Optimal control problem for a generalized sixth order Cahn-Hilliard type equation with nonlinear diffusion

R E S E A R C H

Open Access

Optimal control problem for a generalized

sixth order Cahn-Hilliard type equation with

nonlinear diffusion

Gongcao Shi

_{, Changchun Liu}

_{and Zhao Wang}

*_{Correspondence: [email protected]} 1_{Department of Mathematics, Jilin}

University, Changchun, 130012, China

Full list of author information is available at the end of the article

Abstract

In this paper, we study the initial-boundary-value problem for a generalized sixth order Cahn-Hilliard type equation, which describes the separation properties of oil-water mixtures when a substance enforcing the mixing of the phases is added. The optimal control under boundary condition is given and the existence of optimal solution is proved.

MSC: Primary 49J20; secondary 35K35; 35K55

Keywords: sixth order Cahn-Hilliard; existence; optimal control

1 Introduction

We consider the equation

ut=D

γDu–a(u)Du–a _(u)

 |Du|

_{+f(u) +ku}

t–γDut

, (.)

in×(,T), where= (, ),γ > ,k> , andγ>  with the initial and boundary

conditions

u(x, ) =u, in, (.)

u(x,t) =Du(x,t) =Du(x,t) = , on∂. (.)

The functionf(u) stands for the derivative of a potentialF(u) withF(u),a(u) approxi-mated, respectively, by a sixth and a second order polynomial

F(u) =

u



f(s)ds= (u+ )u+h

(u– ), (.)

a(u) =au+a, (.)

wherea> .

The free energy functional proposed by Gompperet al.[–] has the form

ψ(u) =

ϕ(u,∇u,u)dx,

with the density given by

ϕ(u,∇u,u) =f(u) +

a(u)|∇u|

₊

γ(u)

_.

Hereuis the scalar order parameter, which is proportional to the local difference between oil and water concentrations. The properties of the amphiphile and its concentration enter model (.) implicitly via (.) and (.).F(u) has three minima atu= –,u= , andu= , which describe the oil, water and disordered microemulsion phases. In [–], the coeffi-cienta(u) is approximated by the quadratic function (.) with constantsaof arbitrary

sign andapositive.

Like in the classical Cahn-Hilliard the theory the order parameteruis a conserved quan-tity. Thus it satisfies the conservation law

ut+∇j= , (.)

with the mass fluxjgiven by the constitutive equation

–j= ∂D

∂∇μ=M∇μ, (.)

andμrepresenting the chemical potential

μ=δψ

δu + δD δut

, (.)

whereD≥, the dissipation potential, has the form

D(ut,∇ut,∇μ) =

 k(ut)

₊

γ|∇ut|

₊

M|∇μ|

_{, (.)}

andMis the mobility,k,γare the viscosity coefficients corresponding to the rate of the

order parameter and its spatial gradient.

The first variation δψ_δu is defined by the condition that

d dλ

ψ(u+λζ,∇u+λ∇ζ,u+λζ)dx|λ==:

δψ

δuζdx (.)

most hold for all test functionsζ ∈C∞_(). In the case of free energy this leads to the following expressions:

δψ

δu =f(u) –a(u)u– a(u)

 |∇u|

_+γ _{u, (.)}

δD δut

=kut–γut. (.)

From the above discussions we know that

μ=f(u) –a(u)u–a _(u)

 |∇u|

_+γ _u+ku

Combining (.)-(.) we get the following conserved evolution system:

ut–∇(M∇μ) = ,

μ=f(u) –a(u)u–a _(u)

 |∇u|

_+γ _u+ku

t–γut,

where⊂R_{is a bounded domain with the boundary∂, occupied by the ternary}

mix-ture, and (,T) is the time interval. We endow this system with the initial and boundary condition (.) and (.), in this paper we consider the one-dimensional case withM= .

Schimperna and Pawłow [] studied (.) whenγ=  with logarithmic potential

F(r) = ( –r)log( –r) + ( +r)log( +r) – σ r

_{, σ> .}

They investigated the behavior of the solutions to the sixth order system as the parameter

γ tended to , the uniqueness and regularization properties of the solutions have been discussed.

Pawłow and Zaja¸czkowski [] proved that the problem (.)-(.) withk=γ=  under

consideration is well posed in the sense that it admits a unique global smooth solution which depends continuously on the initial datum.

