Global small amplitude solutions for two-dimensional nonlinear   Klein-Gordon systems in the presence of mass resonance
  We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses satisfying the resonance relation. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate $O(|t|^{-1})$. 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. 
Yuichiro Kawahara and Hideaki Sunagawa
math.AP