Long-time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

In this paper, we consider the hyperbolic relaxation of the nonisothermal Cahn–Hilliard equation based on either Fourier law or Maxwell–Cattaneo law for heat conduction. In the Maxwell–Cattaneo case, we reformulate the problem by using enthalpy instead of relative temperature. In both cases, we prove the existence of the global attractor for the weak solutions of the problem. Moreover, for both laws, we establish that every full trajectory in the global attractor converges to a single stationary point as t → ∞ and another single stationary point as t → −∞. Consequently, we infer that the global attractor is equal to a union of the unstable manifolds emanating from the stationary points.