Space-time resonances and the null condition for first-order systems of wave equations (Q2852539)

The first order system on \(\mathbb R \times\mathbb R^3\) with initial data is reduced to a single scalar equation with quadratic nonlinearities, and the paper focuses on global existence and scattering of small solutions for this system. One proves that if system is nonresonant in the sense defined in the paper and the initial data -- elements of a low weighted Sobolev space -- satisfy a specific inequality, then there exist a unique global small solution to the stated problem that scatters to a linear one as time tends to infinity. For the proof the space-time resonance method is used. The nonresonancy of a system is defined by a set of quite restrictive conditions. Some examples of nonresonant systems are shown and discussed.