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Wave breaking, global existence and persistent decay for the Gurevich-Zybin system. (English) Zbl 1448.35505

Summary: 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.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
35Q86 PDEs in connection with geophysics
35Q31 Euler equations
35Q60 PDEs in connection with optics and electromagnetic theory
35L60 First-order nonlinear hyperbolic equations
35B44 Blow-up in context of PDEs
83C35 Gravitational waves
86A05 Hydrology, hydrography, oceanography
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