= Spherical reaction-diffusion growth threshold
{title2=$\gamma>\pi^2D/a^2$}
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 $n(r,t)=n_c\sin(\pi r/a)/(\pi r/a)\exp[(\gamma-\pi^2D/a^2)t]$. Substitution of $R=f/r$ in the radial <Laplacian> gives $f''+k^2f=0$, 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.
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