Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation

Abstract

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger 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 $\Vert \nabla u_0{\Vert }_{L^2}^{s_c}\Vert u_0{\Vert }_{L^2}^{1-s_c}<\Vert \nabla Q{\Vert }_{L^2}^{s_c}\Vert Q{\Vert }_{L^2}^{1-s_c}$, 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.

arXiv comment39 pages