Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 42, pp. 1–23.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
PERIODIC SOLUTIONS OF A MULTI-DIMENSIONAL
CAHN-HILLIARD EQUATION
JI LIU, YIFU WANG, JIASHAN ZHENG
Abstract. This article concerns a multi-dimensional Cahn-Hilliard equation
subject to Neumann boundary condition. We show existence of the periodic
solutions by using the viscosity approach. By applying the Schauder fixed
point theorem, we show existence of the solutions to the suitable approximate
problem and then obtain the solutions of the considered periodic problem using
a priori estimates. Our results extend those in [20].
1. Introduction
In 1958, Cahn and Hilliard [3] derived the Cahn-Hilliard equation
uτ − ∆(−κ∆u + g(u)) = f,
(1.1)
which is a model of phase separation in binary material. Here g(u) is the derivative
of free energy F (u). If F (u) is a smooth function, (1.1) can be used to characterize
the spread of populations and the diffusion of an oil film over a solid surface, see
[4, 16]. While F (u) is not smooth, (1.1) is used to describe the phase separation
with constraints, see for example [2].
Because of the applications of Cahn-Hilliard equation (1.1) in physics, there has
been a great interest in studying the qualitative properties of solutions to the CahnHilliard equation. For example, we can refer to [6, 19] for existence, uniqueness and
regularity of the solutions, and [7, 13] for asymptotic behavior of the solutions.
In addition, using the techniques of subdifferential operator, Kenmochi et al [9]
investigated the Cahn-Hilliard equation with constraints. More recently, Kubo [11]
considered the strong solution and weak solution to the Cahn-Hilliard equation
with a time-dependent constraint and also discussed the relation between these
solutions.
It is well known that one of the most interesting topics of the higher-order parabolic equations, from a theoretical and practical point of view, is existence of the
periodic solutions, which has been considered in several works [12, 14, 18, 20, 22].
Zhao et al [22] studied existence and uniqueness of the time-periodic generalized
solutions to a fourth-order parabolic equation by the Galerkin method. Moreover,
2010 Mathematics Subject Classification. 35K25, 35B10, 35A01.
Key words and phrases. Periodic solutions; Cahn-Hilliard equation; viscosity approach;
Schauder fixed point theorem.
c
2016
Texas State University.
Submitted August 7, 2014. Published January 29, 2016.
1
2
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
[12, 14] are concerned with the existence, uniqueness and attractivity of the timeperiodic solutions to the Cahn-Hilliard equations with periodic gradient-dependent
potentials and sources. It should be remarked that [12, 14, 22] are all in the case of
one spatial dimension. Also in one spatial dimension, Yin et al [20] used the qualitative theory of parabolic equations to prove existence of the periodic solutions in
the classical sense to the following equation
uτ + κuxxxx = (A(τ )u3 − B(τ )u)xx + f (x, τ ),
where A(τ ) and B(τ ) are positive, continuous and periodic functions with the period
R1
ω > 0, and f (τ ) is also a smooth ω-periodic function satisfying 0 f (x, τ )dx = 0
for any τ ∈ [0, ω]. As for the case of higher dimensions, Wang and Zheng [18]
recently showed the existence of periodic solutions to the Cahn-Hillard equation
with a constraint by applying the viscosity approach.
Motivated by the above works, the purpose of this paper is to show existence of
the periodic solutions to the problem
uτ (x, τ ) − ∆(−κ∆u(x, τ ) + g(u(x, τ ))) = f (x, τ )
in Qω := Ω × (0, ω),
∂
∂u
(x, τ ) =
(−κ∆u(x, τ ) + g(u(x, τ ))) = 0 on Σω := ∂Ω × (0, ω),
∂ν
∂ν
u(x, 0) = u(x, ω) in Ω,
(1.2)
(1.3)
(1.4)
∂
where Ω is a bounded domain in RN (1 ≤ N ≤ 3) with smooth boundary, ∂ν
stands for the outward normal derivative on ∂Ω, f is a ω-periodic function and
g(u) = a3 u3 + a2 u2 + a1 u + a0 with constants a3 > 0 and ai ∈ R (i = 0, 1, 2).
In this case, the free energy F (u) = a43 u4 + a32 u3 + a21 u2 + a0 u + C, where C is
a constant. Particularly, if a2 = 0 and a1 < 0, F (u) is called double-well form
potential. Since the principle part of (1.2) is a fourth-order operator, we take the
viscosity approach in order to use the standard theory of the second order parabolic
equations. More precisely, we study the approximate problem
uτ (x, τ ) − ∆(εuτ (x, τ ) − κ∆u(x, τ ) + g(u(x, τ ))) = f (x, τ )
∂u
∂
(x, τ ) =
(−κ∆u(x, τ ) + g(u(x, τ ))) = 0
∂ν
∂ν
u(x, 0) = u(x, ω) in Ω,
in Qω ,
on Σω ,
(1.5)
where 0 < ε < 1. In order to apply the Schauder fixed point theorem to show
existence of the periodic solutions of (1.5), we need to establish some a priori
estimates on the solutions of (1.5) (cf. Lemma 3.4 below).
The plan of this article is as follows. In Section 2, we state some basic results
in functional analysis and give the main results. In Section 3, we first establish
some estimates of the solutions for (1.5), and then obtain existence of the periodic
solutions for (1.5) by the Schauder fixed point theorem. In Section 4, based on the
a priori estimates in Section 3, we can take the limit as ε → 0 and then obtain the
periodic solutions of (1.2)–(1.4).
2. Preliminaries
The notation and the basic results that we will use here are stated as follows.
(1) We denote by (·, ·) and | · |2 the usual inner product and the norm in L2 (Ω),
respectively. Also, we denote the Hilbert space L2 (Ω) by H.
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MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
3
(2) We denote H 1 (Ω) by V and its inner product by (·, ·)V , where (η1 , η2 )V =
(η1 , η2 )+(∇η1 , ∇η2 ) for any η1 , η2 ∈ H 1 (Ω). As a result, the norm in H 1 (Ω) can be
1/2
denoted by |η|V = (η, η)V . V ∗ denotes the dual space of V and h·, ·iV ∗ ,V stands
for the duality pairing between VR ∗ and V .
(3) We define H0 := {η ∈ H| Ω η(x)dx = 0} which is the closed subspace of H.
We choose the notation πR0 to denote the projection operator from H onto H0 , that
1
η(y)dy. Also, we denote the inner product on H0 by
is, π0 [η](x) = η(x) − |Ω|
Ω
(·, ·)0 .
(4) We denote by V0 the space V ∩ H0 with the inner product (·, ·)V0 and the
norm | · |V0 , where (η1 , η2 )V0 = (∇η1 , ∇η2 ) for any η1 , η2 ∈ V0 . Furthermore, F0−1
and h·, ·iV0∗ ,V0 denote the duality mapping from V0∗ onto V0 and the duality pairing
between V0∗ and V0 , respectively. Thus, we see that V0∗ is a Hilbert space and its
inner product can be defined as
(η1 , η2 )V0∗ = hη1 , F0−1 η2 iV0∗ ,V0 = hF0−1 η1 , η2 iV0 ,V0∗
for any η1 , η2 ∈ V0∗ .
(2.1)
It is observed that the Hilbert spaces stated above satisfy the following relations
V ⊂ H ⊂ V ∗,
V0 ⊂ H0 ⊂ V0∗ ,
where all the injections are compact and densely defined. Throughout this article,
we denote by Cj > 0(j = 1, 2, . . .) the constants induced by injection. Therefore,
from the above injections, we have
|η|V ∗ ≤ C1 |η|2 for any η ∈ H,
(2.2)
|η|2 ≤ C2 |η|V0 for any η ∈ V0 .
(5) Let ∆N be the Laplace operator with homogeneous Neumann boundary
condition in H0 with its domain
∂η
= 0 a.e. on ∂Ω .
D(∆N ) = η ∈ H 2 (Ω) ∩ H0 :
∂ν
Specially, ∆N η = ∆η a.e. on Ω for any η ∈ D(∆N ). We note that −∆N is invertible
in H0 and the inverse (−∆N )−1 is linear, continuous, positive and selfadjoint in H0
as well as its fractional power (−∆N )1/2 [21, Chapter 9, Section 11]. In addition,
we have
|(−∆N )1/2 η|H0 = |(−∆N )−1 η|V0 = |η|V0∗ , ∀η ∈ H0 .
