  We introduce and analyze the nonlocal variants of two Cahn-Hilliard type
equations with reaction terms. The first one is the so-called
Cahn-Hilliard-Oono equation which models, for instance, pattern formation in
diblock-copolymers as well as in binary alloys with induced reaction and type-I
superconductors. The second one is the Cahn-Hilliard type equation introduced
by Bertozzi et al. to describe image inpainting. Here we take a free energy
functional which accounts for nonlocal interactions. Our choice is motivated by
the work of Giacomin and Lebowitz who showed that the rigorous physical
derivation of the Cahn-Hilliard equation leads to consider nonlocal
functionals. The equations also have a transport term with a given velocity
field and are subject to a homogenous Neumann boundary condition for the
chemical potential, i.e., the first variation of the free energy functional. We
first establish the well-posedness of the corresponding initial and boundary
value problems in a weak setting. Then we consider such problems as dynamical
systems and we show that they have bounded absorbing sets and global
attractors.
