Time-periodic solutions for a driven sixth-order Cahn-Hilliard type equation
Liu et al. Boundary Value Problems 2013, 2013:73
http://www.boundaryvalueproblems.com/content/2013/1/73
RESEARCH
Open Access
Time-periodic solutions for a driven
sixth-order Cahn-Hilliard type equation
Changchun Liu* , Aibo Liu and Hui Tang
*
Correspondence:
Department of Mathematics, Jilin
University, Changchun, 130012,
China
Abstract
We study a driven sixth-order Cahn-Hilliard type equation which arises naturally as a
continuum model for the formation of quantum dots and their faceting. Based on the
Leray-Schauder fixed point theorem, we prove the existence of time-periodic
solutions.
MSC: 35B10; 35K55; 35K65
Keywords: sixth-order Cahn-Hilliard equation; time-periodic solution; existence;
Campanato space
1 Introduction
In this paper, we are concerned with the following problem for the sixth-order CahnHilliard type equation:
∂u
– γ D u = D ψ(u, t) + νuDu + f (x, t),
∂t
u|x=, = D u|x=, = D u|x=, = ,
u(x, ) = u(x, T),
x ∈ (, ),
t ≥ ,
(x, t) ∈ Q,
(.)
(.)
(.)
∂
where Q ≡ (, ) × (, +∞), D = ∂x
, ψ(u, t) = –a(t)u + b(t)u, a(t) and b(t) are Hölder
α
+
continuous functions defined on R with period T, f (x, t) belongs to the space C α, (Q)
for some α ∈ (, ) with f (x, ) = f (x, T). Furthermore, we assume that M ≤ a(t) ≤ M,
|b(t)| ≤ N , |a (t)| ≤ L, |b (t)| ≤ , where γ , ν, M, M, N , L and are positive constants.
Equation (.) with f (x, t) = arises naturally as a continuum model for the formation of
quantum dots and their faceting; see []. It can also be used to describe competition and
exclusion of biological population []. If we consider that the perturbation function f (x, t)
(for example, source) has the influence, then we obtain equation (.).
Korzec et al. [] studied equation (.) with f (x, t) = . New types of stationary solutions
of one-dimensional driven sixth-order Cahn-Hilliard type equation (.) are derived by
an extension of the method of matched asymptotic expansions that retains exponentially
small terms. Liu et al. [] proved that equation (.) with f (x, t) = possesses a global
attractor in the H k (k ≥ ) space, which attracts any bounded subset of H k () in the
H k -norm.
During the past years, many authors have paid much attention to other sixth-order thin
film equations such as the existence, uniqueness and regularity of the solutions [–].
© 2013 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Liu et al. Boundary Value Problems 2013, 2013:73
http://www.boundaryvalueproblems.com/content/2013/1/73
Page 2 of 17
However, as far as we know, there are few investigations concerned with the time-periodic
solutions of equation (.), even though there is some literature for population models
and Cahn-Hilliard [, ]. In fact, it is natural to consider the time-periodic solutions of
equation (.) when it is used to describe the models of the growth and dispersal in the
population which is sensitive to time-periodic factors (for example, seasons). In this paper, we prove the existence of time-periodic solutions of problem (.)-(.) based on the
framework of the Leray-Schauder fixed point theorem which can be found in any standard
textbook of PDE (see, for example, []). For this purpose, we first introduce an operator
L by considering a linear sixth-order equation with a parameter σ ∈ [, ]. After verifying
the compactness of the operator and some necessary a priori estimates for the solutions,
we then obtain a fixed point of the operator in a suitable functional space with σ = , which
is the desired solution of problem (.)-(.).
The main difficulties for treating problem (.)-(.) are caused by the nonlinearity of
both the fourth-order term and the convective factors. The main method that we use is
based on the Schauder-type a priori estimates, which here are obtained by means of a
modified Campanato space. We note that the Campanato spaces have been widely used
for the discussion of partial regularity of solutions of parabolic systems of second order and
fourth order. So, in the following section we give a detailed description and the associated
properties of such a space, and subsequently, in the next section we prove the existence of
classical time-periodic solutions of problem (.)-(.).
2 Hölder norm estimates
Let QT = (, ) × (, T), y = (x , t ) ∈ QT . For any fixed R > , we define
BR = BR (x ) = (x – R, x + R),
IR = IR (t ) = t – R , t + R ,
QR = QR (y ) = IR (t ) × BR (x ),
S R = QR ∩ QT ,
ER = ER (x ) = BR (x ) ∩ (, ),
JR = JR (t ) = IR (t ) ∩ (, T).
Let u be a function defined on QT , and set
uR = uy ,R =
|SR |
ûR = ûy ,R =
u dx dt,
SR
⎧
⎨u
if QR ∩ ∂p QT = ∅,
⎩
if QR ∩ ∂p QT = ∅,
R
where ∂p QT denotes the parabolic boundary of QT and |SR | denotes the area of SR .
For any u ∈ C(QT ) and λ > , define
λ
y ∈QT <R≤R R
M [u] = sup
sup
SR (y )
u(x, t) – ûy ,R dx dt,
where R = diam QT . By the space L,λ (QT ) we mean the subset of C(QT ), each element of
which satisfies M[u] < +∞. For u ∈ L,λ
, its norm is defined as
u L,λ (QT ) = supu(x, t) + M[u].
QT
Now, we give some useful lemmas.
Liu et al. Boundary Value Problems 2013, 2013:73
http://www.boundaryvalueproblems.com/content/2013/1/73
Page 3 of 17
Lemma . [] Let λ > ,
u
α
α,
C
≤ C(λ) u L,λ (QT ),
where α = λ–
.
Now we consider the following linear periodical problem:
∂u
– γ D u = (x, t), (x, t) ∈ QT = (, ) × (, T),
∂t
u|x=, = D u|x=, = D u|x=, = , t ∈ (, T),
(.)
x ∈ (, ).
(.)
u(x, ) = u(x, T),
(.)
Here we simply assume that (x, t) is sufficiently smooth. Our main purpose is to find the
relation between the Hölder norm of the solution u and (x, t).
Let y = (x , t ) ∈ QT be a fixed point and define
u(x, t) – ûρ + ρ D u(x, t) dx dt (ρ > ).
ϕ(u, ρ) =
Sρ
Let u be an arbitrary solution of problem (.)-(.). We split u on SR = SR (y ) as u = u + u
so that u solves the problem
∂u
– γ D u = , (x, t) ∈ SR ,
∂t
u (x, t) dx =
u(x, t) dx, t ∈ (, T),
ER
(.)
(.)
ER
u |∂ JR – u |∂ JR = u|∂ JR – u|∂ JR ,
Pi (x, D)u |∂ER = Pi (x, D)u|∂ER ,
(.)
and u solves the problem
∂u
– γ D u = (x, t), (x, t) ∈ SR ,
∂t
u (x, t) dx = , t ∈ (, T),
(.)
(.)
ER
u |∂ JR – u |∂ JR = Pi (x, D)u |∂ER = ,
(.)
where
⎧
⎨Di
Pi (x, D) =
⎩Di+
if x = , ,
i = , ,
if x = , ,
and ∂ JR , ∂ JR are the down-side and up-side points of JR , and ∂ER is the boundary of ER .
Some essential estimates on u and u are based on the following lemmas.
Lemma . For the solution u of problem (.)-(.), we have
i
i+
D u dx dt + R
D u dx dt ≤ CR–i
SR
SR
where C is a positive constant, i (...truncated)
