Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation

Abstract

In this paper, we consider the following inhomogeneous nonlinear Schr\"odinger equation (INLS) \[ i\partial_t u + \Delta u + \mu |x|^{-b} |u|^\alpha u = 0, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d \] with $b, \alpha>0$. First, we revisit the local well-posedness in $H^1(\mathbb{R}^d)$ for (INLS) of Guzm\'an [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. $\mu=-1$ when $0<\alpha<\alpha^\star$ where $\alpha^\star = \frac{4-2b}{d-2}$ for $d\geq 3$, and $\alpha^\star = \infty$ for $d=1, 2$ by assuming that the initial data belongs to the weighted $L^2$ space $\Sigma =\{u \in H^1(\mathbb{R}^d) : |x| u \in L^2(\mathbb{R}^d) \}$. Finally, we combine the local theory and the decaying property to show the scattering in $\Sigma$ for the defocusing (INLS) in the case $\alpha_\star<\alpha<\alpha^\star$, where $\alpha_\star = \frac{4-2b}{d}$.