Book chapter
Scattering and blow up for the two-dimensional focusing quintic nonlinear Schrödinger equation
Recent Advances in Harmonic Analysis and Partial Differential Equations, Vol.581
Contemporary Mathematics, American Mathematical Society
27 Mar 2012
Abstract
Contemporary Mathematics, Volume 581, 2012 117-153 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.
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Details
- Title
- Scattering and blow up for the two-dimensional focusing quintic nonlinear Schrödinger equation
- Creators
- Cristi GuevaraFernando Carreon
- Publication Details
- Recent Advances in Harmonic Analysis and Partial Differential Equations, Vol.581
- Series
- Contemporary Mathematics
- Publisher
- American Mathematical Society
- Resource Type
- Book chapter
- Language
- English
- Academic Unit
- Mathematics
- Identifiers
- 991020534942704721