论文标题
Hartree方程的阈值解决方案
Threshold solutions for the Hartree equation
论文作者
论文摘要
我们考虑$ 5 $ d的hartree方程，即$ l^2 $ superitical，具有有限的能量初始数据，并在质量能阈值下调查了解决方案。在Duyckaerts-Roudenko [11]的工作中，我们建立了特殊解决方案的存在。特别是，除了基础状态解决方案$ q $外，它是全球但非散制的，还有特殊解决方案$ q^+$和$ q^ - $，它在一个时间方向上呈$ q $ quide $ q $，而在另一个时间方向上，$ q^+$在有限的时间内和$ q^ - $ q^ - $ q^ - 存在，展示散射行为。然后，我们将所有径向阈值溶液表征为散射和在两个时间方向上的溶液（类似于质量能阈值下的溶液，请参见Arora-Roudenko [3]），或者像上面描述的特殊溶液一样。为了获得存在和分类结果，在本文中，我们对与Hartree方程相关的线性化操作员的光谱特性进行了彻底而细致的研究。
We consider the focusing $5$d Hartree equation, which is $L^2$-supercritical, with finite energy initial data, and investigate the solutions at the mass-energy threshold. We establish the existence of special solutions following the work of Duyckaerts-Roudenko [11] for the $3$d focusing cubic nonlinear Schrödinger equation (NLS). In particular, apart from the ground state solution $Q$, which is global but non-scattering, there exist special solutions $Q^+$ and $Q^-$, which in one time direction approach $Q$ exponentially, and in the other time direction $Q^+$ blows up in finite time and $Q^-$ exists for all time, exhibiting scattering behavior. We then characterize all radial threshold solutions either as scattering and blow up solutions in both time directions (similar to the solutions under the mass-energy threshold, see Arora-Roudenko [3]), or as the special solutions described above. To obtain the existence and classification result, in this paper we perform a thorough and meticulous investigation of the spectral properties of the linearized operator associated to the Hartree equation.
