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Volume 2010, Article ID 748789, 15 pagesdoi:10.1155/2010/748789 Research Article Exponential Stability and Global Attractors for a Thermoelastic Bresse System Zhiyong Ma College of Scien

Volume 2010, Article ID 748789, 15 pages

doi:10.1155/2010/748789

Research Article

Exponential Stability and Global Attractors for

a Thermoelastic Bresse System

Zhiyong Ma

College of Science, Shanghai Second Polytechnic University, Shanghai 201209, China

Correspondence should be addressed to Zhiyong Ma,mazhiyong1980@hotmail.com

Received 13 September 2010; Accepted 29 October 2010

Academic Editor: E Thandapani

Copyrightq 2010 Zhiyong Ma This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We consider the stability properties for thermoelastic Bresse system which describes the motion

of a linear planar shearable thermoelastic beam The system consists of three wave equations and two heat equations coupled in certain pattern The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations We use multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of the longitudinal or

of the shear angle displacement Moreover, the existence of the global attractor is firstly achieved

1 Introduction

In this paper, we will consider the following system:

ρhw 1tt  Ehw 1x − kw3 − αθ 1tx − kGhφ2 w 3x  kw1



ρhw 3tt  Ghφ2 w 3x  kw1



x  kEhw 1x − kw3 − kαθ 1t , 1.2

ρIφ 2tt  EIφ 2xx − Ghφ2 w 3x  kw1



together with initial conditions

w1x, 0  u0x, w 1t x, 0  v0x, φ2x, 0  φ0x,

φ 2t x, 0  ψ0x, w3x, 0  w0x, w 3t x, 0  ϕ0x,

θ1x, 0  θ0x, θ 1t x, 0  η0x, θ3x, 0  ξ0x

1.6

and boundary conditions

w1x, t  w 3x x, t  φ2x, t  θ1x, t  θ3x, t  0, for x  0, 1, 1.7

where w1, w3, and φ2 are the longitudinal, vertical, and shear angle displacement, θ1and θ3

are the temperature deviations from the T0 along the longitudinal and vertical directions, E,

G, ρ, I, m, k, h, and c are positive constants for the elastic and thermal material properties.

From this seemingly complicated system, very interesting special cases can be obtained In particular, the isothermal system is exactly the system obtained by Bresse1 in

1856 The Bresse system,1.1–1.3 with θ1, θ3removed, is more general than the well-known

Timoshenko system where the longitudinal displacement w1is not considered If both θ1and

w1 are neglected, the Bresse thermoelastic system simplifies to the following Timoshenko thermoelastic system:

ρhw 3tt  Ghφ2 w 3x



x , ρIφ 2tt  EIφ 2xx − Ghφ2 w 3x



− αθ 3tx , ρcθ 3tt  θ 3xxt  θ 3xx − αT0φ 2tx ,

1.8

which was studied by Messaoudi and Said-Houari2 For the boundary conditions

w3x, t  φ2x, t  θ 3x x, t  0, at x  0, l, 1.9

they obtained exponential stability for the thermoelastic Timoshenko system 1.8 when

E  G; later, they proved energy decay for a Timoshenko-type system with history in

thermoelasticity of type III3, and this paper is similar to 2 with an extra damping that comes from the presence of a history term; it improves the result of2 in the sense that the case of nonequal wave speed has been considered and the relaxation function g is allowed to decay exponentially or polynomially We refer the reader to4 10 for the Timoshenko system with other kinds of damping mechanisms such as viscous damping, viscoelastic damping of

Boltzmann type acting on the motion equation of w3 or φ2 In all these cases, the rotational

displacement φ2of the Timoshenko system is effectively damped due to the thermal energy dissipation In fact, the energy associated with this component of motion decays

exponen-tially The transverse displacement w3is only indirectly damped through the coupling, which can be observed from1.2 The effectiveness of this damping depends on the type of coupling and the wave speeds When the wave speeds are the sameE  G, the indirect damping is

actually strong enough to induce exponential stability for the Timoshenko system, but when the wave speeds are different, the Timoshenko system loses the exponential stability This phenomenon has been observed for partially damped second-order evolution equations We would like to mention other works in11–15 for other related models

