The effect of time-varying delay damping on the stability of porous elastic system

P roblems in porous media are essential in the field of petroleum engineering, soil mechanics, power technology, biology, material science, etc. Thus, this has attracted the attention of scientists and mathematicians in particular, see for instance the results in [1–8] and the references cited therein for related theory of porous-elastic materials. The basic equations describing the motion of a classical porous system are given by { ρφtt − Sx = 0, in (0, L)×R+, Jψtt − Gx −Q = 0, in (0, L)×R+, (1)


Introduction
P roblems in porous media are essential in the field of petroleum engineering, soil mechanics, power technology, biology, material science, etc. Thus, this has attracted the attention of scientists and mathematicians in particular, see for instance the results in [1][2][3][4][5][6][7][8] and the references cited therein for related theory of porous-elastic materials. The basic equations describing the motion of a classical porous system are given by ρϕ tt − S x = 0, in (0, L) × R + , where ϕ = ϕ(x, t) and ψ = ψ(x, t) are the displacements of solid elastic material and the volume fraction, respectively. The physical parameters ρ and J are respectively, mass density and product of the equilibrated inertia by the mass density. The constitutive laws S, G and Q are: stress tensor, equilibrated stress vector and equilibrated body force, respectively. Time delays occur in systems modeling different types of phenomena in areas such as: biosciences, medicine, physics, chemical and structural engineering. These phenomena depend naturally on the present state and past history of the system. It is a well known fact that the presence of a delay term in a system, which is a priori a stable system, might cause an instability in the system, see for instance, the result of Nicaise and Pignotti [9]. In past decades, a great number of researchers have investigated the effect of delay on the stability of various systems or wave equations (with or without memory), see for example [10][11][12][13][14][15] and references therein. Back to system (1), we should mention that, there are very few results in literature that studied the effect of delay on this system. With memory and time varying delay dampings, the constitutive laws in (1) are given by where the constitutive physical parameters, k, b, δ, a satisfies µ 1 , µ 2 are real constants, τ(t) > 0 is the time-dependent delay and g is a given function to be specified later. For simplicity, we set L = 1, then substituting (2) into (1), we arrive at the following porous-viscoelastic system with varying time dependent delay; When g = µ 1 = µ 2 = 0, Quintanilla [16] investigated where µ 1 = γ > 0 and showed the lack of exponential stability. However, he established a slow non-exponential decay result. Casas and Quintanilla [17] studied and improved the result in [16] (exponential stability). Soufyane et al., [18] considered (5) with viscoelastic damping on the boundaries and proved a general decay estimate. Recently, Apalara [19] looked at where the memory g satisfies g (t) ≤ −ξ(t)g(t) and established a general decay estimate. Feng and Apalara [20] improved the result in [19] when the relaxation function g satisfies g (t) ≤ −ξ(t)H(g(t)). For results in porous systems with delay damping, not much has been done in this direction. we refer the reader to the result of Khochemane et al., [21], where they considered a porous elastic system with weak internal damping and constant delay damping. Precisely, they studied and proved a general decay result provided g satisfies Recently, Borges Filho and Santos [22] considered and showed that the system is exponentially stable. More related results can be found in [23][24][25][26][27] and references therein. The novelty of this work is to study the stability of system (7). In fact, we show that the solution energy has an optimal decay estimate even in the presence of time varying delay term, from which the results in [21,22] are particular cases. To the best of our knowledge, system (7) has not been considered before in the literature. The rest of work is organized as follows: In Section 2, we recall some preliminaries and assumptions on the memory term. In Section 3, we state and prove several lemmas needed for establishing our main results. In Section 4, we establish the uniform stability result and in Section 5, we give some examples in support of our results.

Problem setting and assumptions
In this work, we consider the following system: In addition to (3), we need the following:
From (A1) and (A2), we can deduce the following: (I) It follows from (A1) that lim t→∞ g(t) = 0. Thus, there exists t 0 ≥ 0 large enough, such that (II) Since g and ξ are positive, non-increasing and continuous functions, in addition to M being a positive continuous function, it follows that, for all t ∈ [0, t 0 ], for some positive constants β 1 and β 2 . Hence, (III) M has an extension M, which is a strictly increasing and strictly convex C 2 function on (0, ∞). As an example, given that M(α) = a 1 , M (α) = a 2 and M (α) = a 3 , then we can define M by From now on, C denotes a positive constant that may change within lines or from line to line. We denote by . 2 the usual norm in L 2 (0, 1) and define the following spaces: We have the following well-possedness result, which is obtained by using the Classical Faedo-Galerkin method.
We recall the following useful lemmas that will be applied repeatedly throughout this article.
+∞ t g(s)ds, along the solution of (7) satisfies the estimate Proof. Differentiation of F 5 and using the fact that f (t) = −g(t) lead to Cauchy-Schwarz inequality and condition (A1) give Therefore, Since f (t) = −g(t) ≤ 0, it follows that f (t) ≤ f (0) = δ − l. Thus, we have For the next lemma, we consider the Lyapunov functional K defined by where N, N j , j = 1, 2, 3, 4 are positive constants to be specified later.
Next, we choose and (54) becomes Now, we choose the remaining constants: First, we select N 4 so that Then, we choose N 3 large enough such that Hence N 3 and N 4 are fixed, we choose N 1 large so that We have that . Thus, using the dominated convergence theorem, we get Thus, there exist 0 < α 0 < 1 such that for all 0 < α ≤ α 0 , we have Finally, we choose N so large and take α = 1 2N Such that The analysis from (55) − (61) yields (52). Applying Young's, Cauchy-Schwarz, and Poincaré's inequalities, we obtain (53) easily. This completes the proof.

Main stability result
The main stability result of the work is the following: Theorem 2. Assume k ρ = δ J and (A1) − (A5) hold. Then, there exist λ 1 > 0, λ 2 > 0 such that the solution energy (20) satisfies where M 1 (t) = r t 1 sM (s) ds and M 1 is a strictly decreasing and strictly convex function on (0, r] with lim t→0 M 1 (t) = +∞.