Aissa GUESMIASalim MESSAOUDI1.Department of Mathematics and Statistics,College of Sciences,King Fahd University of Petroleum and Minerals,P.O.Box 5005,Dhahran 31261,Saudi Arabia2.Institut Elie Cartan de Lorraine,UMR 7502,Université de Lorraine,Bat.A,Ile du Saulcy,57045 Metz Cedex 01,France

?

SOME STABILITY RESULTS FOR TIMOSHENKO SYSTEMS WITH COOPERATIVE FRICTIONAL AND INFINITE-MEMORY DAMPINGS IN THE DISPLACEMENT?

Aissa GUESMIA1,2Salim MESSAOUDI1,?
1.Department of Mathematics and Statistics,College of Sciences,King Fahd University of Petroleum and Minerals,P.O.Box 5005,Dhahran 31261,Saudi Arabia
2.Institut Elie Cartan de Lorraine,UMR 7502,Université de Lorraine,Bat.A,Ile du Saulcy,57045 Metz Cedex 01,France

E-mail:guesmia@kfupm.edu.sa;aissa.guesmia@univ-lorraine.fr;messaoud@kfupm.edu.sa

AbstractIn this paper,we consider a vibrating system of Timoshenko-type in a onedimensional bounded domain with complementary frictional damping and in finite memory acting on the transversal displacement.We show that the dissipation generated by these two complementary controls guarantees the stability of the system in case of the equal-speed propagation as well as in the opposite case.We establish in each case a general decay estimate of the solutions.In the particular case when the wave propagation speeds are different and the frictional damping is linear,we give a relationship between the smoothness of the initial data and the decay rate of the solutions.By the end of the paper,we discuss some applications to other Timoshenko-type systems.

Key wordswell-posedness;decay;damping;Timoshenko;thermoelasticity

2010 MR Subject Classi fi cation35B37;35L55;74D05;93D15;93D20

?Received September 9,2014;revised February 26,2015.

?Corresponding author:Salim MESSAOUDI.

1 Introduction

In this work,we are concerned with the long-time behavior of the solution of the following Timoshenko system:for(x,t)∈]0,L[×R+,where R+=[0,+∞[,a,b:[0,L]→R+,g:R+→R+and h:R→R are given functions(to be speci fied later),L,ρi,ki(i=1,2)are positive constants,?0,?1,ψ0and ψ1are given initial and history data,and(?,ψ):]0,L[×R+→R2is the state of(P).

Our aim is the study of the asymptotic behavior of the solutions of(P)in case of the equal-speed propagation

as well as in the opposite case.

Timoshenko[39],in 1921,introduced the following model to describe the transverse vibration of a beam:

where t denotes the time variable and x is the space variable along the beam of length L,in its equilibrium configuration, u is the transverse displacement of the beam and ? is the rotation angle of the filament of the beam. The coefficients ρ, Iρ,E, I and K are, respectively,the density (the mass per unit length), the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section, and the shear modulus. Since then, this model attracted the attention of many researchers and an important amount of work was devoted to the issue of the stabilization and the search for the minimum dissipation by which the solutions decay uniformly to the stable state as time goes to infinity. To achieve this goal, diverse types of dissipative mechanisms were used and several stability results were obtained. We mention some of these results (for more results, we refer the reader to the list of references of this paper, which is not exhaustive, and the references therein).

In the case of presence of controls on both the rotation angle and the transverse displacement,investigations showed that the weak solutions of(P)are stable without any restriction on the constants ρ1,ρ2,k1and k2.In this regards,many decay estimates were obtained[14,18,23,26,34].However,in the case of only one control on the rotation angle,the rate of decay depends heavily on the constants ρ1,ρ2,k1and k2and the regularity of the initial data.Precisely,if(1.1)holds,the results obtained are similar to those established for the case of the presence controls in both equations.We quote in this regard[4,7,12,13,14,16,17,24,25,29,30,31,38].But,if(1.1)does not hold,a situation which is more interesting from the physics point of view,then it has been shown that the Timoshenko system is not exponentially stable even for exponentially decaying relaxation functions and only weak decay estimates can be obtained for regular solutions in the presence of dissipation.This was demonstrated in[1],for the case of an internal feedback,in[7,14,16,17,27],for the case of finite and in finite memory,and in[10,13],for complementary internal feedback and finite or in finite memory acting on the rotation angle equation.

