Small data scattering of the inhomogeneous cubic–quintic NLS in 2 dimensions

• Published in: Nonlinear analysis Vol. 188; pp. 142 - 157
• Main Authors: Cho, Yonggeun, Lee, Kiyeon
• Format: Journal Article
• Language: English
• Published: Elmsford Elsevier Ltd 01-11-2019; Elsevier BV
• Subjects: 2D inhomogeneous NLS; Angular regularity; Scattering; Small data scattering; Sobolev space; 35Q55; Small data scattering; 35Q53; 2D inhomogeneous NLS; Angular regularity
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2019.05.021

The aim of this paper is to show the small data scattering for 2D ICQNLS: iut=−Δu+K1(x)|u|2u+K2(x)|u|4u.Under the assumption that |∂jKl|≲|x|bl−j for j=0,1,2,l=1,2 and 0≤bl≤l−23, we prove the small data scattering in an angularly regular Sobolev space Hθ1,1. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space Hθ1,1⊂H1 with angular regularity such that ‖∂θf‖H1<∞, and also use the recently developed angularly averaged Strichartz estimates (Tao, 2000; Cho and Lee, 2013; Guo et al., 2018). In addition, we suggest a sufficient condition for non-existence of scattering.