Smooth solutions to the nonlinear wave equation can blow up on Cantor sets

Abstract

We construct $C^{\infty }$ solutions to the one-dimensional nonlinear wave equation $$u_{tt}-u_{xx}-\frac{2(p+2)}{p^2}|u{|}^{p}u=0\text{with}p>0$$ that blow up on any prescribed uniformly space-like $C^{\infty }$ hypersurface. As a corollary, we show that smooth solutions can blow up (at the first instant) on an arbitrary compact set. We also construct solutions that blow up on general space-like $C^k$ hypersurfaces, but only when $4/p$ is not an integer and $k>(3p+4)/p$.