Exponential Stability of Swelling Porous Elastic with a Viscoelastic Damping and Distributed Delay Term

Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, B.P. 789, El Oued 39000, Algeria Department of Mathematics, College of Sciences and Arts, Ar Rass, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria Laboratory of Pure and Applied Mathematics, Amar Teledji Laghouat University, Algeria Preparatory Institute for Engineering Studies in Sfax, Tunisia Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt


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][6][7][8]). From there, studies began to eliminate or reduce the damage, as in ( [9][10][11][12][13]), where the basic field equations of the linear theory of swelling porous elastic soils were presented by ρ u u tt = P 1x + G 1 + H 1 , ð1Þ where u, ϕ are the displacement of the fluid and the elastic solid material. And ρ u , ρ ϕ > 0 are the densities of each constituent. The functions (P 1 , G 1 , H 1 ) represent the partial tension, internal body forces, and eternal forces acting on the displacement, respectively. Similarly (P 2 , G 2 , H 2 ), it works on the elastic solid. In addition, the constitutive equations of partial tensions are given by where a 1 , a 3 > 0 and a 2 ≠ 0 is a real number. A is a matrix positive definite in the sense that a 1 a 3 > a 2 2 .
Quintanilla [10] investigated (1) by taking where ξ > 0; they obtained that the stability is exponential. Similarly, in [14], the authors considered (1) with different conditions where γðxÞ is an internal viscous damping function with a positive mean. They established the exponential stability result (see ( [10][11][12][13][14][15][16][17][18][19][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][21][22][23][24][25][26][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 with the initial data and the boundary conditions Here, ρ u , ρ ϕ , a 1 , a 3 , β 1 are positive constants and a 2 is a real number, with a 1 , a 2 , a 3 satisfying a = a 3 − a 2 2 /a 1 > 0. The integrals represent the memory and the distributed delay terms with τ 1 , τ 2 > 0 are a time delay, β 2 is an L ∞ function, and the kernel g is the relaxation function, under the following assumptions.
(H1) g ∈ C 1 ðℝ + , ℝ + Þ is a nonincreasing function satisfying where a = a 3 − a 2 2 /a 1 > 0. (H2) There exists a ϑ ∈ ðℝ + , ℝ + Þ positive nonincreasing differentiable function, such that 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 ðu, ϕ, YÞ 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 c is a positive constant.

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

Lemma 4. The functional
Proof. Direct computation using integration by parts and Young's inequality, for ε 1 > 0, 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 δ 1 , δ 2 > 0, and (18).

Lemma 6. The functional
satisfies Proof. Direct computations give Estimate (42) easily follows by using Young's inequality and a 1 a 3 > a 2 2 .
Now, let us introduce the following functional used.
On the other hand, if we let Exploiting Young's, Cauchy-Schwartz, and Poincaré inequalities, we obtain L t ð Þ j j≤ c On the other hand, from (22), we can write where a 4 = a 3 − ð t 0 g s ð Þds: ð61Þ Since a 1 a 3 > a 2 2 and (16), we deduce that Consequently, we find that is,