Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 50, 1-16; http://www.math.u-szeged.hu/ejqtde/

Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 50, 1-16; http://www.math.u-szeged.hu/ejqtde/

ON A HIGHER ORDER TWO DIMENSIONAL

THERMOELASTIC SYSTEM COMBINING A LOCAL

AND NONLOCAL BOUNDARY CONDITIONS

SAIDMESLOUB

Abstrat. Due to theirimportane and numerousappliations,

evolutionmixed problemswithnonloalonstraintsin thebound-

ary onditions havebeenextensively studied during the two last

deades. Inthispaper,weonsideraninitialboundaryvalueprob-

lemforahigherorderthermoelastisystemarisingin linearther-

moelastiitywhihombinessomeDirihletandweightedintegral

boundary onditions. The studied systemmodelizes in general a

Kirho plate. We prove the well posedness of the given prob-

lem. Ourproofsaremainlybasedonsomeaprioriboundsinsome

Sobolevtypespaefuntionsandonsomedensityarguments.

1. Introdution

During the few last deades, many researhers have studied linear

and nonlinear systems of thermoelasti equations and many results

havebeenpublished. Mostof theseresultsweredealingwiththestudy

of existene, asymptoti behavior, regularity, ontrollability, propa-

gation of singularities and blow up of solutions. For example Assila

[1℄, has studied global existene and asymptoti behavior of solutions

for a purely linear multidimensional system of nonhomogeneous and

anisotropi thermoelastiity, assoiated with nonlinear boundary on-

ditions. Dafermos and Hsiao [4℄, Hrusa and Messaoudi [6℄, Munoz-

Rivera [15℄, Rake [16℄, Rake and Shibata [17℄ and Slemrod[19℄ have

studiedandobtained someresultsaboutthe existene, regularity,on-

trollabilityandlong-timebehaviorofsomesystemsofthermoelastiity.

Also Rake and Wang [18℄ have onsidered some linear and semilin-

ear Cauhy problems and desribed the propagation of singularities.

Mixedproblems forthermoelastisystemsbeomeveryhardtohandle

in ase of the presene of nonloal onstraints in the boundary (suh

Key words and phrases. Thermoelasti System, integral ondition, Energy In-

equality,Existene,Uniqueness,Weaksolution

2000MathematisSubjetClassiation: 35L70,35L20,35K,35R.

ondition). Thereare two types ofnonloalproblems, spatialnonloal

problems and time nonloalproblems. These type of mixed problems,

arise mainly when the data on the boundary annot be measured di-

retly,butitanbereplaedbyanonloalonditionsuhasanintegral

ondition. Physially, this kind of ondition an represent a mean, a

total energy ora total mass. Boundary value problems with nonloal

onstraintshave many importantappliationssuhasinunderground-

water ow, population dynamis, hemial diusion, thermoelastiity,

heatondution proesses,ertainbiologialproesses, nulearreator

dynamis,ontroltheory, medialsiene, biohemistry,and transmis-

sion theory. Nonloal problems were rst investigated by using the

method of separation of variables and the orresponding eigenvalues

and eigenfuntions were onsidered. Later on, other methods suh as

the funtional analysis method,the energeti method and the method

ofsingularintegralequations wereappliedtomixednonloalproblems

but with great diulties. Forsome nonloalmixed problems for par-

aboli and hyperboli equations the reader should refer to Mesloub

[7,8,9,10℄, Mesloub and Mesloub.F [12℄, Mesloub and Messaoudi [13℄

and Mesloub and Bouziani [11℄.

Motivated by the previous studies, we onsider the following ini-

tialboundaryvalue problemfor a fourth order two dimensional linear

thermoelastisystemwithDirihletandnonloalonstraintsofintegral

type:



 

 

 



 

 

 



L 1 u = ^∂ _∂t ^{2 2 u} + △ ² u − α△θ + c 1 u + c 2 u t = f(x

y

t)

⁽

x

y

t) ∈ Q, L 2 θ = β ^∂θ _∂t − η△θ + σθ + α△u t = g(x

y

t)

⁽

x

y

t) ∈ Q, u(x

y

0) = u _◦ (x, y), u _t (x

y

0) = u ₁ (x

y), θ(x

y

0) = θ _◦ (x

y),

u(0

y

t) = 0

u(a

y

t) = 0

0 < y < b

0 < t < T, u(x

0

t) = 0

u(x

b

t) = 0

0 < x < a

0 < t < T, R a

0 x ^k udx = 0

R b

0 y ^k udy = 0

^,

R a

0 x ^k θdx = 0

R b

0 y ^k θdy = 0

^,

k = 0, 1.

(1)

where

Q = Ω × [0

T ]

^with

Ω = (0

a) × (0

b)

T < ∞

a < ∞

^and

b < ∞

^.

The given data satisfy the onsisteny onditions

u _◦ (0

y) = u _◦ (a

y) = u _◦ (x

0) = u _◦ (x

b) = 0,

⁽²⁾

u ₁ (0

y) = u ₁ (a

y) = u ₁ (x

0) = u ₁ (x

b) = 0,

Z a 0

x ^k u _◦ dx = 0

Z b

0

y ^k u _◦ dy = 0

Z a

0

x ^k u ₁ dx = 0

⁽³⁾

Z b 0

y ^k u 1 dy = 0

Z a

0

x ^k θ _◦ dx = 0

Z b

0

y ^k θ _◦ dy = 0, k = 0, 1

^.

