Global existence of solutions for quadratic quasi-linear Klein-Gordon systems in one space dimension (Q2386851)

The authors prove the global existence of small amplitude solutions to the Cauchy problem of the system of quasi-linear Klein-Gordon equations with quadratic nonlinearity in \(\mathbb{R}^{1+1}_+=\mathbb{R}^{1}_x \times \mathbb{R}^{+}_t\). In \(\mathbb{R}^{n+1}_+\) \((n \geq 2)\) several authors proved the global existence theorems of scalar or system of Klein-Gordon equations with quadratic nonlinarity. Using the technique of \textit{J.-M. Delort} [Ann. Sci. École Norm. Supér. 34, No. 1, 1--61 (2001; Zbl 0990.35119)], they get a convenient null condition. Thanks to this condition, the global existence theorems can be obtained. Their method can be applied to the semi-linear system in \(\mathbb{R}^{1+1}_+\) under the non-resonance assumption, which provides the result of \textit{H. Sunagawa} [J. Differ. Equations 192, No. 2, 308--325 (2003; Zbl 1028.35128)].