Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case
Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case
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DOI：
10.1016/j.jde.2008.11.011
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发表时间：
2008-07
影响因子：
2.4
通讯作者：
C. Miao;Guixiang Xu;Lifeng Zhao
中科院分区：
文献类型：
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作者：
C. Miao;Guixiang Xu;Lifeng Zhao
考虑四阶聚焦能量临界非线性薛定谔方程iut+Δ2u=| u| 8d−4u，d 5。证明了如果最大寿命径向解u：I×Rd→C服从supt∈I <$Δu（t）<$2 <<$ΔW <$2，则它是全局的，并且在时间上向前和向后都是离散的.这里W表示基态，它是方程的稳态解。特别是，如果一个解在某个时间点的能量和动能都小于基态W，那么这个解是全局的并且是分散的。
We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order iut+Δ2u=|u|8d−4u, d⩾5. We prove that if a maximal-lifespan radial solution u:I×Rd→C obeys supt∈I‖Δu(t)‖2<‖ΔW‖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.