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 trightarrow+∞ of solutions x(t) to the second order differential equation [ xprimeprime(t)+fracKtxprime(t)+nablaPhi(x(t))=0, ] where K>3 and Phi is a smooth convex function defined on an Hilbert Space H. Moreover, we improve slightly their result on the rate of convergence of Phi(x(t))-minPhi.