Exponential and polynomial decay results for a swelling porous elastic system with a single nonlinear variable exponent damping: theory and numerics

We consider a swelling porous elastic system with a single nonlinear variable exponent damping. We establish the existence result using the Faedo–Galerkin approximations method, and then, we prove that the system is stable under a natural condition on the parameters of the system and the variable exponent. We obtain exponential and polynomial decay results by using the multiplier method, and these results generalize the existing results in the literature. In addition, we end our paper with some numerical illustrations.


Introduction
It is well known that swelling of soils, drying of fibers, wood, paper, plants, etc., are problems related to porous media theory. There have been several studies introducing continuum theories for fluids infiltrating elastic porous media, see Payne et al. [39] and other references therein. Toward the end of the nineteenth century, Eringen [21] advanced a continuum theory for a mixture of elastic solids, gas and viscous fluid. Moreover, the author highlighted the balance laws for each component of the mixture and got the field equations for a heat-conducting mixture. For detailed historical development/review related to the general theory of the mixtures, see [17]. In this paper, we deal with stability result for swelling soils under porous media theory. It is to be recognized that swelling soils contain clay minerals that attract and absorb water, which may lead to increased pressure (see [20]). In civil engineering and architectural, swelling soils are considered harmful. If the pressure of the soil is higher compared to the main structure, it could result in heaving [27]. As established by Ieşan [24] and simplified by Quintanilla [40], the basic field equations for the linear theory of swelling porous elastic soils are mathematically given by where the constituents z and u represent the displacement of the fluid and the elastic solid material, respectively. The positive constant coefficients ρ z and ρ u are the densities of each constituent. The functions (F 1 , ψ 1 , φ 1 ) represent the partial tension, internal body forces and external forces acting on the displacement, respectively. Similar definition holds for (F 2 , ψ 2 , φ 2 ) but acting on the elastic solid. In addition, the constitutive equations of partial tensions are given by where a 1 , a 3 are positive constants and a 2 = 0 is a real number. The matrix A is positive definite in the sense that a 1 a 3 > a 2 2 . Quintanilla [40] investigated (1.1) by taking where ξ is a positive coefficient, with initial and homogeneous Dirichlet boundary conditions and obtained an exponential stability result. Similarly, Wang and Guo [47] considered (1.1) with initial and some mixed boundary conditions, taking and obtained an exponential decay result provided the internal viscous damping function γ(x) has a positive mean. Ramos et al. [42] used and proved the exponential decay result provided the wave speeds of the system are equal. Al-Mahdi et al. [42] and Apalara [7] studied the case and obtained general decay results for different classes of the relaxation functions. Youkana et al. [13] considered the case and obtained general decay result irrespective of the wave speed of the system. Apalara et al. [48] considered System (1.1) with the case and established the general stability result. For more results on the stability and well-posedness for porous elastic systems, we refer the reader to see [6,8,[10][11][12]18,22,23,25,26,30,31,37,38,40,44,45,47]. In recent years, there has been an increasing interest in treating equations with variable exponent of nonlinearity. This great interest is motivated by the applications to the mathematical modeling of non-Newtonian fluids. One of these fluids is the electro-rheological fluids which have the ability to drastically change when applying some external electromagnetic field. The variable exponent of nonlinearity is a given function of density, temperature, saturation, electric field, etc. For more information about the mathematical models of electro-rheological fluids, we refer to [1,43]. The list of references concerning existence, blow-up and stability of viscoelastic problems with variable exponents is very long, so we recall here a few of them [4,15,16,32,33,35,46]. In particular, Messaoudi [34] considered the following equation and established an exponential and polynomial decay results under some conditions on the variable exponents m(·) and r(·). Li et al. [28] discussed the following problem and a blow-up result has been established for solutions with negative initial energy. Recently, Al-Gharabli et. al [5] investigated the following problem in Ω, (1.5) where Ω is a bounded domain, u 0 and u 1 are given data, g is a relaxation function, and m(.) is a variable exponent. The authors proved a global existence result using the well-depth method and established explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function.
In this paper, we consider the following swelling porous elastic system (1.6) We aim to prove the stability of System (1.6) and provide some numerical tests to illustrate the decay theory results. To the best of our knowledge, the stability of (1.6) has not been discussed in the current nature (with variable exponents) and no theoretical and numerical decay results are available. Our result substantially improves earlier related ones in the literature.

