Time periodic solutions for a Cahn–Hilliard type equation

The viscous Cahn–Hilliard equation with periodic potentials and sources

“…The proof of this theorem is complete. As in [18], we need only prove the existence of weak solutions in the space C 2+α,α/4 (Q ω ). We show the existence of the solutions by the LeraySchauder fixed point theorem.…”

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

“…But, to the best of our knowledge, only a few papers deal with time periodic solutions of fourth-order diffusion equations. In [20], the existence of time periodic solutions for the Cahn-Hilliard type equation with periodic concentrationdependent potentials and sources has been investigated. Here, we will consider the Cahn-Hilliard type equation (1) with periodic gradient-dependent 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 .…”

“…Indeed, by using the qualitative theory of parabolic equation in , Yin et al . mainly studied the existence of periodic solutions of the following Cahn–Hilliard‐type equation in one spatial dimension: where ϕ ( u , t ) = a ( t ) u 3 − b ( t ) u for some positive, continuous, and ω ‐periodic functions a and b and f ( x , t ) is ω ‐periodic with respect to the second argument and satisfies for all t ∈[0, ω ].…”

“…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:…”