(2.3)
In this article, we always assume that the following condition holds
RωR
(H1) f ∈ L∞ (0, ω; H) is a ω−periodic function and satisfies 0 Ω f (x, τ ) dx dτ =
0.
Now, we give the notion of the solution for (1.2)–(1.4).
Definition 2.1. A function u is called a solution of (1.2)–(1.4), if the conditions
below hold:
∂u
1,2
∗
(H2) u ∈ L2 (0, ω; H 2 (Ω)) ∩ L∞ (0,
ω; V ) ∩ W (0, ω; V ), ∂ν = 0 a.e. on Σω .
∂η (H3) For all η ∈ H 2 (Ω) with ∂ν
= 0,
∂Ω
Z ω
Z ω
Z ω
huτ (τ ), ηiV ∗ ,V dτ + κ
(∆u(τ ), ∆η)dτ −
(g(u(τ )), ∆η)dτ
0
0
0
Z ω
=
(f (τ ), η)dτ.
0
(H4) u(0) = u(ω)
in H.
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J. LIU, Y. WANG, J. ZHENG
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R
1
Now, we subtract |Ω|
f (x, τ )dx from (1.2) and obtain
Ω
Z
Z
τ
1
dh
u(x, τ ) −
f (x, s) dx ds − ∆(−κ∆u(x, τ ) + g(u(x, τ )))
dτ
|Ω| 0 Ω
= π0 [f (x, τ )].
Let
w(x, τ ) = u(x, τ ) −
1
|Ω|
Z
τ
(2.4)
Z
f (x, s) dx ds.
0
Ω
Then (2.4) can be rewritten as
Z τZ
i
h
1
f (x, s) dx ds
wτ (x, τ ) − ∆ − κ∆w(x, τ ) + g w(x, τ ) +
|Ω| 0 Ω
(2.5)
= π0 [f (x, τ )].
R
1
Therefore |Ω|
w(x, τ )dx = m0 for some constant m0 . Further, putting v(x, τ ) =
Ω
w(x, τ ) − m0 , we can rewrite (2.5) as
vτ (x, τ ) − ∆N (−κ∆N v(x, τ )) − ∆N π0 [g (v(x, τ ) + m(τ ))] = π0 [f (x, τ )], (2.6)
Rτ R
R
1
with Ω v(x, τ )dx = 0 for all τ > 0, where m(τ ) = m0 + |Ω|
f (x, s) dx ds.
0 Ω
−1
Now for any function z ∈ H0 , we can take (−∆N ) z as η in (H3). Hence by
the arguments in[5, Proposition 1.1], for any z ∈ H0 , it holds that
Z ω
Z ω
−1
((−∆N ) vτ (τ ), z)0 dτ + κ
(−∆N v(τ ), z)0 dτ
0
0
Z ω
(2.7)
+
(π0 [g(v(τ ) + m(τ ))], z)0 dτ
0
Z ω
=
((−∆N )−1 π0 [f (τ )], z)0 dτ.
0
From (2.3), (2.7) and the definition of F0−1 , we obtain an equivalent form of (1.2),
that is,
F0−1 vτ (τ ) − κ∆N v(τ ) + π0 [g((v(τ ) + m(τ )))] = F0−1 π0 [f (τ )].
(2.8)
Similarly, (1.5) is equivalent to
(F0−1 + εI)vε0 (τ ) − κ∆N vε (τ ) + π0 [g(vε (τ ) + m(τ ))] = F0−1 π0 [f (τ )]
in Qω ,
∂vε
(x, τ ) = 0 on Σω ,
∂ν
vε (x, 0) = vε (x, ω) in Ω,
(2.9)
d
where ε ∈ (0, 1), vε0 (τ ) = dτ
vε (τ ) and I is identity operator in H0 .
The main result of this article can be stated as follows.
Theorem 2.2. Assume that (H1) holds. Then for any constant m0 , (1.2)–(1.4)
admits a solution u(x, τ ) with
Z
Z τZ
1
1
u(x, τ )dx = m0 +
f (x, s) dx ds.
|Ω| Ω
|Ω| 0 Ω
To prove this theorem, we use the viscosity approach. Therefore, we need to
investigate (2.9) first. We have the following result which is proved in next section.
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MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
5
Theorem 2.3. Under the hypothesis of Theorem 2.2, (2.9) admits a solution which
has the following properties:
ε
(H2’) vε ∈ L2 (0, ω; H 2 (Ω) ∩ H0 ) ∩ L∞ (0, ω; V0 ) ∩ W 1,2 (0, ω; H0 ), ∂v
∂ν = 0 a.e. on
Σω .
(H3’) For any η ∈ D(∆N ) and 0 < τ < ω,
Z ω
((F0−1 + εI)vε0 (τ ) − κ∆N vε (τ ) + π0 [g(vε (τ ) + m(τ ))], η)0 dτ
0
Z ω
=
(F0−1 π0 [f (τ )], η)0 dτ in H0 .
0
(H4’) vε (0) = vε (ω)
in H0 .
3. Proof of Theorem 2.3
For this purpose we use the Schauder fixed point theorem. Firstly, we study the
system
(F0−1 + εI)v 0 (τ ) − κ∆N v(τ ) = fb in H0 ,
(3.1)
v(0) = v(ω) in H0 ,
where fˆ ∈ L∞ (0, ω; H0 ).
Theorem 3.1. Let fˆ ∈ L∞ (0, ω; H0 ). Then there exists a unique solution v(x, t)
to problem (3.1).
We prove this theorem using Poincaré’s mapping. Thus, we first introduce the
corresponding Cauchy problem
(F0−1 + εI)v 0 (τ ) − κ∆N v(τ ) = fb, 0 < τ < ω,
(3.2)
v(0) = v0 ∈ H0 .
With the help of the results in [8, 10], we can see that (3.2) admits one and only one
solution v ∈ C([0, ω]; H0 ) ∩ L∞
loc (0, ω; V0 ). Consequently, with the unique solution
v(τ ), we can define a S
single-valued mapping P : v(0) ∈ H0 → v(ω) ∈ H0 .
Define φ : H0 → R {+∞} by
(
κ
|∇v|22 , if v ∈ V0 ,
φ(v) = 2
+∞,
otherwise.
We see that φ is a proper, lower semicontinuous, and convex functional on H0 .
Now, we give two lemmas which play an important role in the proof of Theorem
3.1.
Lemma 3.2. There exists a constant R > 0 such that P is a self-mapping on the
set
BR := {v ∈ D(φ); φ(v) ≤ R},
that is P (BR ) ⊂ BR .
Proof. Multiplying the equation in (3.2) by v 0 , we have
1
ε
κ d
|∇v|22 = (fb, v 0 )0 ≤ |fb|2 |v 0 |2 ≤ |fb|22 + |v 0 |22 ,
|v 0 |2V0∗ + ε|v 0 |22 +
2 dt
2ε
2
i.e.,
ε
κ d
1
|v 0 |2V0∗ + |v 0 |22 +
|∇v|22 ≤ |fb|22 .
2
2 dt
2ε
(3.3)
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J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
We also multiply the equation by v and obtain
κ|∇v|22 = (fb, v)0 − (εv 0 , v)0 − hF0−1 v 0 , viV0 ,V0∗
≤ |fb|2 |v|2 + ε|v 0 |2 |v|2 + |v 0 |V0∗ |v|V0∗
2C22 b 2 κ
2ε2 C22 0 2 κ
C 2C 2
κ
|f |2 + |∇v|22 +
|v |2 + |∇v|22 + 1 2 |v 0 |2V0∗ + |∇v|22
κ
8
κ
8
κ
4
κ
2C22 b 2 2ε2 C22 0 2 C12 C22 0 2
2
|f |2 +
|v |2 +
|v |V0∗ + |∇v|2 ,
=
κ
κ
κ
2
which implies
≤
κ
2C22 b 2 2ε2 C22 0 2 C12 C22 0 2
|∇v|22 ≤
|f |2 +
|v |2 +
|v |V0∗ .
