QIN Yu-ming,LIU Zi-li

(Department of Applied Mathematics,Donghua University,Shanghai,201620,China)

Abstract:In this work,a Timoshenko system of type III of thermoelasticity with frictional versus viscoelastic under Dirichlet-Dirichlet-Neumann boundary conditions was considered.By exploiting energy method to produce a suitable Lyapunov functional,we establish the global existence and exponential decay of type-III case.

Key words:Timoshenko;global existence;energy decay;multiplier method;Lyapunov functional

§1.Introduction

In the present paper,we are concerned with

where g,h are two special functions,ρi,κi,μ,γ,δ,β are positive constants.We aim at investigating the behavior of solution in the case of equal speeds of propagation

Before we state and prove our main result,let us recall some results regarding the Timoshenko system.

In 1921,a simple system was proposed by Timoshenko[1]

which describes the transverse vibration of a beam of length L in its equilibrium configuration.Here t denotes the time variable,x is the space variable along the beam.The coefficients ρ,Iρ,E,I and K are respectively the density,the polar moment of inertia of a cross section and the shear modulus.

Together with boundary conditions of the form

is conservative,and so the total energy of the beam remains constant along the time.

Kim and Renardy[2]considered together with two boundary controls of the form

and used the multiplier techniques to establish an exponential decay result for the natural energy of system(1.2).They also provided numerical estimate to the eigenvalues of the operator which is associated with system(1.2).

Raposo et al.[3]studied following system

with homogeneous Dirichlet boundary conditions,and prove that the associated energy decays exponentially.

Soufyane and Wehbe[4]showed that it is possible to stabilize uniformly by using a unique locally distributed feedback.They studied

and prove that the uniform stability of(1.10)hold if and only if the wave speeds are equal,otherwise only the asymptotic stability has been proved.

Amar-Khodja et al.[5]considered a linear Timoshenko-type system with memory of the form in(0,L)×(0,+∞),together with homogeneous boundary conditions.They used the multiplier techniques and proved that the system is uniformly stable if and only if the wave speeds are equaland g decays uniformly.Precisely,they proved an exponential decay if g decays in an exponential rate and polynomially if g decays in a polynomial rate.They also required some extra technical conditions on both g′and g′′to obtain their results.

For Timoshenko system in thermoelasticity,River and Racke[6]considered

where φ,ψ and θ are functions of(x,t)which model the transverse displacement of the beam,the rotation angle of the filament,and the difference temperature respectively.Under appropriate conditions of σ,ρi,b,κ,γ,they proved several exponential decay results for the linearized system and a non-exponential stability result for the case of different wave speeds.

Messaoudi et al.[7]studied the following problem

where(x,t)∈ (0,L)× (0,+∞)and φ = φ(x,t)is the displacement vector,ψ = ψ(x,t)is the rotation angle of the filament,θ = θ(x,t)is the temperature difference,q=q(x,t)is the heat flux vector,ρ1,ρ2,ρ3,b,κ,γ,δ,τi,μ are positive constants.The nonlinear function σ is assumed to be sufficiently smooth and satisfies

Several exponential decay results for both linear and nonlinear cases have been established.

Guesmia and Messaoudi[8]studied the following system

with Dirichlet boundary conditions and initial data where a,b,g and h are specific functions and ρi,κ1,κ2and L are given positive constants.They established a general stability estimate using multiplier method and some properties of convex functions.Without imposing any growth condition on h at the origin,they showed that the energy of the system is bounded above by a quantity,depending on g and h,which tends to zero as time goes to infinity.

Ouchenane and Rahamoune[9]considered a on-dimensional linear thermoelastic system of Timoshenko system

where the heat flux is given by Cattaneo’s law.They established a general decay estimate where the exponential and polynomial decay rates are only particular cases.

§2.Preliminaries

In order to prove our main results,we formulate the following hypotheses

(H1) h:R → R is a differentiable nondecreasing function such that there exist constants∈′,c′,c′′＞ 0 and a convex and increasing function H:R → R of class C1(R)TC2(0,∞)satisfying H(0)=0 and H is linear on[0,∈′]or H′(0)=0 and H′′＞ 0 on(0,∈]such that

(H2)g:R+→ R+is a differentiable function such that

(H3)There exits a non-increasing differentiable function ξ:R+→ R+satisfying

Except all of the above,we also need the following lemmas to prove our results.See,e.g.,Zheng[10].

Lemma 2.1 Let A be a linear operator defined in a Hilbert space H,D(A)?H→H.Then the necessary and sufficient conditions for A being maximal accretive operator are

(1)Re(Ax,x)≤0, ?x∈D(A);

(2)R(I-A)=H.

Lemma 2.2 Suppose that A is m-accretive in a Banach space B,and U0∈D(A).Then problem(1.1)has a unique classical solution U such that

In proving the stability results of global solutions,the next lemma plays a key role.See,e.g.,Mo?noz Rivera[11].

Lemma 2.3 Suppose that y(t)∈ C1(R+),y(t)≥ 0,?t＞ 0,and satisfies

where 0≤ λ(t)≤L1(R+)and C0is a positive constant.Then we have

Furthermore,

(1) If λ(t)≤ C1e-δ0t,?t＞ 0,with C1＞ 0,δ0＞ 0 being constants,then

with C2＞0,δ＞0 being constants.

