Energy scattering for a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation

In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u - |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1, \] with $b, \alpha>0$. We firstly study the decaying property of global solutions for the equation when $0<\alpha<\alpha^\star$ where $\alpha^\star = \frac{4-2b}{d-2}$ for $d\geq 3$. The proof makes use of an argument of Visciglia. We next use this decay to show the energy scattering for the equation in the case $\alpha_\star<\alpha<\alpha^\star$, where $\alpha_\star = \frac{4-2b}{d}$.