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On the Gevrey well-posedness of the Kirchhoff equation

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Abstract

This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space \(\gamma_{\eta,L^2}^s\). Furthermore, similar results are obtained for the initial-boundary value problems in bounded domains and in exterior domains with compact boundary.

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References

  1. A. Arosio, Asymptotic behaviour as t →+∞of the solutions of linear hyperbolic equations with coefficients discontinuous in time (on a bounded domain), J. Differential Equations 39 (1981), 291–309.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Arosio and S. Spagnolo, Global solutions to the Cauchy problem for a nonlinear hyperbolic equation, in Nonlinear Partial Differential Equations and Their Applications, Collège de France seminar, Vol. VI (Paris, 1982/1983), Research Notes in Mathematics, Vol. 109, Pitman, Boston, MA, 1984, pp. 1–26.

    Google Scholar 

  3. S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 17–27.

    MATH  Google Scholar 

  4. E. Callegari and R. Manfrin, Global existence for nonlinear hyperbolic systems of Kirchhoff type, J. Differential Equations 132 (1996), 239–274.

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Colombini, D. Del Santo and T. Kinoshita, Well-posedness of a hyperbolic equation with non-Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (5) 1 (2002), 327–358.

    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, A class of nonlinear hyperbolic problems with global solutions, Arch. Rational Mech. Anal. 124 (1993), 201–219.

    Article  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. P. D’Ancona and S. Spagnolo, Kirchhoff type equations depending on a small parameter, Chin. Ann. of Math. 16B (1995), 413–430.

    MathSciNet  MATH  Google Scholar 

  10. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Wiley-Interscience, NY, 1988.

    MATH  Google Scholar 

  11. M. Ghisi and M. Gobbino, Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), 613–646.

    MathSciNet  MATH  Google Scholar 

  12. M. Ghisi and M. Gobbino, Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness, Rend. Istit. Mat. Univ. Trieste 42 (2010), 89–110.

    MathSciNet  MATH  Google Scholar 

  13. M. Ghisi and M. Gobbino, Kirchhoff equation from quasi-analytic to spectral-gap data, Bull. London Math. Soc. 43 (2011), 374–385.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. M. Greenberg and S. C. Hu, The initial-value problem for a stretched string, Quart. Appl. Math. 38 (1980), 289–311.

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Heiming, Mapping properties of generalized Fourier transforms and applications to Kirchhoff equations, NoDEA Nonlinear Differential Equations Appl. 7 (2000), 389–414.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Hirosawa, Degenerate Kirchhoff equation in ultradifferentiable class, Nonlinear Anal., Ser. A: Theory Methods 48 (2002), 77–94.

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Hirosawa, Global solvability for Kirchhoff equation in special classes of non-analytic functions, J. Differential Equations 230 (2006), 49–70.

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Kajitani, The global solutions to the Cauchy problemfor multi-dimensional Kirchhoff equation, in Advances in Phase Space Analysis of Partial Differential Equations, Progress in Nonlinear Differential Equations and Their Applications, Vol. 78, Birkhäuser, Boston, MA, 2009, pp. 141–153.

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

    MathSciNet  MATH  Google Scholar 

  20. G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig, 1876.

    MATH  Google Scholar 

  21. T. Kotake and M. Narasimhan, Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France 90 (1962), 449–471.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Manfrin, On the global solvability of symmetric hyperbolic systems of Kirchhoff type, Discrete Contin. Dynam. Systems 3 (1997), 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Manfrin, On the global solvability of Kirchhoff equation for non-analytic initial data, J. Differential Equations 211 (2005), 38–60.

    Article  MathSciNet  MATH  Google Scholar 

  24. T. Matsuyama, Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation, J. Math. Soc. Japan 64 (2010), 1167–1204.

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Matsuyama, The Kirchhoff equation with global solutions in unbounded domains, Rend. Istit. Mat. Univ. Trieste. 42 (2010), 125–141.

    MathSciNet  MATH  Google Scholar 

  26. T. Matsuyama and M. Ruzhansky, Scattering for strictly hyperbolic systems with time-dependent coefficients, Math. Nachr. 286 (2013), 1191–1207.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Matsuyama and M. Ruzhansky, Global well-posedness of Kirchhoff systems, J. Math. Pures Appl. 100 (2013), 220–240.

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Matsuyama and M. Ruzhansky, Global Well-Posedness of the Kirchhoff Equation and Kirchhoff Systems, in Analytic Methods in Interdisciplinary Applications, Springer Proceedings in Mathematics & Statistics, Vol. 116, Springer, Berlin, 2015, pp. 81–96.

  29. K. Mochizuki, Spectral and Scattering Theory for Second Order Elliptic Differential Operators in an Exterior Domain, Lecture Notes Univ. Utah, Winter and Spring, 1972.

    Google Scholar 

  30. T. Nishida, A note on the nonlinear vibrations of the elastic string, Mem. Fac. Engrg. Kyoto Univ. 33 (1971), 329–341.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. S. I. Pohožaev, On a class of quasilinear hyperbolic equations, Math. USSR Sb. 25 (1975), 145–158.

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  34. W. Rzymowski, One-dimensional Kirchhoff equation, Nonlinear Analysis 48 (2002), 209–221.

    Article  MathSciNet  MATH  Google Scholar 

  35. N. A. Shenk, II, Eigenfunction expansions and scattering theory for the wave equation in an exterior region, Arch. Rational Mech. Anal. 21 (1966), 120–150.

    MathSciNet  Google Scholar 

  36. C. H. Wilcox, Scattering Theory for the D’Alembert Equation in Exterior Domains, Lecture Notes in Mathematics, Vol. 442, Springer-Verlag, Berlin–Heidelberg, 1975.

  37. T. Yamazaki, Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations 143 (1998), 1–59.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three, Math. Methods Appl. Sci. 27 (2004), 1893–1916.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension three, J. Differential Equations 210 (2005), 290–316.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Tokio Matsuyama

  2. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Michael Ruzhansky

  3. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, S8, 9000, Gent, Belgium

    Michael Ruzhansky

Authors
  1. Tokio Matsuyama
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  2. Michael Ruzhansky
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Corresponding author

Correspondence to Tokio Matsuyama.

Additional information

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.

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Matsuyama, T., Ruzhansky, M. On the Gevrey well-posedness of the Kirchhoff equation. JAMA 137, 449–468 (2019). https://doi.org/10.1007/s11854-019-0017-7

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  • DOI: https://doi.org/10.1007/s11854-019-0017-7

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