For stabilization of Timoshenko systems via heat effect,we mention the pioneer work[28],where the following system:was considered.In their work,Rivera and Racke established,under appropriate conditions on σ,ρi,b,k and γ,several exponential decay results for the linearized system with several boundary conditions.They also proved a non exponential stability result for the case of different wave speeds and proved an exponential decay result for the nonlinear case.Guesmia et al.[15]discussed a linear version of(1.2)and completed the work of[28]by establishing some polynomial decay results in the case of nonequal speed of propagation.

In(1.2),the heat flux is given by Fourier’s law.As a result,this theory predicts an in finite speed of heat propagation;that is,any thermal disturbance at one point has an instantaneous effect elsewhere in the body.Experiments showed that heat conduction in some dielectric crystals at low temperatures is free of this paradox and disturbances,which are almost entirely thermal,propagate in a finite speed.This phenomenon in dielectric crystals is called second sound.

To overcome this physical paradox,many theories have merged.One of which suggests that we should replace Fourier’s law by Cattaneo’s law.In line with this theory,(1.2),in its linear form,becomes

where q denotes the heat fl ux.Fernández Sare and Racke[8]studied(1.3)and proved that the equal-speed conditionis no longer su ffi cient to obtain exponential stability even in the presence of an extra viscoelastic dissipation of the formg(s)ψxx(t?s)ds in the second equation.Very recently,Santos et al.[37]considered(1.3),introduced a new stability number

and used the semigroup method to obtain an exponential decay result,for χ=0,and a polynomial decay,for χ0.See,also,[14,26,33,35,36].

In all above mentioned works,the stabilization was either via both equation control or the angular rotation equation control.Very recently,Almeida Júnior et al.[2]considered the situation when the control is only on the transverse displacement equation,which is more realistic from the physics point of view.Precisely,they looked into the following system

and showed that the linear frictional damping in the first equation is strong enough to obtain exponential stability provided that(1.1)holds.They,also,proved some non-exponential and polynomial decay results in the case of nonequal speed situation.The same authors considered in[3]with various boundary conditions,and established the exponential decay stability for equalspeed case and nonexponential stability for the opposite case.In the case of lack of exponential stability,they proved some algebraic(polynomial)stability for strong solutions.

Our goal in this paper is to investigate the effect of each control on the asymptotic behavior of the solutions of(P)and on the decay rate of its energy,when both controls are acting cooperatively,allowing each control to vanish on the whole domain.We give an explicit and general characterization of the decay rate depending on the growth of g at in finity and h at zero,by considering the case when(1.1)holds and the opposite case.In the latter case,we give a general decay estimate depending on the smoothness of the initial data and the growth of g at in finity.

The proof is based on the multipliers method and an approach introduced by the first author in[9,11],for a class of abstract hyperbolic systems of single or coupled equations with one in finite memory.In the case when(1.1)does not hold,we use also some ideas given in[10]to get a relation between the decay rate of solutions and the general growth of g at in finity characterized by condition(2.8)below introduced in[9].

The paper is organized as follows.In Section 2,we set up the hypotheses,discuss briefly the well-posedness and present our stability results.The proofs of these stability results will be given in Section 3,for the equal-speed case,in Section 4,for the nonequal-speed case,and in Section 5,when h is linear.Finally,in Section 6,we discuss some applications to other Timoshenko-type systems.

2 Preliminaries

2.1Hypotheses

We consider the following hypotheses:

(H1)a,b:[0,L]→R+are such that

(H2)h:R→R is a differentiable non-decreasing function such that there exist constants ∈1,c′,c′1＞0,and a convex and increasing function H:R+→R+of class C1(R+)∩C2(]0,+∞[)satisfying H(0)=0 and

(H3)g:R+→R+is a non-increasing differentiable function such that g(0)＞0 and

and

(H4)There exist a positive constant c′′and an increasing strictly convex function G:R+→R+of class C1(R+)∩C2(]0,+∞[)satisfying

Remark 2.11.Hypothesis(2.8)was introduced in[9]and it allows a wider class of relaxation functions than the ones considered in[7,27](see examples given in[9,14]).

