Asymptotics for a gradient system with memory term

Given a Hilbert space $H$ and a function $\Phi:H\to\mathbb{R}$ of class $\mathcal{C}^1$, we investigate the asymptotic behavior of the trajectories associated to the following dynamical system:

$$(\mathcal{S})\qquad\qquad\qquad\dot x(t) +\frac{1}{k(t)} \int_{t_0}^t h(s) \nabla \Phi(x(s)) ds=0, \qquad t\geq t_0,\qquad\qquad\quad$$

where $h$, $k: [t_0,+\infty)\to \mathbb{R}_+^*$ are continuous maps. When $k(t) \sim \int_{t_0}^t h(s) ds$ as $t\to +\infty$, this equation can be interpreted as an averaged gradient system. We define a natural energy function $E$ associated to system $(\mathcal{S})$ and we give conditions which ensure that $E(t)$ decreases to $\inf \Phi$ as $t\to +\infty$. When $\Phi$ is convex and has a set of non-isolated minima, we show that the trajectories of $(\mathcal{S})$ cannot converge if the average process does not ``privilege'' the recent past. Special attention is devoted to the particular case $h(t)=t^\alpha$, $k(t)=t^\beta$, which is fully treated.