Uniform forcing of a periodic heat equation ($u(t)=e^{\alpha t\Delta }{u}_0+{\int }_0^{t}Q(s)ds$)

For a spatially constant PDE source term $Q(t)$, subtract its time integral from $u$. The result obeys the homogeneous heat equation with the same initial condition. This proves the displayed formula and shows that uniform heating changes only the zero Fourier mode. In an unnormalised $N_x\times N_y$ discrete Fourier transform, a constant increment $q$ adds $N_x{N}_{y}q$ to the zero-frequency coefficient.