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 $k>(3p+4)/p$.

References in corpus (1)

• Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension

Cited by in corpus (8)

• On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations
• Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension
• Blow-up results for semilinear wave equations in the super-conformal case
• The blow-up rate for strongly perturbed semilinear wave equations
• Solutions blowing up on any given compact set for the energy subcritical wave equation
• Blowup behaviour for the nonlinear Klein--Gordon equation
• The blow-up rate for strongly perturbed semilinear wave equations in the conformal case
• The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption