Spatial mean of a periodic heat equation ($\frac{d\overline{u}}{dt}=\overline{Q}$)

Integrate $u_t=\alpha \Delta u+Q$ over a periodic cell. Opposite boundary contributions to the Laplacian cancel, so the spatial mean obeys ${\overline{u}}^{\prime }=\overline{Q}$. With no source the mean is conserved, although the local temperature changes. The mean is the zero Fourier mode in a normalized Fourier series convention.

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.