The Kirchhoff Equation with Gevrey Data
In this article the Cauchy problem for the Kirchhoff equation is considered, and the almost global existence of Gevrey space solutions is described.
KeywordsGevrey 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.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
- 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
- 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
- 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
- 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.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
- 20.T. Matsuyama, The Kirchhoff equation with global solutions in unbounded domains. Rend. Istit. Mat. Univ. Trieste. 42 (Suppl.), 125–141 (2010)Google Scholar
- 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.T. Matsuyama, M. Ruzhansky, On the Gevrey well-posedness of the Kirchhoff equation. J. Anal. Math. (to appear)Google Scholar
- 24.T. Nishida, A note on the nonlinear vibrations of the elastic string. Mem. Fac. Eng. Kyoto Univ. 33, 329–341 (1971)Google Scholar