Convergence to equilibrium for fully discretizations of nonlocal Cahn-Hilliard equation

The study of long-term dynamics for numerical solutions of nonlinear evolution equations, particularly phase field models, has consistently garnered considerable attention. The Cahn-Hilliard (CH) equation is one of the most important phase field models and is widely applied in materials science. In order to more accurately describe the practical phenomena in material microstructural phase transitions, the Nonlocal Cahn-Hilliard (N-CH) equation incorporates a finite range of spatial nonlocal interactions is introduced, which is a generalization of the classic CH equation. However, compared to its classic counterpart, it is very challenging to investigate the long-term asymptotic behavior of solution to the N-CH equation due to the complexity of the nonlocal integral term and the lack of high-order diffusion term. In this paper, we consider first-order and second-order temporal discretization methods for the N-CH equation, respectively, while utilizing a second-order finite difference method for spatial approximation to construct the energy stable fully discrete numerical schemes. Based on energy stability and the {\L}ojasiewicz inequality, we rigorously prove that the numerical solutions of these fully discrete numerical schemes converge to equilibrium as time goes to infinity.