Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schr\"odinger Equation

TL;DR

This paper analyzes the long-term behavior of solutions to the focusing quintic nonlinear Schrödinger equation in two dimensions, establishing criteria for scattering, blow-up, and global existence based on initial data and ground state properties.

Contribution

It characterizes the threshold between scattering and blow-up for the 2D focusing quintic NLS using concentration-compactness and virial methods, extending prior results to this specific equation.

Findings

Identifies the threshold for global existence versus blow-up based on initial mass and energy.

Proves scattering for solutions below the threshold.

Shows finite or weak blow-up for solutions above the threshold.

Abstract

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.