2.Hypothesis(H2)(with∈1=1)was introduced and used in[20,21]to get the asymptotic behavior of solutions of nonlinear wave equations with nonlinear boundary damping,where they obtained decay estimates depending on the solution of an explicit nonlinear ordinary differential equation.

3.Using the second equation and boundary conditions in(P),we easily verify that

By solving this ordinary differential equation and using the initial data of ψ,we find

Then,one can easily check that

and,hence,Poincaré’s inequality is applicable for.In addition,(?,)satisfies(P)with initial datainstead of ψ0and ψ1,respectively.In the sequel,we work withinstead of ψ,but,for simplicity of notation,we use ψ instead of.

4.Thanks to Poincaré’s inequality(applied for ψ),we have

and obtain

2.2Well-Posedness

We give here a brief idea about the existence,uniqueness and smoothness of solution of(P).Following the idea of[6],let

Then

Let η0(x,s)=η(x,0,s)=?0(x,0)??0(x,s)for(x,s)∈]0,L[×R+,

where

and

endowed with the inner product

The space H is equipped with the inner product de fined,if a≡0,by

for any V=(v1,v2,v3,v4)T∈H and W=(w1,w2,w3,w4)T∈H,and ifinfx∈[0,L]{a(x)}＞0,by

for any V=(v1,v2,v3,v4,v5)T∈H and W=(w1,w2,w3,w4,w5)T∈H.Let

and A is the operator de fined byfor any(v1,v2,v3,v4)T∈ D(A),where

Note that,thanks to(2.4)and the fact that h is continuous,we have

thus h(v3)∈L2(]0,L[)for any v3∈L2(]0,L[).The domain D(A)of A can be characterized by

if a≡0,and

As in[10]where the frictional damping and in finite memory were considered on the second equation of(P),we can prove that the operator A is maximal monotone;that is?A is dissipative and Id+A is surjective.Then we deduce that A is an in finitesimal generator of a contraction semigroup on H,which implies the following results of existence,uniqueness and smoothness of the solution of(P)(see[19,32]).

Theorem 2.01.For any U0∈H,one has a unique solution

2.If U0∈D(A),then the solution

3.If h is linear(then A is linear)and U0∈D(An)(for n∈N),then the solution

2.3Stability

The energy functional associated with(P)is de fined by

where

for any v:R→L2(]0,L[)and φ:R+→R+.

Now,we give our first main stability result which concerns case(1.1).

Theorem 2.1Assume that(1.1)and(H1)-(H4)are satis fied and let U0∈H such that a≡0 or(2.7)holds or

Then there exist positive constantsfor which E satisfies

and

2.If a≡0 or(2.7)holds,and b≡0 or H is linear near zero,then

which is the best decay rate given by(2.12).For speci fi c examples of decay rates given by(2.12),see[10].

When(1.1)does not hold,we consider the following additional hypothesis:

(H5)Assume that(H2)is satis fied such that H is linear,

Theorem 2.2Assume that(H1)-(H5)hold and U0∈D(A)such that a≡0 or(2.7)holds or

Then there exist positive constants∈0and c1such that

where

Remark 2.3If a≡0 or(2.7)holds,then(2.18)becomes

which is the best decay rate given by(2.18).

In the particular case where h is linear and the initial data are more regular,we prove a more general stability result than(2.18).

Theorem 2.3Assume that h is linear,and(H1)-(H4)are satis fied.Let n∈N?and U0∈D(An)such that a≡0 or(2.7)holds or

Then there exist positive constant∈0and cnsuch that

where Gm(s)=G1(sGm?1(s))for m=2,···,n and s∈R+,and G0is de fined in(2.19).

Remark 2.4If n=1,then(2.18)and(2.21)are the same.On the other hand,if a≡0 or(2.7)holds,then(2.21)becomes

which is the best decay rate given by(2.21).For speci fi c examples of decay rates given by(2.21),see[11].

3 Proof of Teorem 2.1

We will use c(sometimes cτwhich depends on some parameter τ),throughout this paper,to denote a generic positive constant.Before starting the proofs of our stability resuls,we give the following identity on the derivative of E.

