Skip to main content
SpringerLink
Log in

Geometric optics for the Kirchhoff-type equations

  • Published:
Journal d’Analyse Mathématique Aims and scope

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

References

  1. A. Arosio,Global (in time) solution of the approximate nonlinear string equation of G. F. Carrier and R. Narasimha, Comment. Math. Univ. Carolin.26 (1985), 169–172.

    MathSciNet  MATH  Google Scholar 

  2. A. Arosio and S. Panizzi,On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc.348 (1996), 305–330.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Bateman and A. Erdelyi,Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.

    Google Scholar 

  4. S. Bernstein,Sur une classe d'équations fonctionnelles aux dérivées partielles, Izv. Akad. Nauk SSSR, Sér. Mat.4 (1940), 17–26.

    Google Scholar 

  5. E. Callegari and R. Manfrin,Global existence for nonlinear hyperbolic systems of Kirchhoff type, J. Differential Equations132 (1996), 239–274.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. D'Ancona and S. Spagnolo,Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math.108 (1992), 247–262.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. D'Ancona and S. Spagnolo,Kirchhoff type equations depending on a small parameter, Chinese Ann. Math. Ser. B16 (1995), 413–430.

    MathSciNet  MATH  Google Scholar 

  8. P. D'Ancona and S. Spagnolo,Nonlinear perturbations of the Kirchhoff equation, Comm. Pure Appl. Math.47 (1994), 1005–1029.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. W. Dickey,Infinite systems of nonlinear oscillation equations, J. Differential Equations8 (1970), 16–26.

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Hirosawa,Global solvability for the degenerate Kirchhoff equation, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4)26 (1998), 75–95.

    MathSciNet  MATH  Google Scholar 

  11. L. Hörmander,The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators, Springer-Verlag, Berlin, 1985.

    MATH  Google Scholar 

  12. K. Kajitani and K. Yamaguti,On global real analytic solutions of the degenerate Kirchhoff equation, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4)21 (1994), 279–297.

    MathSciNet  MATH  Google Scholar 

  13. G. Kirchhoff,Vorlesungen über Mechanik, Teubner, Leipzig, 1883.

    Google Scholar 

  14. R. Manfrin,On the global solvability of symmetric hyperbolic systems of Kirchhoff type, Discrete Contin. Dynam. Systems3 (1997), 91–106.

    MathSciNet  MATH  Google Scholar 

  15. R. Manfrin,On global small solutions of nonlinear Timoshenko's type equations, Tsukuba J. Math.22 (1998), 747–768.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Menzala,On classical solutions of a quasilinear hyperbolic equation, Nonlinear Anal.3 (1979), 613–627.

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Nishihara,On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math.7 (1984), 437–459.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. I. Pohozaev,A certain class of quasilinear hyperbolic equations. Mat. Sb. (N.S.)96 (138) (1975), 152–166.

    MathSciNet  Google Scholar 

  20. R. Racke,Generalized Fourier transforms and global, small solutions to Kirchhoff equations, Appl. Anal.58 (1995), 85–100.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Rauch and M. Keel,Lectures on Geometric Optics, inHyperbolic Equations and Frequency Interactions (Park City, UT, 1995), IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999, pp. 383–466.

    Google Scholar 

  22. R. Rodriguez,On local strong solution of a non-linear partial differential equation, Appl. Anal.8 (1980), 93–104.

    Article  Google Scholar 

  23. S. Spagnolo,The Cauchy problem for the Kirchhoff equation, Rend. Sem. Mat. Fis. Milano62 (1992), 17–51.

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Yagdjian,The Lax-Mizohata theorem for the Kirchhoff type equations, J. Differential Equations171 (2001), 346–369.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Acad. Sci. of Armenia, Marshal Bagramian Ave. 24B, 375019, Yerevan, Armenia

    Karen Yagdjian

  2. Department of Mathematics, Kansas State University, 138 Cardwell Hall, 66506-2602, Manhattan, KS, USA

    Karen Yagdjian

Authors
  1. Karen Yagdjian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karen Yagdjian.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yagdjian, K. Geometric optics for the Kirchhoff-type equations. J. Anal. Math. 89, 57–112 (2003). https://doi.org/10.1007/BF02893077

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02893077

Keywords

Search

Navigation