Blow up for critical wave equations on curved backgrounds

TL;DR

This paper constructs finite-energy blow-up solutions for semilinear wave equations on curved backgrounds, extending previous flat-space results and highlighting differences in blow-up rates due to curvature effects.

Contribution

It extends slow blow-up solutions to wave equations on curved manifolds, demonstrating existence with a continuous blow-up rate unlike the Minkowski case.

Findings

Existence of finite-energy blow-up solutions on curved backgrounds.

Blow-up rates are bounded above, contrasting with Minkowski space.

Extension of prior flat-space blow-up results to curved geometries.

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 - \Delta_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.