Boundary Stabilization of Memory Type for the Porous-Thermo-Elasticity System

and Applied Analysis 3 and the boundary conditions u 0, t v 0, t θ 0, t θ L, t 0, t ≥ 0, 1.7 u L, t − ∫ t 0 g1 t − s μux L, s bv L, s ds, 1.8 v L, t − ∫ t 0 g2 t − s αvx L, s ds. 1.9 Our main interest concerns the asymptotic behavior of the solution of the system above. That is, whether the dissipation given by the boundary memory effect is strong enough to stabilize the whole system. And what type of rate of decay may we expect exponential decay or polynomial decay? . We obtain an exponential decay or polynomial decay result under some conditions on gi i 1, 2 . Our proof is based on the multiplier techniques. This work is divided into four sections. In Section 2 we introduce some notations and some material needed for our work. In Section 3 we state and prove the exponential decay of the solutions of our studied system. Section 4 is devoted to the polynomial decay. 2. Preliminaries In this section we introduce some notations and we study the existence of regular and weak solutions to system 1.5 – 1.9 . First, we will use 1.8 and 1.9 to estimate the boundary terms ux L, t and vx L, t . Defining the convolution product operator by g ∗ φ t ∫ t 0 g t − s φ s ds, 2.1 and differentiating equation 1.8 we obtain μux L, t bv L, t 1 g1 0 g ′ 1 ∗ u L, t − 1 g1 0 ut L, t ∀t ≥ 0. 2.2 Applying Volterra’s inverse operator, we get μux L, t bv L, t − 1 g1 0 ut L, t k1 ∗ u L, t , ∀t ≥ 0, 2.3 where the resolvent kernel k1 satisfies k1 t 1 g1 0 g ′ 1 ∗ k1 t − 1 g1 0 g ′ 1 t . 2.4 4 Abstract and Applied Analysis Denoting by η1 1/g1 0 , we arrive at μux L, t bv L, t −η1 ut L, t k1 0 u L, t − k1 t u L, 0 k′ 1 ∗ u L, t , ∀t ≥ 0. 2.5 A similar procedure leads to αvx L, t −η2 vt L, t k2 0 v L, t − k2 t v L, 0 k′ 2 ∗ v L, t , ∀t ≥ 0, 2.6 where η2 1/g2 0 . Reciprocally, taking, in a natural way, the initial data u0 L v0 L 0, the identities 2.5 and 2.6 imply 1.8 and 1.9 . Since we are interested in relaxation functions of exponential or polynomial type and identities 2.5 2.6 involve the resolvent kernels ki i 1, 2 , we want to know if ki has the same properties. The following lemma answers this question. Let h be a relaxation function and k its resolvent kernel, that is, k t − k ∗ h t h t . 2.7 Lemma 2.1 see 11 . If h is a positive continuous function, then k is also positive and continuous. Moreover, 1 If there exist positive constants c0 and γ with c0 < γ such that h t ≤ c0e−γt, 2.8 then the function k satisfies k t ≤ c0 γ − γ − − c0 e − , 2.9 for all 0 < < γ − c0. 2 If cp : sup t∈R ∫ t 0 1 t p 1 t − s −p 1 s −pds < ∞, 2.10 for a given p > 1 and if there exists a positive constant c0 with c0cp < 1, for which h t ≤ c0 1 t −p, 2.11 then the function k satisfies k t ≤ c0 1 − c0cp 1 t −p. 2.12 Abstract and Applied Analysis 5 Based on Lemma 2.1, we will use 2.5 2.6 instead of 1.8 1.9 . We then defineand Applied Analysis 5 Based on Lemma 2.1, we will use 2.5 2.6 instead of 1.8 1.9 . We then define g oφ t : ∫ t 0 g t − s |φ t − φ s |ds, g φ t : ∫ t 0 g t − s φ t − φ s ds. 2.13 By using Hölder’s inequality for 0 ≤ μ ≤ 1, we have | g φ t | ≤ (∫ t 0 |g s | 1−μ ds ) |g|oφ t . 2.14 Lemma 2.2 see 12 . If g, φ ∈ C1 R , then g ∗ φ φt −12g t |φ t | 2 1 2 g ′oφ − 1 2 d dt ( g oφ − (∫ t 0 g s ds ) |φ t | ) . 2.15 3. Exponential Decay In this section we study the asymptotic behavior of the solutions of system 1.5 – 1.9 , when the resolvent kernels ki i 1, 2 satisfy, for γi > 0, the following conditions: ki 0 > 0, ki t ≥ 0, k′ i t ≤ 0, k′′ i t ≥ −γik′ i t . 3.1 These assumptions imply that k′ i converges exponentially to 0, that is, 0 ≤ −k′ i t ≤ Ce−γit. 3.2 We define the first-order energy of system 1.5 – 1.9 by E t : 1 2 ∫L 0 [ ρ|ut x, t | J |vt x, t | c|θ x, t | μ|ux x, t | ] dx 1 2 ∫L 0 [ α|vx x, t | ξ|v x, t | 2bux x, t v x, t ] dx η1 2 k1 t u2 L, t − k′ 1 ou L, t η2 2 k2 t v2 L, t − k′ 2ov L, t . 3.3 In the sequel we define by V1 : {u ∈ H1 0, L : u 0 0}. We are now ready to state our first result. 6 Abstract and Applied Analysis Theorem 3.1. Given u0, u1 , v0, v1 , θ0 ∈ V1 × L2 0, L ×L2 0, L , assume that 3.1 holds with sup t∈ t0,∞ ki t small enough. 3.4 Assume further that b is a small number, then the energy E satisfies the following decay estimates: E t ≤ c1E 0 e−ωt, if u0 L v0 L 0. 3.5


