Abstract
The swelling porous thermoelastic system with the presence of temperatures, microtemperature effect, and distributed delay terms is considered. We will establish the well posedness of the system, and we prove the exponential stability result.
1. Introduction and Preliminaries
Eringen was the first to present a theory in which a mixture of viscous liquid and solids mixed with gas [1]. Then, after studying this heat-resistant mixture, you get to the field equations [2].
Expansive (swelling) soils have also been classified under porous media theory which studies this type of problem. This is why this field is considered fertile for study, as there are many studies to reduce the damage caused by swelling soil, especially in civil engineering and architecture, for more depth (see [3–8]).
The basic field equations of the linear theory of swelling porous elastic soils were presented by where are the displacement of the fluid and the elastic solid material and are the densities of each constituent. And () are the partial tension, internal body forces, and eternal forces acting on the displacement, respectively, similarly (), but acting on the elastic solid. In addition, the constitutive equations of partial tensions are given by where and are real numbers. is matrix positive definite with .
Quintanilla [8] investigated (2) by taking where ; they obtained the stability is exponentially. Similarly, in [9], the authors are considered (2) with a different conditions where is an internal viscous damping function with positive mean. By the spectral method, they obtained the exponential stability result. For more details, see [8–15].
Time delays are very important in most natural phenomena and industrial devices, where the time lag is a source of instability, and it is a problem worthy of attention.
Also, there are many works that have studied this type of problems, of which [11, 15–24].
The basic evolution equations for one-dimensional theories of swelling porous materials with temperature and microtemperature [25–28] are given by
Here, represent the stress, the equilibrated stress, the equilibrated body force, the heat flux vector, the entropy, the first heat flux moment, the mean heat flux, and the first moment of energy. The constitutive equations are where is the microtemperature vector, . As coupling is considered, and satisfies
And the coefficients .
The goal of this work is the thermal effects, so we suppose that the heat capacity , and for more excitement in posing the problem, we suppose that the thermal conductivity is nonexistent .
And by introducing the distributed delay term, form a new problem different from previous studies. Under appropriate suppositions, the well posedness of the system is established, and we prove the exponential stability result by the energy method.
We consider in this work:
Now, by substituting (13)–(17) into (9), we arrive at the following problem: where under the initial and boundary conditions
First, as in [24], we introduce the new variable
Then, we get
Consequently, our problem is written in the form where with the initial data and the boundary conditions
Here, the integral represent the distributed delay terms with are a time delay; is an function satisfying:
(H1) is a bounded function satisfying
Meanwhile, from (23)4 and (26), it follows that
So, by solving (28) and using the initial data of , we get
Consequently, if we let we get
Therefore, the use of Poincare’s inequality for is justified. In addition, simple substitution shows that satisfies system (23). Henceforth, we work with instead of but write for simplicity of notation.
In this paper, we consider to be a solution of system (23)–(26) with the regularity needed to justify the calculations. In Section 2, the well posedness is established, and in Section 3, the exponential stability is proved. In all of the following, we mention that .
Remark 1. The coupling that we have proposed in this work with the presence of microtemperatures and distributed delay in problems of swelling in porous elasticity we believe constitutes a new contribution and differs from the previous studies.
2. Well Posedness
In this section, we established the well posedness of the system (23)–(26).
First, introducing the vector function and the variables , then the system (23) writes as follows: where is the linear operator given by and is the energy space given by where
For any, we equip with the inner product defined by
The domain of is given by
Clearly, is dense in .
Theorem 2. Let and assume that (27) holds. Then, there exists a unique solution of problem (33).
Moreover, if , then
Proof. First, we prove that the operator is dissipative. For any and by using (38), we have
For the third term of the RHS of (41), we have
By using Young’s inequality, we get
Substituting (42), (43) into (41), using and (27), we find
where . Hence, is dissipative operator.
Next, we prove is maximal operator. It is sufficient to show that is surjective operator.
