Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 2 / 3 / Solution

SetDifferentiate, substitute $u_{tt}=\Delta u-u|u{|}^{p-1}$, and integrate every second derivative off $u$. The mixed first-derivative terms cancel because of the correction $(\Delta \psi )u/2$. For radial $u$ and radial $\psi$, the Hessian term is ${\psi }^{\prime \prime }|\nabla u{|}^2$. The potential term uses $u|u{|}^{p-1}\nabla u=\nabla (|u{|}^{p+1})/(p+1)$. One obtains the Morawetz identity for the defocusing wave equation