On the Porous-Elastic System with Thermoelasticity of Type III and Distributed Delay: Well-Posedness and Stability

Abstract

The paper deals with a one-dimensional porous-elastic system with thermoelasticity of type III and distributed delay term. This model is dealing with dynamics of engineering structures and nonclassical problems of mathematical physics. We establish the well posedness of the system, and by the energy method combined with Lyapunov functions, we discuss the stability of system for both cases of equal and nonequal speeds of wave propagation.

1. Introduction

Let . For , we consider the following porous-elastic system: with the initial data and boundary conditions

Here, is the volume fraction of the solid elastic material, is the longitudinal displacement, and is the difference in temperatures. The parameters , are positive constants with . The integral represents the distributed delay term withwhich are time delays, is positive constant, and is an function such that

(Hyp1) is a bounded function satisfying

This type of problem was mainly based on the following equations for one-dimensional theories of porous materials with temperature where .

According to Green and Naghdis theory, the constitutive equations of system (5) are given by where are the thermal conductivity and is the thermal displacement whose time derivative is the empirical temperature , that is .

We substitute (9) in (5) with the condition , which results in

By using in the system (10), we find directly our system (1).

By using the multiplier techniques, the exponential decay results have been established. Next, in [1–3], the authors considered three types of thermoelastic theories based on an entropy equality instead of the usual entropy inequality (see [1–21] for more details).

According to the distributed delay, we mention, as a matter of course, the work by Nicaise and Pignotti in [16], where the authors studied the following system with distributed delay: and proved the exponential stability result with condition

See for example [8, 22, 23]. Hao and Wei [24] considered the following problem: and obtained the well-posedness and stability of system.

There are many other works done by the authors in this context; our work differs from all of them, since we took the delay in the second equation to make the distributed delay in the rotation angle of the filament, which makes the contributions clear and important. In addition, we established the well-posedness of the system, and we obtain the exponential decay rate when and the energy takes the algebraic rate for the case ; these results are mainly stated in Theorem 8.

In order to show the dissipativity of systems (1)–(3), we introduce the new variables and . So, problems (1)–(3) take the form with the initial data and boundary conditions

First, as in [16], taking the following new variable: then we obtain

Consequently, the problem was rewritten as where with the boundary and the initial conditions

Meanwhile, from (19) and (24), it follows that

So, by solving (25) and using (24), we get

Consequently, if we let we get and from (19), we have

So, by solving (29) and using (24), we get

Consequently, if we let we get

Then, the Poincaré’s inequality was used for and which are justified. A simple substitution shows that satisfies system (19) with initial data for and given as

Now, we use instead of and writing for simplicity.

2. Well-Posedness

In this section, we give the existence and uniqueness result of the system (19)–(24) using the semigroup theory.

First, we introduce the vector function and the new dependent variables ; then the system (19) can be written as follows: where is the linear operator defined by and is the energy space given by where

For every we equip with the inner product defined by

The domain of is given by

Clearly, is dense in . Now, we can give the following existence result.

Theorem 1. Let and assume that (4) holds. Then, there exists a unique solution of problem (19).
Moreover, if , then

Proof. First, we prove that the operator is dissipative. For any and by using (40), we have

For the third term of the right-hand side of (43), we have

By using Young’s inequality, we get

Substituting (44) and (45) into (43), using the fact that and (4), we obtained

Hence, the operator is dissipative.

Next, we prove the operator is maximal. It is sufficient to show that the operator is surjective.

Indeed, for any , we prove that there exists a unique such that

That is

We note that the last equation in (48) with has a unique solution given by then we have

Inserting (50) and (51) into (48), (48), and (48), we get where

We multiply (52) by , respectively, and integrate their sum over to get the following variational formulation: where is the bilinear form defined by is the linear functional given by

Now, for , equipped with the norm then, we have we have by assuming , we get then, for some ,

Thus, is coercive. Consequently, using the Lax-Milgram theorem, we conclude that the existence of a unique solution in satisfies

Substituting into (50) and (51), respectively, we have

Let and denote which gives us . Now, we replace by in (54) to obtain

We get which yields

Thus,

Moreover, (52) also holds for any every . Then, by using integration by parts, we obtain

Then, we get for any

Since is arbitrary, we get that . Hence, . Using similar arguments as above, we can obtain

Finally, the application of regularity theory for the linear elliptic equations guarantees the existence of unique such that (47) is satisfied.

