Hyperbolic relaxation of the viscous Cahn-Hilliard equation with a symport term for biological applications.

We consider the hyperbolic relaxation of the viscous Cahn-Hilliard equation with a symport term. This equation is characterized by the presence of the additional inertial term τDΦtt that accounts for the relaxation of the diffusion flux. We suppose that τD is dominated by the viscosity coefficient δ. Endowing the equation with Dirichlet boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space, depending on τD. This system is shown to possess a global attractor that is upper semicontinuous at τD = δ = 0. Then, we construct a family of exponential attractors ℑτD,δ which is a robust perturbation of an exponential attractor of the Cahn-Hilliard equation; namely, the symmetric Hausdorff distance between ℑτD,δ and ℑ0,0 tends to 0 as (τD,δ) tends to (0,0) in an explicitly controlled way. Finally, we present numerical simulations of the time evolution of weak solutions as a function of parameters. [ABSTRACT FROM AUTHOR]