Lemma 3.1The energy functional satisfies

ProofBy multiplying the first two equations in(P),respectively,by ?tand ψt,integrating over]0,L[,and using the boundary conditions,we obtain(3.1)(note that g is non-increasing and sh(s)≥0 for all s∈R,because h is non-decreasing and h(0)=0 thanks to(2.5)).Estimate(3.1)shows that(P)is dissipative,where the entire dissipation is generated by the frictional damping and/or in finite memory.

Lemma 3.2The following inequalities hold:we use the fact that a and a′are bounded and apply H?lder’s and Poincaré’s inequalities to get(3.2)-(3.4).Using again H?lder’s inequality,(3.5)and(3.6)hold.

Lemma 3.3The functional

ProofIf a≡0,(3.2)-(3.4)are trivial.If

satisfies,for any δ＞0,

ProofFirst,note that

Then,by differentiating I1,and using the first equation and boundary conditions in(P),wefind

Therefore,applying H?lder’s and Young’s inequalities,for the last heigh terms of the above equality,and using(3.2)-(3.5),Poincaré’s inequality,for ?,and the fact that a,b and a′are bounded,we get(3.8).

Lemma 3.4The functional

satisfies,for any δ＞0,

ProofBy differentiating I2,and using the first two equations and boundary conditions in(P),we have

Consequently,aplying H?lder’s and Young’s inequalities,for the last two terms of the above equality,and using(3.5),Poincaré’s inequality,for ?,and the fact that a and b are bounded,we find(3.9).

Lemma 3.5The functional

satisfies,for any δ,δ1＞0,

ProofSimilarly to(3.8)and using thatwe see that

Therefore,exploiting the first two equations and boundary conditions in(P),we have

By applying Young’s inequality,for the last four terms,Poincaré’s inequality,for ψ,and using(3.5),(3.6)and the fact that a and b are bounded,(3.10)is established.

Now,as in[4],we use a function w to get a crucial estimate.

Lemma 3.6The function

satisfies the estimates

Then,applying(3.12)to wtand using Poincaré’s inequality,for wt,we arrive at(3.13).

Lemma 3.7The functional

ProofWe just have to note that wx=ψ to get(3.12).On the other hand,

satisfies,for any δ,∈,∈′＞0,

ProofUsing the first two equations and boundary conditions in(P),and exploiting the fact that w(0,t)=w(L,t)=0 and wx=ψ,we find

Applying Young’s inequality,for the last four terms,Poincaré’s inequality,for ? and ψ,and exploiting(3.5),(3.12),(3.13)and the fact that a and b are bounded,we get(3.14).

For N,N1,N2,N3＞0,let

Let a0:=Noting that

Then,by combining(3.1),(3.8),(3.9),(3.10)and(3.14),we obtain

where

and c0＞0,independent of N,Ni,δ,δ1,∈and∈′.At this point,we choose carefully the constants N,Ni,δ,δ1,∈and∈′to get desired signs of li.

Case 1a≡0:the second integral in(3.16)drops,g??x=g′??x=0 and the constants l0,l1,l2and l3do not depent on δ1and∈′.Therefore,we choose

(note that b0＞0 thanks to(2.2)).According to these choices,we get

and then,using(2.9),(2.10)and(3.16),

Case 2a0＞0:we choose

and

Note that∈′and δ1are positive thanks to(2.6)and g0‖a‖∝＞0,N2exists according to the choice of N3,∈exists from the choice of N2,and N1exists because ρ1g0a0+b0＞0.On the other hand,using the de finitions of∈′and δ1,we see that N3exists if and only if

We have

and

Moreover,y1≤y0if and only if k2≤k0k1,and,if k2≤k0k1,

Therefore,f′is positive on]0,y0[,and then f(y)＜f(y0),for any y∈]0,y0[.But f(y0)=0,hence f is negative on]0,y0[.This guarantees the existence of N3.

