  In this paper we aim to minimize the sum of two nonsmooth (possibly also
nonconvex) functions in separate variables connected by a smooth coupling
function. To tackle this problem we chose a continuous forward-backward
approach and introduce a dynamical system which is formulated by means of the
partial gradients of the smooth coupling function and the proximal point
operator of the two nonsmooth functions. Moreover, we consider variable rates
of implicitness of the resulting system. We discuss the existence and
uniqueness of a solution and carry out the asymptotic analysis of its
convergence behaviour to a critical point of the optimization problem, when a
regularization of the objective function fulfills the Kurdyka-Lojasiewicz
property. We further provide convergence rates for the solution trajectory in
terms of the Lojasiewicz exponent. We conclude this work with numerical
simulations which confirm and validate the analytical results.
