  In this paper we carry out an asymptotic analysis of the proximal-gradient
dynamical system \begin{equation*}\left\{ \begin{array}{ll} \dot x(t) +x(t) =
\prox_{\gamma f}\big[x(t)-\gamma\nabla\Phi(x(t))-ax(t)-by(t)\big],\\ \dot
y(t)+ax(t)+by(t)=0 \end{array}\right.\end{equation*} where $f$ is a proper,
convex and lower semicontinuous function, $\Phi$ a possibly nonconvex smooth
function and $\gamma, a$ and $b$ are positive real numbers. We show that the
generated trajectories approach the set of critical points of $f+\Phi$, here
understood as zeros of its limiting subdifferential, under the premise that a
regularization of this sum function satisfies the Kurdyka-\L{}ojasiewicz
property. We also establish convergence rates for the trajectories, formulated
in terms of the \L{}ojasiewicz exponent of the considered regularization
function.
