 Solution

Substitute $C=t^{\alpha }F(\xi )$, $\xi =r{t}^{-\beta }$. The derivatives giveFor a nontrivial spreading similarity solution, equality at all times requires $\boxed{\beta =1/2}$. The conserved total number of molecules isA finite nonzero constant $N$ therefore requires $\alpha +3\beta =0$, giving $\boxed{\alpha =-3/2}$. In one dimension the same diffusion length exponent is $1/2$, but conservation gives amplitude exponent $-1/2$; generally it is $-d/2$ in dimension $d$.

The profile equation is consequentlyMultiplication by ${\xi }^2$ puts it in the integrated form $[D{\xi }^2{F}^{\prime }+{\xi }^{3}F/2{]}^{\prime }=0$. For the regular localized Gaussian heat kernel profile, the constant is zero at the origin, so $D{F}^{\prime }+\xi F/2=0$. Integrating givesThe second expression chooses the optional normalization $A=N/(4\pi D{)}^{3/2}$. Its mass is $N$, and its width tends to zero with $t^{1/2}$ while its mass stays fixed, so it approaches $N$ times a Dirac delta distribution at the origin as $t\downarrow 0$.