Skip to main content
SpringerLink
Log in

Global well-posedness for semilinear hyperbolic equations with dissipative term

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

We study the initial boundary value problem of semilinear hyperbolic equations with dissipative term. By introducing a family of potential wells we derive the invariant sets and vacuum isolating of solutions. Then we prove the global existence, nonexistence and asymptotic behaviour of solutions. In particular we obtain some sharp conditions for global existence and nonexistence of solutions.

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. Ball, J.M.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. 28, 473–486 (1977)

    Article  MATH  Google Scholar 

  2. Levine, H.A.: Instability and non-existence of global solutions to nonlinear wave equations of the form Pu tt =−Au+F(u). Trans. Am. Math. Soc. 192, 1–21 (1974)

    MATH  Google Scholar 

  3. Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273–303 (1975)

    Article  MathSciNet  Google Scholar 

  4. Glassay, R.T.: Blow-up theorems for nonlinear wave equations. Math. Z. 132, 183–203 (1973)

    Article  MathSciNet  Google Scholar 

  5. Sattinger, D.H.: On global solution of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 30, 148–172 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  6. Tsutsumi, M.: On solutions of semilinear differential equations in a Hilbert space. Math. Jpn. 17, 173–193 (1972)

    MATH  MathSciNet  Google Scholar 

  7. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod/Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  8. Liu, Y.: On potential wells and vacuum isolating of solutions for semilinear wave equations. J. Differ. Equ. 192, 155–169 (2003)

    Article  MATH  Google Scholar 

  9. Liu, Y., Zhao, J.: On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal. 64, 2665–2687 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Esquivel-Avila, J.A.: The dynamics of a nonlinear wave equation. J. Math. Anal. Appl. 279, 135–150 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cavalcanti, M.M., Domingos Cavalcanti, V.N.: Existence and asymptotic stability of revolution problems on manifolds with damping and source terms. J. Math. Anal. Appl. 291, 109–127 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zhou, Y.: A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in ℝN. Appl. Math. Lett. 18, 281–286 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gazzola, F., Squassina, M.: Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, 185–207 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, L., Wang, M.: Global solutions and blow-up of solutions for some hyperbolic systems with damping and source terms. Nonlinear Anal. 64, 69–91 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lasiecka, I., Toundykov, D.: Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms. Nonlinear Anal. 64, 1757–1797 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ikehata, R., Nishihara, K., Zhao, H.: Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption. J. Differ. Equ. 226, 1–29 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Toundykov, D.: Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions. Nonlinear Anal. TMA 67, 512–544 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Xu, R., Shen, J.: Some generalized results for global well-posedness for wave equations with damping and source terms. Math. Comput. Simul. 80, 804–807 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hao, J., Zhang, Y., Li, S.: Global existence and blow-up phenomena for a nonlinear wave equation. Nonlinear Anal. TMA 71, 4823–4832 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  22. Rammaha, M.A., Sakuntasathien, S.: Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms. Nonlinear Anal. TMA 72, 2658–2683 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. Messaoudi, S.A., Said-Houari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J. Math. Anal. Appl. 365, 277–287 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Taniguchi, T.: Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms. J. Math. Anal. Appl. 36, 566–578 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Harbin Engineering University, Harbin, 150001, People’s Republic of China

    Xiaoli Jiang & Runzhang Xu

Authors
  1. Xiaoli Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Runzhang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Runzhang Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, X., Xu, R. Global well-posedness for semilinear hyperbolic equations with dissipative term. J. Appl. Math. Comput. 38, 467–487 (2012). https://doi.org/10.1007/s12190-011-0491-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-011-0491-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation