AI 中文总结
研究具有二次非对称相互作用的\(L^{2}\)-超临界非线性克莱因-戈登方程，用辅助泛函、变分法等得到解的有限时间爆破结果、基态驻波存在性、爆破和整体存在阈值以及基态驻波不稳定性。
AI 中文摘要
本文研究具有二次非对称相互作用的\(L^{2}\)-超临界非线性克莱因-戈登方程（LSNKG）的整体存在性、爆破和驻波。首先，通过引入合适的辅助泛函，利用凹性分析和位力估计，得到当初始能量为负时，LSNKG柯西问题解的有限时间爆破结果。其次，通过定义适当的泛函、流形和约束变分问题，采用变分法和拉格朗日乘数法导出相应非线性椭圆系统基态解的存在性，从而建立LSNKG基态驻波的存在性。然后，利用基态解的变分特征并构造LSNKG柯西问题流作用下的不变集，将势阱论证与凹性分析相结合，建立了有限时间爆破和整体存在的精确阈值。最后，通过利用基态的变分性质，引入适当的尺度变换，并基于基态选择合适的初始数据，证明了LSNKG基态驻波的不稳定性。
英文摘要
In this paper we investigate the global existence, blow-up and standing waves for the $L^{2}$-supercritical nonlinear Klein-Gordon equations with quadratic asymmetric interaction (LSNKG). First, by introducing a suitable auxiliary functional, using concavity analysis and virial estimates, we obtain a finite time blow-up result for solutions to the Cauchy problem of (LSNKG) when the initial energy is negative. Next, by defining appropriate functionals, manifolds and a constrained variational problem, we employ variational method and the Lagrange multiplier method to derive the existence of ground state solutions for the corresponding nonlinear elliptic (steady-state) system, thereby to establish the existence of standing wave with the ground state for (LSNKG). Then, using the variational characterization of the ground state solutions and constructing invariant sets under the flow generated by the Cauchy problem for (LSNKG), we combine the potential well argument with concavity analysis to establish a sharp threshold between blow-up in finite time and global existence. Finally, by exploiting the variational characterization of the ground state, introducing appropriate scalings, and choosing suitable initial data based on the ground state, we justify the instability of standing wave with the ground state for (LSNKG).
Comments43 pages
