Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schr\"odinger equation

TL;DR

This paper proves that radial initial data below the ground state threshold leads to global existence and scattering for the 3D focusing inhomogeneous nonlinear Schrödinger equation with specific inhomogeneity parameter.

Contribution

It extends scattering results to the inhomogeneous 3D focusing NLS with radial data, using a Kenig-Merle type approach adapted for inhomogeneity.

Findings

Global existence and scattering for initial data below ground state threshold.

Conditions involving energy and mass determine scattering behavior.

Method adapts Kenig-Merle framework to inhomogeneous NLS.

Abstract

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and $s_c=(1+b)/2$. We show that if the radial initial data $u_0$ belongs to $H^1(\mathbb{R}^3)$ and satisfies $E(u_0)^{s_c}M(u_0)^{1-s_c}<E(Q)^{s_c}M(Q)^{1-s_c}$ and $\| \nabla u_0 \|_{L^2}^{s_c} \|u_0\|_{L^2}^{1-s_c}<\|\nabla Q \|_{L^2}^{s_c} \|Q\|_{L^2}^{1-s_c}$, then the corresponding solution is global and scatters in $H^1(\mathbb{R}^3)$. Our proof is based in the ideas introduced by Kenig-Merle \cite{KENIG} in their study of the energy-critical NLS and Holmer-Roudenko \cite{HOLROU} for the radial 3D cubic NLS.