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.
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Holmer, J., Roudenko, S. A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation. Commun. Math. Phys. 282, 435–467 (2008). https://doi.org/10.1007/s00220-008-0529-y
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DOI: https://doi.org/10.1007/s00220-008-0529-y