## Abstract

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line{A formula is presented} We prove that if the initial data u_{0} ∈ H_{1} ^{0, frac(21,4)} ∩ H_{2}^{1, frac(7, 2)} and the norm {norm of matrix} u_{0} {norm of matrix}_{H}^{1}0, _{frac(21, 4)} + {norm of matrix} u_{0} {norm of matrix}_{H}^{2}1,_{ frac(7, 2)} {less-than or slanted equal to} ε, where ε > 0 is small enough {Mathematical expression}, 〈 x 〉 = sqrt(1 + x^{2}) and λ ∫_{0}^{∞} x u_{0} ( x ) d x = λ θ < 0. Then there exists a unique solution u ∈ C ( [ 0, ∞ ), H_{2}^{1, frac(7, 2)} ) ∩ L^{2} ( 0, ∞ ; H_{2}^{2, 3} ) of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptotics{A formula is presented} for t → ∞ uniformly with respect to x > 0, where η = - 9 θ λ ∫_{0}^{∞} A i^{′ 2} ( z ) d z and A i ( q ) is the Airy function{A formula is presented}.

Original language | English |
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Pages (from-to) | 168-201 |

Number of pages | 34 |

Journal | Journal of Differential Equations |

Volume | 225 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Jun 1 |

## Keywords

- Half-line
- Korteweg-de Vries equation
- Large time asymptotics
- Nonlinear evolution equation