Introduction
An increasing interest has been developed in recent years to determine the decay behavior of the solutions of several elasticity problems.It is known that combining the elasticity equations with thermal effects provokes stability of solutions in the one-dimensional case 1 .Several results concerning the exponential or the polynomial decay of solutions for the thermoelastic systems were obtained by 2-6 .
A sample model describing the one-dimensional porous-thermo-elasticity, which was developed in 7, 8 , is given by the following system: where t denotes the time variable, x is the space variable, the functions uis the displacement, v is the volume fraction of the solid elastic material, and the function θ is the temperature difference.The coefficients ρ, μ, J, α, ξ, τ, c, and κ are positive constants.b is a constant such that b 2 < μξ.
Casas and Quintanilla 7 considered the above system and used the semigroup theory and the method developed by Liu and Zheng 4 to establish the exponential decay of the solution under the boundary conditions of the form u x, t v x x, t θ x x, t 0, x 0, L, t ∈ 0, ∞ .1.2 Soufyane 9 considered the following system: He proved that the solution of 1.3 decays exponentially if the function g decays exponentially, and the solutions 1.3 decay polynomially if the function g decays polynomially.
Recently Pamplona et al. 10 considered the follwing system:

1.4
They proved that the system is not exponential stable, and they showed that the solution decays polynomially.
In this paper we are concerned with the following model: with the initial conditions and the boundary conditions Our main interest concerns the asymptotic behavior of the solution of the system above.That is, whether the dissipation given by the boundary memory effect is strong enough to stabilize the whole system.And what type of rate of decay may we expect exponential decay or polynomial decay? .We obtain an exponential decay or polynomial decay result under some conditions on g i i 1, 2 .Our proof is based on the multiplier techniques.This work is divided into four sections.In Section 2 we introduce some notations and some material needed for our work.In Section 3 we state and prove the exponential decay of the solutions of our studied system.Section 4 is devoted to the polynomial decay.

Preliminaries
In this section we introduce some notations and we study the existence of regular and weak solutions to system 1.5 -1.9 .First, we will use 1.8 and 1.9 to estimate the boundary terms u x L, t and v x L, t .
Defining the convolution product operator by where the resolvent kernel k 1 satisfies Denoting by η 1 1/g 1 0 , we arrive at A similar procedure leads to where η 2 1/g 2 0 .
Reciprocally, taking, in a natural way, the initial data u 0 L v 0 L 0, the identities 2.5 and 2.6 imply 1.8 and 1.9 .
Since we are interested in relaxation functions of exponential or polynomial type and identities 2.5 -2.6 involve the resolvent kernels k i i 1, 2 , we want to know if k i has the same properties.The following lemma answers this question.Let h be a relaxation function and k its resolvent kernel, that is, for a given p > 1 and if there exists a positive constant c 0 with c 0 c p < 1, for which 11

2.12
Abstract and Applied Analysis 5 Based on Lemma 2.1, we will use 2.5 -2.6 instead of 1.8 -1.9 .We then define

Exponential Decay
In this section we study the asymptotic behavior of the solutions of system 1.5 -1.9 , when the resolvent kernels k i i 1, 2 satisfy, for γ i > 0, the following conditions: These assumptions imply that k i converges exponentially to 0, that is, We define the first-order energy of system 1.5 -1.9 by

