The viscous Cahn–Hilliard equation with periodic potentials and sources

Abstract

We investigate the existence, uniqueness and asymptotic behavior of solutions to the viscous Cahn-Hilliard equation with timeperiodic potentials and sources. Mathematics Subject Classification (2010). Primary 35B10; Secondary 35B40.

Time periodic solutions of Cahn-Hilliard systems with dynamic boundaryconditions

“…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].…”

A Viscous Cahn–hilliard Equation With Periodic Gradient Dependent Potentials and Sources

“…It has been only proved that the solutions to the initial boundary value problem of the above equation can be bounded by a suitable upper bound of the time periodic solutions for all large times, see [11,21]. Moreover, for the Cahn-Hilliard equation with gradient dependent potentials, our research also disclose that for the viscous case, the attractivity of periodic solution is under the H 1 norm, which is different from that of the nonviscous case, where the attractivity is under the L 2 norm [12].…”

“…Furthermore, we discuss the limiting process of time periodic solutions and the solutions of initial boundary value problems as the viscous coefficient k tends to 0 (for the case k = 0 we refer readers to [12]), and there is another difference between the characters of the solutions to the Cahn-Hilliard equations with periodic concentration dependent potentials and with periodic gradient dependent potentials. In fact, when the viscous coefficient k tends to zero, for the case of concentration dependent potentials, the time periodic solutions and the solutions of the initial boundary value problem are almost everywhere convergent to the corresponding solutions of the problems with k = 0 (see [11]), while for the case of gradient dependent potentials, the time periodic solutions and the solutions of the initial boundary value problem are uniformly convergent to the corresponding solutions of the problems with k = 0 (see Theorem 4 and Theorem 5).…”