Scattering threshold for the focusing nonlinear Klein–Gordon equation
Scattering threshold for the focusing nonlinear Klein–Gordon equation
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DOI：
10.2140/apde.2011.4.405
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发表时间：
2010-01
期刊：
影响因子：
2.2
通讯作者：
S. Ibrahim;N. Masmoudi;K. Nakanishi
中科院分区：
文献类型：
--
作者：
S. Ibrahim;N. Masmoudi;K. Nakanishi
按照KenigMerle关于H临界波和薛定谔方程的精神，我们给出了聚焦非线性Klein-Gordon方程在基态能量以下的散射和爆破二分法。我们的结果包括H临界情况，其中阈值由无质量方程的基态给出，以及2D平方指数情况，其中基态的质量可以根据尖锐的Trudinger-Moser不等式中的常数而改变。主要的困难在于线性项和非线性项都缺乏标度不变性。
We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of KenigMerle for the H critical wave and Schrödinger equations. Our result includes the H critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger-Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.