Problem

(1)

^{arises inlinear}thermoelastiity. It modelizesaKirho plate, where

u

^{is the} displaement

θ

^{is the thermal damping. The}

integralonditionsmaybeinterpretedastheaverageandthe weighted

average ofthe displaementand the thermaldamping and

α

β

η, σ, c ₁

^and

c ₂

^{are positiveonstants.}

Onthebasisofsomeaprioribounds,energyestimatesandsomeden-

sity arguments,weprovethe existene, uniqueness and the ontinuous

dependene of the solution on the data of the given problem

(1) .

^The

paperisorganizedas follows: Insetion 1,westart byanintrodution

about previous and related results onerning the subjet. In setion

2, we introdue some notations, funtion spaes and reformulation of

the studied problem. Setion 3 is devoted to the study of uniqueness

of the solution of the stated problem. In the fourth and last setion,

weestablishtheexisteneofthe weaksolutionofour problemand give

some remarks. At the end of the paper, we give a list of some used

referenes.

2. Notations, funtional frame, some auxiliary inequalities

and reformulation of the problem

Wedenoteby

L ² (Q)

^{theusualsquareintegrablefuntionsspae}

,

^and

by

W ₂ ^1,0 (Q)

^{the Sobolev spaehaving the inner produt}

(u

υ) _W ¹ , 0

(Q) = (u

υ) _L ² _(Q) + (u x

υ x ) _L ² _(Q) .

Let

B ₂ ^m (0, a)

^{(See [2,3℄) be the spae onstituted of funtions}

u ∈ L ² (0, a),

^if

m = 0

^{and of funtions}

u

^{suh that}

ℑ ^m _x u ∈ L ² (0, a),

^if

m ≥ 1,

^where

ℑ ^m _x V = 1 (m − 1)!

x

Z

0

(x−ξ) ^m−1 V (ξ, t)dξ =

x

Z

0 ξ 1

Z

0

...

ξ m− 1

Z

0

V (η, t)dηdξ _m−1 ...dξ ₁ ,

withinnerprodut

(u, v) B ^m 2 (0,a) =

a

Z

0

ℑ ^m _x u.ℑ ^m _x vdx,

^{andassoiatednorm}

kuk _B m

2 (0,a) = kℑ ^m _x uk _L ² _(0,a) ,

^for

m ≥ 1.

Wealsousethefuntionspaes

C(I, B ₂ ^m (Ω)), C (I, B ₂ ^1,x (Ω)), C(I, B ₂ ^1,y (Ω))

and

C(I, B ₂ ^1,x,y (Ω))

^{of ontinuous mappings from}

I = [0, T ]

^{onto the}

Hilbert spaes

B ₂ ^m (Ω) , B ₂ ^1,x (Ω) , B ₂ ^1,y (Ω),

^and

B ₂ ^1,x,y (Ω)

respetively, and with inner produts (respetively) given by

(u, v) B 2 ^m (Ω) = Z

Ω

ℑ ^m _x u.ℑ ^m _x vdxdy, (u, v) _B ^{1 ,x}

2 (Ω) =

Z

Ω

ℑ x u.ℑ x vdxdy,

(u, v) _B ^{1 ,y}

2 (Ω) =

Z

Ω

ℑ y u.ℑ y vdxdy, (u, v) _B ^{1 ,x,y}

2 (Ω) =

Z

Ω

ℑ xy u.ℑ xy vdydx.

The followinginequalitiesare used inour paper:

A) For every

u ∈ L ² (Λ)

⁽

Λ

^{is either}

(0, d)

^or

Ω

^{) and all}

m ∈ IN ^∗

^,

we have the inequalities

khk ² _B ^m

2 (0,d) ≤ d ²

2 khk ² _B ^{m− 1}

2 (0,d)

^(1*)

and

khk ² _B m 2 (0,d) ≤

d ² 2

m

. khk ² _L ² _(0,d) ,

^(2*)

kℑ x uk ² _L ² _(Ω) ≤ a ²

2 kuk ² _L ² _(Ω)

^(3*)

khk ² _B ^{1 ,x,y}

2 (Ω) = kℑ _xy hk ² _L ² _(Ω) ≤ (ab) ²

4 khk ² _L ² _(Ω) ,

^(4*)

B) Growall's Lemma [5, Lemma 7.1℄. If

f 1 (s)

f 2 (s)

^and

f 3 (s)

^are

nonnegativefuntions on

(0

T )

f 1 (s)

^and

f 2 (s)

^{integrablefuntions,}

and

f 3 (s)

^is nondereasing on (

0

T )

^{then if}

R s

0 f 1 (t) dt + f 2 (s) ≤ c R s

0 f ₂ (t) dt + f ₃ (s) ,

^then

R s

0 f ₁ (t) dt + f ₂ (s) ≤ exp (cs) .f ₃ (s)

^.

C)Cauhy

ε−

inequality: Forall

ε > 0

^{, and for arbitrary}

a

b

ⁱⁿ

R

we have

|ab| ≤ ε

2 |a| ² + 1 2ε |b| ² .

Let us now formulate problem

(1)

^{in its operator form. Problem}

(1)

^{an be viewed as the problem of solving the operator equation:}

JU = G,

^with

U = (u

θ)

JU = (J 1 u

J ₂ θ)

^and

G= (G 1

G ₂ ) =

({f

u _◦

u ₁ } , {g

θ _◦ }) ,

^where

J ₁ u = {L 1 u

ℓ ₁ u

ℓ ₂ u}

J ₂ θ = {L 2 θ

ℓ ₃ θ}

^.

