## Abstract

We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions (∂_{t}^{2} - Δ + m^{2}) u = λ u^{2}, (t, x) ∈ R × R^{2}, where λ ∈ C. We prove that if the final states u_{1}^{+} ∈ H_{frac(q, q - 1)}^{4 - frac(4, q)} (R^{2}) ∩ H^{frac(5, 2), 1} (R^{2}) ∩ H_{1}^{2} (R^{2}),u_{2}^{+} ∈ H_{frac(q, q - 1)}^{3 - frac(4, q)} (R^{2}) ∩ H^{frac(3, 2), 1} (R^{2}) ∩ H_{1}^{1} (R^{2}), and {norm of matrix} u_{1}^{+} {norm of matrix}_{H12} + {norm of matrix} u_{2}^{+} {norm of matrix}_{H11} is sufficiently small, where 4 < q ≤ ∞, then there exists a unique global solution u ∈ C ([T, ∞) ; L^{2} (R^{2})) to the nonlinear Klein-Gordon equations such that u (t) tends as t → ∞ in the L^{2} sense to the solution u_{0} (t) = u_{1}^{+} cos (〈 i ∇ 〉_{m} t) + (〈 i ∇ 〉_{m}^{- 1} u_{2}^{+}) sin (〈 i ∇ 〉_{m} t) of the free Klein-Gordon equation.

Original language | English |
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Pages (from-to) | 3826-3833 |

Number of pages | 8 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 71 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2009 Nov 1 |

## Keywords

- Nonlinear Klein-Gordon equations
- Quadratic nonlinearity
- Two space dimensions

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics