The Zakharov system in 4D radial energy space below the ground state
The Zakharov system in 4D radial energy space below the ground state
复制标题
DOI：
10.1353/ajm.2021.0039
复制
发表时间：
2018-10
影响因子：
1.7
通讯作者：
Zihua Guo;K. Nakanishi
中科院分区：
文献类型：
--
作者：
Zihua Guo;K. Nakanishi
摘要：利用Aubin-Talenti函数证明了在能量小于基态的四维空间能量空间中，Zakharov系统的所有径向解在弱意义上分为散射和爆破。这种二分法的特征是基态波分量的临界质量。这一结果与Kenig和Merle对能量临界非线性薛定谔方程(NLS)的结果相似.然而，与NLS不同的是，证明中最困难的相互作用来自于自由波分量.为了控制它，我们发展的主要新成分是解波动方程的具有亚临界质量势的线性薛定谔方程的全局一致的Strichartz估计.这一估计和证据可能是独立的利益所在。对于散射证明，我们遵循Dodson和Murphy的想法。
Abstract:We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than the ground state, which is written using the Aubin-Talenti function. The dichotomy is characterized by the critical mass of the wave component of the ground state. The result is similar to that by Kenig and Merle for the energy-critical nonlinear Schr\"odinger equation (NLS). Unlike NLS, however, the most difficult interaction in the proof stems from the free wave component. In order to control it, the main novel ingredient we develop in this paper is a uniform global Strichartz estimate for the linear Schr\"odinger equation with a potential of subcritical mass solving a wave equation. This estimate, as well as the proof, may be of independent interest. For the scattering proof, we follow the idea by Dodson and Murphy.