The Cahn–Hilliard type equations with periodic potentials and sources

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:…”

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].…”

Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions

“…Liu and Wang [6] proved the existence of time-periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions. Yin et al [9] considered the existence of time periodic solutions for the Cahn-Hilliard type equation in one dimension, see also [10]. Using the Galerkin method and the Leray-Schauder fixed point theorem, Wang et.al.…”