In past decades, the optimal control of distributed parameter system had received much attention in the academic field. A wide spectrum of problems in applications can be solved by methods of optimal control, such as chemical engineering and vehicle dynamics. Mod-ern optimal control theories and applied models are not only represented by ODEs, but also by PDEs. Kunisch and Volkwein solved open-loop and closed-loop optimal control problems for the Burgers equation [], Armaou and Christofides studied the feedback con-trol of Kuramto-Sivashing equation [].

Recently, many authors studied the optimal control problem for the pseudo-parabolic equation, such as Tianet al.[–], Zhao and Liu [].

In this paper, we consider the optimal control problem for the following equation:

u–kDu+γDu

t–

γ γ

Du–kDu+γDu

+ γ

γ Du

+D

a(u) –γk

γ

Du+a

_(u)  |Du|



=Df(u) +B∗ω, (.)

with (.)-(.).

Wheny=u–kD_u+γ

Du, we take the distributed optimal control problem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

minJ(y,ω) =_Cy–z

S+

δ

ωL_(,T;Q ), s.t.yt–γγD

_y+ γ γD

_u

+D_{((a(u) –}γk

γ)D

_u+a(u)

 |Du|) –Df(u) =B∗ω, y(x, ) =y=u–kDu(x, ) +γDu(x, ),

u(x,t) =D_{u(x,t) =D}_{u(x,t) = .}

(.)

For fixedT> , we set= (, ) andQ=×(,T). LetQ⊂Qbe an open set with

LetV=H

(, ),H=L(, );V∗=H–(, ), andH∗=L(, ) are dual spaces,

respec-tively, and we have

V→H=H∗→V∗.

The extension operatorB∗∈L(L_(,T;Q

),L(,T;V∗)) is given by

B∗q=

⎧ ⎨ ⎩

q, q∈Q,

, q∈Q/Q.

(.)

The spaceW(,T;V) is defined by

W(,T;V) =y,y∈L(,T;V),yt∈L

,T;V∗,

which is a Hilbert space endowed with the common inner product.

The plan of the paper is as follows. In Section , we prove the existence of the weak solution in a special space. The optimal control is discussed in Section , and the existence of an optimal solution is proved.

2 Existence of weak solution

Consider the following the sixth order Cahn-Hilliard type equation:

u–kDu+γDu

t–

γ γ

Du–kDu+γDu

+ γ

γ Du

+D

a(u) –γk

γ

Du+a

_(u)  |Du|



=Df(u) +B∗ω, (.)

under the initial value

u(x, ) =u,

and boundary condition

u(x,t) =Du(x,t) =Du(x,t) = ,

whereB∗ω∈L(,T;V∗) and the control itemω∈L(,T;Q).

Lety=u–kD_u+γ

Du; the above problem is rewritten as

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yt–_γγDy+_γγ Du+D((a(u) –γ_γk)Du+a

_(u)

 |Du|) –Df(u) =B∗ω, y(x, ) =y=u–kDu+γDu,

u(x,t) =D_{u(x,t) =D}_{u(x,t) = ,}

(.)

with (.)-(.).

Now, we give the definition of the weak solution to the problem (.) in the space

Definition . A functiony(x,t)∈W(,T;V) is called a weak solution to problem (.), if

d dt(y,φ) +

γ γ

(Dy,Dφ) – γ

γ

(Du,Dφ)

–

D

a(u) –γk

γ

Du+a

_(u)  |Du|

_,Dφ

+Df(u),Dφ=B∗ω,φ

V∗,V,

for allφ∈V, a.e.t∈[,T] andy∈Hare valid.

Theorem . The problem(.)admits a weak solution y(x,t)∈W(,T;V)in the interval

[,T],if B∗ω∈L_(,T;V∗_{)and y}

∈H.

Proof Employ the standard Galerkin method.

The differential operatorA= –∂_xis a linear unbounded self-adjoint operator inHwith

D(A) dense inH, whereHis a Hilbert space with a scalar product (·,·) and norm · . There exists an orthogonal basis{ψi}ofH. Let{ψi}∞i= be the eigenfunctions of the

op-eratorA= –∂

x with

Aψj=λjψj,  <λ≤λ≤ · · ·, asj→ ∞.

Forn∈N, we define the discrete ansatz space by

Vn=span{ψ,ψ, . . . ,ψn} ⊂V.

Setyn(t) =yn(x,t) =

n

i=yni(t)ψi(x) and requireyn(,·)→yinHholds true.

