Skip to main content
SpringerLink
Log in

On nonlinear hyperbolic equation in unbounded domain

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The following nonlinear hyperbolic equation is discussed in this paper:

$$u_{tt} + A^2 u + M\left( {x,\left\| {A^{\frac{1}{2}} u} \right\|\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} } \right)Au = 0$$

where A=Δ+l and x∈Rn. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as Mdepends on x.

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. Woinowsky-Krieger, S., The effect of axial force on the vibration of hinged bars,J. Appl. Mech.,17 (1950), 35–36.

    MATH  MathSciNet  Google Scholar 

  2. Medeiros, L. A., On a new class of nonlinear wave equation,J. Math. Appl.,69 (1979), 252–262.

    Article  MATH  MathSciNet  Google Scholar 

  3. Menzala, G. P., On global classical solutions of a nonlinear wave equation,J. Appl. Anal.,10 (1980), 179–195.

    MATH  MathSciNet  Google Scholar 

  4. Biler, P., Remark on the decay for damped string and beam,Nonlinear Analysis,10 (1986), 839–842.

    Article  MATH  MathSciNet  Google Scholar 

  5. Brito, E. H., Decay estimates for generalized damped extensible string and beam equation,Nonlinear Analysis,8 (1984), 1489–1496.

    Article  MATH  MathSciNet  Google Scholar 

  6. Brito, E. H., Nonlinear initial-boundary value problems,Nonlinear Analysis,11 (1987), 125–137.

    Article  MATH  MathSciNet  Google Scholar 

  7. Pereira, D. C., Exitence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation,Nonlinear Analysis,14 (1990), 613–623.

    Article  MATH  MathSciNet  Google Scholar 

  8. Vasconcellos, C. F., On a nonlinear wave equation in unbounded domains,Internat. J. Math. & Math. Sci.,11, 2 (1988), 335–342.

    Article  MATH  MathSciNet  Google Scholar 

  9. Goldstein, J., Time dependent hyperbolic equation,J. Func. Anal.,4 (1969), 31–49.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lanzhou University, Lanzhou

    Geng Di & Qu Chang-zheng

Authors
  1. Geng Di
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qu Chang-zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Zhang Shi-sheng

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di, G., Chang-zheng, Q. On nonlinear hyperbolic equation in unbounded domain. Appl Math Mech 13, 255–261 (1992). https://doi.org/10.1007/BF02457371

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02457371

Key words

Search

Navigation