2
κ
κ
κ
Letting µ > 0 and performing (3.3) × µ + (3.4), we obtain
(3.4)
κ
d κ
( |∇v|22 ) + |∇v|22
dt 2
2
2C22 b 2
C 2C 2
µε 0 2
µ
2ε2 C22
)|f |2 + ( 1 2 − µ)|v 0 |2V0∗ + (
−
)|v |2 .
≤( +
2ε
κ
κ
κ
2
C 2 C 2 4C 2 Choosing µ = max 1κ 2 , κ 2 , from 0 < ε < 1 we have
µ
d
1
1
2C22 b 2
φ(v) + φ(v) ≤
+
|f |2 .
dt
µ
2ε
κµ
It follows from the Gronwall inequality that
ω
ω
φ(v(ω)) ≤ e− µ φ(v(0)) + (1 − e− µ )
µ
2C22 b 2
+
kf kL∞ (0,ω;H0 ) .
2ε
κ
2
2C
µ
Set R = ( 2ε
+ κ 2 )kfbk2L∞ (0,ω;H0 ) . Then φ(v(ω)) ≤ R provided that φ(v(0)) ≤ R.
The proof is complete.
Lemma 3.3. The mapping P is continuous in H0 .
Proof. Let v0,n ∈ H0 be such that v0,n → v0 in H0 . We denote the unique solution
of (3.2) by vn and v corresponding to the initial data v0,n and v0 , respectively.
Then we have
F0−1 (vn0 − v 0 ) + ε(vn0 − v 0 ) − κ∆N (vn − v) = 0.
(3.5)
Multiplying (3.5) by vn − v and using integration by parts, we obtain
1 d
ε d
|vn − v|2V0∗ +
|vn − v|22 + κ|∇(vn − v)|22 = 0.
2 dt
2 dt
It can be easy to see that
1 d
ε d
|vn − v|2V0∗ +
|vn − v|22 ≤ 0.
2 dt
2 dt
Therefore,
1
ε
1
ε
|vn (ω) − v(ω)|2V0∗ + |vn (ω) − v(ω)|22 ≤ |vn0 − v0 |2V0∗ + |vn0 − v0 |22 ,
2
2
2
2
which implies vn (w) → v(ω) in H0 as n → ∞. Hence, P is continuous in H0 .
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MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
7
Proof of Theorem 3.1. On the one hand, it follows from the definition of BR and
the convexity of φ that BR is compact and convex in H0 . On the other hand,
Lemmas 3.2 and 3.3 ensure that P maps BR to BR and is continuous in H0 . Thus,
the Schauder fixed point theorem admits a fixed point v0∗ ∈ BR such that P v0∗ = v0∗ ,
which implies that the solution v(x, t) of (3.2) with v0 = v0∗ is the desired solution
of (3.1).
Now, we prove that the solution for (3.1) is unique. To this end, let v1 and v2
be two solutions of (3.1). Then we have
F0−1 (v10 − v20 ) + ε(v10 − v20 ) − κ∆N (v1 − v2 ) = 0.
(3.6)
We multiply (3.6) by v1 − v2 and then get that
1 d
ε d
|v1 − v2 |2V0∗ +
|v1 − v2 |22 + κ|∇(v1 − v2 )|22 = 0.
2 dt
2 dt
Integrating the equation over (0, ω) and by the periodic property, we obtain
Z ω
|∇(v1 (τ ) − v2 (τ ))|22 dτ ≤ 0,
0
which, together with (2.2), implies that
Z ωZ
|v1 − v2 |2 dx dτ ≤ 0.
0
Ω
Hence, v1 = v2 and the proof is complete.
To apply the Schauder fixed point theorem to (2.9), we need to establish a priori
estimates for vε .
Lemma 3.4. Let vε be a solution of (2.9). Then
Z ω
Z ω
ε
|vε0 (τ )|22 dτ +
|vε0 (τ )|2V0∗ dτ ≤ ωC12 kf k2L∞ (0,ω;H) ,
0
sup |vε (τ )|2V0 ≤
τ ∈[0,ω]
(3.7)
0
3C12 + 4ω 2
2
3A1 + 4A2 +
C1 kf k2L∞ (0,ω;H) ,
κ
2
k − ∆N vε k2L2 (0,ω;H0 )
4ω a22
≤ 2
+ |a1 | (3A1 + 4A2 )
κ 2a3
a2
i
ωh
+ 2 (3C12 + 4ω)C12 2 + 2|a1 | + C14 kf k2L∞ (0,ω;H)
κ
a3
kvε (τ ) + m(τ )k6C([0,ω];L6 (Ω)) ≤ A33 ,
where ai (i = 0, 1, 2, 3) are the coefficients of g(·),
h 93 4 3a2
ω
3a2
1
A1 : = |Ω|
a3 |m0 | +
kf kL∞ (0,ω;H) +
+ 2 +1
1/2
4
2a3
a3
|Ω|
2
ω
× |m0 | +
kf kL∞ (0,ω;H)
|Ω|1/2
a4 9 3 a3
3a2 + 3 i
+ 2
+
+ 1
,
4 a3
12
a3
4/3
h 6a4
a2
a
6 1/3 i
A2 := 6|Ω| 32 + 1 + 0
,
a3
2a3
4 a3
(3.8)
(3.9)
(3.10)
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J. LIU, Y. WANG, J. ZHENG
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n2
A3 : = C3
(2ωC12 C22 + 2C22 + 1)(3A1 + 4A2 ) + 4m20 |Ω|(ωC12 + 1)
κ
3C 4 C 2 + 4C 2 + 2
h
2C 4 C 2
1 2
2
1 2
+ 1 + 2C12 ω
+2
+ 4ω 3 C12 + 4ω 2
κ
κ
o
3C14 (2C22 + 1) i
+
kf k2L∞ (0,ω;H) .
κ
Proof. From (2.2), the definition of π0 and (H1), we know that
kπ0 [f ]k2L2 (0,ω;V0∗ ) ≤ C12 kπ0 [f ]k2L2 (0,ω;H0 ) ≤ ωC12 kf k2L∞ (0,ω;H) .
It follows from the Hölder inequality and (H1) that for any τ ∈ [0, ω]
Z τZ
1
f (x, s) dx ds
|m(τ )| = m0 +
|Ω| 0 Ω
ω
≤ |m0 | +
kf kL∞ (0,ω;H) .
|Ω|1/2
(3.11)
(3.12)
We multiply the equation in (2.9) by vε0 , and obtain
Z
κ d
2
0 2
0 2
|vε |V0∗ + ε|vε |2 +
|∇vε |2 +
vε0 π0 [g(vε (τ ) + m(τ ))]dx
2 dt
Ω
= hF0−1 π0 [f ], vε0 iV0 ,V0∗
(3.13)
≤ |π0 [f ]|V0∗ |vε0 |V0∗
1
1
≤ |π0 [f ]|2V0∗ + |vε0 |2V0∗ .
2
2
By the definition of π0 and g(·), we have
Z h
a3
d
1 0 2
κ d
2
0 2
|v | ∗ + ε|vε |2 +
|∇vε |2 +
(vε (τ ) + m(τ ))4
2 ε V0
2 dt
dt Ω 4
i
a2
a1
(3.14)
+ (vε (τ ) + m(τ ))3 + (vε (τ ) + m(τ ))2 + a0 (vε (τ ) + m(τ )) dx
3
2
1
≤ |π0 [f ]|2V0∗ .
2
From the periodic property, we integrate (3.14) over (0, ω) and then obtain
Z ω
Z ω
Z ω
2ε
|vε0 (τ )|22 dτ +
|vε0 (τ )|2V0∗ dτ ≤
|π0 [f (τ )]|2V0∗ dτ.
(3.15)
0
0
0
Combining this inequality with (3.11), we have
Z ω
Z ω
0
2
ε
|vε (τ )|2 dτ +
|vε0 (τ )|2V0∗ dτ ≤ ωC12 kf k2L∞ (0,ω;H) ,
0
0
which is (3.7).
