I found the following theorem in Helmke & Moore, Optimization and Dynamical Systems (Appendix C.12):
"Let Φ : M → R be a smooth function on a Riemannian manifold with compact sublevel sets, i.e. for all c ∈ R the sublevel set {x ∈ M | Φ(x) ≤ c} is a (possibly empty) compact subset of M. Then every solution x (t) ∈ M of the gradient flow on M exists for all t ≥ 0."
They do not leave any references. I am looking for a version of this theorem that would relax smoothness ($C^\infty$) to continuously differentiable, along with whatever other assumptions that may be necessary.
I have found such theorems elsewhere, but they embed $\Phi$ in a Hilbert space (and require it to be convex), which is something that I would like to avoid.
I presume you mean the ODE system \begin{equation} \tag{$*$} \dot{x} = {\color{red}-} {\nabla}\Phi(x). \end{equation}
Assume that $\Phi \colon M \to \mathbb{R}$ is a $C^2$ function having compact sublevel sets. Fix $x_0 \in M$, and put $c := \Phi(x_0)$. Let $x \colon [0, \tau_{\mathrm{max}}) \to M$ be the maximally defined (to the right) solution of ($*$) satisfying the initial condition $x(0) = x_0$. We have $\Phi(x(t)) \le c$, that is, $\{\, x(t) | t \in [0, \tau_{\mathrm{max}})\, \} \subset \{\, \xi \in M | \Phi(\xi) \le c \,\}$. As the latter is compact, the standard extension theorem for ODE systems (see, e.g., Corollary 2.15 on pp. 52-53 in Teschl's Ordinary Differential Equations and Dynamical Systems) gives $\tau_{\mathrm{max}} = \infty$.
I think that the smoothness assumption on $\Phi$ can be relaxed to $C^1$, but then the vector field $-\nabla\Phi$ is continuous only, and there can be difficulties with the uniqueness of solutions.