Spherical reaction-diffusion growth threshold ($\gamma >{\pi }^{2}D/a^2$)

For $n_t=D\Delta n+\gamma n$ in a sphere of radius $a$, with zero surface density and regularity at the origin, the lowest spherically symmetric normal mode is $n(r,t)=n_c\sin (\pi r/a)/(\pi r/a)\exp [(\gamma -{\pi }^{2}D/a^2)t]$. Substitution of $R=f/r$ in the radial Laplacian gives $f^{\prime \prime }+k^{2}f=0$, and regularity removes the cosine solution. The Dirichlet boundary condition gives $k=m\pi /a$; a radial mode with no interior zeros requires $m=1$. Thus diffusion leakage suppresses growth below the displayed threshold, while the mode is stationary at equality. The expression at $r=0$ uses its continuous limit. In a linear neutron multiplication model this is the competition between bulk production and surface loss.