# Yukawa-type system

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

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

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

Global well posedness in the energy class ${\displaystyle (H^{1},H^{1}xL^{2})}$ 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 ${\displaystyle (H^{s},H^{m}xH^{m-1})}$ whenever ${\displaystyle 1\geq s,m>7/10}$ and ${\displaystyle s+m>3/2}$Pe-p2.