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.