Choose any s, t ∈ [0, ω] which satisfy s < t. Integrating (3.14) on (s, t), we have
Z
Z t
Z
a3
κ
1 t 0
|vε (τ )|2V0∗ dτ + ε
|vε0 (τ )|22 dτ + |∇vε (t)|22 +
(vε (t) + m(t))4
2 s
2
s
Ω 4
a2
a1
+ (vε (t) + m(t))3 + (vε (t) + m(t))2 + a0 (vε (t) + m(t)) dx
3Z
2
(3.16)
Z
a3
1 ω
κ
2
≤
|π0 [f (τ )]|V0∗ dτ + |∇vε (s)|22 +
(vε (s) + m(s))4
2 0
2
Ω 4
a2
a1
3
+ (vε (s) + m(s)) + (vε (s) + m(s))2 + a0 (vε (s) + m(s)) dx.
3
2
EJDE-2016/42
MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
9
From Young’s inequality, we obtain
a2
a3
18a4
(vε (τ ) + m(τ ))3 ≤
(vε (τ ) + m(τ ))4 + 3 2 ,
3
24
a3
a1
a3
3a2
(vε (τ ) + m(τ ))2 ≤
(vε (τ ) + m(τ ))4 + 1
2
24
2a3
4/3 a3
6 1/3
3a
a0 (vε (τ ) + m(τ )) ≤
(vε (τ ) + m(τ ))4 + 0
.
24
4
a3
(3.17)
It follows from (3.16) and (3.17) that
1
2
Z
t
|vε0 (τ )|2V0∗ dτ
Z
+ε
t
|vε0 (τ )|22 dτ +
κ
|∇vε (t)|22
2
s
s
Z
a3
+
(vε (t) + m(t))4 dx
8 Ω
Z
3a3
κ
2
(vε (s) + m(s))4 dx
≤ |∇vε (s)|2 +
2
8 Ω
Z
1 ω
+
|π0 [f (τ )]|2V0∗ dτ
2 0
+ A2 .
(3.18)
Deleting the first two terms on the left-hand side of (3.18) and integrating it on
(0, t) with respect to s, we have
Z
a3 t
κt
|∇vε (t)|22 +
(vε (t) + m(t))4 dx
2
8 Ω
Z
Z Z
Z
κ ω
3a3 ω
ω ω
≤
|∇vε (s)|22 ds +
(vε (s) + m(s))4 dx ds +
|π0 [f (τ )]|2V0∗ dτ
2 0
8 0 Ω
2 0
+ A2 ω.
Letting t = ω, one sees that
Z
a3 ω
κω
2
|∇vε (ω)|2 +
(vε (ω) + m0 )4 dx
2
8 Ω
Z
Z Z
Z
κ ω
3a3 ω
ω ω
2
4
≤
|∇vε (s)|2 ds +
(vε (s) + m(s)) dx ds +
|π0 [f (τ )]|2V0∗ dτ
2 0
8 0 Ω
2 0
+ A2 ω,
i.e.,
Z
a3
κ
|∇vε (ω)|22 +
(vε (ω) + m0 )4 dx
2
8 Ω
Z ω
Z Z
Z
κ
3a3 ω
1 ω
≤
|∇vε (s)|22 ds +
(vε (s) + m(s))4 dx ds +
|π0 [f (τ )]|2V0∗ dτ
2ω 0
8ω 0 Ω
2 0
+ A2 .
10
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
From the periodic property, we have
Z
κ
a3
|∇vε (0)|22 +
(vε (0) + m0 )4 dx
2
8 Ω
Z ω
Z Z
3a3 ω
κ
2
|∇vε (s)|2 ds +
(vε (s) + m(s))4 dx ds
≤
2ω 0
8ω 0 Ω
Z
1 ω
+
|π0 [f (τ )]|2V0∗ dτ + A2 .
2 0
(3.19)
Multiplying the equation in (2.9) by vε and performing a proper arrangement,
we obtain
ε d
1 d
|vε |2 ∗ +
|vε |22 + κ|∇vε |22
2 dt Z V0
2 dt
+ a3
(vε (τ ) + m(τ ))4 − m(τ )(vε (τ ) + m(τ ))3 dx
ZΩ
Z
3
2
+ a2
[vε (τ ) + m(τ )] dx + [a1 − a2 m(τ )]
[vε (τ ) + m(τ )] dx
Ω
Ω
Z
− a1 m(τ )
[vε (τ ) + m(τ )] dx
(3.20)
Ω
= hF0−1 π0 [f ], vε iV0 ,V0∗
≤ |π0 [f ]|V0∗ |vε |V0∗
≤ C1 |π0 [f ]|V0∗ |vε (τ ) + m(τ )|2 + C1 |π0 [f ]|V0∗ |Ω|1/2 |m(τ )|.
From Young’s inequality, we obtain
93
1
4
[vε (τ ) + m(τ )] + |m(τ )|4 ,
12
4
4
a3
a 9 3
3
4
a2 [vε (τ ) + m(τ )] ≤
[vε (τ ) + m(τ )] + 2
,
12
4 a3
a3
3
2
4
2
[a1 − a2 m(τ )] [vε (τ ) + m(τ )] ≤
[vε (τ ) + m(τ )] +
[a1 − a2 m(τ )]
6
2a3
a3
3a2
3a2
4
≤
[vε (τ ) + m(τ )] + 1 + 2 |m(τ )|2 ,
6
a3
a3
a3
3 2
2
a1 m(τ ) [vε (τ ) + m(τ )] ≤
[vε (τ ) + m(τ )] +
a |m(τ )|2
6
2a3 1
a3
a3
3 2
4
≤
[vε (τ ) + m(τ )] +
+
a |m(τ )|2 ,
12
12 2a3 1
C2
|π0 [f ]|V0∗ |vε (τ ) + m(τ )|2 ≤ 1 |π0 [f ]|2V0∗ + |vε (τ ) + m(τ )|22
4
Z
C12
a3
3
4
≤
|π0 [f ]|2V0∗ +
[vε (τ ) + m(τ )] dx + |Ω|
4
12 Ω
a3
3
m(τ ) [vε (τ ) + m(τ )] ≤
and
|π0 [f ]|V0∗ |Ω|1/2 |m(τ )| ≤
C12
|π0 [f ]|2V0∗ + |Ω||m(τ )|2 .
4
(3.21)
EJDE-2016/42
MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
11
In light of (3.12), (3.20) and (3.21), we have
1 d
ε d
a3
|vε |2V0∗ +
|vε |22 + κ|∇vε |22 +
2 dt
2 dt
2
2
C
≤ 1 |π0 [f ]|2V0∗ + A1 .
2
Z
4
[vε (τ ) + m(τ )] dx
Ω
From the periodic property, we integrate (3.22) over (0, ω) and obtain
Z Z
Z ω
a3 ω
2
(vε (τ ) + m(τ ))4 dx dτ
κ
|∇vε (τ )|2 dτ +
2
0
Ω
0
Z
C12 ω
2
≤
|π0 [f ]|V0∗ dτ + A1 ω.
2 0
Combining (3.19) with (3.23), we have
Z
a3
κ
2
|∇vε (0)|2 +
(vε (0) + m0 )4 dx
2
8 Ω
Z
Z
C12 ω
1 ω
2
≤
|π0 [f (τ )]|V0∗ dτ + A1 +
|π0 [f (τ )]|2V0∗ dτ + A2
2ω 0
2 0
ω + C12
kπ0 [f ]k2L2 (0,ω;V0∗ ) .
≤ A1 + A2 +
2ω
(3.22)
(3.23)
(3.24)
Letting s = 0 in (3.18) and deleting the first two terms on the left-hand side, we
obtain
Z
κ
a3
|∇vε (t)|22 +
(vε (t) + m(t))4 dx
2
8 Ω
Z
Z
3a3
1 ω
κ
2
4
(vε (0) + m0 ) dx +
≤ |∇vε (0)|2 +
|π0 [f (τ )]|2V0∗ dτ + A2
(3.25)
2
8 Ω
2 0
Z
Z
i 1 ω
hκ
a3
(vε (0) + m0 )4 dx +
≤ 3 |∇vε (0)|22 +
|π0 [f (τ )]|2V0∗ dτ + A2 .
2
8 Ω
2 0
It follows from (3.24) and (3.25) that for any t ∈ [0, ω],
Z
Z
κ
a3
3C12 + 4ω ω
|∇vε (t)|22 +
(vε (t) + m(t))4 dx ≤ 3A1 + 4A2 +
|π0 [f (τ )]|2V0∗ dτ,
2
8 Ω
2ω
0
which together with (3.11) yields (3.8).