Recently, Liu and Rao16 considered a similar system; they used semigroup method and showed that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement Otherwise, only a polynomial-type decay rate can be obtained; their main tools are the frequency-domain characterization of exponential decay obtained by

Pr ¨uss17 and Huang 18 and of polynomial decay obtained recently by Mu˜noz Rivera and Fern´andez Sare5 For the attractors, we refer to 19–24

In this paper, we consider system1.1–1.5; that is, we use multiplier techniques to

prove the exponential stability result only for E  G However, from the theory of elasticity,

E and G denote Young’s modulus and the shear modulus, respectively These two elastic

moduli are not equal since

where ν ∈ 0, 1/2 is the Poisson’s ratio Thus, the exponential stability for the case of E  G is

only mathematically sound However, it does provide useful insight into the study of similar models arising from other applications

Here we state and prove a decay result in the case of equal wave speeds propagation Define the state spaces

H  H1

0× H1

∗× H1

0 × H1

0×L25

where

H1

∗ 



f ∈ H10, 1 |

1

0

f x  0



The associated energy term is given by

E t  1

2

1

0

Eh w 1x − kw32 Ghφ2 w 3x  kw1

2 EIφ2

2x

 ρh

w 1t2  w2

3t



 ρIφ2

2t

ρc

T0



θ 1t2  θ2

1x  θ2 3



dx.

2.3

By a straightforward calculation, we have

dE t

dt  −1

T0



θ 1xt2 θ 3x2

From semigroup theory25,26, we have the following existence and regularity result; for the explicit proofs, we refer the reader to16

Lemma 2.1 Let u0x, w0x, ϕ0x, θ0x, v0x, φ0x, ψ0x, η0x, ξ0x ∈ H be given Then

problem1.1–1.5 has a unique global weak solution ϕ, ψ, θ verifying

w3x, t ∈ CR, H1

∗0, 1∩ C1

R, L20, 1,



w1x, t, φ2x, t, θ1x, t, θ3x, t∈ CR, H1

00, 1∩ C1

R, L20, 1. 2.5

We are now ready to state our main stability result

Theorem 2.2 Suppose that E  G and u0x, w0x, ϕ0x, θ0x, v0x, φ0x, ψ0x, η0x,

ξ0x ∈ H Then the energy Et decays exponentially as time tends to infinity; that is, there exist two

positive constants C and μ independent of the initial data and t, such that

The proof of our result will be established through several lemmas

Let

I1

1

0

where f is the solution of

Lemma 2.3 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has, for all ε1> 0,

I1t

dt ≤ −EI

2 φ 2x2 ρIφ 2t2 ε1



w 3t2 w 1x − kw32

 Cε1θ 3x2 θ 1xt2 φ 2t2

.

2.9

Proof.

dI1

dt  −EIφ 2x2 ρIφ 2t2−

1

0

αθ 3x φ2dx

− kEh

1

0

w 1x − kw3fdx − kα

1

0

θ 1t fdx  ρh

1

0

w 3t f t dx,

2.10

By using the inequalities

1

0

f x2dx ≤

1

0

φ22dx ≤

1

0

φ 2x2 dx,

1

0

f t2dx ≤

1

0

f tx2dx ≤

1

0

φ22t dx,

2.11

and Young’s inequality, the assertion of the lemma follows

I2 ρcρh

1

0

x

0

θ 1t dy

Lemma 2.4 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has, for all ε2> 0,

dI2t

dt ≤ −αρhT0

2

1

0

w 1t2dx  C ε2θ 1xt2 w 3t2

 ε2



w 1x − kw32 φ2 w 3x  kw12

.

