Periodic solutions to the Cahn–Hilliard equation with constraint

Abstract

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.

Time periodic solutions of Cahn-Hilliard systems with dynamic boundaryconditions

“…for all r ∈ R, respectively. We establish the above cut-off function by referring to [31]. From now, we show the next proposition.…”

“…In order to show the Proposition 3.1, we use the method in [31], that is, we employ the viscosity approach. Conforming to the method, we consider the following problem: for each ε ∈ (0, 1] and f ∈ L 2 (0, T ; V 0 ),…”

“…Moreover, focusing on (1.6), the study respect to existence of time periodic solutions of the Cahn-Hilliard equation is not much. For example, [26][27][28]31]. In particular, Wang and Zheng discuss the existence of time periodic solution of the Cahn-Hilliard equation with Neumann boundary condition [31].…”

“…For example, [26][27][28]31]. In particular, Wang and Zheng discuss the existence of time periodic solution of the Cahn-Hilliard equation with Neumann boundary condition [31]. They employ the method of [4].…”

“…In this paper, following the method of [31], we apply the abstract theory of evolution equations by using the viscosity approach and the Schauder fixed point theorem in the level of approximate problem. Moreover, by virtue of the viscosity approach, we can apply the abstract result [4].…”