A note on the null condition for quadratic nonlinear Klein-Gordon   systems in two space dimensions
  We consider the Cauchy problem for quadratic nonlinear Klein-Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.-M. Delort, D. Fang, R. Xue (2004), we show the global existence of asymptotically free solutions if the initial data are sufficiently small in some weighted Sobolev space. Our proof is based on an algebraic characterization of nonlinearities satisfying the null condition. 
Soichiro Katayama, Tohru Ozawa, Hideaki Sunagawa
math.AP