Suppose $\nabla^2 V$ has a negative eigenvalue $-\lambda_0$ at point $x_0$, with eigenvector $v_0$, consider the modified flow 

$$ Y' = - \nabla V(Y) + \nabla V(x_0) $$

which has $x_0$ as a stationary point. This new flow is a constant shift of the original flow and has the same Lipschitz properties. 

In a neighborhood of $x_0$, consider the modified flow with initial data $x_0 + \epsilon v_0$ for some sufficiently small $\epsilon\ll \lambda_0$. The flow looks like

$$ (Y - x_0)' = -\nabla^2 V(x_0) \cdot (Y - x_0) + O(|Y-x_0|^2) $$

and so for short times the solution is well-approximated by 

$$ x_0 + \epsilon v_0 e^{\lambda_0 t} + O (\epsilon^2) $$

and so violates the 1-Lipschitz assumption. 

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The case where $\nabla^2 V \geq \lambda \mathrm{Id}$ is similar.