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The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 132, Number 2, April 2010
- pp. 361-424
- 10.1353/ajm.0.0107
- Article
- Additional Information
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We consider the focusing energy-critical nonlinear Schr\"odinger equation
$iu_t+\Delta u = - |u|^{4\over{d-2}}u$ in dimensions $d\geq 5$. We prove
that if a maximal-lifespan solution $u\colon \ I\times {\Bbb R}^d\to {\Bbb C}$ obeys $\sup_{t\in I}\|\nabla u(t)\|_2<\|\nabla W\|_2$, then it is
global and scatters both forward and backward in time. Here $W$ denotes
the ground state, which is a stationary solution of the equation. In
particular, if a solution has both energy and kinetic energy less than
those of the ground state $W$ at some point in time, then the solution is
global and scatters. We also show that any solution that blows up with
bounded kinetic energy must concentrate at least the kinetic energy of the
ground state. Similar results were obtained by Kenig and Merle for
spherically symmetric initial data and dimensions $d=3,4,5$.