Global Behavior of Solutions to the Focusing 3d Cubic Nonlinear Schrodinger Equation

We consider solutions u to the 3d nonlinear Schrodinger equation i partial derivative(t)u + Delta u + vertical bar u vertical bar(2)u = 0. In particular, we are interested in finding criteria on the initial data u(0) that predict the asymptotic behavior of u(t): whether u(t) blows-up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. We review how this question has been resolved for H-1 data when M[u]E[u] <= M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u, and Q denotes the ground state solution to -Q+Delta Q+vertical bar Q vertical bar(2)Q = 0. Then we consider the complementary case M[u]E[u] > M[Q]E[Q], for which few analytical results are currently available. We start with presenting an analytical result due to Lushnikov [8] that gives a sufficient condition for blow-up, different from the previously known blow up criteria, and then present an alteration to his argument that in some cases improves upon his condition. The last condition is also extended to radial initial-data of infinite variance.