A forward-backward dynamical approach for nonsmooth problems with block structure coupled by a smooth function

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 problem choose continuous forward-backward approach and introduce dynamical system which is formulated means partial gradients function proximal point operator functions. Moreover, consider variable rates implicity resulting system. We discuss existence uniqueness solution carry out asymptotic analysis its convergence behaviour critical optimization problem, when regularization objective fulfills Kurdyka-Łojasiewicz property. further provide for trajectory terms Łojasiewicz exponent. conclude work with numerical simulations confirm validate analytical results.