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In this short note, we recover by a different method the new result due toAttouch, Peyrouqet and Redont concerning the weak convergence as$t\rightarrow+\infty$ of solutions $x(t)$ to the second order differentialequation \[ x^{\prime\prime}(t)+\frac{K}{t}x^{\prime}(t)+\nabla\Phi(x(t))=0, \]where $K>3$ and $\Phi$ is a smooth convex function defined on an Hilbert Space$\mathcal{H}.$ Moreover, we improve slightly their result on the rate ofconvergence of $\Phi(x(t))-\min\Phi.$

Asymptotic for a second order evolution equation with convex potential and vanishing damping term