Blow up for critical wave equations on curved backgrounds

Abstract

We extend the slow blow up solutions of Krieger, Schlag, and Tataru to semilinear wave equations on a curved background. In particular, for a class of manifolds $(M,g)$ we show the existence of a family of blow-up solutions with finite energy norm to the equation {equation} \partial_t^2 u - Δ_g u = |u|^4 u, \notag {equation} with a continuous rate of blow up. In contrast to the case where $g$ is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.

17 pages

References in corpus (3)

• Smooth type II blow up solutions to the four dimensional energy critical wave equation
• Full range of blow up exponents for the quintic wave equation in three dimensions
• Renormalization and blow up for wave maps from $S^2\times \RR$ to $S^2$