## Abstract

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation u_{t} + uu_{x} ? u_{xx} + u _{xxx} = 0, x ε R, t > 0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u _{0} ε H^{s} (R) ∩ L ^{1} (R), where s > - 1/2, then there exists a unique solution u (t, x) ε C^{∞} ((0,∞) ;H ^{∞} (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t) = t^{-1/2} f_{M} (( ·)t^{-1/2}) + o(t ^{-1/2}) as t → ∞, where f _{M} is the self-similar solution for the Burgers equation. Moreover if xu _{0} (x) L ^{1} (R) , then the asymptotics are true u(t) = t^{-1/2} f _{M} (( ·)t^{-1/2}) + O(t^{-1/2-γ}) where γ ε (0,1/2).

Original language | English |
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Pages (from-to) | 1441-1456 |

Number of pages | 16 |

Journal | Acta Mathematica Sinica, English Series |

Volume | 22 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2006 Sep |

Externally published | Yes |

## Keywords

- Asymptotics for large time
- Korteweg-de Vries-Burgers equation
- Large initial data

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics