Convergence of Solutions to a Convective Cahn–Hilliard-Type Equation of the Sixth Order in Case of Small Deposition Rates

We show stabilization of solutions to the sixth-order convective Cahn–Hilliard equation. The problem has the structure of a gradient flow perturbed by a quadratic destabilizing term with coefficient . Through application of an abstract result by Carvalho, Langa, and Robinson we show that for small the equation has the structure of gradient flow in a weak sense. On the way we prove a kind of Liouville theorem for eternal solutions to parabolic problems. Finally, the desired stabilization follows from a powerful theorem due to Hale and Raugel.