On the long time behavior of second order differential equations with asymptotically small dissipation

We investigate the asymptotic properties as $ t\to \infty$ of the following differential equation in the Hilbert space $ H$: $$(\mathcal{S})\qquad\qquad\qquad\ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0$$ where the map $ a:\mathbb{R}_+\to \mathbb{R}_+$ is nonincreasing and the potential $ G:H\to \mathbb{R}$ is of class $ \mathcal{C}^1$. If the coefficient $ a(t)$ is constant and positive, we recover the so-called ``Heavy Ball with Friction'' system. On the other hand, when $ a(t)=1/(t+1)$ we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function $ G$ is convex, the condition $ \int_0^\infty a(t) dt =\infty$ guarantees that the energy function converges toward its minimum. The more stringent condition $ \int_0^{\infty} e^{-\int_0^t a(s) ds}dt<\infty$ is necessary to obtain the convergence of the trajectories of $ (\mathcal{S})$ toward some minimum point of $ G$. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function $ G$. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.