In this paper, the blow-up phenomenon, global existence and persistent decay of the solutions to the Gurevich–Zybin system are studied. We show that the system possesses a so called critical threshold phenomena, that is, global smoothness versus finite time breakdown depends on whether the initial configuration crosses an intrinsic critical threshold. We prove that a finite maximal life span for a solution necessarily implies wave breaking for this solution, and show some conditions which are local-in-space on the initial data to ensure wave-breaking for this system by making use of the characteristics method, otherwise, the system has a global smooth solution. Furthermore, we establish the persistence properties for the system in weighted \(L^p\) spaces.

中文翻译：

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Gurevich-Zybin系统的破波，整体存在和持久衰减

本文研究了Gurevich-Zybin系统解的爆炸现象，整体存在性和持续衰减。我们表明，系统具有所谓的临界阈值现象，即全局平滑度与有限时间分解取决于初始配置是否超过固有的临界阈值。我们证明了解决方案的最大使用寿命有限意味着该解决方案必定会发生断波，并通过使用特征方法显示初始数据在空间上局部存在的条件，以确保该系统的断波，否则，系统将具有全局平滑解决方案。此外，我们在加权\（L ^ p \）空间中建立了系统的持久性。