  In this paper, we consider the Laplacian G_2 flow on a closed
seven-dimensional manifold M with a closed G_2-structure. We first obtain the
gradient estimates of positive solutions of the heat equation under the
Laplacian G_2 flow and then we get the Harnack inequality on spacetime. As an
application, we prove the monotonicity for positive solutions of the heat
equation with bounded Ricci curvature, and get the integral-type Harnack
inequality. Besides, we prove the monotonicity of parabolic frequency for
positive solutions of the linear heat equation with bounded Bakry-Emery Ricci
curvature, and then obtain the backward uniqueness.
