The divergence set for the wave equation in higher dimensions

Abstract

It is shown that if $u$ solves the wave equation in ${\mathbb{R}}^{4+1}$ with initial data $u(\cdot ,0)=u_0(\cdot )\in H^s$ and ${\partial }_{t}u(\cdot ,0)=u_1(\cdot )\in H^{s-1}$, where $0.5<s\le 0.55$, then $u(x,t)\to u_0(x)$ and ${\partial }_{t}u(x,t)\to u_1(x)$ pointwise as $t\to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in ${\mathbb{R}}^{n+1}$ is obtained for $n\ge 4$ and $1/2<s<n/4$.

31 pages