To prove the existence of a unique weak solution to the problem (.), we are going to analyze the limiting behavior of sequences of smooth functions{yn}and{un}.

Performing the Galerkin procedure for the problem (.), we have

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yn,t–_γγDyn+_γγDun

+D((a(un) –γγk)D

_u

n+a

_(u

n)  |Dun|

_{) –D}_f(u

n) =B∗ω,

yn(x, ) =yn,=un,–kDun(x, ) +γDun(x, ),

un(x,t) =Dun(x,t) =Dun(x,t) = .

(.)

According to ODE theory, there is a unique solution to (.) in the interval [,tn]. We

should show that the solution is uniformly bounded whentn→T.

As a first step, multiplying the first equation of (.) by

μn=γDun–a(un)Dun–

a(un)

 |Dun|

_+f(u

n) +kun,t–γDun,t,

and integrating with respect tox, we obtain

d

dtE(un) +Dμn _+ku

n,t+γDun,t=

B∗ω,μn

where

E(un) =





γ

D

_u

n+

a(un)

 |Dun|

_+F(u

n)

dx (.)

and

F(un) =

u_n+ (h– )un+ ( – h)un+h

. (.)

Applying a simple calculation, we have

F(un)≥Cun–C, (.)

whereC>  andC≥.

SinceB∗ω∈L_(,T;V∗_{) is a control item, we assume}

B∗ω_V∗≤M. (.)

Taking into account (.), (.), (.), (.), and integrating (.) with respect to time from  tot, we know





γ

D

_u

n



+a u



n|Dun|+Cun

dx

+

t



Dμndt+k

t



un,tdt+γ

t



Dun,tdt

≤





|a|

 |Dun|

_dx+E(u

n,) +C+

t



B∗ω,μn

V∗,Vdt

≤ε





|a|

 D

_u

n



dx+C(ε)



 u_ndx

+E(un,) +C+

t



B∗ω_V∗μnVdt

≤ε





|a|

 D

_u

n



dx+C(ε)ε



 u

ndx+C(ε)

+E(un,) +C+C(ε)

t



B∗ω_V∗dt+ε

t



Dμndt

=ε





|a|

 D

_u

n



dx+C(ε)ε





u_ndx+C(ε)

+E(un,) +C+C(ε)

t



B∗ω_V∗dt+ε

t



un,tdt.

Choosingε,ε, andεsufficiently small, from the above inequality and the Poincaré

in-equality, we have





Dun



dx≤C, (.)









u_ndx≤C, (.)

QT

|un,t|dx dt≤C. (.)

From (.), we know





u_ndx≤C. (.)

By virtue of (.), (.), and (.), we obtain

unH≤C. (.)

By Sobolev’s imbedding theorem it follows from (.) that

unL∞≤C, DunL∞≤C. (.)

As a second step, multiplying (.) byD_u

nand integrating with respect tox, we obtain

 

d dt





|Dun|dx+k





Dundx+γ





Dundx

+γ





Dundx

= –





Df(un)Dundx+





a(un)DunDundx

+



 a(un)

 |Dun|

_D_u

ndx–

B∗ω,Dun

V∗,V. (.)

From a simple calculation, we have

a(un) = aun, (.)

Df(un) =f(un)Dun+f(un)(Dun), (.)

where

f(un) =

u_n+ (h– )un+ ( – h)

≥–C, C> , (.)

f(u) = u_n+ (h– )un. (.)

Thus it follows from (.), (.), and (.) that

 

d dt





(Dun)dx+k





_D_u

n

 dx+γ





_D_u

n

 dx

+γ





_D_u

n

 dx

≤–





f(un)Dun+f(un)|Dun|

Dundx

+





aun+a

DunDundx

+



 a(un)

 |Dun|

_D_u

≤C





_D_u

n



dx+CunL∞+unL∞

DunL∞





DunDundx

+|a|unL∞





DunDundx+|a|





DunDundx

+|a|





un|Dun|Dundx+C(ε)B∗ω_V∗+ε





Dundx

≤γ 





Dun



dx+C, (.)

whereεis sufficiently small.

By the Gronwall inequality, (.) implies

QT _D_u

n



dx dt≤C, (.)

Dundx≤C. (.)

