In this work, we consider a swelling porous system where the damping terms are on the boundary. We establish an explicit and general decay result, without imposing restrictive growth assumption near the origin on the damping terms. Our result allows a larger class of damping terms, and the usual exponential and polynomial decay estimates are only special cases. We also give some illustrative examples.

1. Introduction

Soil swelling is one of the problems which has been studied under the porous media theory. The soils with highly swelling index have high swelling properties. In the swelling soils, clay minerals are commonly found there, and they attract and absorb the water which may lead to increase in pressure. When the swelling soil is exposed to the water, the water molecules are attracted into the interstices of the soil matrix. With the water drawn is getting increasing, the soil plates are forced apart due to the rise in pressure within the soil pores leading to swelling or heaving of the soils. So, swelling soils are considered to be one of the sources of problems in foundation design and construction. For more information in the soil swelling problems, we refer the reader to [14] and the references therein. The mathematical model of linear theory of swelling porous elastic soils was established by Ieşan [5] and simplified by Quintanilla [6] as follows: where the functions represent the partial tension, internal body forces, and external forces acting on the displacement, respectively. Similar definitions hold for but acting on the elastic solid. The constituent represents the displacement of the fluid and is the coefficient density of . Here, is the elastic solid material and its coefficient density is . Moreover, the constitutive equations of partial tensions are given by where are positive constants and is a real number. The matrix is positive definite in the sense that .

Quintanilla [6] investigated (1) by taking and established an exponential stability result where . Wang and Guo [7] considered (1) by taking and they establish an exponential stability result by using spectral method, where is an internal viscous damping function with a positive mean. Ramos et al. [8] studied (1), with a damping acting on the domain; that is,

They established an exponential decay rate. Recently, Apalara [9] looked into (1), with viscoelastic damping acting on the domain. So, he took and established a general decay rate irrespective of the wave speed of the system. Very recently, Al-Mahdi et al. [10] also considered (1) with and established explicit and general decay results under a wider class of relaxation functions, and they also performed several numerical tests to illustrate their theoretical results. The reader can consult [1120] and the references therein for some other interesting related results.

In this paper, we are concerned with the following swelling system together with initial and boundary conditions: where and are specific functions, are positive constants, and is a real number.

The use of dampings at the boundary is well known in mechanical structure. It is employed to stabilize motions and absorb shock. This can be achieved by adding some controllers at the boundary (see [21, 22]). In this paper, we aim to establish an explicit and general decay rate result for system (8). We obtain our result without imposing any restrictive growth assumption near the origin on the damping functions. The result in this paper allows a larger class of functions and , from which the energy decay rates are not necessarily of exponential or polynomial types (see the examples in Section 3).

The proofs of our results are carried out, using the multiplier method and benefit from [22, 23] with necessary modifications dictated by the nature of our problem. To the best of our knowledge, this is the first work of this nature. For more works used the frictional damping acting in a part/whole domain or in the boundary, we point out to the work of [2427].

The rest of the paper is organized as follows: in Section 2, we present some hypotheses and material needed for our work. Some essential lemmas and the statement with the proof of the decay result are given in Section 3.

2. Preliminaries

In this section, we present some material needed for the proofs of our main results.

In the sequel, we assume that system (8) has a unique solution where This result can be proved, for initial data in suitable function spaces, using standard arguments such as the Galerkin method.

We consider the following hypotheses:

A1. (for ) is a nondecreasing function such that where and are strictly increasing functions on and are positive constants.

A2. The coefficients are positive constants and is a real number such that

Remark 1. Hypothesis A1 implies that , for all

Remark 2. Hypothesis A1 was introduced by Lasiecka and Tataru [28].

The following lemma will be of essential use in establishing the main result.

Lemma 3 (see [23]). Let be a nonincreasing function and be a strictly increasing function, with as Assume that there exist and such that Then, there exist two positive constants and such that The energy functional associated with problem (8) is

Lemma 4. The energy functional satisfies, along the solution of (8),

Proof. By multiplying the two equations of (8) by and , respectively, and then integrating over with using both boundary and initial conditions (9) and (10), the estimate (17) is established.

Throughout this paper, we will use to denote a generic positive constant.

3. Stability

In this section, we state and prove our main stability result which reads as follows:

Theorem 5. Let be the solution of (8) and assume that A1 and A2 hold. Then, there exists a constant such that, for large, the solution of (8) satisfies where

Proof of Theorem 8 will be carried out through several lemmas.

Lemma 6. Let be the solution of (8); assume that A1 and A2 hold, and be a concave nondecreasing function. Then, for and some positive constant , the energy functional satisfies

Proof. Multiplying the first equation in (8) by and the second by , integrating over , and adding the results and recalling (16), we get Using Young’s inequality, (16), and the properties of and , we have Exploiting Young’s and Poincaré’s inequalities and (16), we obtain Using the properties of and , we conclude As in the above calculations, we handle the third term in the right hand side of (21) as follows: Using Cauchy-Schwarz’s inequality, we conclude that Similarly, we have Now, using (26) and Young’s inequality, we find that Similarly, using (27) and Young’s inequality, we see that From (9), we have Therefore, we get where (as assumed in A2). Using (31) and Young’s inequality, we have We use (16), (28), (29), and (32), to estimate the last five terms in (21) as follows: Combining (21)–(25) and (28)–(33), then (20) is established.

Lemma 7. Let be the solution of (8); assume that A1 and A2 hold. Then, the energy functional satisfies where is a strictly increasing function, with as , , and .

Proof. First, we define the following function: for some Then, and is strictly increasing. Thus, is a convex and strictly increasing function, with as If we let then it is easy to check that is strictly increasing and is strictly decreasing. So, is a concave function, with as We use this particular function and take such that , to estimate the last integral in the right hand side of (20), for .
To estimate , we consider the following cases: Using A1, Remark 2, (17), and the first case in (38), we have which gives Now, using A1, the properties of and , and the second case in (38), we get Finally, using A1, (17), the properties of , and the last case in (38), we obtain Combining (40)–(42), we get Repeating the same above calculations, we also find Then, (20) becomes Recalling and letting , then for , we have Using (46) and Lemma 3 with , , and , we obtain Hence, (34) is established.

Proof of Theorem 8. To prove (18), let us define and take such that
Since is increasing, then we have . Therefore, we obtain So, letting , we can see that Then, using (48) and (49), we see that Hence, Now, it is easy to see that Therefore, using (34), estimate (18) is established.

Example 9. As in [22], we consider the following examples to illustrate our decay result: (1)Let near zero. Then, (18) gives (2)Let near zero. Then, (18) implies (3)If near zero, then, using (18), we have (4)If , then, using (18), we have (5)If , near zero. Then,then, By approximating near , we have Hence, which implies that . Then, using (18), we have

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest.


The authors thank King Fahd University of Petroleum and Minerals (KFUPM) and University of Sharjah for their continuous supports. The authors also thank the referees for their very careful reading and valuable comments. This work was initialed and finished during the visits of the second author to KFUPM in February 2022. This work is funded by KFUPM, Grant No. SB201005.