Abstract: In this paper, we consider a kind of quasilinear hyperbolic systems with inhomogeneous terms satisfying dissipative condition or matching condition. For the Cauchy problem of this kind of systems, we prove that, if the initial data is small and satisfies some decay condition, and the system is weakly linearly degenerate, then the Cauchy problem admits a unique global classical solution on t ≥ 0.

Key words: quasilinear hyperbolic system, cauchy problem, classical solution, strongly
dissipative condition, matching condition