A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger 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∂ _t u + Δu + |u|² u = 0 scatter, i.e., approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities ${\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}}$ 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 ${\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} < \|Q\|_{L^2}\|\nabla Q\|_{L^2}}$, 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 ${\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} > \|Q\|_{L^2}\|\nabla Q\|_{L^2}}$, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS.