We discuss the global existence of small solutions to the Cauchy problem for systems of quasilinear wave equations in three space dimensions, when their nonlinear terms have quadratic nonlinearity. A global existence theorem is established on the null condition which is extended to the condition for systems of wave equations with different propagation speeds.
Kazuyoshi YOKOYAMA. "Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions." J. Math. Soc. Japan 52 (3) 609 - 632, July, 2000. https://doi.org/10.2969/jmsj/05230609