Small data scattering of the inhomogeneous cubic–quintic NLS in 2 dimensions. (November 2019) - Cho, Yonggeun; Lee, Kiyeon
Abstract: The aim of this paper is to show the small data scattering for 2D ICQNLS: i u t = − Δ u + K 1 ( x ) | u | 2 u + K 2 ( x ) | u | 4 u . Under the assumption that | ∂ j K l | ≲ | x | b l − j for j = 0, 1, 2, l = 1, 2 and 0 ≤ b l ≤ l − 2 3, we prove the small data scattering in an angularly regular Sobolev space H θ 1, 1 . We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space H θ 1, 1 ⊂ H 1 with angular regularity such that ‖ ∂ θ f ‖ H 1 < ∞, 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.