Creating geodesics on manifolds

Suppose I have two points on a Riemannian manifold $M$, called $p_0$ and $p_1$. I have a family of curves $\gamma:[0,\infty)\times[0,L]\to M$ such that $\gamma(t,0) = p_0$ and $\gamma(t,L) = p_1$. As a function of $t$, let $\gamma$ evolve as if it were made of damped elastic, so that the curve tries to pull itself into a geodesic. That is, denoting $\gamma'(t,x) = \frac{\partial}{\partial x} \gamma(t,x)$, it obeys the PDE: $$ \frac{\partial}{\partial t} \gamma(t,x) = \frac{\partial}{\partial x} \gamma'(t,x) + \nabla_{\gamma'(t,x)} \gamma'(t,x)$$ where $\nabla_u v$ denotes the usual covariant derivative of $v$ with respect to $u$.

I am interested in whether $\gamma(t,\cdot)$ converges to a geodesic on $M$ at an exponential rate. I was a little surprised at how nice a proof I was able to obtain, almost as if the definition of sectional curvature was designed with this result in mind. Basically if the sectional curvature is non-positive, then it does converge with exponential speed, and if the sectional curvature is positive then the hypothesis seems to exactly exclude the case when the curve is on the sphere going from the north pole to the south pole.

My question is this: does this result already exist in the literature, and where? (Maybe it is even an exercise in some book.)

Note this PDE is a kind of heat equation. I do know about a paper by Eells, J.; Sampson, J.H. (1964), "Harmonic mappings of Riemannian manifolds", Amer. J. Math. 86: 109–160, JSTOR 2373037. But I think my problem is a little simpler than theirs. (On the other hand, I haven't looked at this paper in any real detail, so maybe it is hidden in there somewhere.)

I consider this to be the answer to my question:

http://www.math.msu.edu/~parker/ChoiParkerGeodesics.pdf

Thank you, Anthony Carapetis.