By vertue of these choices,we notice that

and then,as in Case 1,using(2.9),(2.10)and(3.16),we find

Choosing δ＞0 small enough in(3.17),we deduce in both cases a≡0 andthat

Now,by the de finitions of the functionals I1?I4and E,there exists a positive constant β satisfying

which implies that

To estimate the last two integrals of(3.18),we use some ideas from[19,20,22].Let

where∈1is de fined in(H2).Using(2.4),we get(note that sh(s)≥0)

Then we choose N large enough so that c?N≤0(so the right hand side of the above inequality is non-positive),N2?c≥0(so the last term of(3.18)is non-positive)and N＞β(that is I5～E),we get from(3.18)

Case 1H is linear on[0,∈1]:then(2.4)is satis fied on R,and therefore

So,with the same choice of N,we get from(3.20),for H0=Id in this case,

Case 2H′(0)=0 and H′′＞0 on]0,∈1]:without loss of generality,we can assume that H′de fines a bijection from R+to R+.Let H?denote the dual function of the convex function H given by

For t∈R+,the function s?→ts?H(s)reaches its maximum on R+at the unique point(H′)?1(t).Therefore

Because H is convex and H(0)=0,then,for any s0∈R+,

which implis that,for s0=H?1(?th(?t)),

Thus,using(2.5),

Therefore,using Jensen’s inequality and(3.1),we find

Consequently,recalling(3.20),we get

Hence,Young’s inequality gives

and the fact that H?(t)≤t(H′)?1(t)and H′(τ0E)is non-increasing leads to

where H0(t)=tH′(τ0t)in this case.By choosing τ0small enough and τ′large enough,we arrive at

Let

where H0is de fined by(2.14)(I6=I5if H is linear on[0,∈1]).The functional I6satid fi es I6～E(because I5～E andis non-increasing)and,using(3.21)and(3.22),

Now,we estimate the term g??xin(3.23).

Case 1a≡0 or(2.7)holds:then,using(3.1),

Case 2a0＞0,(2.8)holds and(2.7)does not hold:we apply here the approach introduced in[9,11]and we get this lemma.

Lemma 3.8For any∈0＞0,we have

ProofBecause E is non-increasing,

Let∈0,τ1(t,s),τ2(t,s)＞0 andfor s∈R+.The function K is non-decreasing,and therefore,

Using this inequality,we get

Let G?denote the dual function of G de fined by

Thanks to(H4),G′is increasing and de fines a bijection from R+to R+,and then,for any t∈R+,the function s?→ts?G(s)reaches its maximum on R+at the unique point(G′)?1(t).Therfore

Using the general Young’s inequality:t1t2≤G(t1)+G?(t2),for

and

we get

Using the fact that G?(t)≤t(G′)?1(t),we get

Condition(2.8)implies that

Then,using the fact that(G′)?1is non-decreasing(thanks to(H4)),we get,for

(thanks to(2.8),(2.11)and the de finition of M(t,s)),we obtain

thus(3.25)holds.

Using(3.23),(3.24)and(3.25),we see that,in both cases,

Let τ＞0 and

We have F～E(because I6～E andis non-increasing)and,using(3.26),

Thanks to(1.1),the last term of(3.28)vanishes.Then,for τ＞0 such that

4 Proof of Teorem 2.2

In this section,we treat the case when(1.1)does not hold which is more realistic from the physics point of view.We will estimate the last term of(3.28)using the system(P2)resulting from differentiating(P)with respect to time

System(P2)is well posed for initial data U0∈D(A).Let E2be the second-order energy(the energy of(P2))de fined by E2(t)=E1(?t,ψt)(t),where E1(?,ψ)(t)=E(t),de fined by(2.10).A simple calculation(as for(3.1))implies that

Let τ=1 in(3.27).Thanks to(H5),H is linear and then(3.28)holds for H0=Id.Thus,

where G0is de fined in(2.19).Now,we proceed as in[7]and we use some ideas of[10].

Lemma 4.1For any∈＞0,we have

ProofWe distinguish two cases(corresponding to hypothesis(2.3)).

Using Young’s inequality and(3.5)(for ?xtinstead of ?x),we get for all∈＞0

On the other hand,by integrating by parts and using(3.6),we obtain

Inserting these last two inequalities into(4.5),multiplying by,integrating over[S,T],noting thatis non-increasing and using(3.1),we obtain(4.4).

Case 2a≡0:accordingto(2.2),we haveand then,by integration with respect to t and using the de finition of,E and E2and their non-increasingness,we get

Using the fact that(by vertue of Poincaré’s inequality)

Therefore,by integrating by parts the last integral with respect to x and noting thatis non-increasing,we have

Therefore,using Young’s inequality and(4.2),we estimate the last integral as follows:

This implies(4.4).

