Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the riemannian gradient). Suppose that our manifold is geodesically convex in the sense that $\exp$ and $\exp^{-1}$ are well defined for the whole manifold and not just locally. I would like to prove that the equation has a unique global solution under some suitable assumptions, given an initial condition for $X$ and $\dot X$ at $0$. The procedure I have in mind is firstly modifying the initial equation making it $\nabla \dot X+\frac{3}{\max(\delta,t)} \dot X + gradf(X)=0$, in order to use Arzela-Ascoli theorem at the end, taking $\delta \rightarrow 0$. Writing each of the modified ODEs in local coordinates, we get the system $\ddot c^k + \sum_{i,j=1}^m \Gamma_{ij}^k(c)\dot c^i\dot c^j+\frac{3}{\max(\delta,t)} \dot c^k+ \sum_{i=1}^m g^{ik}\frac{\partial(f o \psi)}{\partial x^i}(c)=0$, for $k=1,...,m$, where $\psi=\phi^{-1}$ is a parametrization and $c=\phi o X$. By Picard-Lindelof theorem this equation has a local solution around $0$, but I need a global one. Suppose that everything in the equation in local coordinates is Lipschitz continuous. Intuitively there is no global solution in general, because geodesics, which are defined by a similar equation, are not defined globally in general. Do you have any idea about an extra assumtion on the manifold $M$ which could provide a global solution? I believe that completeness could work, but I do not know how.

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