Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation
Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation
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耗散非线性薛定谔方程解的大时间行为
DOI：
10.1017/s0308210500000561
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
P. Naumkin
中科院分区：
文献类型：
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作者：
N. Hayashi;E. Kaikina;P. Naumkin
本文研究了具有耗散项的非线性薛定谔方程的柯西问题，其中L是一个具有耗散符号REL(ξ)≥C1|ξ|2/(1+ξ2)和|L‘(ξ)|≤C2(|ξ|+|ξ|n))的线性拟微分算子，对所有的ξ∈R.这里，C1，C2>；0，n≥1.此外，我们假定L(ξ)=αξ2+O(|ξ|2+γ)对于所有的|ξ|<；1，其中γ>；0，Reα>；0，Imα≥0.当L(ξ)=αξ2)时，方程(A)是带耗散的非线性薛定谔方程，−αuxx+i|u|2u=0.我们的目的是在u0∈hn，0∩H0，1具有平均值和范数‖u0‖hn，0+‖u0‖H0，1=ε充分小的条件下证明(A)的解满足时间衰减估计，其中σ=1当Imα>；0，σ=2当Imα=0时，因此方程(A)被认为是大时间渐近性态的临界情况，因为方程Ut−αuxx+i|u|p−1U=0，其中p>；3具有与线性方程解相同的时间衰减估计‖u‖L∞=O(t−ç)。另一方面，注意柯西问题(A)的解有额外的对数时间衰减。我们证明解的大时间渐近性的策略是将(A)转化为另一个非线性方程，其中的非线性项的平均值始终为零。
We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrödinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and 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 ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) 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.