Multiplying the equation of (2.9) by −∆N vε and integrating it by parts, we have
Z
1 d
ε d
2
2
2
|vε |2 +
|∇vε |2 + κ|∆N vε |2 + 3a3 (vε (τ ) + m(τ ))2 |∇vε |2 dx
2 dt
2 dt
Z
Z Ω
2
+ 2a2 (vε (τ ) + m(τ ))|∇vε | dx + a1
|∇vε |2 dx
=
Ω
−1
hF0 π0 [f ], −∆N vε iV0 ,V0∗
≤ C1 |π0 [f ]|V0∗ |∆N vε |2
≤
C12
κ
|π0 [f ]|2V0∗ + |∆N vε |22 .
2κ
2
Ω
12
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
After a proper arrangement, we obtain
1 d
ε d
κ
|vε |22 +
|∇vε |22 + |∆N vε |22 + 3a3
2 dt
2 dt
2
Z
+ 2a2 (vε (τ ) + m(τ ))|∇vε |2 dx
Ω
Z
C12
2
|π0 [f ]|V0∗ − a1
|∇vε |2 dx.
≤
2κ
Ω
Z
(vε (τ ) + m(τ ))2 |∇vε |2 dx
Ω
(3.26)
From Young’s inequality, we have
2a2 (vε (τ ) + m(τ )) ≤ 2a3 (vε (τ ) + m(τ ))2 +
a22
.
2a3
(3.27)
It follows from (3.26) and (3.27) that
Z
1 d
ε d
κ
|vε |22 +
|∇vε |22 + |∆N vε |22 + a3 (vε (τ ) + m(τ ))2 |∇vε |2 dx
2 dt
2 dt
2
Ω
a2
Z
C12
2
2
2
≤
+ |a1 |
|∇vε | dx +
|π0 [f ]|V0∗ .
2a3
2κ
Ω
(3.28)
With the help of the periodic property, we integrate (3.28) over (0, ω) and then get
that
Z ω
Z
Z ω Z
2 a22
C2 ω
|π0 [f (τ )]|2V0∗ dτ.
|∆N vε (τ )|22 dτ ≤
+ |a1 |
|∇vε (τ )|2 dx dτ + 21
κ
2a
κ
3
0
0
0
Ω
Substituting (3.8) and (3.11) into the above inequality, we obtain (3.9).
By (3.7) and (3.8), we know that vε ∈ W 1,2 (0, ω; V0∗ ) ∩ L∞ (0, ω; V0 ). Therefore,
vε (τ ) + m(τ ) ∈ W 1,2 (0, ω; V ∗ ) ∩ L∞ (0, ω; V ). Since
W 1,2 (0, ω; V ∗ ) ∩ L∞ (0, ω; V ) ,→ C([0, ω]; L6 (Ω)),
it is clear that there exists a positive constant C3 such that
kvε (τ ) + m(τ )k2C([0,ω];L6 (Ω))
≤ C3 (kvε (τ ) + m(τ )k2W 1,2 (0,ω;V ∗ ) + kvε (τ ) + m(τ )k2L∞ (0,ω;V ) ).
(3.29)
Now, we establish the estimates for kvε (τ ) + m(τ )kW 1,2 (0,ω;V ∗ ) and kvε (τ ) +
m(τ )kL∞ (0,ω;V ) , respectively. Since
Z ω
Z ω
|vε (τ ) + m(τ )|2V ∗ dτ ≤
(|vε (τ )|2V0∗ + 2|vε (τ )|V0∗ |m(τ )|V ∗ + |m(τ )|2V ∗ )dτ
0
0
Z ω
Z ω
≤ 2(C12 C22
|vε (τ )|2V0 dτ + C12
|m(τ )|22 dτ )
0
0
≤ 2ωC12 (C22 kvε (τ )k2L∞ (0,ω;V0 ) + |Ω||m(τ )|2 ),
we obtain
Z ω
h 2C 2
2
2
(3A1 + 4A2 ) + 2m20 |Ω|
|vε (τ ) + m(τ )|V ∗ dτ ≤ 2ωC12
κ
0
3C 4 C 2 + 4ωC 2 C 2
i
1 2
1 2
+
+ 2ω 2 kf k2L∞ (0,ω;H) .
κ
(3.30)
EJDE-2016/42
MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
Similarly, we have
Z ω
|vε0 (τ ) + m0 (τ )|2V ∗ dτ
0
Z ω
|vε0 (τ )|2V0∗ + 2|vε0 (τ )|V0∗ |m0 (τ )|V ∗ + |m0 (τ )|2V ∗ dτ
≤
0
Z ω
Z ω
0
2
2
|m0 (τ )|22 dτ .
≤2
|vε (τ )|V0∗ dτ + C1
13
(3.31)
0
0
Moreover,
Z
ω
0
|m
0
(τ )|22 dτ
Z
=
0
ω
1 Z
2
f (x, τ )dx dτ ≤ ωkf k2L∞ (0,ω;H) .
|Ω| Ω
2
Thus, together with (3.7), (3.31) can be written as
Z ω
2
d
(vε (τ ) + m(τ )) ∗ dτ ≤ 4ωC12 kf k2L∞ (0,ω;H) .
dt
V
0
(3.32)
It follows from (3.30) and (3.32) that
kvε (τ ) + m(τ )k2W 1,2 (0,ω;V ∗ )
h 2C 2
2
≤ 2ωC12
(3A1 + 4A2 ) + 2m20 |Ω|
κ
3C 4 C 2 + 4ωC 2 C 2
i
1 2
1 2
+
+ 2ω 2 + 2 kf k2L∞ (0,ω;H) .
κ
(3.33)
Also, since
kvε (τ ) + m(τ )k2L∞ (0,ω;V )
Z
hZ
i
= ess supτ ∈[0,ω]
|vε (τ ) + m(τ )|2 dx +
|∇(vε (τ ) + m(τ ))|2 dx
Ω
ZΩ
Z
2
2
≤ 2 ess supτ ∈[0,ω] [vε (τ ) + |m(τ )| ]dx + ess supτ ∈[0,ω]
|∇vε (τ )|2 dx
Ω
Ω
Z
2
2
≤ (2C2 + 1)kvε kL∞ (0,ω;V0 ) + 2 ess supτ ∈[0,ω]
|m(τ )|2 dx,
Ω
from (3.8) and (3.12) we have
kvε (τ ) + m(τ )k2L∞ (0,ω;V )
2
≤ (2C22 + 1)(3A1 + 4A2 ) + 4m20 |Ω|
κ
h (2C 2 + 1)(3C 2 + 4ω)
i
2
1
+
C12 + 4ω 2 kf k2L∞ (0,ω;H) .
κ
(3.34)
Combining (3.33) with (3.34), we obtain
kvε (τ ) + m(τ )k2C([0,ω];L6 (Ω)) ≤ A3 .
(3.35)
kvε (τ ) + m(τ )k6C([0,ω];L6 (Ω)) ≤ A33 ,
(3.36)
Thus,
which is (3.10). The proof is complete.
14
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
Define a set
n
Y1 := v̄ ∈ L∞ (0, ω; V0 ) ∩ W 1,2 (0, ω; H0 )|v̄(0) = v̄(ω),
o
kv̄(τ ) + m(τ )k6C([0,ω];L6 (Ω)) ≤ A33 .
(3.37)
Now, for any v̄ ∈ Y1 , we study the problem
F0−1 v 0 (τ ) + εv 0 (τ ) − κ∆N v(τ ) = −π0 [g(v̄(τ ) + m(τ ))] + F0−1 π0 [f (τ )]
in H0 ,
0 < τ < ω,
v(0) = v(ω)
(3.38)
in H0 .
For convenience, we denote the above system by (Eε , v̄).
