Scattering and Blow up for the Two Dimensional Focusing Quintic   Nonlinear Schr\"odinger Equation
  Using the concentration-compactness method and the localized virial type arguments, we study the behavior of $H^1$ solutions to the focusing quintic NLS in $\R^2$, namely, $$i \partial_t u+\Delta u+|u|^4u=0,\quad\quad (x, t) \in \R^2\times\R.$$   Denoting by $M[u]$ and $E[u]$, the mass and energy of a solution $u,$ respectively, and $Q$ the ground state solution to $-Q+\Delta Q+ |Q|^4Q=0$, and assuming $M[u]E[u] <M[Q]E[Q]$, we characterize the threshold for global versus finite time existence. Moreover, we show scattering for global existing time solutions and finite or "weak" blow up for the complement region. This work is in the spirit of Kenig and Merle and Duyckaerts, Holmer, and Roudenko. 
Cristi Guevara, Fernando Carreon
math.AP