Semilinear hyperbolic systems violating the null condition
  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 free solution in the large time. 
Soichiro Katayama, Toshiaki Matoba, Hideaki Sunagawa
math.AP