Lemma 3.5. Let v(t) be the solution of (Eε , v̄). Then the following estimates can
be established
Z ω
Z ω
|v 0 (τ )|2V0∗ dτ + ε
|v 0 (τ )|22 dτ
0
0
(3.39)
ω 3 2
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 ) + ωC12 kf k2L∞ (0,ω;H) ,
≤
Z ω ε
|v(τ )|22 dτ
0
(3.40)
i
2ωC24 h 3 2
2
2
2
2
4
2
3A
(a
+
a
+
a
)
+
|Ω|(a
+
2a
)
+
C
kf
k
≤
,
3 3
2
1
2
1
1
L∞ (0,ω;H)
κ2
ω C2 + 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
κ|∇v(t)|22 ≤ 2
ε
κ
(3.41)
C 2C 2 + 2C12 ω + 1 2 kf k2L∞ (0,ω;H)
κ
for t ∈ [0, ω] and
Z ω
|∆N v(τ )|22 dτ
0
(3.42)
i
2ω h 3 2
2
2
2
2
4
2
≤ 2 3A3 (a3 + a2 + a1 ) + |Ω|(a2 + 2a1 ) + C1 kf kL2 (0,ω;V0∗ ) ,
κ
where A3 is the same as that in Lemma 3.4.
Proof. For any τ ∈ [0, ω], we have
Z
π0 [a3 (v̄(τ ) + m(τ ))3 ]2 dx
Ω
Z
Z
i2
a2 h
(v̄(τ ) + m(τ ))3 dx
= a23 (v̄(τ ) + m(τ ))6 dx − 3
|Ω| Ω
ZΩ
≤ a23 (v̄(τ ) + m(τ ))6 dx.
(3.43)
Ω
Similarly, for any τ ∈ [0, ω], we have
Z
Z
π0 [a2 (v̄(τ ) + m(τ ))2 ]2 dx ≤ a22 (v̄(τ ) + m(τ ))4 dx ,
ZΩ
ZΩ
2
|π0 [a1 (v̄(τ ) + m(τ ))]| dx ≤ a21 (v̄(τ ) + m(τ ))2 dx.
Ω
Ω
(3.44)
(3.45)
EJDE-2016/42
MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
It follows from Young’s inequality that
Z
Z
4a2
2
4
2
a2 (v̄(τ ) + m(τ )) dx ≤ a2 (v̄(τ ) + m(τ ))6 dx + 2 |Ω| ,
27
Ω
Ω
Z
Z
2
2a
a21 (v̄(τ ) + m(τ ))2 dx ≤ a21 (v̄(τ ) + m(τ ))6 dx + 31 |Ω|
32
Ω
Ω
Z
2
2a
< a21 (v̄(τ ) + m(τ ))6 dx + 1 |Ω|.
3
Ω
By (3.46)-(3.47), for any τ ∈ [0, ω], we have
Z
π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ ))]2 dx
Ω
Z
4a2
2
+ 2a21
≤ 3(a23 + a22 + a21 ) (v̄(τ ) + m(τ ))6 dx + |Ω|
9
Ω
3 2
2
2
2
2
≤ 3A3 (a3 + a2 + a1 ) + |Ω|(a2 + 2a1 ).
15
(3.46)
(3.47)
(3.48)
Multiplying the equation in (3.38) by v 0 and integrating by parts, we have
κ d
|∇v|22
2 dt
= (−π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ )) + a0 ], v 0 )0
|v 0 |2V0∗ + ε|v 0 |22 +
+ hF0−1 π0 [f ], v 0 iV0 ,V0∗
≤ π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ ))]2 |v 0 |2
+ |π0 [f ]|V0∗ |v 0 |V0∗
2
1 ε
π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ ))]2
≤ |v 0 |22 +
2
2ε
1 02
1
+ |v |V0∗ + |π0 [f ]|2V0∗ ,
2
2
i.e.,
κ d
1 02
ε
|v | ∗ + |v 0 |22 +
|∇v|22
2 V0
2
2 dt
1
(3.49)
≤ π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2
2ε
2 1
+ a1 (v̄(τ ) + m(τ ))]2 + |π0 [f ]|2V0∗ .
2
It follows from (3.48) that
1 02
ε
κ d
|v |V0∗ + |v 0 |22 +
|∇v|22
2
2
2 dt
1
1 3 2
≤
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 ) + |π0 [f ]|2V0∗ .
2ε
2
Integrating (3.50) on (0, ω) and from (3.11), we obtain
Z
Z
1 ω 0
ε ω 0
2
|v (τ )|V0∗ dτ +
|v (τ )|22 dτ
2 0
2 0
ωC12
ω 3 2
≤
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 ) +
kf k2L∞ (0,ω;H) ,
2ε
2
which is (3.39).
(3.50)
16
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EJDE-2016/42
Multiplying the equation of (3.38) by v and integrating by parts, we have
1 d 2
ε d 2
|v|V0∗ +
|v| + κ|∇v|22
2 dt
2 dt 2
= (−π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2
+ a1 (v̄(τ ) + m(τ )) + a0 ], v)0 + hF0−1 π0 [f ], viV0 ,V0∗
≤ π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ ))]2 |v|2
(3.51)
+ |π0 [f ]|V0∗ |v|V0∗
κ
C 2 ≤ |∇v|22 + 2 π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2
2
κ
2 C 2 C 2
+ a1 (v̄(τ ) + m(τ ))] + 1 2 |π0 [f ]|2V0∗ ,
κ
2
where the last two inequality signs follow from Young’s inequality and (2.2). Combining (3.51) with (3.48), we obtain
1 d 2
ε d 2 κ
|v| ∗ +
|v| + |∇v|22
2 dt V0
2 dt 2 2
C 2C 2
C2 ≤ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 ) + 1 2 |π0 [f ]|2V0∗ .
κ
κ
Integrating (3.52) on (0, ω), we obtain
Z
ωC22 3 2
κ ω
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 )
|∇v(τ )|22 dτ ≤
2 0
κ
C12 C22
+
kπ0 [f ]k2L2 (0,ω;V0∗ ) .
κ
(3.52)
(3.53)
Therefore, from (2.2) and (3.11), we have
Z
Z
κC22 ω
κ ω
|v(τ )|22 dτ ≤
|∇v(τ )|22 dτ
2 0
2 0
o
ωC24 n 3 2
≤
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 ) + C14 kf k2L∞ (0,ω;H) ,
κ
which is (3.40).
For any s, t ∈ [0, ω] satisfying s < t, we integrate (3.50) on (s, t) and obtain
Z
Z
1 t 0
ε t 0
κ
|v (τ )|2V0∗ dτ +
|v (τ )|22 dτ + |∇v(t)|22
2 s
2 s
2
Z
1 ω
ω 3 2
κ
≤
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 ) +
|π0 [f (τ )]|2V0∗ dτ + |∇v(s)|22 .
2ε
2 0
2
(3.54)
Deleting the first two terms on the left-hand side of (3.54) and integrating it on
(0, t) with respect to s, we have
Z
ω ω
κt
ω2 3 2
|∇v(t)|22 ≤
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 ) +
|π0 [f (τ )]|2V0∗ dτ
2
2ε
2 0
Z
κ ω
+
|∇v(s)|22 ds.
2 0
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MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
17
In view of (3.53), we obtain
ω
κt
C2 |∇v(t)|22 ≤ ω
+ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
2
2ε
κ
(3.55)
ω C 2C 2 + 1 2 kπ0 [f ]k2L2 (0,ω;V0∗ ) .
+
2
κ
Let t = ω, then (3.55) can be rewritten as
ω
C2 κω
|∇v(ω)|22 ≤ ω
+ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
2
2ε
κ
ω C 2C 2 +
+ 1 2 kπ0 [f ]k2L2 (0,ω;V0∗ ) ,
2
κ
i.e.,
ω
κ
C2 |∇v(ω)|22 ≤
+ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
2
2ε
κ
(3.56)
1 C 2C 2 1 2
2
+
kπ0 [f ]kL2 (0,ω;V0∗ ) .
+
2
κω
It follows from the periodic property that
ω
κ
C2 |∇v(0)|22 ≤
+ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
2
2ε
κ
(3.57)
1 C 2C 2 1 2
2
+
kπ0 [f ]kL2 (0,ω;V0∗ ) .
+
2
κω
Choosing s = 0 in (3.54), by (3.57) we obtain
Z
Z
1 t 0
κ
ε t 0
|v (τ )|2V0∗ dτ +
|v (τ )|22 dτ + |∇v(t)|22
2 0
2 0
2
ω C2 (3.58)
≤
+ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
ε
κ
C 2C 2 + 1 + 1 2 kπ0 [f ]k2L2 (0,ω;V0∗ ) .
