Wave Breaking, Global Existence and Persistent Decay for the Gurevich–Zybin System

Abstract

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.