Blow up of solutions to semilinear wave equations

Abstract

This work shows the absence of global solutions to the equation $$ u_{tt} = Delta u + p^{-k}|u|^m, $$ in the Minkowski space $mathbb{M}_0=mathbb{R}imesmathbb{R}^N$, where $ m > 1$, $(N-1)m < N+1$, and $p $ is a conformal factor approaching 0 at infinity. Using a modification of the method of conformal compactification, we prove that any solution develops a singularity at a finite time.

Keywords

• Blow up
• conformal compactification.