I have a reaction-diffusion PDE of the form
$\frac{dp}{dt} = D \nabla p - \frac{a\Omega p}{p + k}$
where $D$, $a$, $k$ and $\Omega$ are all known constants. Initially, in the region of interest, $p(r,0) = 0$ . There is a boundary at zero at a constant value of $p_{o}$ (Dirichlet condition?) so I suspect I write $p(0,t) = p_{o}$ for one of my boundary conditions. The problem is the other one; from my understanding, I could need two boundary conditions to specify the problem and apply a numerical solution, and I only have one. I know approximately where the other boundary $L$ is (such that $p(L) = 0$) but I wonder if there's a better way to do this than just guess?
Specifically, as the solution diffuses into the region of interest, it is consumed at a rate $\frac{ap}{p + k}$ , and eventually reaches a point $L$ where $p(L) = 0$ and it can no longer diffuse - is it possible to use this knowledge to specify the second boundary, or is trial and error the only way? Any insight people can share would be greatly appreciated!
Note: The other constants are all known; $D = 2 \times 10^{-9}$ , $a = 7.5 \times 10^{-7}$, $\Omega = 3.0318 \times 10^{7}$, $k = 1$ and $p_{o} = 100$.