Compact radial cubic-diffusion profile ($F(\xi )=\sqrt{(1-{\xi }^2{)}_{+}}$)

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 $(\xi F^2{F}^{\prime }{)}^{\prime }=-({\xi }^{2}F{)}^{\prime }$ when ${\lambda }^5\dot{\lambda }=D_0/r_0^2$. Zero central flux and $F(0)=1$ give the compact profile shown above, with ${\lambda }^6=6D_{0}t/r_0^2$. Its height decays as $t^{-1/3}$ and its radius grows as $t^{1/6}$; its total mass is $2\pi n_0{r}_0^2/3$.