论文标题
聚焦广义的Hartree方程的阈值解决方案
Threshold solutions for the focusing generalized Hartree equations
论文作者
论文摘要
我们研究了对聚焦概括的Hartree方程的全球行为，并在范围内的质量能量阈值下使用$ H^1 $数据。在Arora-Roudenko [Arora-Roudenko 2021]的较早作品中，将质量能阈值以下的解决方案的行为分类。在本文中，我们首先展示了三个特殊解决方案：$ e^{it} q $，$ q^\ pm $，其中$ q^\ pm $指数级用于$ e^{it} q $在积极的时间方向上，$ q^+$ blows up up和$ q^ - $ q^ - $ $ q^ - $ $ $ scatter of负时间方向。然后，我们在此阈值下对解决方案进行了分类，表明它们的行为完全像以上三个特殊解决方案，直到对称性为对称性，或者在两个时间方向上散射或爆炸。该论点依赖于基态的独特性和非分类性，我们认为这是一般情况的假设。
We study the global behavior of solutions to the focusing generalized Hartree equation with $H^1$ data at mass-energy threshold in the inter-range case. In the earlier works of Arora-Roudenko [Arora-Roudenko 2021], the behavior of solutions below the mass-energy threshold was classified. In this paper, we first exhibit three special solutions: $e^{it} Q$, $Q^\pm$, where $Q^\pm$ exponentially approach to the $e^{it} Q$ in the positive time direction, $Q^+$ blows up and $Q^-$ scatters in the negative time direction. Then we classify solutions at this threshold, showing that they behave exactly as the above three special solutions up to symmetries, or scatter or blow up in both time directions. The argument relies on the uniqueness and non-degeneracy of ground state, which we regard as an assumption for the general case.
