Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3-D

We consider a modified version of the viscous Cahn-Hilliard equation governing the relative concentration u of one component of a binary system. This equation is characterized by the presence of the additional inertial term that accounts for the relaxation of the diffusion fl ux. The inertial parameter omega is supposed to be dominated from above by the viscosity coefficient delta. Endowing the equation with suitable boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space depending on omega. This system is shown to possess a global attractor that is upper semicontinuous at omega= delta = 0. Then, we construct a family of exponential attractors E(omega,delta) , which is a robust perturbation of an exponential attractor of the Cahn-Hilliard equation, namely the symmetric Hausdorff distance between E(omega,delta) and E(0,0) goes to 0 as (omega, delta) goes to (0, 0) in an explicitly controlled way. This is done by using a general theorem which requires the construction of another dynamical system, strictly related to the original one, but acting on a different phase-space depending on both omega and delta.