Affiliations: Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, 20133 Milano, Italy. E-mail: [email protected] | Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: [email protected] | Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133 Milano, Italy. E-mail: [email protected]
Abstract: We consider a modified Cahn–Hiliard equation where the velocity of the order parameter u depends on the past history of Δμ, μ being the chemical potential with an additional viscous term αut, α≥0. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual no-flux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the so-called past history space. Existence of a variational solution is obtained. Then, in the case α>0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case α=0, we only establish the existence of a trajectory attractor.
