Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three

Abstract

In this article, by following the strategies in dealing with supercritical cubic and quintic wave equations in (J. Eur. Math. Soc. (JEMS) 16 (2014) 1–30) and (J. Math. Pures Appl. (9) 105 (2016) 342–366), we obtain that, the equation \begin{equation*}(\partial^{2}_{t}-\Delta)u+|u|^{p-1}u=0,\quad3<p<5\end{equation*} is almost surely global well-posed with initial data $(u(0),\partial_{t}u(0))\in H^{s}(\mathbb{T}^{3})\times H^{s-1}(\mathbb{T}^{3})$ 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 $3<p<5$.