Now,exploiting(4.3)and(4.4)and choosing∈small enough,we get

and recalling(3.1)and the fact that F～E andis non-increasing,we have

To estimate the last term in(4.7),we distingish two cases.

Case 1a≡0 or(2.7)holds:we have G0=Id.Using(4.2),we get

Then we get in both cases

Inserting this inequality into(4.7)and choosing∈0small enough,we deduce that

Choosing S=0 in(4.8)and using the fact that G0(E)is non-increasing,we get

5 Proof of Theorem 2.3

We prove(2.21)by induction on n.For n=1,condition(2.20)coincides with(2.17),and(2.21)is exactly(2.18).

Now,suppose that(2.21)holds and let U0∈D(An+1)satisfying(2.20),for n+1 instead of n.We have Ut(0)∈D(An)(thanks to Theorem 2.0-3),Ut(0)satisfies(2.20)(because U0satisfies(2.20),for n+1)and Utsatisfies the first two equations and the boundary conditions of(P),and then the energy E2of(P2)(de fined in Section 4)also satisfies,for some positive constant?cn,

This proves(2.21),for n+1.The proof of Theorem 2.3 is completed.

Remark 5.1One important system related to(P)is the following system:

which results from the governing equations

taking into account the action on two tensors

This system looks more realistic than(P)from the physics point view.However the energy given by(2.10)is not dissipative.

We believe that such a system is worth looking at and a“modi fied”energy needs to be de fined,as well the functionals used to prove stability.

6 Applications

In this section,we give applications of our results of Section 2 to some Timoshenko-type systems.

6.1Timoshenko-Heat

We start by considering coupled Timoshenko-heat system on]0,L[under Fourier’s law of heat conduction and in the presence of an in finite memory acting on the first equation.That is,

where ?,ψ and θ are functions of(x,t)and denote the transverse displacement of the beam,the rotation angle of the fi lament,and the di ff erence temperature,respectively,ρi,ki,γ,κ,L are positive constants,and the functions a and g are as in Section 2.

From the third equation in(6.1)and the boundary conditions,we easily verify that

By solving this ordinary differential equation and using the initial data of θ,we get

So,we set

instead of θ0,and more importantly

which implies that Poincaré’s inequality is applicable forIn the sequel,we work withinstead of θ,but,for simplicity of notation,we use θ instead of

6.1.1Well-Posedness

By combining arguments from the Subsection 2.2 above and Subsection 6.1 of[14],one can easily establish the well-posedness of(6.1).For this purpose,we de fi ne η as in Subsection 2.2 and set

where Lgand its inner product are given in Subsection 2.2,and

If a≡0,the space H is equipped with the inner product

for any V=(v1,v2,v3,v4,v5)T,W=(w1,w2,w3,w4,w5)T∈H,and ifwe equip H with the inner product

for any V=(v1,v2,v3,v4,v5,v6)T,W=(w1,w2,w3,w4,w5,w6)T∈H.By letting

and

problem(6.1)can be written as

where,if a≡0,

for any V=(v1,v2,v3,v4,v5)T∈D(A)and,if

for any V=(v1,v2,v3,v4,v5,v6)T∈D(A).By noting that(6.2)is linear and exploiting the semigroup theory[19,32],one can easily prove the following:

Theorem 6.1For any n∈N and U0∈D(An),problem(6.2)has a unique solution

6.1.2Stability

Similarly to(P),we establish a general stability result for solutions of(6.1),under the hypotheses(H3)and(H4).we de fi ne the first-order energy of(6.1)by

Straightforward computations yield

Now,we give our first stability result.

Theorem 6.2Assume(1.1),(2.1),(2.3),(H3)and(H4)hold,and let U0∈H such that a≡0 or(2.7)or(2.11)is satis fied.Then,the energy E satisfies(2.12)withand G0is de fined in(2.19).

In order to prove our main result,we adopt several functionals from Section 2 and prove several lemmas.

Lemma 6.3The functional

satisfies,for any δ＞0,

ProofBy using equations(6.1),a simple integration leads to

Exploiting Young’s and Poincaré’s inequalities,(6.5)follows.

