Scattering for the $2d$ NLS with inhomogeneous nonlinearities

TL;DR

This paper proves large-data scattering for the 2D inhomogeneous nonlinear Schrödinger equation across all powers, employing concentration-compactness and Morawetz estimates to handle inhomogeneity and decay conditions.

Contribution

It extends scattering results to inhomogeneous nonlinearities in 2D NLS for all powers, using novel adaptation of concentration-compactness and Morawetz methods.

Findings

Established scattering for all powers p>0 in 2D inhomogeneous NLS

Developed decay conditions for p≤2 cases

Precluded compact solutions via Morawetz estimates

Abstract

We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in two space dimensions for all powers $p>0$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in the case $0<p\leq 2$. We use the method of concentration-compactness and contradiction. We preclude the existence of compact solutions using a Morawetz estimate in the style of Nakanishi.