We analyze a diffuse interface model that couples a viscous
Cahn-Hilliard equation for the phase variable with a diffusion-reaction
equation for the nutrient concentration. The system under consideration
also takes into account some important mechanisms like chemotaxis,
active transport as well as nonlocal interaction of Oono’s type. When
the spatial dimension is three, we prove the existence and uniqueness of
global weak solutions to the model with singular potentials including
the physically relevant logarithmic potential. Then we obtain some
regularity properties of the weak solutions when t>0. In
particular, with the aid of the viscous term, we prove the so-called
instantaneous separation property of the phase variable such that it
stays away from the pure states ±1 as long as t>0.
Furthermore, we study long-time behavior of the system, by proving the
existence of a global attractor and characterizing its ω-limit set.