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 tau D phi tt$$ {\tau}_D{\phi}_{tt} $$ that accounts for the relaxation of the diffusion flux. We suppose that tau D$$ {\tau}_D $$ is dominated by the viscosity coefficient delta$$ \delta $$. Endowing the equation with Dirichlet boundary conditions, we show that it generates a dissipative dynamical system acting on a certain phase-space, depending on tau D$$ {\tau}_D $$. This system is shown to possess a global attractor that is upper semicontinuous at tau D=delta=0$$ {\tau}_D equal to \delta equal to 0 $$. Then, we construct a family of exponential attractors I tau D,delta$$ {\Im}_{\tau_D,\delta } $$ which is a robust perturbation of an exponential attractor of the Cahn-Hilliard equation; namely, the symmetric Hausdorff distance between I tau D,delta$$ {\Im}_{\tau_D,\delta } $$ and I0,0$$ {\Im}_{0,0} $$ tends to 0 as (tau D,delta)$$ \left({\tau}_D,\delta \right) $$ tends to (0,0)$$ \left(0,0\right) $$ in an explicitly controlled way. Finally, we present numerical simulations of the time evolution of weak solutions as a function of parameters.