Global attractors for a three-dimensional conserved phase-field system with memory

We consider a conserved phase-field system on a tridimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature [\vartheta] . These effects are represented through a convolution integral whose relaxation kernel [k] is a summable and decreasing function. Therefore the system consists of a linear integrodifferential equation for [\vartheta] which is coupled with a viscous Cahn-Hilliard type equation governing the order parameter [\chi] . The latter equation contains a nonmonotone nonlinearity [\phi] and the viscosity effects are taken into account by the term [-\alpha \Delta\chi_t] , for some [\alpha \geq 0] . Thus, we formulate a Cauchy-Neumann problem depending on [\alpha ] . Assuming suitable conditions on [k] , we prove that this problem generates a dissipative strongly continuous semigroup [S^\alpha (t)] on an appropriate phase space accounting for the past histories of [\vartheta] as well as for the conservation of the spatial means of the enthalpy [\vartheta+\chi] and of the order parameter. We first show, for any [\alpha \geq 0] , the existence of the global attractor [\mathcal A_\alpha ] . Also, in the viscous case ( [\alpha > 0] ), we prove the finiteness of the fractal dimension and the smoothness of [\mathcal A_\alpha ] .