The long time behavior of a class of second-order gradient-like systems with vanishing dissipative term and non-convex analytic potential

In this paper, a class of second order dissipative system equation(1)ẍ(t)+a(t)ẋ(t)+∇f(x(t))=0 is studied, where f:RN→Rf:RN→R is analytic and non-convex, a:R+→R+a:R+→R+ is continuous and nonincreasing with limt→∞a(t)=0limt→∞a(t)=0. We give a sufficient condition for the convergence of global and bounded solutions of (1). The condition shows that the rate of convergence of damping coefficient a(t)a(t) is related to the Lojasiewicz exponent of the analytic function ff.