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

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

2011-03-27

arxiv.org/abs/1103.5257v1

Mathematics

Analysis of PDEs

Scientific paper

We construct $C^\infty$ solutions to the one-dimensional nonlinear wave equation $$ u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with} \quad p>0 $$ 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 $k > (3p+4)/p$.

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