In the diffusion equation $v_t-D\Delta v=f$, there can be several different boundary conditions and properties assumed about $v$(2nd space derivative continuous, 1st time derivative continuous in the space-time cylinder $\Omega\times (0,T)$, and $v$ continuous up to the boundary of that cylinder) such that the problem is well-posed.
I'm interested in knowing what those properties may imply for $f$.
P.S: I'm new to this subject, so it's probable I'm being too relaxed with the notation or formality. Any helpful contributions to improve the question are welcome.