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In a real Hilbert space $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order evolution equation $$ \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = 0, $$ where $\nabla \Phi$ is the gradient of a convex continuously differentiable function $\Phi : \mathcal H \rightarrow \mathbb R$, and $\alpha$ is a positive parameter.
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