Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 2 / 6 / Solution

Write $q=w_t+w_r$. The equation in part 5 saysDifferentiate $J_{\psi }$, replace $q_t$ by $q_r-|w{|}^{p-1}w/r^{p-1}$, and integrate the $q{q}_r$ and $|w{|}^{p-1}w{w}_r$ terms by parts. This gives the outgoing-energy identity for a radial defocusing wave