Indeed, for any , we prove that there exists a unique such that
That is,
We note that Equation (46)7 with has a unique solution defined by
then
and we have
Inserting (48) and (49) in (46)2, (46)4, (46)5, and (46)6, we get
where
We multiply (50) by , respectively, and integrate their sum over to find the following variational formulation:
where
is the bilinear form given by
and
is the linear functional defined by
Now, for , equipped with the norm,
then we have
On the other hand, we can write
Since (14), we deduce
then, for some Thus, is coercive. Hence, we use the Lax-Milgram theorem to conclude that (52) has a unique solution:
Substituting , and into (46)1,3, we have
Similarly, the compensation of in (47) with (46)7, gives
Moreover, if we take in (54), we get
which implies
that is
Consequently
Similarly, we get
and, if we let in (54), we get
which implies
Thus, using integration by parts, we get
Therefore,
Consequently,
Finally, the application of regularity theory for the linear elliptic equations guarantees the existence of unique such that (45) is satisfied.
Consequently, we conclude that is a maximal dissipative operator. Hence, by Lumer-Philips theorem (see [29]), we have the well-posedness result. This completes the proof.☐
3. Exponential Decay
In this section, we prove our stability result of the system (23)–(26).
For this, we have the following lemmas.
Lemma 3. The energy functional , defined by satisfies where .
Proof. Multiplying Equation (23)1,2,3,4 by , and , integrating by parts over , and using (26), we get Now, multiplying Equation (23)5 by , and integrating the result over Now, by substituting (78) into (77), and using Young’s inequality, we have then, by (27), so that then we obtain (76) ( is a nonincreasing function).☐
Remark 4. Using (14), we conclude that satisfies where Then, the function is nonnegative.
Lemma 5. The functional satisfies, for any
Proof. Direct computation using integration by parts and Young’s inequality yields We use Cauchy-Schwartz, Young’s, and Poincare’s inequalities; for , we obtain Bearing in mind (14), and letting , we obtain the estimate (84).☐
Lemma 6. The functional satisfies,
Proof. By differentiating , then using (23), integration by parts, and (26), we obtain Now, we estimate the last six terms in the RHS of (89), using Young’s, Cauchy-Schwartz, and Poincare’s inequalities. For , we have and By letting , and substituting into (89), we get (88).☐
Lemma 7. The functional satisfies
Proof. Direct computations give Estimating (93) easily follows by using Young’s inequality and (14).☐
Lemma 8. The functional satisfies
Proof. Direct computations give Estimate (96) easily follows by using Young’s and Cauchy-Schwartz inequalities.☐
Now, let us introduce the following functional used by.
Lemma 9. The functional satisfies where .
Proof. By differentiating , with respect to and using the last equation in (23), we have Using the fact that , and , for all , we obtain ☐
Because is a increasing function, we have , for all .
Finally, setting and recalling (27), we find (99). We are now ready to prove the main result.
Theorem 10. Assume (27) holds. Then, there exist such that the energy functional given by (75) satisfies
Proof. We define the functional of Lyapunov
where we will assign them later.
By differentiating (103) and using (75), (84), (88), (93), (96), and (99), we have
By setting
we obtain
At this point, we choose our constants.
We choose large enough so that
then we pick large enough such that
then we select and large enough such that
Thus, we arrive at
where .
On the other hand, if we let
then
According Young’s, Cauchy-Schwartz, and Poincaré inequalities, we find
On the other hand, we can write
Since and (27), we deduce that
Hence, we get
that is,
At this point, we choose large enough such that
and exploiting (75), the estimates (110) and (117), respectively, gives
for some
Consequently, for some , we find
Integration of (120) over gives
Consequently, (102) is established by virtue of (117) and (121).☐
Data Availability
No data were used to support the study.
Conflicts of Interest
This work does not have any conflicts of interest.
Acknowledgments
The sixth author extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research group program under grant (R.G.P.1/3/42).