Global well-posedness and scattering for a nonlinear Klein-Gordon system in low dimensions

This article investigates the global well-posedness and the scattering for a nonlinear Klein-Gordon system in spatial dimensions 1 and 2. We establish a Morawetz estimate for this system which is similar to the Morawetz estimate established by Nakanishi [K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169(1), pp. 201-225], combining this Morawetz estimate with the induction on energy argument developed by Bourgain [J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, J. Am. Math. Soc. 12 (1999), pp. 145-171], the bound of a certain space-time norm and scattering result are obtained.