The unbounded operator

A

^{is onsidered on the Banah spae}

B

^into

a Hilbert spae

H

^{with domain}

D(J)

^{dened by}

D (J) =

U = (u

θ) ∈ (L ² (Q)) ²

^{suh that}

u t

θ t

u tt

∂ ⁱ u

∂x ⁱ

∂ ⁱ u

∂y ⁱ

u tx

u _txx

u _ty

u _tyy ∈ L ² (Q) , i = 1.4

(4)

and the funtions

U = (u

θ)

^{satisfy boundary onditions in}

(1).

^Here

B

^{istheBanah spaeobtained byenlosing}

D (J )

^{withrespet tothe}

nite norm

kU k _B =

ku t (.

.

τ )k ² _C(I,B ^{1 ,x,y}

2 (Ω)) + ku(.

y

τ)k ² _C(I,B ^{1 ,x}

2 (Ω))

+ ku(x

.

τ)k ² _C(I,B ^{1 ,y}

2 (Ω)) + kθ(.

.

τ)k ² _C(I,B ^{1 ,x,y}

2 (Ω))

1/2

,

⁽⁵⁾

where

ℑ x u = Z x

0

u(ξ, y, t)dξ, ℑ xy u = Z x

0 y

Z

0

u(ξ, η, t)dηdξ.

Theelements

U = (u

θ) ∈ B

^{arethe setof ontinuous funtions}

u

^and

θ

^on

I = [0, T ],

^{wherefuntions}

u

^{have valuesin}

B ₂ ^1,x (Ω) , B ₂ ^1,y (Ω)

^and

havederivatives

u t

^{whihareontinuouson}

I

^{withvalues in}

B ₂ ^1,x,y (Ω)

and

θ

^{have values in}

B ₂ ^1,x,y (Ω) .

Let

H = H 1 ×H 2

^{betheHilbertspae}

{L ² (Q) × W ₂ ¹ (Ω) × L ² (Ω)}×

{L ² (Q) × L ² (Ω)}

^{havingthe nite norm}

kGk ² _H = kf k ² _L ² _(Q) + ku _◦ k ² _W ¹

2 (Ω) + ku ₁ k ² _L ² _(Ω) + kg k ² _L ² _(Q) + kθ _◦ k ² _L ² _(Ω) .

⁽⁶⁾

3. Uniqueness of solution

We now state the rst result onerning the uniqueness of solution

of problem

(1)

^.

Theorem 1. For any funtion

U = (u

θ) ∈ D (J)

^{there exists a}

positiveonstant

C > 0

independent of

U

^{suh that}

kU k ² _B − C kJU k ² _H ≤ 0

⁽⁷⁾

Proof. Taking the inner produt in

L ² (Q τ )

^{of equations}

L 1 u = f

^,

L ₂ u = g

^{and the} intgrodierential operators

ℑ ² _xy u _t

^and

ℑ ² _xy θ,

respetively, where

Q ^τ = (0

τ) × Ω

^with

0 ≤ τ ≤ T,

^{we have}

c ₁ u

ℑ ² _xy u _t

L ² (Q ^τ ) + c ₂ u _t

ℑ ² _xy u _t

L ² (Q ^τ ) + u _tt

ℑ ² _xy u _t

L ² (Q ^τ )

+ ∂ ⁴ u

∂x ⁴

ℑ ² _xy u t

L ² (Q ^τ )

+ 2

∂ ⁴ u

∂x ² ∂y ²

ℑ ² _xy u t

L ² (Q ^τ )

+ ∂ ⁴ u

∂y ⁴

ℑ ² _xy u t

L ² (Q)

−α ∂ ² θ

∂x ²

ℑ ² _xy u t

L ² (Q ^τ )

− α ∂ ² θ

∂y ²

ℑ ² _xy u t

L ² (Q ^τ )

+ β θ t

ℑ ² _xy θ

L ² (Q ^τ )

−η ∂ ² θ

∂x ²

ℑ ² _xy θ

L ² (Q)

− η ∂ ² θ

∂y ²

ℑ ² _xy θ

L ² (Q)

+σ θ

ℑ ² _xy θ

L ² (Q ^τ ) + α

∂ ³ u

∂x ² ∂t

ℑ ² _xy θ

L ² (Q ^τ )

+ α

∂ ³ u

∂y ² ∂t

ℑ ² _x θ

L ² (Q ^τ )

= f

ℑ ² _xy u _t

L ² (Q ^τ ) + g

ℑ ² _xy θ

L ² (Q ^τ ) .

⁽⁸⁾

By suessive integration by parts of eah term of

(8),

^{and using}

boundary onditions in

(1),

^{we derivethe following} equalities:

c 1 u

ℑ ² _xy u t

L ² (Q ^τ ) = c ₁

2 kℑ xy u(.

.

τ)k ² _L ² _(Ω) − c ₂

2 kℑ xy u 0 k ² _L ² _(Ω) ,

⁽⁹⁾

c 2 u t

ℑ ² _xy u t

L ² (Q ^τ ) = c 2 kℑ xy u t k ² _L ² _(Q τ ) ,

⁽¹⁰⁾

u tt

ℑ ² _xy u t

L ² (Q ^τ ) = 1

2 kℑ xy u t (.

.

τ )k ² _L ² _(Ω) − 1

2 kℑ xy u 1 k ² _L ² _(Ω)

^{, (11)}

∂ ⁴ u

∂x ⁴

ℑ ² _xy u _t

L ² (Q ^τ )

= 1

2 kℑ y u _x (x

.

