Cahn-Hilliard equations with memory and dynamic boundary conditions

We consider a modified Cahn\u2013Hiliard equation where the velocity of the order parameter u depends on the past history of \u394\u3bc, \u3bc being the chemical potential with an additional viscous term \u3b1ut, \u3b1 650. 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 \u3b1>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 \u3b1=0, we only establish the existence of a trajectory attractor