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We consider a nonlinear system of two-dimensional Klein-Gordon equations withmasses satisfying the resonance relation. We introduce a structural conditionon the nonlinearities under which the solution exists globally in time anddecays at the rate $O(|t|^{-1})$. In particular, our new condition includes theYukawa type interaction, which has been excluded from the null condition in thesense of J.-M.Delort, D.Fang and R.Xue.

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