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 $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0,u(0)=u_0\in H^1,$$ 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 $d\ge 3$. 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 ${\alpha }_{\star }=\frac{4-2b}{d}$.