November 4, 2005 17:43 Dissipative Phase Transitions P. Colli, N. Kenmochi, J. Sprekels GLOBAL ATTRACTOR FOR THE WEAK SOLUTIONS OF A CLASS OF VISCOUS CAHN-HILLIARD EQUATIONS Riccarda Rossi Dipartimento di Matematica, Universit`a di Brescia Via Valotti 9, I–25133 Brescia, Italy e-mail: riccarda.rossi @ ing.unibs.it We address the long-time behaviour of a class of viscous Cahn-Hilliard equations, modelling phase separation in mixtures and alloys. Specifi- cally, we prove the existence of (a suitable notion of) the global attractor for the weak solutions of the so-called generalized viscous Cahn-Hilliard equation. 1. Introduction This paper is concerned with the analysis of the long-time behavior of the (weak) solutions of the following fourth-order equation ∂ t χ − Δ(α(∂ t χ − Δχ + χ 3 − χ)) = 0 in Ω × (0, +∞). (1) Here, Ω a bounded, connected domain in R N , N =1, 2, 3, with smooth boundary Γ; α : D(α) ⊂ R → R is a (strictly) increasing, differentiable function, while the term χ 3 − χ is the derivative of the double-well potential W(x)= (x 2 − 1) 2 4 , x ∈ R. (2) In fact, (1) models the evolution of a phase separation process, to which a two-phase material (for instance, a binary alloy or a mixture), occupying the domain Ω, is subject. In this connection, the variable χ, usually referred to as order parameter, stands for the local concentration of one of the two components. Equation (1) is indeed a generalized viscous Cahn-Hilliard equation: in fact, the viscous Cahn-Hilliard equation ∂ t χ − κΔ(∂ t χ − Δχ + χ 3 − χ)=0 in Ω × (0, +∞) (3) 1