I am studying the following PDE for a function $c(r,t)$:
$$ \frac{\partial c}{\partial t} = D \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{ \partial c}{\partial r} \right) - \gamma c + \eta \delta(r-R) $$
The steady-state solution of this equation has the form
$$ c(r) = \frac{c_R R}{r} \exp{\left(\frac{R-r}{\lambda}\right)}, $$
where $\lambda=\sqrt{\frac{D}{\gamma}}$ and $c_R = \frac{\eta \gamma R}{\lambda (\lambda+R)}$.
I am looking for the time-dependent solution $c(r,t)$ on the domain $(r,t) \in (R,\infty) \times (0, \infty)$ without resorting to numerical computation.
As boundary conditions I would like to take
$$ -D\frac{\partial c}{\partial r} = \eta, \\ \lim\limits_{r\rightarrow \infty} c(r) = 0 $$
I still have to think about the initial condition, but let's suppose it is $c(r,0) = f(r)$ for some function $f(r)$ that satisfies the first boundary condition.
- Is there a general solution to $c(r,t)$ for this problem, i.e. is this a well-posed problem?
- What are general approaches that will help me in studying this problem? I have only basic background in solving PDEs, so any suggestion for how to approach this problem will be appreciated!
- It is perhaps possible to leave out the last term $\eta \delta(r-R)$ since production is already implied by taking the first boundary condition $-D \frac{\partial c}{\partial t}$.
Note
Without the degradation term $-\gamma c$, this reduces to the heat equation with a constant source. The solution of this equation with the given boundary conditions is known, see e.g. this paper (in the context of signaling molecules too). However, I couldn't find any straightforward generalization to the system with degradation term.
Some physical background
The equation is a so-called reaction-diffusion equation meant to describe the concentration of a molecule secreted by a cell. The cell has radius $R$ and secretes with a rate $\eta$. The diffusion constant is $D$ and the molecules are degraded at a rate $\gamma$.
$$ \frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c) - \gamma c + \eta \delta(r-R) $$
Since the problem is spherically symmetric, we can limit ourselves to finding a solution for $c(r,t)$. Here is a paper for a (complicated) system of interacting cells that uses the steady-state solution I gave above.