Spherical nonlinear-diffusion source profile ($r_0=(75Mkt{)}^{1/5}$)

For $C_t=k\nabla \cdot (C\nabla C)$ in three dimensions and total mass $4\pi M$, the nonnegative radial source solution is $C=M^{2/5}(kt{)}^{-3/5}f(r/(Mkt{)}^{1/5})$, with $f(\xi )=(75^{2/5}-{\xi }^2{)}_{+}/10$. Integrating the similarity equation once gives $f{f}^{\prime }=-\xi f/5$, and normalization ${\int }_0^{\infty }{\xi }^{2}fd\xi =1$ fixes the compact-support radius. The flux vanishes at the moving front, so the profile solves the equation weakly and approaches the prescribed point mass as time tends to zero.