τ )k ² _L ² _(Ω) − 1 2

ℑ y

∂u _◦

∂x

2 L ² (Ω)

,

⁽¹²⁾

2

∂ ⁴ u

∂x ² ∂y ²

ℑ ² _xy u t

L ² (Q ^τ )

= ku(x

y

τ )k ² _L ² _(Ω) − ku _◦ k ² _L ² _(Ω)

⁽¹³⁾

∂ ⁴ u

∂y ⁴

ℑ ² _xy u _t

L ² (Q ^τ )

= 1

2 kℑ _x u _y (.

y

τ)k ² _L ² _(Ω) − 1 2

ℑ _x ∂u _◦

∂y

2 L ² (Ω)

(14)

−α ∂ ² θ

∂x ²

ℑ ² _xy u _t

L ² (Q ^τ )

= α (ℑ y θ

ℑ y u _t ) _L ² _(Q τ )

⁽¹⁵⁾

−α ∂ ² θ

∂y ²

ℑ ² _xy u _t

L ² (Q ^τ )

= α (ℑ _x θ

ℑ _x u _t ) _L ² _(Q τ )

⁽¹⁶⁾

β θ _t

ℑ ² _xy θ

L ² (Q ^τ ) = β

2 kℑ xy θ (.

.

τ)k ² _L ² _(Ω) − β

2 kℑ xy θ _◦ k ² _L ² _(Ω)

⁽¹⁷⁾

−η ∂ ² θ

∂x ²

ℑ ² _xy θ

L ² (Q ^τ )

= η kℑ y θk ² _L ² _(Q τ )

⁽¹⁸⁾

−η ∂ ² θ

∂y ²

ℑ ² _xy θ

L ² (Q ^τ )

= η kℑ _x θk ² _L ² _(Q τ )

⁽¹⁹⁾

σ θ

ℑ ² _xy θ

L ² (Q ^τ ) = σ kℑ xy θk ² _L ² _(Q τ )

⁽²⁰⁾

α

∂ ³ u

∂x ² ∂t

ℑ ² _xy θ

L ² (Q ^τ )

= −α (ℑ y u t

ℑ y θ) _L ² _(Q τ )

⁽²¹⁾

α

∂ ³ u

∂y ² ∂t

ℑ ² _xy θ

L ² (Q ^τ )

= −α (ℑ x u t

ℑ x θ) _L ² _(Q τ )

⁽²²⁾

f

ℑ ² _xy u t

L ² (Q ^τ ) = (ℑ xy f

ℑ xy u t ) _L ² _(Q τ )

⁽²³⁾

g

ℑ ² _xy θ

L ² (Q ^τ ) = (ℑ xy g

ℑ xy θ) _L ² _(Q τ )

^{. (24)}

Ifweuse Cauhy-

ε

^{-inequalitygiven inC),Poinare'inequalityoftype}

(4*)

,

^{and equalities}

(9) − (24),

^{then equation}

(8)

^{redues to}

kℑ _xy u(.

.

τ)k ² _L ² _(Ω) + kℑ _xy u _t (.

.

τ )k ² _L ² _(Ω) + kℑ _x u (.

y

τ )k ² _L ² _(Ω) + kℑ x u y (.

y

τ )k ² _L ² _(Ω) + kℑ y u (x

.

τ )k ² _L ² _(Ω) + kℑ xy u t k ² _L ² _(Q τ )

+ kℑ y u x (x

.

τ)k ² _L ² _(Ω) + kℑ xy θ (.

.

τ)k ² _L ² _(Ω)

≤ C ₁









ku 1 k ² _L ² _(Ω) + ku _◦ k ² _L ² _(Ω) + ^{∂u ◦}

∂x

2

L ² (Ω) + kℑ xy u 0 k ² _L ² _(Ω) +

∂u ◦

∂y

2

L ² (Ω) + kθ _◦ k ² _L ² _(Ω) + kf k ² _L ² _(Q τ )

+ kgk ² _L ² _(Q τ ) + kℑ xy u t k ² _L ² _(Q τ ) + kℑ xy θk ² _L ² _(Q τ )









(25)

where

C 1 =

max n

1

^a ₄ ²

^b ₄ ²

^,

^(ab) ₈ ²

^,

^β(ab) ₈ ² , ^c ₂ ¹ o min _a ²

4

b ²

4 , ^c ₂ ² .

Applying the Gronwall's lemma given in B) to inequality

(25)

^and

disarding the fourth, sixth and seventh term of its left-hand, we

obtain

kℑ _xy u(.

.

τ)k ² _L ² _(Ω) + kℑ _xy u _t (.

.

τ )k ² _L ² _(Ω) + kℑ _x u (.

y

τ )k ² _L ² _(Ω) + kℑ _y u (x

.

τ)k ² _L ² _(Ω) + kℑ _xy θ (.

.

τ)k ² _L ² _(Ω)

≤ C 1 e ^{C 1 T} ku 1 k ² _L ² _(Ω) + ku _◦ k ² _W ¹

2 (Ω) + kθ _◦ k ² _L ² _(Ω) + kf k ² _L ² _(Q) + kgk ² _L ² _(Q)

!

.

⁽²⁶⁾

Eah term on the left-hand side of

(26)

^{is bounded and sine the}

right-handoftheaboveinequality

(26)

^isindependentof

τ

^,weantake

the least upper bound of eah term of the left-hand side with respet

to

τ

^over

[0

T ]

^{,weget thedesiredestimate}

(7)

^with

C = C ₁ e ^{C 1 T}

^{. This}

ompletes the proof of Theorem 1.

