Heat equation energy identity ($\frac{d}{dt}\Vert u{\Vert }_2^2=-2\Vert du{\Vert }_2^2$)

On a closed manifold with a Riemannian metric, a smooth solution of ${\partial }_{t}u+\Delta u=0$ for the positive Laplace-Beltrami operator satisfies the displayed identity by integration by parts. Zero initial data force the solution to vanish, proving uniqueness and the semigroup property for heat evolution.