  This article is concerned with the existence and the long time behavior of
weak solutions to certain coupled systems of fourth-order degenerate parabolic
equations of gradient flow type. The underlying metric is a Wasserstein-like
transportation distance for vector-valued functions, with nonlinear mobilities
in each component. Under the hypothesis of (flat) convexity of the driving free
energy functional, weak solutions are constructed by means of the variational
minimizing movement scheme for metric gradient flows. The essential regularity
estimates are derived by variational methods.
