 Spatial mean of a periodic heat equation

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.