In this article, we follow the strategies, listed in \cite{ Burq2011} and \cite{ OhPo}, in dealing with supercritical cubic and quintic wave equations, we obtain that, the equation \begin{ equation*} \left\{ \begin{ split} &(\partial^2_t-\Delta)u+|u|^{ p-1}u=0,\ \ 3<p<5 &\big(u,\partial_tu\big)|_{ t=0}=(u_0,u_1)\in H^{ s}\times H^{ s-1}=:\mathcal{ H}^s, \end{ split} \right. \end{ equation*} is almost surely global well-posed in the sense of Burq and Tzvetkov\cite{ Burq2011} for any $s\in (\frac{ p-3}{ p-1},1)$. The key point here is that $\frac{ p-3}{ p-1}$ is much smaller than the critical index $\frac{ 3}{ 2}-\frac{ 2}{ p-1}$ for $3<p<5$.
Categories: math.AP