The Kirchhoff Equation with Gevrey Data

  • Tokio MatsuyamaEmail author
  • Michael Ruzhansky
Conference paper
Part of the Trends in Mathematics book series (TM)


In this article the Cauchy problem for the Kirchhoff equation is considered, and the almost global existence of Gevrey space solutions is described.


Gevrey spaces Kirchhoff equation 

Mathematics Subject Classification (2010)

Primary 35L40 35L30 Secondary 35L10 35L05 35L75 



The first author was supported by Grant-in-Aid for Scientific Research (C) (No. 15K04967), Japan Society for the Promotion of Science. The second author was supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Research Grant RPG-2014-02. The authors were also supported in part by EPSRC Mathematics Platform grant EP/I019111/1. No new data was collected or generated during the course of the research.


  1. 1.
    A. Arosio, S. Spagnolo, Global solutions to the Cauchy problem for a nonlinear hyperbolic equation. Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. VI (Paris, 1982/1983), pp. 1–26, Res. Notes in Math., vol. 109 (Pitman, Boston, 1984)Google Scholar
  2. 2.
    S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles. Izv. Akad. Nauk SSSR Ser. Mat. 4, 17–27 (1940)zbMATHGoogle Scholar
  3. 3.
    E. Callegari, R. Manfrin, Global existence for nonlinear hyperbolic systems of Kirchhoff type. J. Differ. Equ. 132, 239–274 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    F. Colombini, D. Del Santo, T. Kinoshita, Well-posedness of a hyperbolic equation with non-Lipschitz coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1, 327–358 (2002)Google Scholar
  5. 5.
    P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    P. D’Ancona, S. Spagnolo, A class of nonlinear hyperbolic problems with global solutions. Arch. Ration. Mech. Anal. 124, 201–219 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P. D’Ancona, S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation. Commun. Pure Appl. Math. 47, 1005–1029 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    P. D’Ancona, S. Spagnolo, Kirchhoff type equations depending on a small parameter. Chin. Ann. Math. 16B, 413–430 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    M. Ghisi, M. Gobbino, Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness. Rend. Istit. Mat. Univ. Trieste 42 (Suppl.), 89–110 (2010)Google Scholar
  10. 10.
    M. Ghisi, M. Gobbino, Kirchhoff equation from quasi-analytic to spectral-gap data. Bull. Lond. Math. Soc. 43, 374–385 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J.M. Greenberg, S.C. Hu, The initial-value problem for a stretched string. Q. Appl. Math. 38, 289–311 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    C. Heiming (=C. Kerler), Mapping properties of generalized Fourier transforms and applications to Kirchhoff equations. Nonlinear Differ. Equ. Appl. 7, 389–414 (2000)Google Scholar
  13. 13.
    F. Hirosawa, Global solvability for Kirchhoff equation in special classes of non-analytic functions. J. Differ. Equ. 230, 49–70 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    K. Kajitani, The global solutions to the Cauchy problem for multi-dimensional Kirchhoff equation, in Advance in Phase Space Analysis of Partial Differential Equations, ed. by A. Bove, D. Del Santo, M.K.V. Murthy. Progress in Nonlinear Differential Equations and Their Applications, vol. 78 (Birkhäuser, Boston 2009), pp. 141–153Google Scholar
  15. 15.
    K. Kajitani, K. Yamaguti, On global analytic solutions of the degenerate Kirchhoff equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 21, 279–297 (1994)Google Scholar
  16. 16.
    G. Kirchhoff, Vorlesungen über Mechanik (Teubner, Leibzig, 1876)zbMATHGoogle Scholar
  17. 17.
    R. Manfrin, On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete Contin. Dyn. Syst. 3, 91–106 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Manfrin, On the global solvability of Kirchhoff equation for non-analytic initial data. J. Differ. Equ. 211, 38–60 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    T. Matsuyama, Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation. J. Math. Soc. Jpn. 64, 1167–1204 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    T. Matsuyama, The Kirchhoff equation with global solutions in unbounded domains. Rend. Istit. Mat. Univ. Trieste. 42 (Suppl.), 125–141 (2010)Google Scholar
  21. 21.
    T. Matsuyama, M. Ruzhansky, Global well-posedness of Kirchhoff systems. J. Math. Pures Appl. 100, 220–240 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    T. Matsuyama, M. Ruzhansky, Global well-posedness of the Kirchhoff equation and Kirchhoff systems. Analytic Methods in Interdisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol. 116 (Springer, Berlin, 2015)Google Scholar
  23. 23.
    T. Matsuyama, M. Ruzhansky, On the Gevrey well-posedness of the Kirchhoff equation. J. Anal. Math. (to appear)Google Scholar
  24. 24.
    T. Nishida, A note on the nonlinear vibrations of the elastic string. Mem. Fac. Eng. Kyoto Univ. 33, 329–341 (1971)Google Scholar
  25. 25.
    K. Nishihara, On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 7, 437–459 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    S.I. Pohožhaev, On a class of quasilinear hyperbolic equations. Math. USSR Sb. 25, 145–158 (1975)CrossRefGoogle Scholar
  27. 27.
    R. Racke, Generalized Fourier transforms and global, small solutions to Kirchhoff equations. Asymptot. Anal. 58, 85–100 (1995)MathSciNetzbMATHGoogle Scholar
  28. 28.
    W. Rzymowski, One-dimensional Kirchhoff equation. Nonlinear Anal. 48, 209–221 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    T. Yamazaki, Scattering for a quasilinear hyperbolic equation of Kirchhoff type. J. Differ. Equ. 143, 1–59 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three. Math. Methods Appl. Sci. 27, 1893–1916 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension three. J. Differ. Equ. 210, 290–316 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsChuo UniversityTokyoJapan
  2. 2.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations