= Spherical-shell heat equation via radial rescaling
{title2=$\psi(t,r)=r^{-1}\sum_{n\geq1}b_ne^{-\kappa(n\pi/L)^2t}\sin[n\pi(r-r_-)/L]$}
On a shell $0<r_-<r<r_+$ with homogeneous <Dirichlet boundary conditions>, put $L=r_+-r_-$ and $u=r\psi$. The <radial heat equation in three dimensions> becomes $u_t=\kappa u_{rr}$, with zero values at both endpoints. Expanding $u(0,r)=r\psi(0,r)$ in a <Fourier sine series> gives the displayed solution, with $b_n=(2/L)\int_{r_-}^{r_+}r\psi(0,r)\sin[n\pi(r-r_-)/L]dr$. Each radial mode decays at the square of its wave number. The positive inner radius makes division by $r$ harmless throughout the shell.
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