Itan beprovedinastandard way thatthe operator

J : B −→ H = H 1 × H 2

^{is losable.}

Proposition 2. The operator

J : B → H = H 1 × H 2

^{has alosure}

See

[15].

Let

J

^{be the losure of}

J

^and

D(J)

^{its domain of denition}

.

^We

denethe stronggeneralizedsolution ofproblem

(1)

^{as the solutionof}

the operator equation

JU = (J 1 u

J ₂ θ),

^with

J ₁ u = {L 1 u

ℓ ₁ u

ℓ ₂ u}

J ₂ θ = {L 2 θ

ℓ ₃ θ} , (J 1 u

J ₂ θ) ∈ H.

^{If we pass to the limit in}

(7)

^we

havethe result

Corollary 3. There exists a positive onstant

C

^{suh that}

kU k ² _B − C

JU

2

H ≤ 0, ∀U ∈ D J

.

⁽²⁷⁾

Wededue fromtheaprioriestimate

(27)

^thatastronggeneralizedso- lutionof

(1)

^{ifitexists isuniqueanddepends}ontinuouslyon

G= (G ₁

G 2 ) = ({f

u _◦

u 1 } , {g

θ _◦ }) ∈ H

^{, and that the range}

R J

of

J

^is

losed in

H

^and

R J

= R (J ).

Existene of the solution of the stated problem.

Theorem 4. Problem

(1)

^{has aunique strong solutionverifying}

u ∈ C(I, B ₂ ^1,x (Ω)), u ∈ C(I, B ₂ ^1,y (Ω)), θ ∈ C(I, B ₂ ^1,x,y (Ω)), ∂u

∂t ∈ C(I, B ₂ ^1,x,y (Ω)).

Moreover, the funtions

ℑ x u, ℑ y u, , ℑ xy θ, ℑ xy θ

^depend ontinuously on the free terms

f ∈ L ² (Q)

^,

g ∈ L ² (Q) ,

^{and on the initial data}

u _◦ ∈ W ₂ ¹ (Ω)

u ₁ ∈ L ² (Ω)

θ _◦ ∈ L ² (Ω)

^{that is}

ku(.

.

τ )k ² _C(I,B ^{1 ,x,y}

2 (Ω)) ≤ C kJU k _H ,

ku(x

.

τ)k _C(I,B ^{1 ,y}

2 (Ω)) ≤ C kJU k _H ,

ku(.

y

τ )k _C(I,B ^{1 ,x}

2 (Ω)) ≤ C kJU k _H ,

ku _t (.

.

τ )k _C(I,B ^{1 ,x,y}

2 (Ω)) ≤ C kJU k _H ,

kθ(.

.

τ )k ² _C(I,B ^{1 ,x,y}

2 (Ω)) ≤ C kJU k _H .

⁽²⁸⁾

Proof. To establish the existene of the solutionof problem

(1)

^{, it}

is suientto prove that the imageof the operator

J

^{is dense in}

H.

General ase for density: Sine

H

^{is a Hilbert spae,}

R (J) = H

^is

equivalenttotheorthogonalityofavetor

Φ = (G 1

G 2 ) = ({σ 1

σ ₂

σ ₃ }

{σ 4

σ ₅ }) ∈ H

^{tothe range}

R (J ) ,

^{that is the equality}

+ (ℓ 2 u

σ ₄ ) _L ² _(Ω) + (ℓ 3 θ

σ ₅ ) _L ² _(Ω)

+ (L 1 u

σ 1 ) _L ² _(Q) + (L 2 θ

σ 2 ) _L ² _(Q) + (ℓ 1 u

σ 3 ) _W ¹

2 (Ω)

= 0,

⁽²⁹⁾

for all

U = (u

θ) ∈ D (J) ,

^{implies that}

Φ = 0.

^Let

D ₀ (J)

^{bea subset}

of

D(J )

^{for whih}

ℓ ₁ u = 0

ℓ ₂ u = 0, ℓ 3 θ = 0.

^If

U ∈ D ₀ (J),

^then

(29)

redues to

(L 1 u

σ 1 ) _L ² _(Q) + (L 2 θ

σ 2 ) _L ² _(Q) = 0.

⁽³⁰⁾

We have to prove that

Ψ = (σ 1

σ 2 ) = 0

^{everywhere in}

Q.

^{Thus we}

must prove the followingspeial ase (ofdensity) and then gobak to

the general ase.

Proposition5. If,forsomefuntion

Ψ = (σ ₁

σ ₂ ) ∈ (L ² (Q)) ²

^and

for allelements

U ∈ D _◦ (J )

^{we have}

(L 1 u

σ ₁ ) _L ² _(Q) + (L 2 θ

σ ₂ ) _L ² _(Q) = 0

⁽³¹⁾

then

Ψ

^{vanishesalmost everywhere in}

Q.

Proof. Sinerelation

(31)

^{holdsforany elementof}

D _◦ (J)

^wethen

take anelement

U = (u

θ)

^{with a speial formgiven by}

U =

( (0

0)

0 ≤ t ≤ s, R t

s (τ − t) u _{τ τ} dτ

R t s θ _τ dτ

s ≤ t ≤ T

^,

(32)

suh that

(u tt

θ _t )

^{isa solutionof the system}

ℑ ² _xy u _tt = E ₁ (r

t) =

Z T t

σ ₁ (r

τ) dτ

ℑ ² _xy θ _t = E ₂ (r

t) = Z T

t

σ ₂ (r

τ) dτ

(33)

where

E ₁ (x

t) = R T

t σ ₁ (r

τ ) dτ

^and

E ₂ (x

t) = R T

t σ ₂ (r

τ) dτ.

