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

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