I'm having trouble getting started on this problem. Here's the question:

My issue is in setting up the governing partial differential equation in 3 dimensions. What I've tried so far is setting du/dt equal to the Laplacian of u(x,y,z,t) + g(x,y). I'm not entirely sure if I can do this, though. It leads me to assume u(x,y,z,t) = X(x)Y(y)Z(z)T(t), but I'm not sure how to cancel out the inhomogeneous boundary conditions.

I know that if g were a constant I could let v = u - g, but since it isn't its derivative with respect to x and y won't go away and v will stay inhomogeneous.

Sorry if that sounds like word vomit - its late and I've been up reading textbooks that only seem to have one dimensional examples. If anybody could shed some light on how to tackle this problem I would be very thankful - I'm mostly looking for a starting point and hopefully I can take it from there.