κω
Dropping the first two terms on the left-hand side of (3.58) and from (3.11), we
obtain
ω C2 κ
|∇v(t)|22 ≤
+ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 )
2
ε
κ
C 2C 2 + C12 ω + 1 2 kf k2L∞ (0,ω;H) ,
κ
which is (3.41).
Now, multiplying the equation of (3.38) by −∆N v and integrating by parts, we
have
1 d 2 ε d
|v| +
|∇v|22 + κ|∆N v|22
2 dt 2 2 dt
≤ π0 a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ )) 2 |∆N v|2
+ |π0 [f ]|V0∗ |∆N v|V0∗
2
1 ≤ π0 a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ )) 2
κ
C2
κ
+ 1 |π0 [f ]|2V0∗ + |∆N v|22 .
κ
2
18
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
After a proper arrangement and with the aid of (3.48), we obtain
1 d 2 ε d
κ
|v| +
|∇v|22 + |∆N v|22
2 dt 2 2 dt
2
(3.59)
C12
1 3 2
2
2
2
2
2
≤
3A3 (a3 + a2 + a1 ) + |Ω|(a2 + 2a1 ) +
|π0 [f ]|V0∗ .
κ
κ
Integrating (3.59) on (0, ω), we have
Z
κ ω
ω 3 2
|∆N v(τ )|22 dx ≤
3A3 (a3 + a22 + a21 ) + |Ω|(a22 + 2a21 )
2 0
κ
Z
C2 ω
|π0 [f (τ )]|2V0∗ dτ,
+ 1
κ 0
i.e.,
Z ω
i
2ω h
|∆N v(τ )|22 dx ≤ 2 3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 ) + C14 kf k2L∞ (0,ω;H) .
κ
0
Therefore, we obtain (3.42). The proof is complete.
Now, we prove Theorem 2.3 by the Schauder fixed point theorem.
Proof of Theorem 2.3. Based on Lemma 3.5, we define a set
Y2 := {v̄ ∈ Y1 : sup κ|∇v̄|22 + ε|v̄|2W 1,2 (0,ω;H0 ) + |v̄|2W 1,2 (0,ω;V0∗ ) ≤ M },
t∈[0,w]
where
h
i
2C14 C22
2ωC14 C24
2
2
+
(ε
+
C
)
M : = 3ωC12 +
1 kf kL∞ (0,ω;H)
κ
κ2
h 3ω 2C 2
i
2ωC24
2
2
3A33 (a23 + a22 + a21 ) + |Ω|(a22 + 2a21 ) .
+
+
+
(ε
+
C
)
1
ε
κ
κ2
We can see that Y2 is a non-empty compact convex subset of L2 (0, ω; H0 ). Since
for any τ ∈ [0, ω], we have
Z
Z
2
1 f (x, τ )dx ≤ kf k2L∞ (0,ω;H) .
|π0 [f (τ )]|2H0 =
|f (x, τ )|2 dx −
|Ω| Ω
Ω
Thus, it follows from (3.48) and (H1) that
−π0 [a3 (v̄(τ ) + m(τ ))3 + a2 (v̄(τ ) + m(τ ))2 + a1 (v̄(τ ) + m(τ ))] + π0 [f ] ∈ L∞ (0, ω; H0 ).
As a result, by Theorem 3.1, there exists a unique solution vε for each v̄ ∈ Y2 . From
Lemma 3.5, it is clear that vε ∈ Y2 . Hence, the mapping S defined by S(v̄) = vε
maps Y2 into itself.
Next, we show that vε is a solution of (3.38) and that S is continuous in Y2 with
respect to the topology of L2 (0, ω; H0 ). Let {v̄n } be any convergent sequence in Y2
with respect to the topology of L2 (0, ω; H0 ). We denote the limit of {v̄n } by v̄. Let
{vn } be the sequence of solutions corresponding to {v̄n }. It follows from lemma
3.5 that vn (n = 1, 2, . . .) satisfy (3.39) and (3.40). Thus, we can find a vε and a
subsequence of {vn } which is denoted by {vnk } such that
vnk → vε
weakly in W 1,2 (0, ω; H0 ).
(3.60)
in L2 (0, ω; H0 ),
(3.61)
Therefore, by (3.41),
vnk → vε
vn0 k
→
vε0
2
weakly in L (0, ω; H0 ).
(3.62)
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MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
19
Since W 1,2 (0, ω; H0 ) ∩ L∞ (0, ω; V0 ) ,→ C([0, ω]; H0 ) [15, Section 8, Corollary 5] and
the embedding is compact, it is clear that
vnk → vε
in C([0, ω]; H0 ).
(3.63)
Thus,
vnk (0) → vε (0),
vnk (ω) → vε (ω)
in H0 ,
which implies vε satisfies the periodic condition
vε (0) = vε (ω).
(3.64)
Since F0−1 is linear and selfadjoint, it holds for any η ∈ D(∆N ) that
Z ω
Z ω
(F0−1 vn0 k (τ ) − F0−1 vε0 (τ ), η)0 dτ =
(F0−1 (vn0 k (τ ) − vε0 (τ )), η)0 dτ
0
0
Z ω
=
(vn0 k (τ ) − vε0 (τ ), F0−1 η)0 dτ.
(3.65)
0
Therefore, from (3.62), we know that
F0−1 vn0 k → F0−1 vε0
weakly in L2 (0, ω; H0 ) as k → ∞.
Also, for any η ∈ D(∆N ), we have
Z ω
Z
(−∆N vnk − (−∆N vε ), η)0 dτ = 0
ω
0
(3.66)
(vnk − vε , −∆N η)0 dτ ≤ kvnk − vε kL2 (0,ω;H0 ) |∆N η|2 ω 1/2 .
In view of (3.61), we obtain
− ∆N vnk → −∆N vε
weakly in L2 (0, ω; H0 ) as k → ∞.
(3.67)
For any v(τ ) ∈ L2 (0, ω; H0 ), let ψ(v(τ )) = π0 [a3 (v(τ ) + m(τ ))3 ]. Then for any
v1 (τ ), v2 (τ ) ∈ L2 (0, ω; H0 ),
Z ω
(ψ(v1 (τ )) − ψ(v2 (τ )), v1 (τ ) − v2 (τ ))0 dτ
0
Z ωZ
i2
nh
1
2
(3.68)
= a3
[v1 (τ ) − v2 (τ )]
(v1 (τ ) + m(τ )) + (v2 (τ ) + m(τ ))
2
0
Ω
o
3
+ [v2 (τ ) + m(τ )]2 dx dτ ≥ 0.
4
Therefore, ψ is a monotone operator in L2 (0, ω; H0 ). Since a3 (v(τ ) + m(τ ))3 is
continuous with respect to v(τ ), it is easy to prove that ψ(·) is hemicontinuous [1,
Chapter II, Definition 1.3] in L2 (0, ω; H0 ). In addition, it is clear to see that ψ is
everywhere defined in L2 (0, ω; H0 ). Thus, by [1, Chapter II, Theorem 1.3], ψ(·) is
maximal monotone in L2 (0, ω; H0 ).
It follows from the definition of Y2 and (3.43) that
kπ0 [a3 (v̄nk + m(τ ))3 ]k2L2 (0,ω;H0 ) ≤ a23 A33 ω.
Thus, there exist a γ(τ ) ∈ L2 (0, ω; H0 ) and a subsequence of {v̄nk } which is still
denoted by {v̄nk } such that
π0 [a3 (v̄nk (τ ) + m(τ ))3 ] = ψ(v̄nk (τ )) → γ(τ )
weakly in L2 (0, ω; H0 )
(3.69)
20
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
as k → ∞. It follows from (3.69), the maximal monotonicity of ψ and the [17,
Theorem A], we obtain ψ(v̄) = γ, i.e.,
π0 [a3 (v̄nk + m(τ ))3 ] → π0 [a3 (v̄ + m(τ ))3 ]
weakly in L2 (0, ω; H0 ).
