Regular attractors for the Cahn-Hilliard equation with memory

The Cahn-Hilliard equation is a parabolic differential equation of forth order
which plays an essential role in material sciences since 1958, when it was introduced by J.W. Cahn and J.E Hilliard. This talk concerns
with its memory relaxation, namely an integro-differential version
of the original equation which arises as a model for the phenomenological description of phase transition based on the relaxation of the chemical potential.
We first deal with the viscous version of the model and present results on
the existence of a global attractor of optimal regularity and its stability
with respect to the physical sensible parameters involved in the equation.
We finally discuss very recent results on the asymptotic behavior of the non-viscous 2D-model; remarkably, in absence of viscosity and instantaneous diffusion effects, even the well-posedness of the 3D-model is an open question.