Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions
Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions
复制标题
DOI：
10.1090/tran/7636
复制
发表时间：
2017-06
影响因子：
1.3
通讯作者：
Satoshi Masaki;H. Miyazaki;Kota Uriya
中科院分区：
文献类型：
--
作者：
Satoshi Masaki;H. Miyazaki;Kota Uriya
本文考虑具有临界阶齐次非线性项的非线性薛定谔方程的终态问题，其中临界阶齐次非线性项不一定是多项式。在文献[10]中，第一和第二作者考虑了一维和二维情形，给出了相应方程的非线性的一个充分条件，即在有或没有对数相位校正的情况下，方程的解具有自由解的性质。本文致力于研究三维情况，其中要求解以比低维情况更快的速度收敛到给定的渐近轮廓。为了获得必要的收敛速度，我们采用端点的Eschhartz估计，并修改了一个依赖于时间的正则化算子，在[10]中介绍。此外，我们提出了一个候选的第二渐近轮廓的解决方案。
In this paper, we consider the final state problem for the nonlinear Schrodinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [10], the first and the second authors consider one- and two-dimensional cases and gave a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converges to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in [10]. Moreover, we present a candidate of the second asymptotic profile to the solution.