Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS) \[ i\partial_t u + Δu + μ|x|^{-b} |u|^αu = 0, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d \] with $b, α>0$. First, we revisit the local well-posedness in $H^1(\mathbb{R}^d)$ for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. $μ=-1$ when $0<α<α^\star$ where $α^\star = \frac{4-2b}{d-2}$ for $d\geq 3$, and $α^\star = \infty$ for $d=1, 2$ by assuming that the initial data belongs to the weighted $L^2$ space $Σ=\{u \in H^1(\mathbb{R}^d) : |x| u \in L^2(\mathbb{R}^d) \}$. Finally, we combine the local theory and the decaying property to show the scattering in $Σ$ for the defocusing (INLS) in the case $α_\star<α<α^\star$, where $α_\star = \frac{4-2b}{d}$.