  In a Hilbert setting we study the convergence properties of a second order in
time dynamical system combining viscous and Hessian-driven damping with time
scaling in relation with the minimization of a nonsmooth and convex function.
The system is formulated in terms of the gradient of the Moreau envelope of the
objective function with time-dependent parameter. We show fast convergence
rates for the Moreau envelope and its gradient along the trajectory, and also
for the velocity of the system. From here we derive fast convergence rates for
the objective function along a path which is the image of the trajectory of the
system through the proximal operator of the first. Moreover, we prove the weak
convergence of the trajectory of the system to a global minimizer of the
objective function. Finally, we provide multiple numerical examples which
illustrate the theoretical results.
