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In this paper, we first give a lower bound of the lifespan and some estimatesof classical solutions to the Cauchy problem for general quasi-linearhyperbolic systems, whose characteristic fields are not weakly linearlydegenerate and the inhomogeneous terms satisfy Kong's matching condition. Afterthat, we investigate the lifespan of the classical solution to the Cauchyproblem and give a sharp limit formula. In this paper, we only require that theinitial data are sufficiently small in the $L^1$ sense and the BV sense.

Lifespan of Classical Solutions to Quasi-linearHyperbolic Systems with Small BV Normal Initial Data