelastic equation with Balakrishnan–Taylor damping and acoustic boundary conditions, Evol. Equ. Control Theory, 7(2):281–291, 2018.

    MathSciNet  MATH  Google Scholar 

  10. J. Hao and Y. Hou, Stabilization for wave equation of variable coefficients with Balakrishnan–Taylor damping and source term, Comput. Math. Appl., 76:2235–2245, 2018.

    MathSciNet  Google Scholar 

  11. J. Hao and F. Wang, General decay rate for weak viscoelastic wave equation with Balakrishnan–Taylor damping and time-varying delay, Comput. Math. Appl., 78:2632–2640, 2019.

    MathSciNet  Google Scholar 

  12. D. Kim and I. H. Jung, Asymptotic behavior of a nonlinear Kirchhoff type equation with spring boundary conditions, Comput. Math. Appl., 62:3004–3014, 2011.

    MathSciNet  MATH  Google Scholar 

  13. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.

    MATH  Google Scholar 

  14. W. Liu, G. Li B. Zhu, and D. Wang, General decay for a viscoelastic Kirchhoff equation with Balakrishnan–Taylor damping, dynamic boundary conditions and a time-varying delay term, Evol. Equ. Control Theory, 6(2):239–260, 2017.

    MathSciNet  MATH  Google Scholar 

  15. C. Mu and J. Ma, On a system of nonlinear wave equations with Balakrishnan–Taylor damping, Z. Angew. Math. Phys., 65:91–113, 2014.

    MathSciNet  MATH  Google Scholar 

  16. K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal., 216:321–342, 1997.

    MathSciNet  MATH  Google Scholar 

  17. F.R.D. Silva and J.M.S. Pitot, Existence, uniqueness and exponential decay of solutions to Kirchhoff equation in Rn, J. Differ. Equations, 2016(247):1–27, 2016.

    Google Scholar 

  18. N.A. Triet, V.T.T. Mai, L.T.P. Ngoc, and N.T. Long, A Dirichlet problem for a nonlinear wave equation of Kirchhoff–Love type, Nonlinear Funct. Anal. Appl., 22(3):595–626, 2017.

    MATH  Google Scholar 

  19. N.A. Triet, L.T.P. Ngoc, and N.T. Long, On a nonlinear Kirchhoff–Carrier wave equation associated with Robin conditions, Nonlinear Anal., Real World Appl., 11:3363–3388, 2010.

  20. Z. Yang and Z. Gong, Blow-up of solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differ. Equ., 2016(332):1–8, 2016.

    MathSciNet  MATH  Google Scholar 

  21. A. Zarai and N. Tatar, Global existence and polynomial decay for a problem with Balakrishnan–Taylor damping, Arch. Math., 46(3):157–176, 2010.

    MathSciNet  MATH  Google Scholar 

  22. Y. Zhijian and L. Xiao, Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl., 375:579–593, 2011.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam

    Le Thi Phuong Ngoc

  2. Nguyen Tat Thanh University, 300A Nguyen Tat Thanh Str., 13 Ward, Dist. 4, Ho Chi Minh City, Vietnam

    Nguyen Huu Nhan

  3. Ho Chi Minh City University of Food Industry, 140 Le Trong Tan Str., Tay ThanhWard, Tan Phu Dist, Ho Chi Minh City, Vietnam

    Bui Duc Nam

  4. Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam

    Bui Duc Nam & Nguyen Thanh Long

  5. Vietnam National University, Ho Chi Minh City, Vietnam

    Bui Duc Nam & Nguyen Thanh Long

Authors
  1. Le Thi Phuong Ngoc
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Huu Nhan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bui Duc Nam
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nguyen Thanh Long
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Le Thi Phuong Ngoc.

Additional information

Publisher’s note

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

∗This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under the project “The properties of the solutions of the nonlinear boundary value problems and integrodifferential equations” (B2020-18-xx).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ngoc, L.T.P., Nhan, N.H., Nam, B.D. et al. Existence and exponential decay of the Dirichlet problem for a nonlinear wave equation with the Balakrishnan–Taylor term*. Lith Math J 60, 225–247 (2020). https://doi.org/10.1007/s10986-020-09469-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-020-09469-7

Keywords

MSC

Search

Navigation