Consequently, we conclude that is a maximal dissipative operator. Hence, by Lumer-Philips theorem (see [25, 26]), we have the well-posedness result. This completes the proof.

3. Stability Results

We prepare the next lemmas (Lemmas 2–7) which will be useful to introduce the Lyapunov function in (104).

Lemma 2. The energy functionalassociated with our problem defined bysatisfies where .

Proof. Multiplying (19) by , (19) by , and (19) by then integration by parts over , we get

Now, multiplying (19) by and integrating the result over , we get

From (75) and (76), we get (73) and (74).

Now, using Young’s inequality, (74) can be written as

Then, by (4), there exists a positive constant such that

Thus, the functional is nonincreasing.

Lemma 3. The functionsatisfies where .

Proof. Direct computation, using integration by parts and Young’s inequality, for , yields

By Cauchy-Schwartz’s inequality, it is clear that

So, estimate (81) becomes where the Cauchy-Schwartz, Young, and Poincaré’s inequalities have been used, for .

By the fact that , we get the desired result (80).

Lemma 4. Assume that ((4)) holds. Then, the functionsatisfies

Proof. By differentiating , then using (19), integration by parts gives

Thanks to Young, Cauchy-Schwartz, and Poincaré’s inequalities to estimate terms in RHS of (86). For , we have

The replacement of (87)–(90) into (86) and settinghelps to obtain (85).

Lemma 5. The functionsatisfies

Proof. Direct computations give

Estimate (92) easily follows by using Young’s and Poincaré’s inequalities setting to obtain (92).

Lemma 6. The functionsatisfies

Proof. Direct computations give

By using Young and Poincaré’s inequalities, we get (96).

Lemma 7. The functionsatisfies where is a given positive constant.

Proof. By differentiating with respect to and using the last equation in , we have

Using the fact that and , for all , we obtain

We have . Set , and by (4), we get (99).

We state and prove the decay result in Theorem 8.

Theorem 8. Let ((4)) hold. Then, there exist positive constantsandsuch that the function ((73)) satisfies, for any

Proof. We define a class of an appropriate Lyapunov function as where , , , and are positive constants to be selected later.

Differentiating (104) and by (74), (80), (85), (92), (96), and (99), we have

By setting , we obtain

Next, we carefully choose the constants, starting by to be large enough such that and so that and large enough such that

We arrive at where .

Now, let us define the related function then

Thanks to Young, Cauchy-Schwartz, and Poincaré’s inequalities, we get

Then,

Thus,

One can nowlarge enough such that

We get and using (73), (110), and (116), and the fact that which gives for some .

Case 1. If, in this case, ((120)) takes the form

The combination of (118) and (121) gives

Finally, by integrating (122) and recalling (118), we obtain the first result of (103).

Case 2. If, then

Let be denoted by

Then, we have

The last term in (120), by using (19), and Young’s inequality, and by setting , we have

Let then (120) where

Let

If , indeed, where . By (118), we obtain

It is not hard to prove where . By using (129) and (128), we obtain

Choosing such that we have

Integrating (137), we get using the fact that

We get that which is desired to be the second result of (103). This completes the proof.

4. Conclusion

This paper studied the asymptotic behavior of a one-dimensional thermoelastic system with distributed time delay; namely, an integral damping term on a time interval is taken into account. Beside the distributed delay term, a standard undelayed damping is included in the model . We established the well-posedness of the system, and we proved stability estimates by means of appropriate Lyapunov functions. Exponential decay estimates are proved by nonclassical condition between the delay damping coefficient and the coefficient of the undelayed one which is satisfied. Several papers have been proposed for models including both undelayed and delayed damping of the same form, and exponential stability results have been obtained if the coefficient of the delay is smaller than the one of the undelayed term. This analysis has been extended to the case of a distributed delay in [16]. Also in this case, there are now a few literature, dealing with different PDE models, including thermoelastic systems. Typically, under the assumption (4), the system keeps the same properties, the one without delay but only with a standard frictional damping , for some coefficient . Then, this paper introduced a considerable novelties different from those of [15].

Data Availability

No data were used to support the study.

Conflicts of Interest

This work does not have any conflicts of interest.

Acknowledgments

The third author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant No. R.G.P-1/3/42.