  In a Hilbert setting we aim to study a second order in time differential
equation, combining viscous and Hessian-driven damping, containing a time
scaling parameter function and a Tikhonov regularization term. The dynamical
system is related to the problem of minimization of a nonsmooth convex
function. In the formulation of the problem as well as in our analysis we use
the Moreau envelope of the objective function and its gradient and heavily rely
on their properties. We show that there is a setting where the newly introduced
system preserves and even improves the well-known fast convergence properties
of the function and Moreau envelope along the trajectories and also of the
gradient of Moreau envelope due to the presence of time scaling. Moreover, in a
different setting we prove strong convergence of the trajectories to the
element of the minimal norm from the set of all minimizers of the objective.
The manuscript concludes with various numerical results.
