Focusing semilinear wave equation ($u_{tt}-\Delta u=u|u{|}^{p-1}$)

The focusing power wave equation is $u_{tt}-\Delta u=u|u{|}^{p-1}$. Its conserved energy has the negative potential term $-|u{|}^{p+1}/(p+1)$ and is not positive definite. Localized ordinary differential equation blowup for a wave equation produces smooth compactly supported data with finite maximal lifespan for suitable focusing powers.

A spatially constant blowup profile for a wave equation solves an ordinary differential equation. Choose smooth compactly supported initial data which agree with the profile on a ball larger than its blowup time times the wave speed. Finite propagation speed preserves agreement in an interior light cone, forcing finite-time breakdown of the localized solution. For $u_{tt}-\Delta u=u^3$, the profile $u(t)=\sqrt{2}/(1-t)$ is one example.