Preliminary and assumptions
In this section, we present some preliminaries about the Lebesgue and Sobolev spaces with variable exponents (see [14,19,41]). Throughout this paper, c is used to denote a generic positive constant. Let p : Ω → [1, ∞] be a measurable function, where Ω is a domain of R n . The Lebesgue space with a variable exponent p(·) is given by Equipped with the following Luxembourg-type norm v p(·) := inf λ > 0 : the space L p(·) (Ω) is a Banach space (see [19]), separable if p(·) is bounded and reflexive if 1 < p 1 ≤ p 2 < ∞, where The variable exponent Sobolev space is defined as follows: This is a Banach space with respect to the norm v W 1,p(·) (Ω) = v p(·) + ∇v p(·) , and it is separable (Ω) to be the closure of is said to be satisfying the log-Hölder continuity condition, if there exists a constant A > 0 such that, for all δ with 0 < δ < 1, where the positive constant c ρ depends on p 1 , p 2 and Ω only. In particular, the space W 1,p(·) 0 (Ω) has an equivalent norm given by v W 1,p(·) 0 (Ω) = ∇v p(·) .

Lemma 2.2. [19][Embedding Property]
Let Ω be a bounded domain in R n with a smooth boundary ∂Ω. Assume that p, k ∈ C(Ω) such that ZAMP Exponential and polynomial decay results for a swelling... Page 5 of 28 72

Existence of the solution
In this section, we present and establish the existence result by using the Faedo-Galerkin approximations method. Let us first denote L 2 (0, 1) and H m 0 (0, 1), for m = 1, 2, to be the usual Lebesgue and Sobolev spaces with their usual norms.
Here, denotes a derivative with respect to x.

Theorem 3.2. For any given initial data
provided that Conditions (A1-A2) hold.
Proof. Let {v j } j≥1 be an orthogonal basis for H 1 0 (0, 1) which is orthonormal in L 2 (0, 1) with Introduce a sequence of finite dimensional space V n by We consider the following approximate problem in V n × V n : The approximate problems (3.1) and (3.2) become a system of linear ordinary differential equations (ODEs) with the unknown functions a j,n and b j,n . Standard ODE theory ensures the existence of a unique solution (z n , u n ) on the maximal interval [0, t n ) for each n ≥ 1.
Next, we show that t n = T for any n ≥ 1. Setting (v, ϕ) = (z n t , u n t ) in (3.1) and then adding the resultants, we obtain, for any t ≥ 0, d dt This implies, for any t ≥ 0, where C > 0 which does not depend on n and t. Thus, we conclude from estimates (3.3) that Whence, we can extract a subsequence of z n , u n , still denoted by z n , u n , such that Since (z n ZAMP Exponential and polynomial decay results for a swelling... Page 7 of 28 72 Integrating over (0, t) in (3.1) for any t > 0, we obtain, for 1 ≤ j ≤ n, Passing limit as n approaches infinity, we realize, for any j ≥ 1, Now, our task is to show that ψ = |z t | m(.)−2 z t . For this purpose, we define This is true by the following elementary inequality (see Theorem 6.1, p. 222 [29]): So, by using (3.3), we get Taking n → +∞, we obtain (3.10) Replacing v by z t in (3.7) and integrating over (0, T ), we obtain Combining (3.10) and (3.11), we arrive at ZAMP Exponential and polynomial decay results for a swelling... Page 9 of 28 72 As λ → 0, we get Similarly, for λ < 0, we get Thus, (3.12) and (3.13) imply that ψ = |z t | m−2 z t . We need to show that (z, u) satisfy the initial conditions. Using Aubin-Lions lemma, it follows from (3.5) that . A combination of these and (3.2) yields z(·, 0), u(·, 0) = (z 0 , u 0 ) and (z t (·, 0), u t (·, 0) = (z 1 , u 1 ).
Consequently, the triplet (z, u) is a weak solution of (1.6).

Stability result
In this section, we state and prove our main decay result. Our proofs of stability theorems will be carried out through several Lemmas.
and satisfies the following Proof. By multiplying (1.6) by z t and u t , respectively, and integrating over (0, 1), using integration by parts and some manipulations, we obtain (4.2).

