Instability of standing wave, global existence and blowup for the Klein–Gordon–Zakharov system with different-degree nonlinearities
Instability of standing wave, global existence and blowup for the Klein–Gordon–Zakharov system with different-degree nonlinearities
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DOI：
10.1016/j.jde.2009.03.003
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发表时间：
2009-05
影响因子：
2.4
通讯作者：
Z. Gan;B. Guo;Jian Zhang
中科院分区：
文献类型：
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作者：
Z. Gan;B. Guo;Jian Zhang
本文讨论了二维和三维空间中具有不同程度非线性的Klein-Gordon-Zakharov系统。首先，用复杂的变分方法证明了具有基态的驻波的存在性。其次，通过引入辅助泛函和等效最小化问题，我们得到了Klein-Gordon-Zakharov系统Cauchy问题解流下的两个不变流形。在此基础上，利用上述两种不变流形构造了一类约束变分问题，并应用势阱论证和凹性方法，得到了全局存在和爆破的一个尖锐阈值。然后，结合上述结果，我们利用膨胀变换得到了解全局存在的初始数据有多小的两个结论。最后，对所研究的系统证明了一种修正的驻波不稳定性。
This paper discusses the Klein–Gordon–Zakharov system with different-degree nonlinearities in two and three space dimensions. Firstly, we prove the existence of standing wave with ground state by applying an intricate variational argument. Next, by introducing an auxiliary functional and an equivalent minimization problem, we obtain two invariant manifolds under the solution flow generated by the Cauchy problem to the aforementioned Klein–Gordon–Zakharov system. Furthermore, by constructing a type of constrained variational problem, utilizing the above two invariant manifolds as well as applying potential well argument and concavity method, we derive a sharp threshold for global existence and blowup. Then, combining the above results, we obtain two conclusions of how small the initial data are for the solution to exist globally by using dilation transformation. Finally, we prove a modified instability of standing wave to the system under study.