Radial heat equation in three dimensions
= Radial heat equation in three dimensions
{title2=$u_t=D r^{-1}(ru)_{rr}$}
For a spherically symmetric field $u(r,t)$ in three dimensions,
$$
\nabla^2u=\frac1{r^2}\frac\partial{\partial r}\left(r^2u_r\right)
=\frac1r\frac{\partial^2}{\partial r^2}(ru).
$$
Thus the <heat equation> becomes $u_t=D r^{-1}(ru)_{rr}$. Introducing $w=ru$ reduces its spatial part to the one-dimensional second derivative $w_{rr}$, while regularity at the centre requires $w(0,t)=0$.