As a third step, multiplying (.) byD_u

nand integrating with respect tox, we obtain

d dt  

_D_u

n

 dx+δ

_D_u

n

 dx+γ

_D_u

n  dx +γ

_D_u

n  dx = –

f(un)DunDundx

–

Da(un)DunDundx–

D

a(un)

 |Dun|



Dundx

= –

f(un)DunDundx

+

Da(un)DunDundx+

D

a(un)

 |Dun|



Dundx

=I+I+I. (.)

On account of (.) and (.), we know

I≤C

C(ε)

|Dun|dx+ε

_D_u

n

 dx

.

On the other hand, by the Nirenberg inequality, we have

_D_u

n_∞≤Dun

 _Du

n



_{. (.)}

Hence, by the Hölder and Young inequalities, we obtain

|I|=

Daun+a

Dun

Dundx

≤aun∞Dun∞

Dun

 dx

 

Dun

 dx

+CDu_∞un∞

u_ndx 



Dundx

 

≤C(ε)C

Dun

 dx.

Similarly,

|I|=

D

a(un)

 |Dun|



Dundx

≤aDun∞

|Dun|dx

 

Dundx

 

+ aun∞Dun∞

Dun

 dx

 

Dun

 dx

 

≤C(ε)C

Dun

 dx+C,

theεis sufficiently small.

Therefore, by the Gronwall inequality, we have

sup

<t<T

Dun



dx≤C, (.)

QT Dun



dx dt≤C. (.)

From a simple calculation, we have

ynV =un–kDun+γDun_V

≤Cun+Dun+Dun+Dun+Dun+Dun.

From (.), (.), (.), and (.), we obtain

ynL_(,T;V)≤C. (.)

As a fourth step, from (.), (.), (.), and the Sobolev embedding theorem, we have

yn,tV∗≤B∗ω_V∗+Dun+

D

a(un) –

γk γ

Dun+

a(un)

 |Dun|



+Df(un)

≤B∗ω_V∗+Dun+CunL∞Dun+CunL∞Dun

+CunL∞Dun+C

≤CDun+C.

Then

Thus, we have:

(i) For everyt∈[,T], the sequence{yn}n∈Nis bounded inL(,T;H)as well as in

L_{(,T;V), which is independent of the dimension of the ansatz spacen.}

(ii) For everyt∈[,T], the sequence{yn,t}n∈Nis bounded inL(,T;V∗), which is

independent of the dimension of the ansatz spacen.

Hence, we get{yn,t}n∈N⊂W(,T;V), and{yn,t}n∈N weak inW(,T;V), weak star in

L∞(,T;H) and strong inL_{(,T;H) to a function y(x,t)∈W(,T;V). Obviously, the}

uniqueness of the solution is easy to obtain []. We omit it here.

To ensure that the norm of weak solution in the spaceW(,T;V) can be controlled by the initial value and the control item, we need the following theorem.

Theorem . If B∗ω∈L_(,T;V∗_{)and y}

∈H,then there exist constants C> and C>

,such that

y_W(,T;V)≤C

yH+ωL_(,T;Q )

+C. (.)

Proof Similar to the proof of Theorem ., we obtain

u ≤C, Du ≤C, uV≤C, Du≤C. (.)

Multiplying the equation byyand integrating the equation with respect tox, we obtain

 

d dty



H+

γ γ

Dy_H

= γ

γ





DyDu dx+



 D

a(u) –γk

γ

Du+a

_(u)  |Du|



Dy dx

–





DyDf(u)dx+B∗ω,y_V∗_,V. (.)

From the Hölder and Young inequalities, we have

γ γ





DyDu dx≤C(ε)Du+εDy. (.)

From (.), we have



 D

a(u) –γk

γ

Du+a

_(u)  |Du|



Dy dx

≤u_L∞+CDyDu+Cu_L∞DuL∞Du_L∞Dy +DuDy ≤εDy+C

and

–





DyDf(u)dx≤C





Note that

B∗ω,y_V∗_,V≤B∗ω_V∗yV. (.)

From (.)-(.), we have

 

d dty



H+

γ γ

Dy_H≤εDy_H+CB∗ω_V∗+C. (.)

Integrating the above inequality with respect totyields

y_H≤ yH+CB∗ω



L_(,T;V∗₎+C. (.)

By (.), (.), and (.), we deduce that

ytV∗≤B∗ω



V∗+

γ γ

y_V+ γ

γ Du

+D

a(u)Du+a _(u)

 |Du|

_+Df(u)

≤B∗ω_V∗+CyV+C

≤ yH+CB∗ω



L_(,T;V∗₎+C. (.)

