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

Abstract

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 that accounts for the relaxation of the diffusion flux. We suppose that is dominated by the viscosity coefficient . Endowing the equation with Dirichlet boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase‐space, depending on . This system is shown to possess a global attractor that is upper semicontinuous at . Then, we construct a family of exponential attractors which is a robust perturbation of an exponential attractor of the Cahn–Hilliard equation; namely, the symmetric Hausdorff distance between and tends to 0 as tends to in an explicitly controlled way. Finally, we present numerical simulations of the time evolution of weak solutions as a function of parameters.