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

Abstract

The aim of this paper is to show the small data scattering for 2D ICQNLS: $$i{u}_t=-\Delta u+K_1(x)|u{|}^{2}u+K_2(x)|u{|}^{4}u.$$ Under the assumption that $\left|{\partial }^j{K}_l\right|\lesssim |x{|}^{b_l-j}$ for $j=0,1,2,l=1,2$ and $0\le b_l\le l-\frac{2}{3}$, we prove the small data scattering in an angularly regular Sobolev space $H_{\theta }^{1,1}$. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space $H_{\theta }^{1,1}\subset H^1$ with angular regularity such that $\Vert {\partial }_{\theta }f{\Vert }_{H^1}<\infty$, and also use the recently developed angularly averaged Strichartz estimates \cite{stri2, cholee, ghn}. In addition, we suggest a sufficient condition for non-existence of scattering.

15 pages, to appear in Nonlinear Analysis

References in corpus (1)

• Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities

Cited by in corpus (1)

• On the GWP of focusing energy-ciritical inhomogeneous NLS