From (.) and (.), we have

yW(,T;V) =yL_(,T;V)+y_tL_(,T;V∗₎ ≤C

yH+ωL_(,T;Q)

+C.

The proof is completed.

3 Optimal problem

In this section, we will study the distributed optimal control and the existence of the op-timal solution is obtained based on Lions’ theory.

We study the following problem whenω∈L_(,T;Q ),

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

minJ(y,ω) =_Cy–z

S+

δ

ωL_(,T;Q ), s.t.yt–γγD

_y+ γ γD

_u+D_{((a(u) –}γk

γ)D

_u+a(u)  |Du|

_{) –D}_{f(u) =B}∗_ω,

y(x, ) =y=u–kDu(x, ) +γDu,

u(x,t) =Du(x,t) =Du(x,t) = ,

wherey=u–kD_u+γ Du.

As we know that there exists a weak solutionyto (.), due tou= ( –k∂_x+γ∂x)–y,

we know that there exists a weak solutionuto (.). Let there be given an observation operatorC∈L(W(,T;V),S), in whichSis a real Hilbert space andCis continuous.

We choose a performance index of tracking type

J(y,ω) = 

Cy–z



S+

δ

ω



L_(,T;Q

), (.)

The optimal control problem as regards the further generalized sixth order Cahn-Hilliard equation is

minJ(y,ω), (.)

where (y,ω) satisfies the problem (.). LetX=W(,T;V)×L_(,T;Q

) andY=L(,T;V)×H.

We define an operatore=e(e,e) :X→Yby

e(y,ω) =ee(y,ω),e(y,ω)

,

where

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

e(y,ω) = (D)–(yt–γγ_Dy+ γ γ_Du

+D((a(u) –γ_γk )D

_u+a(u)  |Du|

_{) –D}_{f(u) –B}∗_ω),

e=y(x, ) –y,

andD_{is an operator fromH}_{(, ) toH}–_{(, ).}

Then (.) is rewritten as

minJ(y,ω) subject toe=e(y,ω) = .

Now, we have the following theorem.

Theorem . There exists an optimal control solution to the problem.

Proof Let (y,ω)∈Xsatisfy the equatione=e(y,ω) = . In view of (.), we have

J(y,ω)≥δ

ωL(,T;Q). From Theorem ., we have

yW(,T;V)→ ∞ yields ωL_(,T;Q)→ ∞.

Hence

J(y,ω)→+∞, wheny,ωX→ ∞. (.)

As the norm is weakly lowered semi-continuous [], we find thatJ is weakly lowered semi-continuous.

SinceJ(y,ω)≥ for all (y,ω)∈Xholds, there exists

η=infJ(y,ω)|(y,ω)∈Xsuch thate(y,ω) = ,

which means that there exists a minimizing sequence{(yn,ωn)}n∈NinXsuch that

η= lim

n→∞J

yn,ωn

and e=eyn,ωn

From (.), there exists an element (y∗,ω∗)∈Xsuch that

yny∗, y∈W(,T;V), (.)

ωn ω∗, ω∈L(,T;Q), (.)

whenn→ ∞. From (.), we have

lim

n→∞

T



yn(t) –y∗(t),φ(t)

V∗,Vdt= , ∀φ∈L

_(,T;V).

SinceW(,T;V) is compactly embedded intoL_(,T;L∞_{) and continuously embedded}

intoC(,T;H), we derive thatyn→y∗ strongly inL(,T;L∞) andyn→y∗ strongly in

C(,T;H), asn→ ∞. Then we also derive thatun→u∗,Dun→Du∗, Dun→Du∗,

Dun→Du∗,Dun→Du∗strongly inC(,T;H), asn→ ∞.

As the sequence{yn}n∈N converges weakly,ynW(,T;V) is bounded. Also, we see that

ynL_(,T;L∞₎is bounded based on the embedding theorem.

