Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

Details Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case

2008-07-04

arxiv.org/abs/0807.0690v2

J. Differential Equations 246 (2009) 3715-3749

Mathematics

Analysis of PDEs

37 pages, no figure

Scientific paper

10.1016/j.jde.2008.11.011

We consider the focusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys $\displaystyle\sup_{t\in I}\|\Delta u(t)\|_{2}<\|\Delta W\|_{2}$, 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.

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