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

## TL;DR

This paper proves energy scattering for a class of inhomogeneous nonlinear Schrödinger equations in two dimensions, extending previous results to a broader range of parameters and initial data types.

## Contribution

It introduces a new approach to establish energy scattering for radially symmetric initial data in 2D inhomogeneous NLS, extending prior work to more general cases.

## Key findings

- Energy scattering established for radially symmetric initial data.
- Extension of results to the full range of parameters where local well-posedness holds.
- Applicable to both focusing and defocusing cases 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 $N\geq 3$.