Advertisement

Global Well-Posedness of the Kirchhoff Equation and Kirchhoff Systems

  • Tokio Matsuyama
  • Michael Ruzhansky
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 116)

Abstract

This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.

References

  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. 109 (Pitman, Boston, MA, 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)Google Scholar
  3. 3.
    E. Callegari, R. Manfrin, Global existence for nonlinear hyperbolic systems of Kirchhoff type. J. Differ. Equ. 132, 239–274 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    P. D’Ancona, S. Spagnolo, A class of nonlinear hyperbolic problems with global solutions. Arch. Ration. Mech. Anal. 124, 201–219 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    P. D’Ancona, S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation. Comm. Pure Appl. Math. 47, 1005–1029 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    P. D’Ancona, S. Spagnolo, Kirchhoff type equations depending on a small parameter. Chin. Ann. Math. 16B, 413–430 (1995)MathSciNetGoogle Scholar
  7. 7.
    M. Ghisi, M. Gobbino, Kirchhoff equation from quasi-analytic to spectral-gap data. Bull. Lond. Math. Soc. 43, 374–385 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    J.M. Greenberg, S.C. Hu, The initial-value problem for a stretched string. Quart. Appl. Math. 38, 289–311 (1980)zbMATHMathSciNetGoogle Scholar
  9. 9.
    C. Heiming, (=Kerler, C.), Mapping properties of generalized Fourier transforms and applications to Kirchhoff equations. Nonlinear Differ. Equ. Appl. 7, 389–414 (2000).Google Scholar
  10. 10.
    F. Hirosawa, Global solvability for Kirchhoff equation in special classes of non-analytic functions. J. Differ. Equ. 230, 49–70 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    K. Kajitani, in The Global Solutions to the Cauchy Problem for Multi-dimensional Kirchhoff Equation, 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, (Birkhäuser, Boston, 2009), pp. 141–153.Google Scholar
  12. 12.
    K. Kajitani, K. Yamaguti, On global analytic solutions of th degenerate Kirchhoff equation. Ann. Scuola Norm. Sup. Pisa Cl Sci. 4(21), 279–297 (1994).Google Scholar
  13. 13.
    G. Kirchhoff, Vorlesungen über Mechanik (Teubner, Leibzig, 1883)Google Scholar
  14. 14.
    R. Manfrin, On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discret. Contin. Dynam. Syst. 3, 91–106 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    R. Manfrin, On the global solvability of Kirchhoff equation for non-analytic initial data. J. Differ. Equ. 211, 38–60 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    T. Matsuyama, Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation. J. Math. Soc. Jpn. 64, 1167–1204 (2010)CrossRefMathSciNetGoogle Scholar
  17. 17.
    T. Matsuyama, The Kirchhoff equation with global solutions in unbounded domains, Rend. Istit. Mat. Univ. Trieste. 42 Suppl., 125–141 (2010).Google Scholar
  18. 18.
    T. Matsuyama, M. Ruzhansky, Scattering for strictly hyperbolic systems with time-dependent coefficients. Math. Nachr. 286, 1191–1207 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    T. Matsuyama, M. Ruzhansky, Global well-pesedness of Kirchhoff systems. J. Math. Pures Appl. 100, 220–240 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    T. Matsuyama, M. Ruzhansky, Asymptotic integration and dispersion for hyperbolic equations. Adv. Differ. Equ. 15, 721–756 (2010)zbMATHMathSciNetGoogle Scholar
  21. 21.
    S. Mizohata, The Theory of Partial Differential Equations, (Cambridge University Press, 1973).Google Scholar
  22. 22.
    T. Nishida, A note on the nonlinear vibrations of the elastic string. Mem. Fac. Eng. Kyoto Univ. 33, 329–341 (1971)MathSciNetGoogle Scholar
  23. 23.
    K. Nishihara, On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 7, 437–459 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    S.I. Pohožhaev, On a class of quasilinear hyperbolic equations. Math. USSR Sb. 25, 145–158 (1975)CrossRefGoogle Scholar
  25. 25.
    R. Racke, Generalized Fourier transforms and global, small solutions to Kirchhoff equations. Asymptot. Anal. 58, 85–100 (1995)zbMATHMathSciNetGoogle Scholar
  26. 26.
    W. Rzymowski, One-dimensional Kirchhoff equation. Nonlinear Anal. 48, 209–221 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    T. Yamazaki, Scattering for a quasilinear hyperbolic equation of Kirchhoff type. J. Differ. Equ. 143, 1–59 (1998)CrossRefzbMATHGoogle Scholar
  28. 28.
    T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three. Math. Methods Appl. Sci. 27, 1893–1916 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension three. J. Differ. Equ. 210, 290–316 (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

Personalised recommendations