Lemma 6.4The functional

satisfies,for any δ,δ1＞0,

ProofDi ff erentiation of I3,using equations(6.1),gives

By using Young’s and Poincaré’s inequalities and recalling(3.2),(3.5)and(3.6),estimate(6.6)follows.

By using w de fined in(3.11)and repeating the proof of Lemma 3.7,we can easily establish this lemma.

Lemma 6.5The functional

satisfies,for any δ,∈,∈′＞0,

ProofDi ff erentiation of I3,using equations(6.1),leads to

Again,Young’s and Poincaré’s inequalities,(3.5),(3.12)and(3.13)give the desired result.

Finally,we need the following lemma:

Lemma 6.6The functional

for any δ＞0,

ProofBy using equations(6.1),a simple integration keeping in mind that θ stands forleads to

By using Young’s and Poincaré’s inequalities and(3.5),(6.8)is established.

For N,N2,N3,N4,we set

Direct calculations,using(6.4)-(6.8),yield

At this point,we distinguish two cases.

Case 1a≡0:in this case(6.9),reduces to

where c is a positive constant.

δ small enough,and N large enough,(6.9)becomes

We then proceed,as in Section 3,to complete the proof.

Remark 6.1When a≡0 or g satisfies(2.7),we obtain the exponential decay.That is,

When(1.1)does not hold,we have the following:

Theorem 6.3Assume(2.1),(2.3),(H3),and(H4)hold and let n∈N?and U0∈D(An)such that a≡0 or(2.7)or(2.20)is satis fied.Then,the energy E satisfies(2.21).

ProofThe proof goes exactly like that of Theorem 2.3.

6.2Timoshenko-Heat Type III

In this subsection,we consider a coupled Timoshenko-thermoelasticity system of type III on]0,L[in the presence of an in finite memory acting on the first equation.That is,

where ?,ψ,and θ are functions of(x,t)and denote the transverse displacement of the beam,the rotation angle of the fi lament,and the temperature displacement,respectively;ρi,ki,γ,κ,δ,L are positive constants and a and g are as in Section 2.We only give brief comments and state the main results and leave the proofs for the reader since they go exactly like the ones done in Subsection 6.1.

From the third equation in(6.13)and the boundary conditions,we easily verify that

By solving this ordinary differential equation and using the initial data of θ,we get

So,we set

and

instead of θ0and θ1,respectively,and more importantly

which implies that Poincaré’s inequality is applicable forIn the sequel,we work withinstead of θ,but,for simplicity of notation,we use θ instead of

6.2.1Well-Posedness

By combining arguments from the Subsection 2.2 above and subsection 6.1 of[14],one can easily establish the well-posedness of(6.13).For this purpose,we de fi ne η as in Subsection 2.2 and set

where Lgand its inner product are given in Subsection 2.2,and

If a≡0,the space H is equipped with the inner product

for any V=(v1,v2,v3,v4,v5,v6)T,W=(w1,w2,w3,w4,w5,w6)T∈H;and ifwe equip H with the inner product

for any V=(v1,v2,v3,v4,v5,v6,v7)T,W=(w1,w2,w3,w4,w5,w6,w7)T∈H.By letting

and

problem(6.13)can be written as

where,if a≡0,

for any V=(v1,v2,v3,v4,v5,v6)T∈D(A)and,if

for any V=(v1,v2,v3,v4,v5,v6,v7)T∈D(A).By noting that(6.14)is linear and exploiting the semigroup theory[19,32],one can easily show that Theorem 6.1 also holds for(6.14).Hence,the well-posedness for(6.13)is established.

6.2.2Stability

Similarly to(6.1),we establish a general stability result for solutions of(6.13),under the hypotheses(H3)and(H4).We de fi ne the first-order energy of(6.13)by

Straightforward computations yield

Remark 6.3By adopting the same functionals used in the subsection 6.1 and repeating the same steps,one can easily show that Theorems 6.2 and 6.3 remain valid for problem(6.13).In particular,we obtain the exponential stability if a≡0 or g decays exponentially.

AcknowledgementsThis work has been funded by KFUPM under the scienti fi c project IN141015.The authors would like to express their sincere thanks to KFUPM for its continuous support.

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