A geometric flow in the space of $G_2$-structures on the cone over $S^3\times S^3$

We consider a flow of $G_2$-structures on a $7$-dimensional manifold admitting a $G_2$-structure. The general solution to this flow is found in the case when the manifold is the cone over $S^3\times S^3$. We prove the convergence of the metric associated with the solution to the conical metric modulo homotheties.