### Abstract

We study the final state problem for the nonlinear Klein-Gordon equation, u_{tt} +u- u_{xx} =μ u^{3}, t ∈ R,x ∈ R, where μR. We prove the existence of solutions in the neighborhood of the approximate solutions 2 Re U (t) w+ (t), where U (t) is the free evolution group defined by U (t) = F^{-1} e-^{it} 〈φ〉 F, 〈x〉 = 1+ x^{2}, F and F-1 are the direct and inverse Fourier transformations, respectively, and w+ (t,x) = F-1 (û_{+} (φ) e^{(3/2) iμ 〈φ〉2} u_{+} (φ) ^{2} log t), with a given final data u+ is a real-valued function and ∥ 〈φ〉^{3} 〈 i∥_{φ}〉 u + (φ) ∥ L^{∞} is small.

Original language | English |
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Article number | 103511 |

Journal | Journal of Mathematical Physics |

Volume | 50 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2009 Nov 10 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Hayashi, N., & Naumkin, P. I. (2009). Final state problem for the cubic nonlinear Klein-Gordon equation.

*Journal of Mathematical Physics*,*50*(10), [103511]. https://doi.org/10.1063/1.3215980