Abstract
We study the Cauchy problem for a system of cubic nonlinear Klein–Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order \(O(t^{-1/2})\) in \(L^\infty \) as t tends to infinity without the condition of a compact support on the Cauchy data which was assumed in the previous works.
Similar content being viewed by others
References
Delort, J.M.: Existence globale et comportement asymptotique pour l’équation de Klein–Gordon quasi linéaire à données petites en dimension 1. Ann. Sci. École Norm. Sup. 34(4), 1–61 (2001). (Erratum: Ann. Sci. École Norm. Sup.(4), 39 (2006), 335–345)
Fang, D., Xue, R.: Global existence of small solutions for cubic quasi-linear Klein–Gordon systems in one space dimension. Acta Math. Sin. 22, 1085–1102 (2006)
Hayashi, N., Naumkin, P.I.: The initial value problem for the cubic nonlinear Klein–Gordon equation. Z. Angew. Math. Phys. 59, 1002–1028 (2008)
Hayashi, N., Naumkin, P.I.: Final state problem for the cubic nonlinear Klein–Gordon equation. J. Math. Phys. 50(103511), 1–14 (2009)
Hayashi, N., Naumkin, P.I.: The initial value problem for the quadratic nonlinear Klein–Gordon equation. Adv. Math. Phys. 2010(504324), 1–35 (2010)
Hayashi, N., Naumkin, P.I.: Quadratic nonlinear Klein–Gordon equation in one space dimension. J. Math. Phys. 53(103711), 1–36 (2012)
Hayashi, N., Naumkin, P.I.: A system of quadratic nonlinear Klein–Gordon equations in 2d. J. Diff. Equ. 254, 3615–3646 (2013)
Katayama, S.: A note on global existence of solutions to nonlinear Klein–Gordon equations in one space dimension. J. Math. Kyoto Univ. 39, 203–213 (1999)
Kim, D., Sunagawa, H.: Remarks on decay of small solutions to systems of Klein–Gordon equations with dissipative nonlinearities. Nonlinear Anal. 97, 94–105 (2014)
Kim, D.: Global existence of small amplitude solutions to one-dimensional nonlinear Klein-Gordon systems with different masses, Preprint, arXiv:1406.3947 [math.AP]
Lindblad, H., Soffer, A.: A remark on asymptotic completeness for the critical nonlinear Klein–Gordon equation. Lett. Math. Phys. 73, 249–258 (2005)
Moriyama, K.: Normal forms and global existence of solutions to a class of cubic nonlinear Klein–Gordon equations in one space dimension. Differ. Integral Equ. 10, 499–520 (1997)
Sunagawa, H.: On global small amplitude solutions to systems of cubic nonlinear Klein–Gordon equations with different mass terms in one space dimension. J. Differ. Equ. 192, 308–325 (2003)
Sunagawa, H.: A note on the large time asymptotics for a system of Klein–Gordon equations. Hokkaido Math. J. 33, 457–472 (2004)
Sunagawa, H.: Large time asymptotics of solutions to nonlinear Klein–Gordon systems. Osaka J. Math. 42, 65–83 (2005)
Sunagawa, H.: Remarks on the asymptotic behavior of the cubic nonlinear Klein–Gordon equations in one space dimension. Differ. Integral Equ. 18, 481–494 (2005)
Sunagawa, H.: Large time behavior of solutions to the Klein–Gordon equation with nonlinear dissipative terms. J. Math. Soc. Jpn. 58, 379–400 (2006)
Acknowledgments
The author would like to express his gratitude to Professor Hideaki Sunagawa for his comments at the beginning of this work. The author also thanks unknown referees for their useful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, D. A Note on a System of Cubic Nonlinear Klein–Gordon Equations in One Space Dimension. Differ Equ Dyn Syst 25, 431–451 (2017). https://doi.org/10.1007/s12591-015-0259-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-015-0259-5