= Compact radial cubic-diffusion profile
{title2=$F(\xi)=\sqrt{(1-\xi^2)_+}$}
For the planar diffusion equation $n_t=D_0\nabla\cdot[(n/n_0)^2\nabla n]$, a mass-preserving radial <similarity solution> has $n=n_0\lambda^{-2}F(r/(r_0\lambda))$. Substitution reduces the equation to $(\xi F^2F')'=-(\xi^2F)'$ when $\lambda^5\dot\lambda=D_0/r_0^2$. Zero central flux and $F(0)=1$ give the compact profile shown above, with $\lambda^6=6D_0t/r_0^2$. Its height decays as $t^{-1/3}$ and its radius grows as $t^{1/6}$; its total mass is $2\pi n_0r_0^2/3$.
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