Semilinear hyperbolic systems violating the null condition

@article{Katayama2012SemilinearHS,
  title={Semilinear hyperbolic systems violating the null condition},
  author={Soichiro Katayama and Toshiaki Matoba and Hideaki Sunagawa},
  journal={Mathematische Annalen},
  year={2012},
  volume={361},
  pages={275-312}
}
We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition related to the weak null condition. For two-component systems satisfying this condition, we also observe a new kind of asymptotic behavior: Only one component is dissipated and the other one behaves like a non-trivial free solution in the large time. 
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18 Citations

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TLDR
Time decay estimates of small solutions to the Schrödinger systems under the mass resonance condition in 2-dimensional space are summarized and the existence of wave operators and modified wave operators of the systems under some mass conditions in n- dimensional space is shown.
Nonlinear Schrödinger systems in 2d with nondecaying final data
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