(2) If λ(t)≤ C3(1+t)-p,?t＞ 0,with p ＞ 1,C3＞ 0 being constants,then

with a constant C4＞0.

Lemma 2.4 If 1≤p≤∞and a,b≥0,then

See,e.g.,Adams[12].

§3.Global Existence

In this section,we establish the global existence,starting with the vector function U=(φ,u,ψ,v,θ,w)T,where u= φt,v= ψt,w= θt.We introduce as in[13]

then by(1.1)3,we have

The problem(1.1)can be written as the following

where the operator A is defined by

Let

which is a Hilbert space.For U=(u1,u2,u3,u4,u5,u6),V=(v1,v2,v3,v4,v5,v6)∈H,defines inner product

The domain of A is

we have the following global existence result.

Theorem 3.1 Let U0∈H,then problem(1.1)has a unique classical solution,that verifies

Proof The result follows from Theorem 3.1 provided we prove that A is a maximal accretive operator.In what follows,we prove that A is monotone.For any U∈D(A),and using the inner product,we obtain

using(H1)-(H3),we have

it follows that Re(AU,U)≤0,which implies that A is monotone.

Next,we prove that the operator I-A is subjective.Given B=(b1,b2,b3,b4,b5,b6)T∈H,we prove that there exists U=(u1,u2,u3,u4,u5,u6)∈D(A)satisfying

that is,

In order to solve(3.1),we consider the following variational formulation

where F:[H10(0,1)×H10(0,1)×H1?(0,1)]2→R is the bilinear form defined by

and G:[H10(0,1)×H10(0,1)×H1?(0,1)]→R is the linear functional given by

Now,for V=H10(0,1)×H10(0,1)×H1?(0,1)equipped with the norm

using local integral,we have,

for a constant α0＞ 0.Thus,B is coercive.

By Cauchy-Schwarz inequality and Poincar? inequality,we can easily get

similarly

According to Lax-Milgram Theorem,we can easily obtain unique

satisfying

Applying the classical elliptic regularity,it follows from(3.1)that

satisfying

The existence result has been proved.

§4.Exponential Stability

In this section,we establish the exponential estimate for the generalized solutions to problem(1.1).

Theorem 4.1 Now,we introduce the energy functional defined by

which satisfies

precisely

where C0and δ1are positive constants.

To prove Theorem 4.1,we will use the energy method to produce a suitable Lyapunov functional.This will be established through several lemmas.We have the following results.

Lemma 4.1 Let(φ,ψ,θ)be the solution of problem(1.1)and assume(H1)-(H3)hold.Then the energy E is non-increasing function and satisfies,?t≥ 0,

Proof Multiplying(1.1)1,(1.1)2and(1.1)3by φt,ψtand θt,respectively,and integrating over(0,1),summing them up,then using integration by parts and the boundary conditions,we obtain

Calculating the term

then we have

Using(H1)-(H3),we get

Thus Lemma 4.1 has been proved.

Lemma 4.2 Let(φ,ψ,θ)be the solution of problem(1.1)and assume(H1)-(H3)hold.The functional defining by

satisfies the estimate

Proof By using(1.1)2,we get

By using Young’s inequality and Poinc?are inequality,we obtain,?∈1＞ 0,

Similarly,we have

and

By using Lemma 2.4,we get

Then we have

There exists c ≥ max{c1,c2,c3,c4,c5,(c′′)2},∈≥ max{∈1, ∈2, ∈3, ∈4, ∈5,2∈6}such that

By combining all the above estimates,Lemma 4.2 is proved.

Lemma 4.3 Let(φ,ψ,θ)be the solution of problem(1.1)and assume(H1)-(H3)hold.Then the functional

satisfies the estimate

Proof By exploiting(1.1)2,(1.1)2and repeating the same procedure as in the above,we have

By using the Young’s inequality,we prove Lemma 4.3.

Lemma 4.4 Let(φ,ψ,θ)be the solution of problem(1.1)and assume(H1)-(H3)hold.Then the functional

satisfies the estimate

Proof By exploiting(1.1)3,we have

By using Young’s inequality,we prove Lemma 4.4.

Lemma 4.5 Let(φ,ψ,θ)be the solution of problem(1.1)and assume(H1)-(H3)hold.Then the functional

satisfies the estimate

Proof By using(1.1)1,(1.1)2and repeating the same procedure as in the above,we have

By using Young’s inequality and Poincar? inequality,we prove Lemma 4.5.

Lemma 4.6 For N sufficiently large,the functional defined by

where N and Niare positive real numbers to be chosen appropriately later,satisfies

Proof It is easily to get,?t≥ 0,

Combining Lemmas 3.1-3.5,(H3),we obtain

At this point,we choose our constants carefully.First,let us take N3＞ 0,then pick N,N2,∈7,c7,∈8so that

then we select N1,∈,c8such that

Finally,we choose c9, ∈9,N4,c′such that

and

Combining all above inequalities,there exists positive δ0＞ 0 such that

Then we have

which gives

Up to now,Lemma 4.6 has been proved.

Exploiting(4.4),we have

Thus,the proof of Theorem 4.1 is co mpleted.