  We prove that a good \dot{H}^{s_{p}} critical theory for the 3D wave equation
\partial_{tt} u - \triangle u = -|u|^{p-1} u can be extended to prove global
well-posedness of smooth solutions of at least one 3D barely \dot{H}^{s_{p}}
supercritical wave equation \partial_{tt} u - \triangle u =- |u|^{p-1} u
g(|u|), with g growing slowly to infinity, provided that a Kenig-Merle type
condition is satisfied. This result extends those obtained for the particular
case s_{p}=1.
