Research Article | Open Access
Boundary Stabilization of Memory Type for the Porous-Thermo-Elasticity System
We consider the one-dimensional viscoelastic Porous-Thermo-Elastic system. We establish a general decay results. The usual exponential and polynomial decay rates are only special cases.
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 . 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 denotes the time variable, is the space variable, the functions is the displacement, is the volume fraction of the solid elastic material, and the function is the temperature difference. The coefficients , , , , , , , and are positive constants. is a constant such that .
Casas and Quintanilla  considered the above system and used the semigroup theory and the method developed by Liu and Zheng  to establish the exponential decay of the solution under the boundary conditions of the formSoufyane  considered the following system: He proved that the solution of (1.3) decays exponentially if the function decays exponentially, and the solutions (1.3) decay polynomially if the function decays polynomially.
Recently Pamplona et al.  considered the follwing system: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 conditionsand 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 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.
Defining the convolution product operator byand differentiating equation (1.8) we obtainApplying Volterra's inverse operator, we getwhere the resolvent kernel satisfiesDenoting by , we arrive atA similar procedure leads towhere .
Since we are interested in relaxation functions of exponential or polynomial type and identities (2.5)-(2.6) involve the resolvent kernels we want to know if has the same properties. The following lemma answers this question. Let be a relaxation function and its resolvent kernel, that is,
Lemma 2.1 (see ). If is
a positive continuous function, then
is also positive and continuous. Moreover,
(1) If there exist positive constants and with such that then the function satisfies for all
(2) If for a given and if there exists a positive constant with , for which then the function satisfies
3. Exponential Decay
In this section we study the asymptotic behavior of the solutions of system (1.5)–(1.9), when the resolvent kernels satisfy, for , the following conditions:These assumptions imply that converges exponentially to that is,We define the first-order energy of system (1.5)–(1.9) byIn the sequel we define by . We are now ready to state our first result.
Theorem 3.1. Given , assume that (3.1) holds with Assume further that is a small number, then the energy satisfies the following decay estimates: Otherwise, where and are positive constants independent of the initial data.
Proof. The main idea is to construct a Lyapunov functional equivalent to . 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.
Proof. Multiplying first equation of (1.5) by , multiplying second equation of (1.5) by and third equation of (1.5) by , and integrating by parts over we obtainBy a summation of these three identities, we getusing (2.5), (2.6), and Lemma 2.2, we obtainwhich ends the proof of Lemma 3.2.
Proof. We multiply first equation of (1.5) by to obtainIntegrating by parts, we getSimilarly, we multiply second equation of (1.5) by and integrate over using integration by parts, to arrive atSumming the above two identities and using Poincare's and Young's inequalities and taking small, we deduce thatThe proof of Lemma 3.3 is completed.
Now, we introduce the Lyapunov functional. So, for large enough, letApplying Young's inequality and Poincaré's inequality to the boundary terms, we have, for By rewriting the boundary conditions (2.5)-(2.6) asand combining all above relations, using the fact that is a small number, the condition 17, taking large enough, very small, and , we obtainApplying inequality (2.14) with , using the trace formula we have, for some positive constant , the following estimate:Also, by direct computations, it is easy to check that, for large, we haveTherefore (3.20) becomesfor some positive constants At this point we distinguish two cases.
Case 1. If then (3.22) reducesA simple integration over yieldsBy using (3.21), estimate (3.5) is proved.Case 2. If or then (3.22) giveswhereIn this case we introduce the following functional:A simple differentiation of , using (3.25), leads toAgain a simple integration over yieldsA combination of (3.21), (3.27), and (3.29) then yields the estimate (3.6). This completes the proof of Theorem 3.1.
4. Polynomial Decay
In this section we study the asymptotic behavior of the solutions of system (1.5)–(1.9) when the resolvent kernels satisfyfor , and some positive constants These assumptions imply that decays polynomially to if That is,The following lemmas will play an important role in the sequel.
Lemma 4.1. (see ) Let , , and Then for , and for , where
Theorem 4.2. Given that , assume that small number and (4.1) hold. Then there exists a positive constant for which the energy satisfies, for all the following decay estimates: Otherwise,whereMoreover, iffor some satisfying (4.8), then (4.7) reduces to (4.6).
Proof. By using (4.1) in (3.7), we easily see
defining the functional as in (3.16), we getApplying
inequality (2.14) for
the above inequalities and taking
large, then for some positive constant
such that and
From (4.1) we get
(4.15) into (4.13) and using (3.21), we deduce thatHere,
we distinguish two cases.
Case 1 (). In this case (4.16) reduces toA simple integration over givesAs a consequence of (4.18) and the fact that , we easily verify thatTherefore, by using Lemma 4.1, (3.3), and (3.21), we havewhich implies thatConsequently we haveInserting (4.22) into (4.13), with we deduce thatA simple integration then leads towhere By using (3.21), estimate (4.6) is established.
Case 2 ( or ). In this case (4.16) gives thatwhereWe then introduce the following functional whereBy using (4.1), it is easy to show that, for some , we havewhereHence, simple calculations giveThanks to (4.25)–(4.30), we haveBy using the fact thatestimate (4.31) givesA simple integration of (4.33) over then leads toTherefore, using (4.27) we get, for all, Again, recalling (3.21) and using the continuity and the boundedness of , estimate (4.7) is established.
If, in addition,for some , satisfying (4.8) then (4.35) takes the formBy repeating (4.18)–(4.24), the desired estimate is established.
The authors are deeply grateful to the referees for their valuable comments.
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Copyright © 2009 Abdelaziz Soufyane et al. 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.