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

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