Well-Posedness and Stability Results for a Nonlinear Damped Porous–Elastic System with Inﬁnite Memory and Distributed Delay Terms

: In the present paper, we consider an important problem from the application perspective in science and engineering, namely, one-dimensional porous–elastic systems with nonlinear damping, inﬁnite memory and distributed delay terms. A new minimal conditions, placed on the nonlinear term and the relationship between the weights of the different damping mechanisms, are used to show the well-posedness of the solution using the semigroup theory. The solution energy has an explicit and optimal decay for the cases of equal and nonequal speeds of wave propagation.


Introduction
As introduced in [1], the one-dimensional porous-elastic model constitutes a system of two partial differential equations with unknown (u, ϕ) given by ρ 0 u tt = µu xx + βϕ x , in (0, l) × (0, L), where l, L > 0 the constant ρ is the mass density, κ is the equilibrated inertia and the constants µ, α, β, τ, ξ are assumed to satisfy the appropriate conditions. This type of problem has been studied by many authors and a lot of results have been shown (please see [1][2][3][4][5][6][7][8][9]). The pioneering contribution was made by [10] for the problem (1). The basic evolution equations for one-dimensional theories of porous materials with memory effect are given by where T is the stress tensor, H is the equilibrated stress vector and G is the equilibrated body force. The variables u and φ are the displacement of the solid elastic material and the volume fraction, respectively. The constitutive equations are g(t − s)φ xx (x, s)ds = 0, in (0, 1) × (0, ∞).
System (4) subjected Neumann-Dirichlet boundary conditions, where g is the relaxation function; the authors obtained a general decay result for the case of equal speeds of wave propagation (See [12,13]). In [14], the authors improved the case of non-equal speed of wave propagation. In [15] the authors considered the following system with memory and distributed delay terms The exponential stability results of systems with memory and distributed delay terms, for the case of equal speeds of wave propagation under a suitable assumptions, are proved.
In [16], the following system was considered The authors proved the global well-posedness and stability results of (6), which has been extended in [17] for the case of nonequal speeds of wave propagation. Very recently, one-dimensional equations of an homogeneous and isotropic porous-elastic solid with an interior time-dependent delay term feedbacks was treated by Borges Filho and M. Santos in [1]. The result in [10] for system (1) was improved by Apalara to exponential stability in [18]. For more papers related to our paper, please see [19][20][21][22].
Motivated by all the above papers, we investigate the well-posedness and stability results with distributed delay for the cases of equal and nonequal speeds of wave propagation, under additional conditions of the following system where (x, , t) ∈ (0, 1) × (τ 1 , τ 2 ) × (0, ∞), with the Neumann-Dirichlet boundary conditions and the initial data u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ (0, 1) Here, ρ, µ, J, b, δ, ξ and µ 1 are positive constants satisfying µξ > b 2 , the term α(t) f (φ t ), where the functions α and f are specified later, represent the nonlinear damping term. The delay that acts only on the porous equation and τ 1 , τ 2 are two real numbers with 0 ≤ τ 1 ≤ τ 2 , where µ 2 is an L ∞ function, and the function g is called the relaxation function. We first state the following assumptions: Hypothesis 2 (H2). There exists a non-increasing differentiable function α, η : R + → R + such that and Hypothesis 3 (H3). f ∈ C 0 (R, R) is non-decreasing such that there exist v 1 , v 2 , ε > 0 and a strictly increasing function G ∈ C 1 ([0, ∞)), with G(0) = 0 and G is a linear or strictly convex C 2 -function on (0, ε], such that which implies that s f (s) > 0 for all s = 0. The function f satisfies where k 0 , β > 0.
Hypothesis 4 (H4). The bounded function µ 2 : [τ 1 , τ 2 ] → R, satisfying Now, as in [23], taking the following new variable then we obtain As in [24], we introduce the following new variable where η t is the relative history of φ satisfies Consequently, the problem (7) is equivalent to where with the following boundary and initial conditions Meanwhile, from (7) 1 and (9), it follows that Therefire, by solving (18) and using the initial data of u, we get we get Therefore, the use of Poincaré's inequality for − u is justified. In addition, a simple substitution shows that ( − u, φ, y, η t ) satisfies system (7). Hence, we work with − u instead of u, but write u for simplicity of notation.
By imposing new appropriate conditions (H3), with the help of some special results, we obtain an unusual, weaker decay result using Lyaponov functiona, extending some earlier results known in the existing literature. The main results in this manuscript are as follows: Theorem 1 for the existence and uniqueness of solution and Theorem 2 for the general stability estimates.

