I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE:

\begin{equation*} \frac{\partial c}{\partial t}+\frac{K}{r^2}\frac{\partial c}{\partial r}+\frac{Da~c}{c+n}=0 \end{equation*}

The initial condition is $c(r,0) = c_{in}$ and boundary condition is $\partial c(R_{in},t) \over \partial r $ $= 0 $, where $R_{in}$ is the radius of an inner sphere (and $R_{out}$ the radius of an outer sphere). $K$ and $Da$ are both constants. $r$ is the radius measured from the centre of the sphere outwards, and $t$ time.

I have attempted to solve this using the method of characteristics but the solution keeps reducing to only depend on time. And even then I don't end up using the second Neumann boundary condition.

Does anyone have any ideas as to where I'm going wrong. Is it just the boundary conditions that are causing the problem?