= Solution
<Fick's first law> gives the diffusive flux
$$
\boxed{\mathbf J=-D(C)\nabla C}.
$$
Local <conservation law>[conservation] says $C_t+\nabla\cdot\mathbf J=0$, and therefore
$$
\boxed{C_t=\nabla\cdot(D(C)\nabla C)}.
$$
For $D(C)=kC$, conservation of the deposited amount requires
$$
2\pi\int_0^\infty rC(r,t)\,dr=2\pi M.
$$
The dimensions satisfy $[M]=[C]L^2$ and $[k]=L^2/([C]T)$. Thus $Mkt$ has dimension $L^4$, so the similarity length is $(Mkt)^{1/4}$. Requiring the concentration scale to have dimension $[C]$ gives
$$
\boxed{\alpha=\beta=\frac12,\qquad\gamma=\frac14}.
$$
Hence
$$
C(r,t)=\sqrt{\frac{M}{kt}}\,F(\xi),
\qquad
\xi=\frac{r}{(Mkt)^{1/4}},
\qquad
\int_0^\infty\xi F(\xi)\,d\xi=1.
$$
In polar coordinates, substitution into
$C_t=k r^{-1}(rCC_r)_r$ gives
$$
-\frac12F-\frac14\xi F'
=\frac1\xi(\xi FF')'.
$$
Multiplying by $\xi$ and integrating once, with regularity and zero radial flux at the origin, yields
$$
\xi FF'=-\frac14\xi^2F.
$$
Where $F>0$, this reduces to $F'=-\xi/4$, so
$$
F(\xi)=A-\frac{\xi^2}{8}.
$$
Nonnegativity and zero flux at the moving front give the compactly supported profile
$$
F(\xi)=\left(A-\frac{\xi^2}{8}\right)_+,
\qquad
\xi_0^2=8A.
$$
The normalization gives
$$
1=\int_0^{\xi_0}\xi\left(A-\frac{\xi^2}{8}\right)d\xi
=2A^2,
$$
and hence $A=1/\sqrt2$ and $\xi_0^4=32$. This is the <Two-dimensional Barenblatt solution with diffusivity proportional to concentration>, and its support radius is
$$
\boxed{r_0(t)=(32Mkt)^{1/4}},
$$
so $N=32$.
Now add the linear reaction term and write
$$
C(r,t)=e^{at}G(r,\tau(t)).
$$
Substitution cancels the terms $aC$ and leaves
$$
e^{at}\tau'G_\tau
=ke^{2at}\nabla\cdot(G\nabla G).
$$
Choosing
$$
\boxed{\tau'=e^{at},\qquad
\tau(t)=
\begin{cases}
(e^{at}-1)/a,&a\ne0,\\
t,&a=0
\end{cases}}
$$
produces $G_\tau=k\nabla\cdot(G\nabla G)$, as in the <linear-reaction time change for quadratic nonlinear diffusion>. Therefore
$$
\boxed{
C(r,t)=e^{at}\sqrt{\frac{M}{k\tau(t)}}
\left(
\frac1{\sqrt2}
-\frac{r^2}{8\sqrt{Mk\tau(t)}}
\right)_+
}.
$$
For $a=0$, the front grows like $t^{1/4}$ while the central concentration decays like $t^{-1/2}$. For $a>0$, $\tau\sim e^{at}/a$, so the front grows like $e^{at/4}$ and the central concentration like $e^{at/2}$. For $a<0$, $\tau\to1/|a|$: the front approaches a finite limiting radius while the concentration decays exponentially to zero.
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