  In the context of hyperbolic systems of balance laws, the Shizuta-Kawashima
coupling condition guarantees that all the variables of the system are
dissipative even though the system is not totally dissipative. Hence it plays a
crucial role in terms of sufficient conditions for the global in time existence
of classical solutions. However, it is easy to find physically based models
that do not satisfy this condition, especially in several space dimensions. In
this paper, we consider two simple examples of partially dissipative hyperbolic
systems violating the Shizuta-Kawashima condition ([SK]) in 3D, such that some
eigendirections do not exhibit dissipation at all. We prove that, if the source
term is non resonant (in a suitable sense) in the direction where dissipation
does not play any role, then the formation of singularities is prevented,
despite the lack of dissipation, and the smooth solutions exist globally in
time. The main idea of the proof is to couple Green function estimates for
weakly dissipative hyperbolic systems with the space-time resonance analysis
for dispersive equations introduced by Germain, Masmoudi and Shatah. More
precisely, the partially dissipative hyperbolic systems violating [SK] are
endowed, in the non-dissipative directions, with a special structure of the
nonlinearity, the so-called Nonresonant Bilinear Form for the wave equation
(see Pusateri and Shatah, CPAM 2013).