2.13

Proof Using1.4 and 1.1, we get

I2t

dt  ρcρh

1

0

x

0

θ 1tt dy

w 1t dx  ρcρh

1

0

x

0

θ 1t dy

w 1tt dx

 ρh

1

0

x

0

θ 1xxt  θ 1xx − αT0w 1tx − kw 3t dy

w 1t dx



1

0

x

0

θ 1t dy

Ehw 1x − kw3 − αθ 1tx − KGhφ2 w 3x  kw1



dx

 ρh

1

0

θ 1xt  θ 1x w 1t dx − ρhαT0

1

0

w21t dx  ρhk

1

0

x

0

w 3t dy

w 1t dx

 ρhEh

1

0



θ 1xt w1 kθ 1t w3 αθ2

1t



dx

− ρckGh

1

0

x

0

θ 1t dy

φ2 w 3x  kw1



dx.

2.14

The assertion of the lemma then follows, using Young’s and Poincar´e’s inequalities

Let

I3 ρcρI

1

0

x

0

θ3dy

Lemma 2.5 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has, for all ε3> 0,

dI3

dt ≤ −αρIT0

2 φ 2t2 Cε3θ 3x2 ε3φ 2x2 ε3φ2 w 3x  kw12. 2.16

Proof Using1.3 and 1.5, we have

dI3

dt  ρcρI

1

0

x

0

θ 3t dy

φ 2t dx  ρcρI

1

0

x

0

θ3dy

φ 2tt dx

 ρI

1

0

x

0



θ 3xx − αT0φ 2xt



dyφ 2t dx

 ρc

1

0

x

0

θ3dy

EIφ 2xx − Ghφ2 w 3x  kw1



− αθ 3x



dx

 ρI

1

0

θ 3x φ 2t dx − αIT0

1

0

φ 2t2dx  ρcEI

1

0

θ3φ 2x dx

− ρcGh

1

0

x

0

θ3dy

φ2 w 3x  kw1



dx − αρc

1

0

θ32dx.

2.17

Then, using Young’s and Poincar´e’s inequalities, we can obtain the assertion

Next, we set

I4 hρI

1

0

φ 2t



φ2 w 3x  kw1



dx  hρI

1

0

Lemma 2.6 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has, for all ε4> 0,

dI4

dt ≤ −Gh2

2

1

0



φ2 w 3x  kw1

2

dx  C ε4θ 3x2 θ 1xt2

khρI 2



φ 2t2 w 1t2

 Cε4φ 2x2 ε4w 1x − kw32.

2.19

Proof Letting 1  I 1

0φ 2t φ2 w 3x  kw1dx, 2  hρI 1

0φ 2x w 3t dx, then using 1.2, 1.3,

we have

1 ρI

1

0

φ 2tt



φ2 w 3x  kw1



dx  hρI

1

0

φ 2t



φ2 w 3x  kw1



t dx

 hEI

1

0

φ 2xx



φ2 w 3x  kw1



dx − Gh2

1

0



φ2 w 3x  kw1

2

dx

− αh

1

0

θ 3x



φ2 w 3x  kw1



dx  hρI

1

0

φ2

2t dx  hρI

1

0

φ 2t w 3x  kw1t dx,

2 Iρh

1

0

φ 2xt w 3t dx  Iρh

1

0

φ 2x w 3tt dx

 −Iρh

1

0

φ 2t w 3xt dx  IGh

1

0

φ 2x



φ2 w 3x  kw1



x dx

 IkEh

1

0

φ 2x w 1x − kw3dx − αIk

1

0

φ 2x θ 1t dx.

2.20

Noticing that E  G, then

I4  1 2

 −Gh2

1

0



φ2 w 3x  kw1

2

dx − αh

1

0

θ 3x



φ2 w 3x  kw1



dx  hρI

1

0

φ 2t2dx

 khIρ

1

0

φ 2t w 1t dx  IkEh

1

0

φ 2x w 1x − kw3dx − αIk

1

0

φ 2x θ 1t dx.

2.21

Then, using Young’s inequality, we can obtain the assertion

We set

I5 −hρ

1

0

w 3t w 1x − kw3dx − hρ

1

0

w 1t



φ2 w 3x  kw1



Lemma 2.7 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has, for all ε5> 0,

dI5

dt ≤ −kEh

2 w 1x − kw32−ρh

2 w 1t2 kρhw 3t2

ρh

2 φ 2t2 Cε5θ 1xt2 kGh  ε5φ2 w 3x  kw1



2.

