On the time-fractional Cahn-Hilliard equation: Well-posedness, global solutions, and convergence to classical dynamics

This paper investigated the Cauchy problem for the time-fractional Cahn–Hilliard equation formulated with the Caputo derivative. By developing sharp estimates for the corresponding Green's functions, we first established the local-in-time existence of mild solutions, which were obtained through an integral formulation of the problem. To address the challenges posed by the nonlocal nature of the fractional derivative, we introduced new weighted function spaces that enabled us to extend these local solutions to a global-in-time framework under suitable smallness conditions on the initial data. Moreover, we rigorously analyzed the asymptotic behavior of the solution and demonstrated that, as the fractional order approached one, the mild solution converged to that of the classical Cahn–Hilliard equation. These results not only clarified the relationship between nonlocal dynamics and traditional phase separation but also suggested that the methods and techniques presented in this paper could be applied to many other models related to the Caputo derivative.