Solution
= Solution
Write $q=w_t+w_r$. The equation in part 5 says
$$
q_t-q_r=-\frac{|w|^{p-1}w}{r^{p-1}}.
$$
Differentiate $J_\psi$, replace $q_t$ by $q_r-|w|^{p-1}w/r^{p-1}$, and integrate the $qq_r$ and $|w|^{p-1}ww_r$ terms by parts. This gives the <outgoing-energy identity for a radial defocusing wave>
$$
\frac{dJ_\psi}{dt}
=-\frac12\int_0^\infty\psi' q^2dr
+\frac1{p+1}\int_0^\infty
\frac{|w|^{p+1}}{r^{p-1}}
\left(\psi'-(p-1)\frac\psi r\right)dr.
$$