A remark on barely $\dot H^{s_{p}}$-supercritical wave equations

Abstract

We prove that a good ${Ḣ}^{s_p}$ critical theory for the $3D$ wave equation ${\partial }_{tt}u-△u=-|u{|}^{p-1}u$ can be extended to prove global well-posedness of smooth solutions of at least one $3D$ barely ${Ḣ}^{s_p}$-supercritical wave equation ${\partial }_{tt}u-△u=-|u{|}^{p-1}ug\left(\left|u\right|\right)$, with $g$ growing slowly to infinity, provided that a Kenig-Merle type condition is satisfied. This result is related to those obtained by Tao and the author for the particular case $s_p=1$, showing global regularity for $g$ growing logarithmically with radial data and for $g$ growing doubly logarithmically with general data.