2.23

Proof Let 1  −hρ 1

0w 3t w 1x −kw3dx, 2  −hρ 1

0w 1t φ2w 3x kw1dx, then using 1.1,

1.2, we have

1 −Gh

1

0



φ2 w 3x  kw1



x w 1x − kw3dx − kEh

1

0

w 1x − kw32dx

 αk

1

0

θ 1t w 1x − kw3dx  kρh

1

0

w23t − ρh

1

0

w 3t w 1xt dx,

2 −Eh

1

0

w 1x − kw3xφ2 w 3x  kw1



dx  α

1

0

θ 1xt



φ2 w 3x  kw1



dx

 kGh

1

0



φ2 w 3x  kw1

2

dx − ρh

1

0

w 1t2dx − ρh

1

0

w 1t φ 2t dx  ρh

1

0

w 1tx w 3t dx.

2.24

Then, noticing E  G, again, from the above two equalities and Young’s inequality, we can

obtain the assertion

Next, we set

I6 −ρh

1

0

w 3t w3dx − ρh

1

0

Lemma 2.8 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has

dI6

dt ≤ −ρhw 3t2 w 1t2

 Cθ 1xt2 Cφ 2x2. 2.26

Proof Using1.1, 1.2, we have

I6  −ρh

1

0

w 3t2dx − ρh

1

0

w21t dx  Eh

1

0

w 1x − kw32

dx

 Gh

1

0



φ2 w 3x  kw1



w 3x  kw1dx − α

1

0

θ 1t w 1x − kw3dx.

2.27

Noticing2.3 and 2.4, we have that ∃C1 > 0 satisfy the following:

−α

1

0

θ 1t w 1x − kw3dx ≤ C1θ 1xt2− Ehw 1x − kw32. 2.28 Similarly,

Gh

1

0



φ2 w 3x  kw1



w 3x  kw1dx

 Ghφ2 w 3x  kw12− Gh

1

0



φ2 w 3x  kw1



φ2dx

≤ C1φ 2x2.

2.29

Then, insert2.28 and 2.29 into 2.27, and the assertion of the lemma follows

Now, we set

I7 ρc

1

0

θ 1t θ1dx  1

Lemma 2.9 Letting w1, w3, φ2, θ1, θ3be a solution of 1.1–1.5, then one has, for all ε7> 0,

dI7

dt ≤ −1

2θ 1x2 ρcθ 3xt2 Cε7w 1t2 w 3t2

Proof Using1.5, we have

dI7

dt  −θ 1x2 αT0

1

0

w 1t θ 1x dx  αT0k

1

0

w 3t θ 1x dx  ρcθ 1t2. 2.32

Then, using Young’s and Poincar´e’s inequalities, we can obtain the assertion

Now, letting N, N1, N2, N3, N4, N5, N6, N7 > 0, we define the Lyapunov functional F

as follows:

F  NE  N1I1 N2I2 N3I3 N4I4 N5I5 N6I6 N7I7. 2.33

By using2.4, 2.9, 2.13, 2.16, 2.19, 2.23, 2.26, and 2.31, we have

dF

dt ≤ Υ1θ 1xt2 Υ2θ 3x2 Υ3φ 2x2 Υ4w 1t2 Υ5φ 2t2

 Υ6φ2 w 3x  kw12 Υ7w 1x − kw32 Υ8w 3t2 Υ9θ 1x2,

2.34

where

Υ1 −N

T0  Cε1N1 N2C ε2  N4C ε4  N5C ε5  N7C1 ρcN7,

Υ2 −N

T0  Cε1N1 N3C ε3  N4C ε4,

Υ3 −N1EI

2  ε3N3 Cε4N4 C1N6,

Υ4 −αρhT0N2

2  khρIN4

2 − ρhN6 N7C ε7,

Υ5 −αρhT0N3

2  khρIN4

2  N1ρI  N1C ε1  ρhN5

Υ6 −Gh2N4

2  kGhN5 N5ε5 N3ε3 N2ε2,

Υ7 −kEhN5

2  N4ε4 N1ε1 N2ε2,

Υ8 −N6ρh  N5kρh  C ε2N2 N1ε1 N7C ε7,

Υ9 −N7

2  N2C ε2.

2.35

We can choose N big enough, ε1, , ε8small enough, and

N1 4, N6,

N2 4,

N3 1, N6,

N4 5,

N6 2, N5, N7,

N7 2.

