Skip to main content
SpringerLink
Log in

Global existence and blowup of solutions for a class of nonlinear higher-order wave equations

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

Abstract

In this paper, we consider a class of nonlinear higher-order wave equation with nonlinear damping

$$u_{tt}+(-\Delta)^mu+a|u_t|^{p-2}u_t=b|u|^{q-2}u$$

in a bounded domain \({\Omega\subset\mathbb{R}^N}\) (N ≥ 1 is a natural number). We show that the solution is global in time under some conditions without the relation between p and q and we also show that the local solution blows up in finite time if q > p with some assumptions on initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blow-up solution are given. This extend the recent results of Ye (J Ineq Appl, 2010).

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Ye Y.J.: Existence and asymptotic behavior of gobal solutions for aclass of nonlinear higher-order wave equation. J. Ineq. Appl. 2010, 1–14 (2010)

    Article  Google Scholar 

  2. Levine H.A.: Instability and nonexistence of global solutions of nonlinear wave equation of the form Du tt  = Au + f(u). Trans. Am. Math. Soc. 192, 1–21 (1974)

    MATH  Google Scholar 

  3. Levine H.A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Georgiev V., Todorova D.: Existence of solutions of the wave equations with nonlinear damping and source terms. J. Diff. Eqns. 109, 295–308 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Messaoudi S.A.: Below up in a nonlinearly damped wave equation. Math. Nachr. 231, 1–7 (2001)

    Article  MathSciNet  Google Scholar 

  6. Ikehata R.: Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Anal. TMA 10, 1165–1175 (1996)

    MathSciNet  Google Scholar 

  7. Nakao M., Ono K.: Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation. Funkcialaj Ekvacioj 38, 417–431 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Aassila M.: Global existence and global nonexistence of solutions to a wave equation with nonlinear damping and source terms. Asymptot. Anal. 30, 301–311 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Guesmia A.: Existence globale et stabilisation interne non linéaire d’un système de Petrovsky. Bell. Belg. Math. Soc. 5, 583–594 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Guesmia A.: Energy decay for a damped nonlinear coupled system. J. Math. Anal. Appl. 239, 38–48 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Aassila M., Guesmia A.: Energy decay for a damped nonlinear hyperbolic equation. Appl. Math. Lett. 12, 49–52 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Komornik, V.: Exact Controllability and Stabilization. The Multiplier Method. Masson, Paris (1994)

  13. Messaoudi S.A.: Global existence and nonexistence in a system of Petrovsky. J. Math. Anal. Appl. 265, 296–308 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen W.Y., Zhou Y.: Global nonexistence for a semilinear Petrovsky equation. Nonlinear Anal. TMA 70, 3203–3208 (2009)

    Article  MATH  Google Scholar 

  15. Wu S.T., Tsai L.Y.: On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system. Taiwan. J. Math. 13, 545–558 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Messaoudi S.A.: Global existence and decay of solutions to a system of Petrovsky. Math. Sci. Res. J. 11, 534–541 (2002)

    MathSciNet  Google Scholar 

  17. Nakao M.: A difference inequality and its application to nonlinear evolution equations. J. Math. Soc. Jpn. 30, 747–762 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Amroun N.E., Benaissa A.: Global existence and energy decay of solutions to a Petrovsky equation with general nonlinear dissipation and source term. Georgian Math. J. 13, 397–410 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Li, G., Sun, Y.N., Liu, W.J.: Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping. Appl. Analysis, 90(2011). doi:10.1080/00036811.2010.550576

  20. Li F.: Global existence and blow up of solutions for a higher-orderKirchhofftype equation with nonlinear dissipation. Appl. Math. Lett. 17, 1409–1414 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Messaoudi S.A., Belkacem S.-H.: A blow-up result for a higherorder nonlinear Kirchhoff-type hyperbolic equation. Appl. Math. Lett. 20(8), 866–871 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Adams, R.A.: Sobolev Space. Academic Press, New York (1975)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Jun Zhou

  2. Department of Mathematics, Yibin College, Yibin, 644007, People’s Republic of China

    Xiongrui Wang

  3. Department of Mathematics, China West Normal University, Nanchong, 637002, People’s Republic of China

    Xiaojun Song

  4. School of Mathematics and Statistics, Chongqing University, Chongqing, 400044, People’s Republic of China

    Chunlai Mu

Authors
  1. Jun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiongrui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaojun Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chunlai Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, J., Wang, X., Song, X. et al. Global existence and blowup of solutions for a class of nonlinear higher-order wave equations. Z. Angew. Math. Phys. 63, 461–473 (2012). https://doi.org/10.1007/s00033-011-0165-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-011-0165-9

Mathematics Subject Classification (2000)

Keywords

Search

Navigation