I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE:
$\partial c \over \partial t$ + $K \over r^2$ $\partial c \over \partial r$ + $Da \ c \over c+n$ $=0$
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 ($c_{in} exp(-Da \ t)$. 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 (or whether it's solvable, or the boundary/initial conditions are even correct).
Thank you
Mathematica suggests the equation is solvable:
$c(t,r) = n W\left(\frac{e^{\frac{c_1\left(\frac{1}{3} \left(r^3-3 K t\right)\right)}{n}-\frac{\text{Da} t}{n}}}{n}\right)$
where $W$ denotes ProductLog function.