  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
$\alpha_\star = \frac{4-2b}{d}$.