^It

is lear that

σ 1 = −ℑ ² _xy u ttt , σ 2 = −ℑ ² _xy θ tt .

⁽³⁴⁾

Proposition 6. The funtion

Ψ = (σ 1

σ ₂ ) ∈ (L ² (Q)) ²

^{dened in}

(34)

^{is in}

(L ² (Q)) ² .

Proof. it an bearried out as in[10℄.

Now replaing the funtions

σ ₁

^and

σ ₂

^{given by}

(34)

ⁱⁿ

(31),

^we

obtain

−c 1 u

ℑ ² _xy u _ttt

L ² (Q) − c ₂ u _t

ℑ ² _xy u _ttt

L ² (Q) − u _tt

ℑ ² _xy u _ttt

L ² (Q)

− ∂ ⁴ u

∂x ⁴

ℑ ² _xy u ttt

L ² (Q)

− 2

∂ ⁴ u

∂x ² ∂y ²

ℑ ² _xy u ttt

L ² (Q)

− ∂ ⁴ u

∂y ⁴

ℑ ² _xy u ttt

L ² (Q)

+α ∂ ² θ

∂x ²

ℑ ² _xy u ttt

L ² (Q)

+ α ∂ ² θ

∂y ²

ℑ ² _xy u ttt

L ² (Q)

−β θ t

ℑ ² _xy θ _tt

L ² (Q) + η ∂ ² θ

∂x ²

ℑ ² _xy θ _tt

L ² (Q)

+η ∂ ² θ

∂y ²

ℑ ² _xy θ _tt

L ² (Q)

+ −σ θ

ℑ ² _xy θ _tt

L ² (Q)

−α

∂ ³ u

∂x ² ∂t

ℑ ² _xy θ _tt

L ² (Q)

− α

∂ ³ u

∂y ² ∂t

ℑ ² _xy θ _tt

L ² (Q)

= 0

⁽³⁵⁾

Taking into aount the speial form of

U

^{given by}

(32)

^and

(33)

using boundary onditions in

(1)

^and integrating by parts eah terms of

(35)

^gives

−c ₁ u

ℑ ² _xy u _ttt

L ² (Q) = c ₁ Z

Q s

ℑ _x u.ℑ _xyy u _ttt dxdydt

= −c 1

Z

Q s

ℑ xy u.ℑ xy u ttt dxdydt

= c 1

2 kℑ xy u t (.

.

T )k ² _L ² _(Ω) ,

⁽³⁶⁾

−c 2 u _t

ℑ ² _xy u _ttt

L ² (Q) = c ₂ Z

Q s

ℑ x u _t .ℑ xyy u _ttt dxdydt

= −c 2

Z

Q s

ℑ xy u t .ℑ xy u ttt dxdydt

= c ₂ kℑ xy u _tt k ² _L ² _(Ω) ,

⁽³⁷⁾

− u tt

ℑ ² _xy u ttt

L ² (Q) = Z

Q s

ℑ x u tt .ℑ xyy u ttt dxdydt

= − Z

Q s

ℑ _xy u _tt .ℑ _xy u _ttt dxdydt

= 1

2 kℑ _xy u _tt (.

.

s)k ² _L ² _(Ω)

⁽³⁸⁾

− ∂ ⁴ u

∂x ⁴

ℑ ² _xy u _ttt

L ² (Q)

= Z

Q s

∂ ³ u

∂x ³ .ℑ _xyy u _ttt dxdydt

= − Z

Q s

∂ ² u

∂x ² .ℑ ² _y u ttt dxdydt

= Z

Q s

u x .ℑ ² _y u tttx dxdydt

= − Z

Q s

ℑ y u _x .ℑ y u _tttx dxdydt

= Z

Q s

ℑ _y u _tx .ℑ _y u _ttx dxdydt

= 1

2 kℑ y u xt (.

.

T )k ² _L ² _(Ω)

⁽³⁹⁾

−2

∂ ⁴ u

∂x ² ∂y ²

ℑ ² _xy u ttt

L ² (Q)

= 2 Z

Q s

∂ ³ u

∂x∂y ² .ℑ xyy u ttt dxdydt

= −2 Z

Q s

∂ ² u

∂y ² .ℑ ² _y u ttt dxdydt

= 2 Z

Q s

u _y .ℑ y u _ttt dxdydt

= −2 Z

Q s

u.u _ttt dxdydt

= 2 Z

Q s

u t .u tt dxdydt

= ku t (x

y

T )k ² _L ² _(Ω)

⁽⁴⁰⁾

− ∂ ⁴ u

∂y ⁴

ℑ ² _xy u _ttt

L ² (Q)

= Z

Q s

∂ ³ u

∂y ³ .ℑ xxy u _ttt dxdydt

= Z

Q s

u _y .ℑ ² _x u _ttty dxdydt

= − Z

Q s

ℑ _x u _y .ℑ _x u _ttty dxdydt

= Z

Q s

ℑ x u ty .ℑ x u tty dxdydt

= 1

2 kℑ x u _ty (.

y

T )k ² _L ² _(Ω)

⁽⁴¹⁾

α ∂ ² θ

∂x ²

ℑ ² _xy u _ttt

L ² (Q)

= −α Z

Q s

θ _x .ℑ _xyy u _ttt dxdydt

= α Z

Q s

θ.ℑ ² _y u _ttt dxdydt

= −α Z

Q s

ℑ y θ.ℑ y u ttt dxdydt

= α (ℑ y θ _t

ℑ y u _tt ) _L ² _{(Q s )}

⁽⁴²⁾

α ∂ ² θ

∂y ²

ℑ ² _xy u ttt

L ² (Q)

