Sharp threshold of blow-up and scattering for the fractional Hartree equation
Sharp threshold of blow-up and scattering for the fractional Hartree equation
复制标题
DOI：
10.1016/j.jde.2017.11.001
复制
发表时间：
2017-05
影响因子：
2.4
通讯作者：
Qing Guo;Shihui Zhu
中科院分区：
文献类型：
--
作者：
Qing Guo;Shihui Zhu
我们考虑分数Hartree方程在L2-超临界的情况下，并找到一个尖锐的阈值的散射与爆破二分法的径向数据：如果M [u 0] s-s c s c E [u 0]< M [Q] s-s c s c E [Q]且M [u 0] s-s c s c u 0 H stecs 2< M [Q] s-s c s c Q H stecs 2，则解u（t）是全局适定的并且是分散的;如果M [u 0] s− s c s c E [u 0]< M [Q] s− s c s c E [Q]并且M [u 0] s− s c s c <$u 0 <$H stecs 2> M [Q] s− s c s c <$Q <$H stecs 2，则解u（t）在有限时间内爆破。这个条件是尖锐的，在这个意义上，孤立波解eit Q（x）是整体的，但不是散射的，这满足上述条件中的等式。这里，Q是分数Hartree方程的基态解。
We consider the fractional Hartree equation in the L 2-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If M [u 0] s− s c s c E [u 0]< M [Q] s− s c s c E [Q] and M [u 0] s− s c s c‖ u 0‖ H˙ s 2< M [Q] s− s c s c‖ Q‖ H˙ s 2, then the solution u (t) is globally well-posed and scatters; if M [u 0] s− s c s c E [u 0]< M [Q] s− s c s c E [Q] and M [u 0] s− s c s c‖ u 0‖ H˙ s 2> M [Q] s− s c s c‖ Q‖ H˙ s 2, the solution u (t) blows up in finite time. This condition is sharp in the sense that the solitary wave solution e i t Q (x) is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.