  We consider a multi-component version of the conserved Allen-Cahn equation
proposed by J. Rubinstein and P. Sternberg in 1992 as an alternative model for
phase separation. In our case, the free energy is characterized by a mixing
entropy density which belongs to a large class of physically relevant entropies
like, e.g., the Boltzmann-Gibbs entropy. We establish the well-posedness of the
Cauchy-Neumann problem with respect to a natural notion of (finite) energy
solution which is more regular under appropriate assumptions and is strictly
separated from pure phases if the initial datum is. We then prove that the
energy solution becomes more regular and strictly separated instantaneously.
Also, we show that any finite energy solution converges to a unique
equilibrium. The validity of a dissipative inequality (identity for strong
solutions) allows us to analyze the problem within the theory of
infinite-dimensional dissipative dynamical systems. On account of the obtained
results, we can associate to our problem a dissipative dynamical system and we
can prove that it has a global attractor as well as an exponential attractor.
