Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions

Consider a quasi-linear system of two Klein–Gordon equations with masses m1, m2. We prove that when m1≠2m2 and m2≠2m1, such a system has global solutions for small, smooth, compactly supported Cauchy data. This extends a result proved by Sunagawa (J. Differential Equations 192 (2) (2003) 308) in the semi-linear case. Moreover, we show that global existence holds true also when m1=2m2 and a convenient null condition is satisfied by the nonlinearities.