I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes:

Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] \longrightarrow M$, consider the flow $$ \frac{\partial}{\partial \varepsilon}c(t, \varepsilon) = \frac{\nabla}{\partial t}\frac{\partial}{\partial t} c(t, \varepsilon), \quad c(t, 0) = c_0(t)$$ What is known here (short-time existence, long-time-existence etc.)? If $c$ is a closed loop, does it (maybe suppose $M$ to be compact) either converge to a closed geodesic or collapse to a point?

What I was thinking initially was that I could fix the endpoints (let's say two that are not cutpoints of each other) and hopefully get the flow to converge to the shortest geodesic between the two. However, having thought about it, it feels that the problem of fixed endpoints is ill-posed, isn't it? What other flow should I consider here?