Long-range scattering for a critical homogeneous type nonlinear Schr?dinger equation with time-decaying harmonic potentials
Long-range scattering for a critical homogeneous type nonlinear Schr?dinger equation with time-decaying harmonic potentials
复制标题
具有时间衰减谐波势的临界齐次非线性薛定谔方程的长程散射
DOI：
10.1016/j.jde.2023.04.009
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发表时间：
2023
影响因子：
2.4
通讯作者：
Miyazaki Hayato
中科院分区：
文献类型：
--
作者：
Kawamoto Masaki;Miyazaki Hayato
本文研究了具有时间衰减谐振势的齐型非线性薛定谔方程的终态问题。非线性具有临界阶，并且不一定是多项式的形式。在规范不变的幂类型非线性的情况下，第一作者证明了该方程存在一个非平凡解，它的行为类似于文[22]中具有对数位相校正的自由解。在本文中，我们利用Masaki和第二作者引入的傅里叶级数展开的技巧，将他的结果推广到具有一般齐次非线性的情形。为了适应上述文章中的论点，我们建立了传播子的因式分解恒等式，并要求由谐势产生的傅立叶系数有一个稍强的衰减条件。此外，在二维或三维上，我们改进了[26]、[29]中最终数据的正则性条件。
This paper is concerned with the final state problem for the homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26], [29].