= −α Z

Q s

θ y .ℑ xxy u ttt dxdydt

= α Z

Q s

θ.ℑ ² _x u ttt dxdydt

= −α Z

Q s

ℑ x θ.ℑ x u _ttt dxdydt

= α (ℑ _x θ _t

ℑ _x u _tt ) _L ² _(Q

s )

⁽⁴³⁾

−β θ t

ℑ ² _xy θ _tt

L ² (Q) = β Z

Q s

ℑ x θ _t .ℑ xyy θ _tt dxdydt

= −β Z

Q s

ℑ xy θ t .ℑ xy θ tt dxdydt

= β

2 kℑ xy θ t (.

.

s)k ² _L ² _(Ω)

⁽⁴⁴⁾

η ∂ ² θ

∂x ²

ℑ ² _xy θ _tt

L ² (Q)

= −η Z

Q s

θ _x .ℑ _xyy θ _tt dxdydt

= η Z

Q s

θ.ℑ ² _y θ tt dxdydt

= −η Z

Q s

ℑ y θ.ℑ y θ tt dxdydt

= η kℑ _y θ _t k ² _L ² _(Q

s )

⁽⁴⁵⁾

η ∂ ² θ

∂y ²

ℑ ² _xy θ _tt

L ² (Q)

= η kℑ _x θ _t k ² _L ² _(Q

s )

⁽⁴⁶⁾

−σ θ

ℑ ² _xy θ _tt

L ² (Q) = σ Z

Q s

ℑ _x θ.ℑ _xyy θ _tt dxdydt

= −σ Z

Q s

ℑ xy θ.ℑ xy θ tt dxdydt

= σ kℑ xy θ _t k ² _L ² _{(Q s )}

⁽⁴⁷⁾

−α

∂ ³ u

∂x ² ∂t

ℑ ² _xy θ _tt

L ² (Q)

= α Z

Q s

∂ ² u

∂x∂t .ℑ xyy θ _tt dxdydt

= −α Z

Q s

u _t .ℑ ² _y θ _tt dxdydt

= α Z

Q s

ℑ y u t .ℑ y θ tt dxdydt

= −α (ℑ y u _tt

ℑ y θ _t ) _L ² _{(Q s )}

⁽⁴⁸⁾

−α

∂ ³ u

∂y ² ∂t

ℑ ² _xy θ _tt

L ² (Q)

= α Z

Q s

∂ ² u

∂y∂t .ℑ xxy θ _tt dxdydt

= −α Z

Q s

u _t .ℑ ² _x θ _tt dxdydt

= α Z

Q s

ℑ _x u _t .ℑ _x θ _tt dxdydt

= −α (ℑ x u tt

ℑ x θ t ) _L ² _{(Q s )}

⁽⁴⁹⁾

Combiningequalities

(36) − (49)

^and

(35)

^{we obtain}

1

2 kℑ xy u tt (.

.

s)k ² _L ² _(Ω) + 1

2 kℑ y u xt (.

.

T )k ² _L ² _(Ω) + ku t (x

y

T )k ² _L ² _(Ω) + 1

2 kℑ x u ty (.

y

T )k ² _L ² _(Ω) + β

2 kℑ xy θ _t (.

.

s)k ² _L ² _(Ω) + η kℑ y θ _t k ² _L ² _{(Q s )} +η kℑ _x θ _t k ² _L ² _(Q

s ) + σ kℑ _xy θ _t k ² _L ² _(Q

s ) + c ₁

2 kℑ _xy u _t (.

.

T )k ² _L ² _(Ω) +c 2 kℑ xy u tt k ² _L ² _(Ω)

= 0

⁽⁵⁰⁾

where

Q _s = Ω × (s

T ) .

Equality

(50)

^impliesthat

ℑ xy u tt (ζ

ς

s) = 0

^on

Ω

^and

ℑ xy θ t = 0

on

Q _s

^{hene we dedue that}

Ψ = (σ 1

σ ₂ ) = (0

0)

^{almost everywhere}

in

Q _s .

Proeeding in this way step by step, we prove that

Ψ = 0

^almost

every where in

Q.

Now bak tothe generalase: Sine

Ψ = (σ 1

σ ₂ ) = 0

^{everywhere in}

Q

^{, thereforeequality}

(29)

^beomes

(ℓ 1 u

σ 3 ) _W ¹

2 (Ω) + (ℓ 2 u

σ 4 ) _L ² _(Ω) + (ℓ 3 θ

σ 5 ) _L ² _(Ω) = 0.

⁽⁵¹⁾

Sine the three quantities in

(51)

^vanish independently and sine the rangesof thetraeoperators

ℓ 1

ℓ 2

^and

ℓ 3

^are respetively everywhere dense in the spaes

W ₂ ¹ (Ω)

L ² (Ω)

^and

L ² (Ω)

^{therefore it follows,}

from

(51)

^that

σ 3 = σ 4 = σ 5 = 0.

^Hene

R (J) = H.

^{This ahieves}

the proof of Theorem 4.

