A Cahn-Hilliard type equation with periodic gradient-dependent potentials and sources

Abstract

This paper is concerned with the existence, uniqueness and attractability of time periodic solutions of a Cahn-Hilliard type equation with periodic gradient-dependent potentials and sources.

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

“…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).…”

“…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.…”

Global well-posedness of solutions for the epitaxy thin film growth model

“…Latterly, the problems of stability, long time behavior and other properties of solutions for the initial boundary value problem of ( 5) have been studied by various authors (see e.g. Liu [15], Zhao, Zhang and Liu [27], Kohn and Yan [11], Li and Liu [12], Zhang and Zhu [25], Li, Yin and Jin [14], Zhao and Liu [26]). It is particularly important to note that the global well-posedness for the Cauchy problem of Eq.…”

Periodic solutions to the Cahn–Hilliard equation with constraint

“…function φ on a real Hilbert space H and the maximal monotone operator A has linear growth, namely there exist some constants C i >0,( i = 1,2,3), such that for all [ u , η ]∈ G ( A ). As far as we know, although the Cahn–Hillard equation can be reformulated as a class of the evolution equations in the dual space of H 1 , there are few results on the periodic solutions for Cahn–Hilliard equations with the periodic source . Indeed, by using the qualitative theory of parabolic equation in , Yin et al .…”

“…This paper is concerned with the following multidimensional Cahn-Hilliard equation: Here is a bounded domain in R N .N 1/ with smooth boundary @ , 4 D P N iD1 @ 2 @x 2 i for all OEu, Á 2 G.A/. As far as we know, although the Cahn-Hillard equation can be reformulated as a class of the evolution equations in the dual space of H 1 , there are few results on the periodic solutions for Cahn-Hilliard equations with the periodic source [19][20][21][22]. Indeed, by using the qualitative theory of parabolic equation in [23], Yin et al [21] mainly studied the existence of periodic solutions of the following Cahn-Hilliard-type equation in one spatial dimension:…”