## Abstract

We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation, u_{t} + ( -1)^{α} uu_{x} - u_{xx} + (-1)^{α} u_{xxx} = 0, (x, t) ε R^{+} × R^{+}, (0.1) u(x, 0) = u_{0}(x), x ε R^{+}, ∂_{x}^{n}(0, t) = 0, n = 0, α, t ε R^{+}, where α = 0, 1. We prove that if the initial data u_{0} ε H^{0,ω} ∩ H^{1,0}, where H^{s,k} = {f ε L^{2}; ∥f∥_{H}^{sk} = ∥ 〈x〉^{k} 〈i∂_{x}〉^{s} f∥_{L}^{2} < ∞}, ω ε (1/2, 3/2), and the norm ∥u_{0}∥_{H}^{0} ω + ∥u_{0}∥_{H}^{1},^{0} is sufficiently small, then there exists a unique solution u ε C([0, ∞), H^{0},^{x}) ∩ C((0, ∞), H^{1},ω) of the initial-boundary value problem (0.1), where x ε (0,1/2). Moreover, if the initial data are such that x^{1+μ} u_{0}(x) ε L^{1}, μ = ω - 1/2, then there exists a constant A such that the solution has the asymptotics for t → ∞ uniformly with respect to x > 0, where α = 0, 1, Φ_{0} (q, t) = (q/π)e^{-q2}, Φ_{1} (q,t) 1/2 π t) (e^{-q2}) (2q t-1) + e^{-2q} ^{t}.

Original language | English |
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Pages (from-to) | 343-370 |

Number of pages | 28 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 265 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2002 Jan 15 |

## Keywords

- Dissipative nonlinear evolution equation
- Half-line
- Korteweg-de Vries-Burgers equation
- Large time asymptotics