### Abstract

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Korteweg-de Vries (gKdV) equationu_{t}+(u^{ρ-1}u)_{x}+13u_{xxx}=0, wherex,t∈Rwhen the initial data are small enough. If the powerρof the nonlinearity is greater than 3 then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solutionu(t) satisfies the decay estimate u(t)_{Lβ}≤C(1+t) ^{-(1/3)(1-1/β)}forβ∈(4,∞], uu_{x}(t)_{L∞}≤Ct^{-2/3}(1+t)^{-1/3}and using these estimates we prove the existence of the scattering stateu_{+}∈L^{2}such that u(t)-U(t)u_{+L2}≤Ct^{-(ρ-3)/3}for any small initial data belonging to the weighted Sobolev spaceH^{1,1}={f∈L^{2}; (1+x^{2})^{1/2}(1-∂^{2}_{x}) ^{1/2}f_{L2}<∞}, whereU(t) is the Airy free evolution group.

Original language | English |
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Pages (from-to) | 110-136 |

Number of pages | 27 |

Journal | Journal of Functional Analysis |

Volume | 159 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1998 Oct 20 |

Externally published | Yes |

### ASJC Scopus subject areas

- Analysis

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## Cite this

*Journal of Functional Analysis*,

*159*(1), 110-136. https://doi.org/10.1006/jfan.1998.3291