3.3
In the sequel we define by V 1 : {u ∈ H 1 0, L : u 0 0}.We are now ready to state our first result.
Assume further that b is a small number, then the energy E satisfies the following decay estimates: Otherwise, where c 1 and ω are positive constants independent of the initial data.
Proof.The main idea is to construct a Lyapunov functional L t equivalent to E t .To do this we use the multiplier techniques.The proof of Theorem 3.1 will be achieved with the help of a sequence of lemmas.
Lemma 3.2.Under the assumptions of Theorem 3.1, the energy of the solution of 1.5 -1.9 satisfies

3.7
Proof.Multiplying first equation of 1.5 by u t , multiplying second equation of 1.5 by v t and third equation of 1.5 by θ, and integrating by parts over 0, L , we obtain

3.8
Abstract and Applied Analysis 7 By a summation of these three identities, we get where ε 0 is a small positive number.
Proof.We multiply first equation of 1.5 by 2xu x 1 − ε 0 u to obtain

3.12
Integrating by parts, we get

3.13
Similarly, we multiply second equation of 1.5 by 2xv x 1 − ε 0 v and integrate over 0, L , using integration by parts, to arrive at

3.14
Summing the above two identities and using Poincare's and Young's inequalities and taking ε 0 small, we deduce that

3.15
The proof of Lemma 3.3 is completed.Now, we introduce the Lyapunov functional.So, for N > 0 large enough, let

3.16
Applying Young's inequality and Poincaré's inequality to the boundary terms, we have, for ε > 0, x L, t .

3.17
By rewriting the boundary conditions 2.5 -2.6 as and combining all above relations, using the fact that b is a small number, the condition 17, taking N large enough, ε 0 very small, and ε ε 0 , we obtain

3.19
Applying inequality 2.14 with μ 1/2, using the trace formula we have, for some positive constant c 0 , the following estimate:

3.20
Also, by direct computations, it is easy to check that, for N large, we have for some positive constants ω.At this point we distinguish two cases.

3.24
By using 3.21 , estimate 3.5 is proved. where

3.26
In this case we introduce the following functional: 2 t e ωs ds.

3.27
A simple differentiation of F, using 3.25 , leads to dF dt t ≤ −ωF t .

3.28
Again a simple integration over 0, t yields A combination of 3.21 , 3.27 , and 3.29 then yields the estimate 3.6 .This completes the proof of Theorem 3.1.

Polynomial Decay
In this section we study the asymptotic behavior of the solutions of system 1.5 -1.9 when the resolvent kernels k i i 1, 2 satisfy for q 1 , q 2 / 0, 0 , 0 ≤ q i < 1/2, and some positive constants γ i .These assumptions imply that k i decays polynomially to 0 if q i > 0. That is, The following lemmas will play an important role in the sequel.
Lemma 4.1.(see [13]) Let p > 1, 0 ≤ r < 1, and t ≥ 0. Then for 0 < r, and for r 0, where , assume that b small number and 4.1 hold.Then there exists a positive constant λ > 0, for which the energy E satisfies, for all t ≥ 0, the following decay estimates: Otherwise, where for some r, satisfying 4.8 , then 4.7 reduces to 4.6 .
Proof.By using 4.1 in 3.7 , we easily see that

4.10
By defining the functional L t as in 3.16 , we get

4.11
Applying inequality 2.14 for k i with μ p 1 2 /2 p 1 1 if i 1 and μ p 2 2 /2 p 2 1 if i 2, we get Using the above inequalities and taking N large, then for some positive constant c 1 , we obtain

4.14
Consequently we have

4.15
Inserting 4.15 into 4.13 and using 3.21 , we deduce that

4.16
Here, we distinguish two cases.
Case 1 u 0 L v 0 L 0 .In this case 4.16 reduces to

4.17
A simple integration over 0, t gives 4.20 which implies that

4.21
Consequently we have −k 1 t 1 q 1 ou L −k 2 t 1 q 2 ov L .

4.22
Inserting 4.22 into 4.13 , with u 0 L v 0 L 0, we deduce that d dt L t ≤ −C L t 1 1/ p 1 .
Case 2 u 0 L / 0 or v 0 L / 0 .In this case 4.16 gives that where

4.26
Abstract and Applied Analysis 15 We then introduce the following functional H t : L t − g t , where g t : C 1 1 t − 1−r p 1 t 0 k 2 1 s 1 s 1−r p 1 ds C 2 1 t − 1−r p 1 t 0 k 2 2 s 1 s 1−r p 1 ds.

4.27
By using 4.1 , it is easy to show that, for some t 0 > 0, we have

4.31
By using the fact that L t 1 1/ 1−r p 1 H t g t 1 1/ 1−r p 1 ≥ H t 1 1/ 1−r p 1 g t