Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions

• Published in: arXiv.org
• Main Authors: Masaki, Satoshi, Miyazaki, Hayato, Kota Uriya
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.06.2017
• Subjects: Asymptotic properties; Convergence; Nonlinear equations; Nonlinearity; Polynomials; Schrodinger equation; Time dependence
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1706.03491

In this paper, we consider the final state problem for the nonlinear Schr\"odinger 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.