Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term

TL;DR

This paper investigates the long-term behavior of solutions to a modified Cahn-Hilliard equation with inertial effects, proving the equivalence of attractors for energy and strong solutions and demonstrating exponential decay of non-smooth parts.

Contribution

It establishes the coincidence of attractors for energy and strong solutions and shows exponential decay of non-smooth components in the modified Cahn-Hilliard equation.

Findings

Attractors for energy and strong solutions coincide.

Non-smooth parts of solutions decay exponentially.

Energy solutions are asymptotically smooth.

Abstract

The paper is devoted to a modification of the classical Cahn-Hilliard equation proposed by some physicists. This modification is obtained by adding the second time derivative of the order parameter multiplied by an inertial coefficient which is usually small in comparison to the other physical constants. The main feature of this equation is the fact that even a globally bounded nonlinearity is "supercritical" in the case of two and three space dimensions. Thus the standard methods used for studying semilinear hyperbolic equations are not very effective in the present case. Nevertheless, we have recently proven the global existence and dissipativity of strong solutions in the 2D case (with a cubic controlled growth nonlinearity) and for the 3D case with small inertial coefficient and arbitrary growth rate of the nonlinearity. The present contribution studies the long-time behavior of rather weak (energy) solutions of that equation and it is a natural complement of the results of our previous papers. Namely, we prove here that the attractors for energy and strong solutions coincide for both the cases mentioned above. Thus, the energy solutions are asymptotically smooth. In addition, we show that the non-smooth part of any energy solution decays exponentially in time and deduce that the (smooth) exponential attractor for the strong solutions constructed previously is simultaneously the exponential attractor for the energy solutions as well.