Skip to main content
Springer Nature Link
Log in

Energy Decay Rate of Solutions for the Wave Equation with Singular Nonlinearities

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper, we study the initial boundary value problem of the wave equation with singular nonlinearities of the form

$$u_{tt}-u_{xx}+\sigma(t)|u|^{-r}g(u_{t})+|u|^{-\alpha}u=0\quad\hbox{in}\ I\times \mathbb{R}_+.$$

We prove decay estimates using multiplier method and weighted integral inequalities. We show that the energy of the system is bounded above by a quantity, depending on σ,g,r and α, which tends to zero (as time goes to infinity). We give many significant examples to illustrate how to derive from our general estimates the polynomial, exponential or logarithmic decay.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Agre, K., Rammaha, M.A.: Quenching and non-quenching for nonlinear wave equations with damping. Can. Appl. Math. Q. 9(3), 203–223 (2001)

    MATH  MathSciNet  Google Scholar 

  2. Barbu, V., Lasiecka, I., Rammaha, M.A.: Blow-up of generalized solutions to wave equations with nonlinear degenerate damping and source terms. Indiana Univ. Math. J. 56(3), 995–1021 (2007)

    Article  MathSciNet  Google Scholar 

  3. Barbu, V., Lasiecka, I., Rammaha, M.A.: Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms. Control Cybern. 34(3), 665–687 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Barbu, V., Lasiecka, I., Rammaha, M.A.: On nonlinear wave equations with degenerate damping and source terms. Trans. Am. Math. Soc. 357(7), 2571–2611 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Haraux, A.: Two remarks on hyperbolic dissipative problems. In: Research Notes in Mathematics, vol. 122, pp. 161–179. Pitman, Boston (1985)

    Google Scholar 

  6. Katayama, S., Sheikh, M.A., Tarama, S.: The Cauchy and mixed problems for semilinear wave equations with nonlinear damping terms. Math. Jpn. 51(3), 459–466 (2000)

    MATH  MathSciNet  Google Scholar 

  7. Katayama, S.: Semilinear wave equations with some kind of nonlinear damping in higher space dimensions. Sci. Math. Jpn. 57(3), 511–518 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Komornik, V.: Differential and integral inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 7, 55–67 (1996)

    MATH  MathSciNet  Google Scholar 

  9. Martinez, P.: A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var. 4, 419–444 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nakao, M.: Periodic solutions and decay for some nonlinear wave equations with sublinear dissipative terms. Nonlinear Anal. 10, 587–602 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nakao, M.: An example of nonlinear wave equation whose solutions decay faster than exponentially. J. Math. Anal. Appl. 122(1), 260–264 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rammaha, M.A., Strei, T.A.: Global existence and nonexistence for nonlinear wave equations with damping and source terms. Trans. Am. Math. Soc. 354(9), 3621–3637 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sheikh, M.A., Matin, M.A.: Global existence of solution for semilinear dissipative wave equation. Vietnam J. Math. 34(3), 295–305 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Sheikh, M.A.: Asymptotic behaviour of solutions for some semilinear dissipative wave equations. Indian J. Pure Appl. Math. 33(12), 1799–1809 (2002)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Mathematics, Djillali Liabes University, P.O. Box 89, Sidi Bel Abbes, 22000, Algeria

    Abbes Benaissa

  2. Ibn Khaldoun University, P.O. Box 78, Zaaroura, Tiaret, Algeria

    Abdelkader Maatoug

Authors
  1. Abbes Benaissa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelkader Maatoug
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abbes Benaissa.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benaissa, A., Maatoug, A. Energy Decay Rate of Solutions for the Wave Equation with Singular Nonlinearities. Acta Appl Math 113, 117–127 (2011). https://doi.org/10.1007/s10440-010-9588-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-010-9588-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation