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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-154/2/5/solution