Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schrödinger Equation

Details Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schrödinger Equation Scattering and Blow up for the Two Dimensional Focusing Quintic Nonlinear Schrödinger Equation

2012-03-27

arxiv.org/abs/1203.6089v2

Mathematics

Analysis of PDEs

37 pages, 2 figures and updated references

Scientific paper

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]

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