Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation
Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation
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DOI：
10.1016/j.jde.2010.10.027
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发表时间：
2010-05
影响因子：
2.4
通讯作者：
K. Nakanishi;W. Schlag
中科院分区：
文献类型：
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作者：
K. Nakanishi;W. Schlag
非线性色散方程的全局动力学分析有着悠久的历史，从小解开始。在本文中，我们利用能量空间中的大量径向数据研究了 R3 中的聚焦三次非线性 Klein-Gordon 方程。该方程有一个唯一的正稳态解 Q，称为基态。 1975 年 Payne 和 Sattinger 证明，能量 E[u,u˙] 严格低于基态能量的解 u(t) 可分为两类，具体取决于合适的函数 K(u)： 如果 K(u)<0，则具有有限时间爆炸，如果 K(u)⩾0 全局存在；而且，这些集合在流下是不变的。最近，Ibrahim、Masmoudi 和第一作者 [22] 通过 Kenig-Merle 方法的变体将 K[u]⩾0 的散射建立为零，从而改进了这一结果（Kenig 和 Merle，2006，2008 [25,26]）。在本文中，我们稍微超出了基态能量，并对这种情况下的演化给出了完整的描述。例如，在 Q 的一个小邻域中，我们会遇到以下三分法：在中心稳定流形的一侧，在 t⩾0 时有有限时间爆炸，在另一侧散射到零，在流形本身上，在 t→+∞ 时，有到 Q 的散射。总的来说，能量至多略高于 Q 的数据类被分为九个不相交的非空集合，每个集合都显示出不同的渐近行为，即 t→±∞，其中包括解在一个时间方向上爆炸，而在另一个时间方向上散射为零。 Duyckaerts 和 Merle (2009, 2008) [13,14] 为能量临界波和薛定谔方程找到的解的类似物在这里显示为唯一的一维稳定/不稳定流形，分别以 t→∞ 或 t→−∞ 指数方式逼近 ±Q。我们证明中的主要技术成分是“单程”定理，该定理排除了 Q（以及 -Q）和连接 Q 与 -Q 的（几乎）异宿轨道之间（几乎）同宿轨道的存在。在一篇配套论文（Nakanishi 和 Schlag，2010 [31]）中，我们为 NLS 方程建立了类似的属性。
The analysis of global dynamics of nonlinear dispersive equations has a long history starting from small solutions. In this paper we study the focusing, cubic, nonlinear Klein–Gordon equation in R3with large radial data in the energy space. This equation admits a unique positive stationary solution Q, called the ground state. In 1975 Payne and Sattinger showed that solutions u(t) with energy E[u,u˙]strictly below that of the ground state are divided into two classes, depending on a suitable functional K(u): If K(u)<0, then one has finite time blow-up, if K(u)⩾0 global existence; moreover, these sets are invariant under the flow. Recently, Ibrahim, Masmoudi and the first author [22] improved this result by establishing scattering to zero for K[u]⩾0 by means of a variant of the Kenig–Merle method (Kenig and Merle, 2006, 2008 [25,26]). In this paper we go slightly beyond the ground state energy and we give a complete description of the evolution in that case. For example, in a small neighborhood of Q one encounters the following trichotomy: On one side of a center-stable manifold one has finite time blow-up for t⩾0, on the other side scattering to zero, and on the manifold itself one has scattering to Q, both as t→+∞. In total, the class of data with energy at most slightly above that of Q is divided into nine disjoint non-empty sets each displaying different asymptotic behavior as t→±∞, which includes solutions blowing up in one time direction and scattering to zero on the other. The analogue of the solutions found by Duyckaerts and Merle (2009, 2008) [13,14] for the energy critical wave and Schrödinger equations appear here as the unique one-dimensional stable/unstable manifolds approaching ±Q exponentially as t→∞ or t→−∞, respectively. The main technical ingredient in our proof is a “one-pass” theorem which excludes the existence of (almost) homoclinic orbits between Q (as well as −Q) and (almost) heteroclinic orbits connecting Q with −Q. In a companion paper (Nakanishi and Schlag, 2010 [31]) we establish analogous properties for the NLS equation.