Global Behavior of Solutions to the Focusing Generalized Hartree Equation
Global Behavior of Solutions to the Focusing Generalized Hartree Equation
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DOI：
10.1307/mmj/20205855
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发表时间：
2019-04
影响因子：
0.9
通讯作者：
A. Arora;S. Roudenko
中科院分区：
文献类型：
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作者：
A. Arora;S. Roudenko
本文研究了非线性广义Hartree方程解的整体性态，其中非线性项是非局部的，并表示为卷积，$$ i u_t + \Delta u +（|X| ^{-（N-\gamma）} \ast| u| ^p）|u| ^{p-2}u=0，\quad x \in \mathbb{R}^N，t\in \mathbb{R}. $$我们的主要目标是了解这个方程的$H^1$（有限能量）解在各种情况下的行为。在这项工作中，我们对这一目标进行了初步尝试。我们首先研究了H^1 $局部适定性和小数据理论。然后，我们在临界区间（$0<s<1$），在质能假设$\mathcal{ME}[u_0]<1$下对H^1 $解的行为进行分类，通过相应卷积型Gagliardo-Nirenberg插值不等式的尖锐常数确定全局解与有限时间解的尖锐阈值（注意，基态的唯一性在一般情况下是未知的）。特别是，根据初始质量和梯度的大小，解要么始终存在并以$H^1$分散，要么在有限时间内爆炸。为了得到H^1 $散射，本文采用了著名的Kenig-Merle [33]的浓度紧致性和刚性方法，其新奇在于研究了通过与$的负幂卷积给出的非局域非线性势。|X|和不同的非线性幂。
We study the global behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of the non-local type and is expressed as a convolution, $$ i u_t + \Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0, \quad x \in \mathbb{R}^N, t\in \mathbb{R}. $$ Our main goal is to understand behavior of $H^1$ (finite energy) solutions of this equation in various settings. In this work we make an initial attempt towards this goal. We first investigate the $H^1$ local wellposedness and small data theory. We then, in the intercritical regime ($0<s<1$), classify the behavior of $H^1$ solutions under the mass-energy assumption $\mathcal{ME}[u_0]<1$, identifying the sharp threshold for global versus finite time solutions via the sharp constant of the corresponding convolution type Gagliardo-Nirenberg interpolation inequality (Note that the uniqueness of a ground state is not known in the general case). In particular, depending on the size of the initial mass and gradient, solutions will either exist for all time and scatter in $H^1$, or blow up in finite time. To obtain $H^1$ scattering, in this paper we employ the well-known concentration compactness and rigidity method of Kenig-Merle [33] with the novelty of studying the nonlocal nonlinear potential given via convolution with negative powers of $|x|$ and different powers of nonlinearities.