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

TL;DR

This paper establishes lower bounds and sharp lifespan estimates for classical solutions to certain quasi-linear hyperbolic systems with small BV initial data, under specific degeneracy and inhomogeneity conditions.

Contribution

It provides new lower bounds and a sharp limit formula for the lifespan of solutions to quasi-linear hyperbolic systems with small BV initial data.

Findings

Lower bound of solution lifespan established

Sharp limit formula for lifespan derived

Results applicable to systems with non-weakly linearly degenerate characteristic fields

Abstract

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.