Global small amplitude solutions for two-dimensional nonlinear Klein–Gordon systems in the presence of mass resonance

Abstract

We consider a nonlinear system of two-dimensional Klein–Gordon equations with masses $m_1$, $m_2$ satisfying the resonance relation $m_2=2m_1>0$. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate $O({|t|}^{-1})$ as $t\to \pm \infty$ in $L^{\infty }$. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M. Delort, D. Fang and R. Xue [J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein–Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004) 288–323].