  We consider a Cahn-Hilliard equation which is the conserved gradient flow of
a nonlocal total free energy functional. This functional is characterized by a
Helmholtz free energy density, which can be of logarithmic type. Moreover, the
spatial interactions between the different phases are modeled by a singular
kernel. As a consequence, the chemical potential $\mu$ contains an integral
operator acting on the concentration difference $c$, instead of the usual
Laplace operator. We analyze the equation on a bounded domain subject to
no-flux boundary condition for $\mu$ and by assuming constant mobility. We
first establish the existence and uniqueness of a weak solution and some
regularity properties. These results allow us to define a dissipative dynamical
system on a suitable phase-space and we prove that such a system has a
(connected) global attractor. Finally, we show that a Neumann-like boundary
condition can be recovered for $c$, provided that it is supposed to be regular
enough.
