  We give sharp conditions for global in time existence of gradient flow
solutions to a Cahn-Hilliard-type equation, with backwards second order
degenerate diffusion, in any dimension and for general initial data. Our
equation is the 2-Wasserstein gradient flow of a free energy with two competing
effects: the Dirichlet energy and the power-law internal energy. Homogeneity of
the functionals reveals critical regimes that we analyse. Sharp conditions for
global in time solutions, constructed via the minimising movement scheme, also
known as JKO scheme, are obtained. Furthermore, we study a system of two
Cahn-Hilliard-type equations exhibiting an analogous gradient flow structure.
