We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers equation

u t +u xxx +u p u x +εv y -νu xx =0,v x =u y ,u(0)=ϕ

in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts. After showing local existence by the contraction principle for initial data ϕ∈H s (ℝ 2 ) such that ℱ -1 (k 2 k 1 ϕ ^)∈H r (ℝ 2 ), 0≤r≤s-1, we extend the solutions for all positive times. Whereas for ε=-1 and 1≤p<4/3 this is done without any assumption on the initial data, we require a smallness condition on the initial data otherwise. In a last part, we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in L 2 (ℝ 2 ) when ε=-1 and 1≤p<4/3.

Molinet, L (2000). *The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation.* Differential and Integral Equations, 13(1-3):189-216.

## Abstract

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers equation

u t +u xxx +u p u x +εv y -νu xx =0,v x =u y ,u(0)=ϕ

in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts. After showing local existence by the contraction principle for initial data ϕ∈H s (ℝ 2 ) such that ℱ -1 (k 2 k 1 ϕ ^)∈H r (ℝ 2 ), 0≤r≤s-1, we extend the solutions for all positive times. Whereas for ε=-1 and 1≤p<4/3 this is done without any assumption on the initial data, we require a smallness condition on the initial data otherwise. In a last part, we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in L 2 (ℝ 2 ) when ε=-1 and 1≤p<4/3.

## Citations

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | local existence; smallness condition on the initial data; local smoothing effect; existence of weak global solutions |

Language: | English |

Date: | 2000 |

Deposited On: | 29 Nov 2010 16:27 |

Last Modified: | 05 Apr 2016 13:26 |

Publisher: | Khayyam |

ISSN: | 0893-4983 |

Official URL: | http://www.aftabi.com/DIE/die13.html |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1811955 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0974.35109 |

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