论文标题
具有反平方势的能量临界NLS的阈值解决方案的动力学
Dynamics of threshold solutions for energy critical NLS with inverse square potential
论文作者
论文摘要
我们考虑了具有逆平方电位的聚焦能量关键NLS $ d = 3、4、5 $，其中$ d = 3 $中给出的详细信息以及其他维度的结果发表了评论。表征了基态能量表面上的溶液。我们证明，具有小于基态的动能的溶液必须散布到零或属于基态的稳定/不稳定的歧管。在后一种情况下，它们以$ t \ to \ infty $或$ t \ to -t to -\ infty $在能量空间中成倍收敛到基态。 （在没有径向假设的3-DIM中，这是在能量表面上非散射溶液的紧凑性假设下。）当动能大于基态的能量时，我们表明，所有径向$ h^1 $解决方案在有限的时间内都爆发出来，而在5 dim的情况下，唯一的两个例外是稳定的稳定/不稳定状态。证明依赖于详细的光谱分析，局部不变的歧管理论和全球病毒分析。
We consider the focusing energy critical NLS with inverse square potential in dimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results in other dimensions. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as $t\to \infty$ or $t\to -\infty$. (In 3-dim without radial assumption, this holds under the compactness assumption of non-scattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial $H^1$ solutions blow up in finite time, with the only two exceptions in the case of 5-dim which belong to the stable/unstable manifold of the ground state. The proof relies on the detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.
