The J-flow is a parabolic flow on Kähler manifolds. It was defined by Donaldson in
the setting of moment maps and by Chen as the gradient flow of the J-functional appearing
in his formula for the Mabuchi energy. It is shown here that under a certain condition on
the initial data, the J-flow converges to a critical metric. This is a generalization to
higher dimensions of the author's previous work on Kähler surfaces. A corollary of this is the lower
boundedness of the Mabuchi energy on Kähler classes satisfying a certain inequality when the first
Chern class of the manifold is negative.