Remark: The following larger lass of problems may handled by

using the previous same tehniques



 

 

 



 

 

 



L ₁ u = ^∂ _∂t ^{2 u} 2 + △ ² u − α△θ + (c ₁ u + c ₂ u _t ) = f(x

y

t

^,

θ, u)

⁽

x

y

t) ∈ Q, L 2 θ = β ^∂θ _∂t − η△θ + σθ + α△u t − c 3 ∆u = g(x

y

t

^,

θ, u)

⁽

x

y

t) ∈ Q,

u(x

y

0) = u _◦ (x, y), u t (x

y

0) = u 1 (x

y), θ(x

y

0) = θ _◦ (x

y), u(0

y

t) = 0

u(a

y

t) = 0

0 < y < b

0 < t < T, u(x

0

t) = 0

u(x

b

t) = 0

0 < x < a

0 < t < T, R a

0 x ^k udx = 0

R b

0 y ^k udy = 0

^,

R a

0 x ^k θdx = 0

R b

0 y ^k θdy = 0

^,

k = 0, 1,

(47)

where the funtions

f

^and

g

^{satisfy the onditions}

|f(x

y

t

^,

θ ₁ , u ₁ ) − f (x

y

t

^,

θ ₂ , u ₂ )| ≤ µ (|θ 1 − θ ₂ | + |u 1 − u ₂ |) ,

|g(x

y

t

^,

θ ₁ , u ₁ ) − g(x

y

t

^,

θ ₂ , u ₂ )| ≤ µ (|θ 1 − θ ₂ | + |u 1 − u ₂ |) .

and

α

β

η

^,

σ, c 1

^and

c 2

^{are positive onstants.}

We rst deal with the assoiated linear problem,that is when

f (x

y

t

^,

θ, u) = f (x

y

t )

^and

g(x

y

t

^,

θ, u) = g(x

y

t )

^{and then,}

on the basis of the obtained results of the linear problem, we apply

an iterative proess to establish the existene and uniqueness of the

nonlinear problem.

Aknowledgment: The author extends his appreiation to the

Deanship of Sienti Researh at King Saud University for funding

the work through the researh groupprojetNo: RGP-VPP-117.

Referenes

[1℄ M. Assila, Nonlinear boundary stabilization of an inhomogeneous and

anisotropithermoelastiitysystem,AppliedMathLetters.13 (2000),71-76.

[2℄ A.Bouziani,OnaninitialboundaryvalueproblemwithDirihletintegralon-

ditionsforahyperboliequationwiththeBesseloperator,JournalofApplied

Mathematis,10(2003)487-502.

paraboliequation,J.Appl.Math.StohastiAnal.9(1996),no3,323-330.

[4℄ C. M. Dafermos, L. Hsiao, Development of singularities in solutions of the

equations on nonlinear thermoelastiity system, Q. Appl. Math 44 (1986),

463-474.

[5℄ L.Garding,Cauhy'sProblemforHyperboliEquations.LetureNotes.Uni-

versityofChiago: Chiago,1957.

[6℄ W.J.Hrusa,S.A.Messaoudi,Onformationofsingularitiesonone-dimensional

nonlinearthermoelastiity,Arh.RationalMeh.Anal3(1990),135-151.

[7℄ S.Mesloub,Anonlinearnonloalmixedproblemforaseondorderparaboli

equation,J.Math.Anal.Appl.316(2006)189-209.

[8℄ S.Mesloub,Onasingulartwodimensionalnonlinearevolutionequationwith

nonloalonditions,NonlinearAnalysis68(2008)2594-2607.

[9℄ S.Mesloub,Onanonlinearsingularhyperboliequation,MathematialMeth-

odsintheAppliedSienes,Vol33,Issue1(2010)57-70.

[10℄ S. Mesloub, On anon loal problem for apluriparaboliequation, Ata Si.

Math.(Szeged)67(2001),203-219.

[11℄ S. Mesloub, A. Bouziani, On a lass of singular hyperboli equation with a

weighted integral ondition, Internat. J. Math. &Math. Si. Vol 22. N 511-

519,(1999).

[12℄ S. Mesloub, and F. Mesloub, Solvability of amixed non loal problem for a

nonlinearSingularVisoelastiequation,Ata.Appl.Mathematiae,Vol110,

Number1,109-129.

[13℄ S.Mesloub,S.Messaoudi,GlobalExistene, Deay,andBlowupofSolutions

ofaSingularNonloalVisoelastiProblem,AtaAppliandaeMathematiae,

Volume: 110Issue: 2(2010),705-724.

[14℄ S. Mesloub,N. Lekrine, Onanon loal hyperbolimixed problem, Ata Si.

Math.(Szeged),70(2004),65-75.

[15℄ J.E. Munoz Rivera, R. K.Barreto,Existene andexponentialdeayin non-

linearthermoelastiity,NonlinearAnalysis31No.1/2(1998),149-162.

[16℄ R. Rake, Blow up in nonlinear three dimensional thermoelastiity, Math.

MethodsAppl.Si. 12No3(1990),273-276.

[17℄ R. Rake, Y. Shibata, Global smooth solutions and asymptoti stability in

one-dimensional nonlinearthermoelastiity, Arh. rational. Meh. Anal. 116

(1991),1-34.

[18℄ R. Rake, Y. G.Wang, Propagationof singularities in one-dimensionalther-

moelastiity,J.Math.Anal. Appl,223(1998),216-247.

[19℄ M.Slemrod,Globalexistene,uniqueness,andasymptotistabilityoflassial

solutionsinone-dimensionalthermoelastiity, Arh. rational.Meh.Anal. 76

(1981),97-133.

(Reeived September 5,2011)

King Saud University, College of sienes Department of Mathe-

matis,P.O.Box2455, Riyadh11451, SaudiArabia

E-mailaddress: mesloubsyahoo.om