On the long time behavior of second order differential equations with asymptotically small dissipation
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 $H$: \begin{equation*}(\mathcal {S})\qquad \qquad \qquad \ddot {x}(t)+a(t)\dot {x}(t)+ \nabla G(x(t))=0, \quad t\geq 0,\qquad \qquad \qquad \qquad \quad \end{equation*} where the map $a:\mathbb {R}_+\to \mathbb {R}_+$ is nonincreasing and the potential $G:H\to \mathbb {R}$ …