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The diffusive flux is $\mathbf{J}=-D\nabla C$ by Fick's first law. Conservation in any fixed control volume $V$ gives $\frac{d}{dt}{\int }_{V}C=-{\int }_{\partial V}\mathbf{J}\cdot \mathbf{n}+{\int }_{V}F$. The divergence theorem and arbitrariness of $V$ yieldAfter the initial point injection, $F=0$ and $D=kC$ with $k>0$, so radial symmetry gives $C_t=k{r}^{-2}(r^{2}C{C}_r{)}_r$. Total mass is $4\pi {\int }_0^{\infty }{r}^{2}Cdr=4\pi M$.

A spreading radius $R$ and typical concentration obey $C{R}^3\sim M$ and $R^2\sim kCt$. Thus $R^5\sim Mkt$. In the proposed ansatz this givesDirect substitution, including the time derivative of $\xi$, givesMultiplying by ${\xi }^2$ makes the left side $-({\xi }^{3}f{)}^{\prime }/5$. Regularity and zero radial flux at the origin set the integration constant to zero, so $f{f}^{\prime }=-\xi f/5$. On the positive region $f^{\prime }=-\xi /5$. Nonnegativity, compact support and continuity at its edge giveIts normalization is $1={\int }_0^{{\xi }_0}{\xi }^{2}fd\xi ={\xi }_0^5/75$. Therefore the spherical nonlinear-diffusion source profile isThe concentration and flux vanish at the free boundary, so there is no distributional flux jump despite the derivative kink. The solution satisfies the diffusion equation weakly across that boundary and classically inside its positive region. Its support shrinks to the origin as $t\downarrow 0$, and testing against a continuous compactly supported function gives the limit $4\pi M$ times its value at zero. This verifies the prescribed point injection as well as the mass normalization.