A Sharp Condition for Scattering of the Radial 3d Cubic Nonlinear Schrödinger Equation

We consider the problem of identifying sharp criteria under which radial H (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂tu + ∆u + |u|2u = 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 ‖u0‖L2‖∇u0‖L2 and M [u]E[u], where u0 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 eQ(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 ‖u0‖L2‖∇u0‖L2 < ‖Q‖L2‖∇Q‖L2, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution eQ(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 ‖u0‖L2‖∇u0‖L2 > ‖Q‖L2‖∇Q‖L2, then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [16] in their study of the energy-critical NLS.