Well-Posedness
In this section, we prove the existence and uniqueness result of the system (16)- (18) using the semigroup theory. To achieve our goal, we first introduce the vector function U = (u, u t , φ, φ t , y, η t ) T , and the new dependent variables v = u t , ψ = φ t , ϕ = η t ; then, the system (16) can be written as follows where A : D(A) ⊂ H :→ H is the linear operator defined by and and H is the energy space given by where the space L g (0, 1) is endowed with the following inner product For any The space H equipped with the inner product is defined by The domain of A is given by Clearly, D(A) is dense in H. Now, we can state and prove the existence result.
Theorem 1. Let U 0 ∈ H and assume that (10)-(15) hold. Then, there exists a unique solution U ∈ C(R + , H) of problem (20). Moreover, if U 0 ∈ D(A), then Proof. First, we prove that the operator A is dissipative. For any U 0 ∈ D(A) and by using (23), we have For the third term of the RHS of (24), we have Using Young's inequality, we obtain By integrating the last term of the right-hand side of (24), we have Substituting (25), (26) and (27) into (24), using the fact that y(x, 0, , t) = ψ(x, t) and (15), we obtained Hence, the operator A is dissipative. Next, we prove that the operator A is maximal. This is enough to show that the operator That is We note that the equation (30) 5 with y(x, 0, , t) = ψ(x, t) has a unique solution, given by then and we infer from (30) 6 that ϕ = e λs s 0 e τ (ψ + f 6 (τ))dτ, Inserting (32), (33) and (34) in (30) 2 and (30) 4 , we get We multiply (35) by u, φ, respectively and integrate their sum over (0, 1) to obtain the following variational formulation where and On the other hand, we can write Since µξ > b 2 , we deduce that Thus, B is coercive, similarly, Consequently, using Lax-Milgram theorem, we conclude that (16) has a unique solution Substituting u, φ into (32), (33) and (34), respectively, we have we get which yields Consequently, (45) takes the following form This give (35) 2 . Similarly, if we take φ = 0 ∈ H 1 0 (0, 1) in (37) to obtain we obtain which yields Consequently, (48) takes the following form This give (35) 1 . Moreover, (48) also holds for any Φ ∈ C 1 ([0, 1]). Then, by using integration by parts, we obtain Then, we obtain for any Since Φ is arbitrary, we obtain that u . Therefore, the application of regularity theory for the linear elliptic equations guarantees the existence of unique U ∈ D(A) such that (29) is satisfied. Consequently, we conclude that A is a maximal dissipative operator. Now, we prove that the operator Γ defined in (22) is locally Lipschitz in H. Let By using (14) and Holder and Poincaré's inequalities, we can obtain Then, the operator Γ is locally Lipschitz in H. Consequently, the well-posedness result follows from the Hille-Yosida theorem. The proof is completed.

Stability Result
In this section, we state and prove our decay result for the energy of the system (16)-(18) using the multiplier technique. We need the following Lemmas. where Proof. Multiplying (16) 1 by u t and (16) 2 by φ t , then integration by parts over (0, 1) and using (17), we get The last term in the LHS of (54) is estimated as follows Now, multiplying the equation (16) Now, using (54), (55), (56) and (57), we have then, by (10), there exists a positive constant η 0 , such that hence, by (11)- (15) we obtain E is a non-increasing function.

Lemma 5. The functional
where η 1 is a positive constant.

Theorem 2. Assume (10)-(15) hold. Let h(t) = α(t). η(t) be a positive non-increasing function.
Then, for any U 0 ∈ D(A), satisfying, for some c 0 > 0 there are positive constants β 1 , β 2 and β 3 such that the energy functional given by (52) satisfies where G 0 (t) = tG (ε 0 t), ∀ε 0 ≥ 0, and (s) = ∞ s g(σ)dσ. (79) Proof. We define a Lyapunov functional where N, N 1 , N 2 , and N 4 are positive constants, to be chosen later. By differentiating (80) and using (53), (62), (67), (75), (76), we have where χ = ( µ ρ − δ J ) and by setting we obtain Next, we carefully choose our constants so that the terms inside the brackets are positive. We choose a N 2 that is large enough that then, we choose a large enough N 1 that then we choose a large enough N 4 that thus, we arrive at where On the other hand, if we let Exploiting Young, Cauchy-Schwartz and Poincaré inequalities, we obtain Consequently, we obtain Now, by choosing a large enough N that and exploiting (52), estimates (81) and (82), respectively, we obtain and for some k 1 , k 2 , k 3 , c 2 , c 3 > 0.
This case is more important from the physical perspective, where waves are not necessarily of equal speeds. Let E (t) = E (u, φ, y, ϕ) = E 1 (t).