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

In this paper, we consider the following inhomogeneous nonlinear Schr"odinger equation (INLS) $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mu \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d$$ with $b, \alpha>0$\mathrm{.}First,werevisitthelocalwell-posednessin$H^1($\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 $\mathrm{\Sigma }$ for the defocusing (INLS) in the case ${\alpha }_{\star }<\alpha <{\alpha }^{\star }$, where ${\alpha }_{\star }=\frac{4-2b}{d}$.