Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrödinger Equation in $H^s(\mathbb{R}^n)$

Abstract

and f (u) is a nonlinear function that behaves like λ |u| u with λ ∈ C and σ > 0. We prove that the Cauchy problem of the INLS equation is globally well–posed in Hs(Rn) if the initial data is sufficiently small and σ0 < σ < σs, where σ0 = 4−2b n and σs = 4−2b n−2s if s < n 2 ; σs = ∞ if s ≥ n 2 . Our global well–posedness result improves the one of Guzmán in (Nonlinear Anal. Real World Appl. 37: 249–286, 2017) by extending the validity of s and b. In addition, we also have the small data scattering result.