Generic ill-posedness for wave equation of power type on 3D torus

Abstract

In this article, we prove that the equation \begin{equation*} \left\{\begin{split} &(\partial^2_t-\Delta)u+|u|^{p-1}u=0,\ \ \ 3\leq p<5 &\big(u(0),\partial_tu(0)\big)=(u_0,u_1)\in H^{s}(\mathbb{T}^3)\times H^{s-1}(\mathbb{T}^3)=:\mathcal{H}^s(\mathbb{T}^3) \end{split}\right. \end{equation*} with $s<\frac{3}{2}-\frac{2}{p-1}$ is everywhere ill-posed. This work also indicates that, only properly regularizing the initial data can we smoothly approximate the solutions constructed in \cite{BT14} and \cite{Xia14}.