Energy scattering for a class of inhomogeneous nonlinear Schr\"odinger equation in two dimensions

Abstract

We consider a class of $L^2$-supercritical inhomogeneous nonlinear Schr\"odinger equations in two dimensions \[ i\partial_t u + \Delta u = \pm |x|^{-b} |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $0<b<1$ and $\alpha>2-b$. By adapting a new approach of Arora-Dodson-Murphy \cite{ADM}, we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah-Guzm\'an \cite{FG-high} to the whole range of $b$ where the local well-posedness is available. In the defocusing case, our result extends the one in \cite{Dinh-scat} where the energy scattering for non-radial initial data was established in dimensions $N\geq 3$.