Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

International audienceIn this paper, we consider a class of the defocusing inhomogeneous nonlinear Schrödinger 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 in [22]. We next use this decay to show the energy scattering for the equation in the case $\alpha_\star < \alpha < \alpha^\star$, where $\alpha_\star= \frac{4-2b}{d}$