How can I describe diffusion in 3D with a continuous production of signal at a distance d? Is there a set of general equations or mathematical tools to describe 3D diffusion of molecules with a continuous production of signal?
This is the specific situation I have in mind to model a biological system I'm working on. 
Task: predict the concentration "C" at distance "d" from the signal source after time "t" has passed.
Source "S" produces 1000 molecules per minute. It's like a pump that keeps pumping out signal at that rate. Signal is assumed not to degrade. 
The signal diffuses in 3D space, with a certain diffusion constant D. 
How can I mathematically describe this situation to calculate the answers to my task? (Ex, I'd like to know if I'm 1cm away from the source, after 1 hr has passed, what would be the effective concentration at that point?)
I can do 2D diffusion, with a constant concentration at source S. I am not sure how I can extend it to 3D diffusion and with the source constantly producing. I'm sure this is not a very hard math problem so I think I can make sense of the answer if I see one, though I cannot come up with one on my own with my current math skills. 
 A: Diffusion of some quantity $u(p,t)$ at point $p$ and time $t$, in any dimension, is governed by the Laplacian in that dimension:
$$\frac{\partial u}{\partial t} = D\Delta u$$
given appropriate initial and boundary conditions (concentration at every point at time $t=0$, and what happens to the signal at the boundary of the domain, respectively.)
Now suppose you are pumping in more of the quantity at a rate $r(p)$ that depends on the position $p$ (for instance, perhaps $r$ is zero outside of a small area around the location of a single pump). This situation corresponds to adding a source term to the PDE:
$$\frac{\partial u}{\partial t} = D\Delta u + r.$$
This equation should make intuitive sense: the rate of change of $u$ at each point is equal to the change due to pumping in more of the quantity, plus/minus whatever amount gets diffused towards/away from that point.
Finally, actually solving this equation, either numerically or analytically, is one of the simplest problems as far as PDEs are concerned, but not trivial if you haven’t studied PDEs before. If you search for tutorials on the heat equation with source term, you should find an abundance of material. The difficulty will vary depending on your domain geometry and boundary conditions.
