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

摘要: We consider solutions u to the 3d nonlinear Schrodinger equation i∂tu+Δu+|u|2u = 0. In particular, we are interested in finding criteria on initial data u0 that predict asymptotic behavior of u(t): whether u(t) blows‐up finite time, exists globally time but behaves like a linear solution for large times (scatters), or does not scatter. review how this question has been resolved H1 when M[u]E[u]⩽M[Q]E[Q], where M[u] and E[u] denote mass energy u, Q denotes ground state −Q+ΔQ+|Q|2Q = 0. Then complementary case M[u]E[u]>M[Q]E[Q], which few analytical results currently available. start with presenting an result due Lushnikov [8] gives sufficient condition blow‐up, different from previously known blow up criteria, then present alteration his argument some cases improves upon condition. The last is also extended radial initial‐data inf...