Abstract: In this paper we investigate the formation of singularities of hyperbolic systems. Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.

Key words:  quasi-linear , systems;strictly , hyperbolic , systems;life , span;blowup , of , cusp type;the envelope of characteristics