Since yn →y∗ strongly in L(,T;L∞), we derive that y∗L_(,T;L∞₎, u∗L_(,T;L∞₎, D_u∗

L_(,T;L∞₎andDu∗_L_(,T;L∞₎are bounded. Notice that

T







Df(un) –Df

u∗ψdx dt

=

T







Df(un) –f

u∗Dψdx dt

≤C

T







(un)Dun+ (un)Dun+Dun

–u∗Du∗–u∗Du∗–Du∗Dψdx dt

≤

T



unL∞Dun–Du∗_HDψHdt

+

T



(un)–

u∗_HDu∗_L∞DψHdt

+

T



_(un)

L∞Dun–Du

∗

HDψHdt

+

T



(un)–

u∗_HDu∗_L∞DψHdt

+

T



_Dun–Du∗_HDψHdt

≤ unC(,T;L∞)Dun–Du∗_L_(,T;H)DψL_(,T;H) +unC(,T;L∞)+u∗



C(,T;L∞)Du

∗

C(,T;L∞)un–u

∗

× Dψ_L_(,T;H)+Du_n–Du∗

C(,T;H)DψL(,T;H)

→, ∀ψ∈L(,T;V). (.)

As we know

_T_D

a(un)Dun+

a(un)

 |Dun|



–D

au∗Du∗+a _(u∗₎

 Du ∗

ψdx dt

=

T







Da(un)Dun–a

u∗Du∗ψdx dt

+ T    D a(un)

 |Dun|

_–a(u∗)

 Du ∗

ψdx dt

=|I+I|.

Note that

|I|=

T    D a(un) –

γk γ

Dun–

au∗–γk

γ

Du∗

ψdx dt

≤ T    D

a(un)+a–

γk γ

Dun

–

a

u∗+a–

γk γ

Du∗

Dψdx dt

=

T







Da(un)Dun–a

u∗Du∗Dψdx dt

+ T    D a–

γk γ

Dun–

a–

γk γ

Du∗

Dψdx dt

=I_+I_.

ForI

, we have

I_ =

T







aunDunDun– au∗Du∗Du∗+a(un)Dun

–a

u∗Du∗Dψdx dt

≤ |a|

T



unL∞DunL∞Dun–Du∗_HDψHdt

+ |a|

T



un–u∗_HDunL∞Du∗_L∞DψHdt

+ |a|

T



_u∗

L∞Dun–Du

∗

HD

_u∗

L∞DψHdt

+|a|

T



+|a|

T



u∗_L∞Dun–Du∗_HDψHdt

≤ |a|unC(,T;L∞)DunC(,T;L∞)Dun–Du∗_L(_,T;H)DψL_(,T;H) + |a|un–u∗_L_(,T;H)DunC(,T;L∞)Du∗_C(,T;L∞₎DψL_(,T;H) + |a|u∗_C(,T;L∞₎Dun–Du∗_L_(,T;H)Du∗_C(,T;L∞₎DψL_(,T;H) +|a|unC(,T;L∞)Dun_L_(,T;L∞₎un–u∗_C(,T;H)DψL_(,T;H) +u∗_C(,T;L∞₎Dun_L_(,T;L∞₎un–u∗_C(,T;H)DψL_(,T;H) +|a|u∗



C(,T;L∞)D _u

n–Du∗_L_(,T;H)DψL_(,T;H) →, ∀ψ∈L(,T;V).

Also we have

I_ =

T





 D

a–

γk γ

Dun–

a–

γk γ

Du∗

Dψdx dt

≤

T



a–

γk γ

Dun–Du∗_HDψHdt

≤

a–

γk γ

Dun–Du∗_C(,T;H)DψL_(,T;H) →, ∀ψ∈L(,T;V).

Further, similar to (.), we have

I→, ∀ψ∈L(,T;V).

From (.), we have

_T_B∗ωn–B∗ω∗ψdx dt→, ∀ψ∈L(,T;V).

In view of the above discussion, we can conclude that

e

y∗,ω∗= , ∀n∈N.

Sincey∗∈W(,T;V), we havey∗()∈H. Fromyny∗inW(,T;V), we can infer that

yn()y∗(). Thus we obtain

yn() –y∗(),ψ

→, ∀ψ∈H,

which means thate(y∗,ω∗) = ,∀n∈N.

Hence, we can derive thate(y∗,ω∗) = ,∀n∈N.

In conclusion, there exists an optimal solution (y∗,ω∗) to the problem. We can infer that there exists an optimal solution (y∗,ω∗) to the viscous generalized Cahn-Hilliard equation

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Author details

1_{Department of Mathematics, Jilin University, Changchun, 130012, China.}2_{School of Basic Science, Changchun}

University of Technology, Changchun, 130012, China.

Acknowledgements

The authors would like to express their sincere thanks to the referee’s valuable suggestions for the revision and improvement of the manuscript. This work is supported by the National Science Foundation of China (No. J1310022).

Received: 19 November 2014 Accepted: 24 March 2015 References

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