# 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.

## 18 Citations

Asymptotic behavior for a class of derivative nonlinear Schrödinger systems

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We consider a two-component system of cubic semilinear wave equations in two space dimensions satisfying the Agemi-type structural condition (Ag) but violating (Ag$_0$) and (Ag$_+$). For this system,…

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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.

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