Abstract

In this paper, we consider a swelling porous elastic system with a viscoelastic damping and distributed delay terms in the second equation. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous elastic soils. The general decay result is established by the multiplier method.

1. Introduction and Preliminaries

In the late 19th century, Eringen [1] proposed a theory in which he presented a mixture of viscous liquids and elastic solids in addition to gas. And he also studied the equilibrium laws for all components of this mixture, and finally, you get the field equations for a heat conductive mixture (for more details, see [2]). In [3], the author has classified expansive (swelling) soils under the classification of porous media theory.

On the other hand, it contains clay minerals that attract and absorb water, which leads to an increase in pressure [4], and this is considered a harmful and dangerous problem in architecture and civil engineering in most countries of the world, especially in foundations, which leads to cracks in buildings and ripples in sidewalks and roads (see [5–8]). From there, studies began to eliminate or reduce the damage, as in ([9–13]), where 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. The functions () represent the partial tension, internal body forces, and eternal forces acting on the displacement, respectively. Similarly (), it works on the elastic solid. In addition, the constitutive equations of partial tensions are given by where and is a real number. is a matrix positive definite in the sense that .

Quintanilla [10] investigated (1) by taking where ; they obtained that the stability is exponential. Similarly, in [14], the authors considered (1) with different conditions where is an internal viscous damping function with a positive mean. They established the exponential stability result (see ([10–20]) for some other interesting results on the swelling porous system).

Time delays arise in many applications because most phenomena naturally depend not only on the present state but also on some past occurrences.

In recent years, the control of PDEs with time delay effects has become an active area of research (see, for example, [15, 20–27]). In many cases, it was shown that delay is a source of instability unless additional condition or control terms are used; the stability issue of systems with delay is of theoretical and practical great importance.

A complement to these works, and by introducing the terms of memory and distributed delay, forms a new problem different from previous studies. Under appropriate assumptions and by using the energy method, we prove the stability results.

In this paper, we are interested in problem (1) with null internal body forces, but the eternal force acting only on the elastic solid is in the form of viscoelastic damping and distributed delay terms, that is,

Remark 1. Regarding the problems of swelling porous elastic, we believe that there are no studies of viscoelasticity (the memory) and the distributed delay conditions that act as a simultaneous dissipation mechanism, and hence, our coupling constitutes a new contribution.
Thus, we are interested in the following problem: where under the initial and boundary conditions First, as in [27], taking the following new variable then we obtain Consequently, the problem is equivalent to where with the initial data and the boundary conditions Here, are positive constants and is a real number, with satisfying . The integrals represent the memory and the distributed delay terms with are a time delay, is an function, and the kernel is the relaxation function, under the following assumptions.
(H1) is a nonincreasing function satisfying where .
(H2) There exists a positive nonincreasing differentiable function, such that (H3) is a bounded function satisfying

Remark 2. The results that we obtained in this work are also correct with other conditions, including Of course, there can be some difficulties with regard to the following boundary conditions: unless we assume respectively.

In this paper, we consider to be a solution of system (12)–(15) with the regularity needed to justify the calculations. In Section 2, we proved our decay result. And we symbolize that is a positive constant.

2. Main Result

In this section, we prove our stability result for the energy of system (12)–(15).

We need the following lemmas.

Lemma 3. The energy functional , defined by satisfies where and

Proof. Multiplying (12)1,2 by and , then integration by parts over , with (15), gives The estimate of the last term in the LHS of (25) is as follows: Now, multiplying ((12))3 by , and by integration over Now, by substituting (26) into (25), and using Young’s inequality, we have then, by (18), there exists so that then we obtain (22) and (23) ( is a nonincreasing function).

Lemma 4. The functional satisfies

Proof. Direct computation using integration by parts and Young’s inequality, for , yields The estimate of the two last terms in the RHS of (32) is as follows: where we have used Cauchy-Schwartz, Young’s, and Poincaré’s inequalities, for , and (18).
By substituting (33) and (34) into (32), we find Bearing in mind that and using (16), we get let , and letting , , gives (31).

Lemma 5. Assume that (16) hold. Then, the functional satisfies,

Proof. By differentiating , then using (12), integration by parts, and (15), we find In what follows, we estimate the different terms in the RHS of (39); we use Young’s, Cauchy-Schwartz, and Poincaré’s inequalities. For , we have By letting and substituting into (39), we get (38).

Lemma 6. The functional satisfies

Proof. Direct computations give Estimate (42) easily follows by using Young’s inequality and .

Now, let us introduce the following functional used.

Lemma 7. The functional satisfies where .

Proof. By differentiating , with respect to and using the last equation in (12), we have By using , and , for all , we find

Because is an increasing function, we have , for all .

Finally, setting and recalling (18) give (45). We are now ready to prove the main result.

Theorem 8. Assume (16)–(18) hold.
Then, there exist such that the energy functional given by (22) satisfies

Proof. We define the functional of Lyapunov where to be selected later.
By differentiating (49) and using (22), (31), (38), (42), and (45), we have By setting we obtain At this stage, we choose our different constants.
First, choosing large enough such that then we pick large enough such that then we select large enough such that Thus, we arrive at where .
On the other hand, if we let then Exploiting Young’s, Cauchy-Schwartz, and Poincaré inequalities, we obtain On the other hand, from (22), we can write where Since and (16), we deduce that Consequently, we find that is, At this point, we choose large enough such that and exploiting (22), estimates (56) and (64), respectively, leads to for some
By multiplying (67) by we get Now, by using (17), we have the following estimate: Thus, (68) becomes which can be rewritten as By using , we have By exploiting (66), we notice that Consequently, for , we get Integrating (74) over gives Consequently, (48) is established by virtue of (66) and (75).

Remark 9. The estimate (48) also remains valid for , thanks to the boundedness and continuity of and .

Data Availability

No data were used to support the study.

Conflicts of Interest

This work does not have any conflicts of interest.

Acknowledgments

The fifth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under Grant No. (G.R.P-2/42).