On the stability of swelling porous elastic soils with a viscoelastic damping

The present work studies a swelling porous-elastic system with viscoelastic damping. We establish a general and optimal decay estimate which generalizes some recent results in the literature. Our result is established without imposing the usual equal-wave-speed condition associated with similar problems in literature.


Introduction
The basic field equations describing the linear theory of swelling porous elastic soils, see Ieşan [?] and Quintanilla [?] are given by where the constituents ϕ = ϕ(x, t) and ψ = ψ(x, t) are respectively, the displacement of the fluid and the elastic solid material. The physical parameters ρ ϕ and ρ ψ are the densities of each constituent. The functions (F 1 , K 1 , L 1 ) and (F 2 , K 2 , L 2 ) are the partial tensions, internal body forces, and external forces acting on the displacement and the elastic solid respectively. In addition, the constitutive equations of partial tensions are given by where b 1 , b 3 are positive constants and b 2 = 0 is a real number. The matrix Λ is positive definite ( that is b 1 b 3 > b 2 2 . Quintanilla [?] studeid (??) with where ξ is a positive coefficient and established an exponential stability result. Wang and Guo [?] investigated (??) with K 1 = K 2 = 0, L 1 = −ρ ϕ γ(x)ϕ t , L 2 = 0, where γ(x) is an internal viscous damping function with positive mean. The authors in [?] used the spectral method to prove an exponential stability result. For more related results, we refere the reader to [?]- [?] and the references cited there in.
The present work aims at studying (??) with null internal body forces, where the external force is acting only on the elastic solid as a viscoelastic force, that is: where g is a given kernel to be specified later (also known as the relaxation function). Substituting (??) into (??), we arrive at g(t − s)ψ xx (x, s)ds = 0, in (0, 1) × (0, ∞). (1.4) We supplement (??) with the following boundary conditions ψ(0, t) = ψ(1, t) = ϕ(0, t) = ϕ(1, t) = 0, t ≥ 0 (1.5) and initia data The novelty of this paper is to improve the work established by Apalara [?], where he considered (??) and proved a general decay result when g satisfied g (t) ≤ −ξ(t)g(t). This present work considers very general condition with minimal assumption on g and prove an optimal decay estimate from which the result in [?] is a particular case. For more related results or background of porous elastic swelling soil theory, we refer the reader to [?]-[?] and the references cited there. The rest of this work is organized as follows: In Section ??, we present preliminary materials which will be helpful in obtaining our results. In Section ??, we establish some useful lemmas. In Section ??, we study the decay rate of the energy functional associated to problem (??)-(??).
Proof. The result follows easily by Cauchy-Schwarz and Poincaré's inequalities.

Essential Lemmas
We state and prove some essential lemmas in this section.
Proof. Differentiating F 6 and observing that f (t) = −g(t), we infer Therefore, Since f is decreasing, so f (t) ≤ f (0) = g 0 . Thus, we get Lemma 3.7. Let (ϕ, ψ) be the solution given by Theorem ??. Then, for suitable positive parameters N, N j , j = 1, 2, 3, 4 to be choosen later, the functional and where c 1 , c 2 and β are some positive constants.

Main decay result
Now, we state the main stability result of this paper.

Concluding Remarks
The present work improves the result in [?], where the author established a general decay result. The decay results in Theorem ?? is optimal in the sense that it agrees with the decay rate of the memory term g, see Remark 2.3 in [?]. This decay result is paramount to the engineers and architects as they might employ it to attenuate the harmful effects of swelling soils. The result in this paper also holds for some other boundary conditions such as ψ x (0, t) = ψ x (1, t) = ϕ x (0, t) = ϕ x (1, t) = 0, ψ(0, t) = ψ x (1, t) = ϕ(0, t) = ϕ x (1, t) = 0, and ψ x (0, t) = ψ(1, t) = ϕ x (0, t) = ϕ(1, t) = 0.
However, there might be some challenges for the following boundary conditions