### 抜粋

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation (Formula Presented) where ℒ is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C_{1}|ξ|^{2}/(1 + ξ^{2}) and |L′(ξ)| ≤ C_{2}(|ξ| + |ξ|^{n}) for all ξ ∈R. Here, C_{1},C_{2} > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ^{2} + O(|ξ|^{2+γ}) for all |ξ| < 1, where γ > 0, Reα > 0, Im α ≥ 0. When L(ξ) = αξ^{2}, equation (A) is the nonlinear Schrödinger equation with dissipation u_{t} - αu_{xx} + i|u|^{2}u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate ∥u(t)∥_{∞} ≤ C(1 + t)^{-1/2}(1 + log(1 + t))^{-1/2σ} under the conditions that u_{0} ∈ H^{n,0} ∩ H^{0,1} have the mean value û_{0}(0) = 1/√2π ∫ u_{0}(x) dx ≠ 0 and the norm ∥u_{0}∥_{Hn,0} + ∥u_{0}∥_{H0,1} = ε is sufficiently small, where σ= 1 if Im α > 0 and σ = 2 if Im α = 0, and H^{m,s} = {φ ∈ S′; ∥φ∥_{m,s} = ∥(1 + x^{2})^{s/2}(1 - ∂_{x}^{2})^{m/2}φ∥ < ∞}, m,s ∈ R. Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation u_{t} - αu_{xx} + i|u|^{p-1}u = 0, with p > 3 have the same time decay estimate ∥u∥_{L∞} = O(t^{-1/2}) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

元の言語 | English |
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ページ（範囲） | 1029-1043 |

ページ数 | 15 |

ジャーナル | Royal Society of Edinburgh - Proceedings A |

巻 | 130 |

発行部数 | 5 |

出版物ステータス | Published - 2000 12 1 |

外部発表 | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

## フィンガープリント Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation' の研究トピックを掘り下げます。これらはともに一意のフィンガープリントを構成します。

## これを引用

*Royal Society of Edinburgh - Proceedings A*,

*130*(5), 1029-1043.