In this paper, we consider the Cauchy problem for systems of quasi-linear wave equations with multiple propagation speeds in spatial dimensions n ≥ 4. The problem when the nonlinearities depend on both the unknown function and their derivatives is studied. Based on some Klainerman- Sideris type weighted estimates and space-time L2 estimates, the results that the almost global existence for space dimensions n = 4 and global existence for n≥ 5 of small amplitude solutions are presented.