(3.70)
From [15, Section 8, Corollary 5], we know that W 1,2 (0, ω; H0 ) ∩ L∞ (0, ω; V0 ) ,→
C([0, ω]; L4 (Ω) ∩ H0 ) and the embedding is compact. Thus,
vnk → vε
in C([0, ω]; L4 (Ω) ∩ H0 ).
(3.71)
Since
Z ωZ
2
a22
(v̄nk (τ ) + m(τ ))2 − (v̄(τ ) + m(τ ))2 dx dτ
0
Z Ω
ωZ
2
= a2
(v̄nk (τ ) − v̄(τ ))2 (v̄nk (τ ) + v̄(τ ) + 2m(τ ))2 dx dτ
0
Ω
Z ω hZ
i1/2 h Z
i1/2
2
4
≤ a2
(v̄nk (τ ) − v̄(τ )) dx
(v̄nk (τ ) + v̄(τ ) + 2m(τ ))4 dx
dτ
0
Ω
ZΩω h Z
i1/2
dτ,
≤ a22 kv̄nk (τ ) − v̄(τ ))k2C([0,ω];L4 (Ω)∩H0 )
(v̄nk (τ ) + v̄(τ ) + 2m(τ ))4 dx
0
Ω
where the last second inequality sign follows from the Hölder inequality. From
(3.71), we see that
a2 (v̄nk (τ ) + m(τ ))2 → a2 (v̄(τ ) + m(τ ))2
in L2 (0, ω; H0 ) as k → ∞.
Then, it is easy to prove that
π0 [a2 (v̄nk (τ ) + m(τ ))2 ] → π0 [a2 (v̄(τ ) + m(τ ))2 ]
in L2 (0, ω; H0 )
(3.72)
as k → ∞. Similarly, (3.63) implies
π0 [a1 (v̄nk (τ ) + m(τ ))] → π0 [a1 (v̄(τ ) + m(τ ))]
in L2 (0, ω; H0 )
(3.73)
as k → ∞.
Consequently, with the help of (3.62), (3.66), (3.67), (3.70), (3.72) and (3.73),
for any η ∈ D(∆N ), we take k → ∞ on both sides of the equality
Z ω
Z ω
Z ω
(F0−1 vn0 k (τ ), η)0 dτ +
(εvn0 k (τ ), η)0 dτ +
(−κ∆N vnk (τ ), η)0 dτ
0
0
0
Z ω
Z ω
=
(π0 [g(v̄nk (τ ) + m(τ ))], η)0 dτ +
(F0−1 f, η)0 dτ,
0
and then
Z
0
ω
Z ω
Z ω
(F0−1 vε0 (τ ), η)0 dτ +
(εvε0 (τ ), η)0 dτ +
(−κ∆N vε (τ ), η)0 dτ
0
0
0
Z ω
Z ω
=
(π0 [g(v̄(τ ) + m(τ ))], η)0 dτ +
(F0−1 f, η)0 dτ.
0
0
This implies that vε = S(v̄) is a unique solution of (2.9). As a result, from (3.61),
S is continuous in Y2 with respect to the topology of L2 (0, ω; H0 ). By the Schauder
fixed theorem, we can see that S has at least one fixed point in Y2 . The proof is
complete.
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MULTI-DIMENSIONAL CAHN-HILLIARD EQUATION
21
4. Proof of main results
In this section, based on Lemma 3.4, we prove Theorem 2.2 by taking the limit
as ε → 0.
Proof of Theorem 2.2. By Lemma 3.4, we can see that the constants on the righthand side of (3.7)-(3.10) are independent of ε. Thus, it follows from (3.7) and (3.8)
that
vε ∈ W 1,2 (0, ω; V0∗ ) ∩ L∞ (0, ω; V0 ),
(4.1)
1,2
∗
∞
and there exists a v ∈ W (0, ω; V0 ) ∩ L (0, ω; V0 ) such that
weakly in W 1,2 (0, ω; V0∗ ) as ε → 0,
vε → v
vε → v
∞
weakly star in L (0, ω; V0 ) as ε → 0,
2
vε → v
vε0
→v
0
in L (0, ω; H0 ) as ε → 0,
2
weakly in L
(0, ω; V0∗ )
as ε → 0.
(4.2)
(4.3)
(4.4)
(4.5)
Since W 1,2 (0, ω; V0∗ ) ∩ L∞ (0, ω; V0 ) ,→ C([0, ω]; H0 ) [15, Section 8, Corollary 5] and
the embedding is compact, it follows from (4.2) and (4.3) that
vε → v
in C([0, ω]; H0 ) as ε → 0.
Thus,
vε (0) → v(0),
vε (ω) → v(ω)
in H0 ,
which implies
v(0) = v(ω).
(4.6)
Similarly as (3.65), for any η ∈ D(∆N ), we have
Z ω
Z ω
−1 0
−1 0
hF0 vε (τ ) − F0 v (τ ), ηiV0 ,V0∗ dτ =
hvε0 (τ ) − v 0 (τ ), F0−1 ηiV0∗ ,V0 dτ.
0
0
By (4.5), we obtain
F0−1 vε0 → F0−1 v 0
weakly in L2 (0, ω; V0 ) as ε → 0.
Furthermore, for any η ∈ D(∆N ), we have
Z ω
Z
(−∆N vε (τ ) − (−∆N v(τ )), η)0 dτ =
0
(4.7)
ω
(vε (τ ) − v(τ ), −∆N η)0 dτ
0
≤ kvε − vkL2 (0,ω;H0 ) |∆N η|2 ω 1/2 .
It follows from (4.4) that
− ∆N vε → −∆N v
weakly in L2 (0, ω; H0 ) as ε → 0.
(4.8)
From (3.10) and the similar arguments as (3.70), we obtain
π0 [a3 (vε (τ ) + m(τ ))3 ] → π0 [a3 (v(τ ) + m(τ ))3 ]
weakly in L2 (0, ω; H0 )
(4.9)
as ε → 0.
By [15, Section 8, Corollary 5], we see that both W 1,2 (0, ω; V0∗ )∩L∞ (0, ω; V0 ) ,→
C([0, ω]; L4 (Ω)∩H0 ) and W 1,2 (0, ω; V0∗ )∩L∞ (0, ω; V0 ) ,→ C([0, ω]; H0 ) are compact.
Therefore, being similar as (3.72) and (3.73), we have
π0 [a2 (vε (τ ) + m(τ ))2 ] → π0 [a2 (v(τ ) + m(τ ))2 ]
in L2 (0, ω; H0 ) as k → ∞ (4.10)
and
π0 [a1 (vε (τ ) + m(τ ))] → π0 [a1 (v(τ ) + m(τ ))]
in L2 (0, ω; H0 ) as k → ∞. (4.11)
22
J. LIU, Y. WANG, J. ZHENG
EJDE-2016/42
With the help of (4.7)-(4.11), for any η ∈ D(∆N ), we take ε → 0 on both sides of
the equation
Z ω
hvε0 (τ ), ηiV0∗ ,V0 dτ
ε
0
Z ω
F0−1 f (τ ) − π0 [g(vε (τ ) + m(τ ))] + κ∆N vε (τ ), η dτ
=
0
0
Z ω
−
hF0−1 vε0 (τ ), ηiV0 ,V0∗ dτ,
0
and then get that for any η ∈ D(∆N ),
Z ω
0=
(F0−1 f (τ ) − π0 [g(v(τ ) + m(τ ))] + κ∆N v(τ ), η)0 dτ
0
Z ω
hF0−1 v 0 (τ ), ηiV0 ,V0∗ dτ
−
0
holds, which together with (4.6) implies that v is a solution of the problem
F0−1 v 0 (τ ) − κ∆N v(τ ) + π0 [g(v(τ ) + m(τ ))] = F0−1 f (τ ),
0 < τ < ω,
v(0) = v(ω).
As a result, from the equivalence of (1.2) and (2.8), we know that u(x, τ ) = v(x, τ )+
m(τ ) is the solution of (1.2)-(1.4).
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Ji Liu (corresponding author)
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081,
China
E-mail address: cau lj@126.com, phone 18811789567
Yifu Wang
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081,
China
E-mail address: wangyifu@bit.edu.cn
Jiashan Zheng
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081,
China
E-mail address: zhengjiashan2008@163.com