2.36

ThenΥ1, , Υ9 are all negative constants; at this point, there exists a constant ω > 0, and

2.34 takes the form

dF

dt ≤ −ωθ 1xt2 θ 1x2 θ 3x2 φ 2x2 w 1t2

φ 2t2 φ2 w 3x  kw12 w 1x − kw32 w 3t2

.

2.37

We are now ready to proveTheorem 2.2

Proof of Theorem 2.2 Firstly, from the definition ofF, we have

which, from2.37 and 2.38, leads to

d

Integrating 2.39 over 0, t and using 2.38 lead to 2.6 This completes the proof of

Theorem 2.2

3 Global Attractors

In this section, we establish the existence of the global attractor for system1.1–1.5

Setting v  w 1t , ϕ  w 3t , ψ  φ 2t , η  θ 1t, then,1.1–1.5 can be transformed into the system

w 1t  v,

w 3t  ϕ,

φ 2t  ψ,

θ 1t  η,

ρhv t  Ehw 1x − kw3 − αθ1x − kGhφ2 w 3x  kw1



, ρhϕ t  Ghφ2 w 3x  kw1



x  kEhw 1x − kw3 − kαθ1, ρIψ t  EIφ 2xx − Ghφ2 w 3x  kw1



− αθ 3x , ρcη t  θ 1xxt  θ 1xx − αT0w 1tx − kw 3t ,

ρcθ 3t  θ 3xx − αT0φ 2tx

3.1

We consider the problem in the following Hilbert space:

H  H1

0× H1

∗× H1

0 × H1

0×L25

Recall that the global attractor of St acting on H is a compact set A ⊂ H enjoying the

following properties:

1 A is fully invariant for St, that is, StA  A for every t ≥ 0;

2 A is an attracting set, namely, for any bounded set R ⊂ H,

lim

where δHdenotes the Hausdorff semidistance on H

More details on the subject can be found in the books23,26,27

Remark 3.1 The uniform energy estimate2.6 implies the existence of a bounded absorbing setR∗⊂ H for the C0semigroup St Indeed, if R∗is any ball ofH, then for any bounded set

R ⊂ H, it is immediate to see that there exists tR ≥ 0 such that

for every t ≥ tR.

Moreover, if we define

R0

t≥0

it is clear that R0 is still a bounded absorbing set which is also invariant for St, that is,

StR0⊂ R0for every t ≥ 0.

In the sequel, we define the operator A as Af  −f xx with Dirichlet boundary

conditions It is well known that A is a positive operator on L2with domainDA  H2∩ H1

0

Moreover, we can define the powers A s of A for s ∈ R The space V 2s  DA s turns out to be

a Hilbert space with the inner product

where· stands for L2-inner product on L2

In particular, V−1  H−1, V0  L2, and V1  H1

0 The injection V s1 → V s2 is compact

whenever s1> s2 For further convenience, for s ∈ R, introduce the Hilbert space

Hs  V1s× V1s× V1s× V1s× V s5

Clearly,H0 H

Now, let z0  u0, w0, φ0, θ0, v0, ϕ0, ψ0, η0, ξ0, where R0 is the invariant, bounded

absorbing set of St given by Remark 3.1, and take the inner product in H0 of 3.1 and

A σ w1, A σ w3, A σ φ2, A σ θ1, A σ v, A σ ϕ, A σ ψ, A σ η, A σ θ3 to get

d

dt

Eh w 1x − kw32

σ  Ghφ2 w 3x  kw1



2

σ  EIφ22

1σ

ρhw 1t2

σ  w 3t2

σ



 ρIφ 2t2

σρc

T0



θ 1t2

σ  θ12

1σ θ32

σ



 − 2

T0



θ 1t2 1σ θ32

1σ



.

3.8

Proof...