A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Abstract

We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) $i{\partial }_{t}u+\Delta u+|u{|}^{2}u=0$ scatter, i.e. approach the solution to a linear Schrödinger equation as $t\to \pm \infty$. The criteria is expressed in terms of the scale-invariant quantities $\Vert u_0{\Vert }_{L^2}\Vert \nabla u_0{\Vert }_{L^2}$ and $M[u]E[u]$, where $u_0$ denotes the initial data, and $M[u]$ and $E[u]$ denote the (conserved in time) mass and energy of the corresponding solution $u(t)$. The focusing NLS possesses a soliton solution $e^{it}Q(x)$, where $Q$ is the ground-state solution to a nonlinear elliptic equation, and we prove that if $M[u]E[u]<M[Q]E[Q]$ and $\Vert u_0{\Vert }_{L^2}\Vert \nabla u_0{\Vert }_{L^2}<\Vert Q{\Vert }_{L^2}\Vert \nabla Q{\Vert }_{L^2}$, then the solution $u(t)$ is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution $e^{it}Q(x)$, for which equality in these conditions is obtained, is global but does not scatter. We further show that if $M[u]E[u]<M[Q]E[Q]$ and $\Vert u_0{\Vert }_{L^2}\Vert \nabla u_0{\Vert }_{L^2}>\Vert Q{\Vert }_{L^2}\Vert \nabla Q{\Vert }_{L^2}$, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle \cite{KM06a} in their study of the energy-critical NLS.

References in corpus (3)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3
• Blow up of the critical norm for some radial L^2 super critical nonlinear Schrodinger equations

Cited by in corpus (6)

• Scattering for the non-radial 3D cubic nonlinear Schroedinger equation
• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions
• Threshold solutions for the focusing 3d cubic Schroedinger equation
• A Critical Centre-Stable Manifold for the Cubic Focusing Schroedinger Equation in Three Dimensions
• Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications