Yukawa-type system

$i\partial _{t}^{}u+\Delta u=-Au$

$\Box A=m_{}^{2}A+|u|2$

for $d=3$. A represents the meson field, while u is the nucleon field.

Global well posedness in the energy class $(H^{1},H^{1}xL^{2})$ is in [Bch1984], [BlChd1978], [FuTs1978], [HaWl1987]. Modified wave operators were constructed for large energy data at infinity in [GiVl-p2].

With positive mass m=1, global well-posedness can be pushed to $(H^{s},H^{m}xH^{m-1})$ whenever $1\geq s,m>7/10$ and $s+m>3/2$[Pe-p2].