For a radial function in three dimensions,
$$
\Delta u=u_{rr}+\frac2r u_r
=\frac1r\partial_{rr}(ru).
$$
Set $w=ru$. Multiplying the wave equation by $r$ gives the <radial reduction of the three-dimensional wave equation>
$$
w_{tt}-w_{rr}+\frac{|w|^{p-1}w}{r^{p-1}}=0,
\qquad r>0,
$$
with the regularity boundary condition $w(t,0)=0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 154 / 2 / 5 / Solution

For a radial function $u(t,r)$ in three dimensions, setting $w=ru$ removes the radial first derivative because $r\Delta u=w_{rr}$. The defocusing power equation becomes
$$
w_{tt}-w_{rr}+r^{1-p}|w|^{p-1}w=0
$$
on the half-line.


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