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, Chbani, Peyrouqet, and Redont concerning the weak convergence as t -> +infinity of solutions x(t) to the second-order differential equation x (t) + k/t x' (t) + del Phi(x(t)) = 0, where K > 3 and Phi is a smooth convex function defined on a Hilbert space H. Moreover, we improve their result on the rate of convergence of Phi(x(t)) min Phi.