Remark 4.2.
It is remarkable that the energy functional E(t) defined by (9) is nonnegative. To establish this, it is enough to show that It is clear that, from the left side of (4.3), we obtain Hence, using the fact that a 1 a 3 > a 2 2 , the nonnegativity is guaranteed.
Proof. By recalling (4.2), it is easy to establish (4.35). To prove (4.36), we set the following partitions Use of Hölder's and Young's inequalities and (9), we obtain
where μ, μ 1 , μ 2 , μ 3 , μ 4 are positive constants to be properly chosen. By taking the derivative of the functional L and using all the above estimates (4.6)-(4.33), we obtain Choosing ε i = μ i , i = 1, 2, and δ 2 = 1 μ2 then, the above estimate becomes First, we select μ 2 such that Then, we chose μ 3 large enough such that Next, we chose μ 1 large enough such that Now, we choose μ 4 such that . ZAMP Exponential and polynomial decay results for a swelling... Page 17 of 28 72 After fixing μ i , where i = 1, 2, 3, 4, we select μ large enough (if needed) such that and L ∼ E. That is, we can find two positive constants α 1 and α 2 such that On the other hand, Young's inequality and (9) allow us to conclude that Hence, estimate (5.3) becomes for any t ≥ 0 and for some positive constant α 3 , Then, from (5.6) and (5.5), we arrive at and thanks to (5.4), we get for any t ≥ 0 Using (4.2), multiplying the above equation by E α (t), α = 2−m1 m1−1 , using the fact E ∼ L, and using Young's inequality, we get: Taking ε small enough and using the non-increasing property of E, (5.8) becomes: where L 1 (t) = E α (t)L(t) + cE(t) ∼ E. Integration over (0, t), using E ∼ L 1 , gives where α = 2−m1 m1−1 > 0. Then, the proof of (5.1) is completed.
We let μ 2 = 1, and then, we chose μ 3 large enough such that Next, we chose μ 1 large enough such that Now, we choose μ 4 such that and L ∼ E. That is we can find two positive constants β 1 and β 2 , In the other hand, Young's inequality and (9) allow us to conclude that Then, from (5.15) and (5.16), we arrive at and thanks to (5.14), we get for any t ≥ 0 Now, we will discuss two cases: Case 1: if m 2 = 2, then by using Lemma (4.7), we have This gives where L 1 = (L + E) ∼ E. Integrating the last estimate over the interval (0, t) and using the equivalence properties L 1 , L ∼ E, the proof of (5.11) is completed. Case 1: if m 2 > 2, then by using Lemma (4.7), we have Multiplying the last equation by E q where q = m2−2 2 , then we obtain  Use of Young's inequality with γ = q+1 q and γ * = m2 m2−2 , we obtain for ε > 0 .
Taking ε small enough and using the non-increasing property of E, the above estimate becomes: Integration over (0, t), using E ∼ L 2 , gives (5.19) where q = m2−2 2 > 0. Then, the proof of (5.12) is completed.

Numerical tests
In this section, we present two numerical tests. We discretize the system (1.   In order to ensure the numerical stability of the implemented method and the executed code, we fix the temporal-spatial condition as Δt < 0.5Δx. This satisfies the stability condition according to the Courant-Friedrichs-Lewy (CFL) inequality, where Δt represents the time step and Δx the spatial step. The spatial interval [0, 1] is subdivided into 500 subintervals, whereas the temporal interval [0, T e ] = [0, 50] is deduced from the stability condition above. We run our code for 50000 time steps using the following initial conditions: For the first numerical Test 1, we examine the polynomial decay case for both u and z for the theoretical result in theorem 4.1. We choose m 1 satisfying 1 < m 1 < 2. Under the initial and boundary conditions 6.1, we plot in Fig. 3a, b the whole wave till time T = 20, where the decay is clearly assured. Moreover, we run our code for T = 50 and we plot three cross sections at x = 0.25, 0.5 and at 0.75 (see Fig. 3c-h).
For the first numerical Test 2, we examine the first result of theorem 4.2, namely the exponential decay case for both u and z. We choose m 1 satisfying m 1 = 2. Under the initial and boundary conditions 6.1, we plot in Fig. 5a, b the whole wave till time T = 20, where the decay is clearly assured. Moreover, we run our code for T = 50 and we plot three cross sections at x = 0.25, 0.5 and at 0.75 (see Fig. 5c-h). For the third numerical Test 2, we check the second result of theorem 4.2, namely the polynomial decay case for both u and z. We choose m 1 satisfying m 1 = 2. Under the initial and boundary conditions 6.1, we plot in Fig. 7a, b the whole wave till time T = 20, where the decay is clearly assured. Moreover, we run our code for T = 50 and we plot three cross sections at x = 0.25, 0.5 and at 0.75 (see Fig. 7c-h).
In Fig. 9, we plot the energies for all three tests. In the light of these numerical results, the theoretical outputs are clearly confirmed. However, even if Fig. 9a, b, c is differentially different, they show some similarities in some parts. This is due to the system formulations and the choice of the used parameters. Moreover, it should be stressed that our intention focusses to show the energy decay proved in Theorems 4.1 and 4.2. We expect that for other choices of initial solution and parameters, we could get clear discrepancy between energy functions as polynomial or as exponential one.

Conclusion
We considered a swelling porous elastic system with a single nonlinear variable exponent damping. We established the existence result using the Faedo-Galerkin approximations method. We obtained exponential and polynomial decay results when m 1 ≥ 2 and the decay results depend on the values of m 2 . We also obtained at least a polynomial decay result when 1 < m 1 < 2 and the decay result depends on the value of m 1 and m 2 . We point out that in the particular case m(.) = 2 we have the swelling porous with frictional linear damping and we obtained the same result as in the literature and in the case if m(.) = m 1 < 2, we obtained the polynomial decay, so our result generalizes the existing results in the literature. It is more interesting to investigate the asymptotic behavior of the solutions in the case swelling porous with variable exponent and with delay term.