Asymptotic stability of porous-elastic system with thermoelasticity of type III

In this work, we investigate a one-dimensional porous-elastic system with thermoelasticity of type III. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation. At the end, we use some numerical approximations based on finite difference techniques to validate the theoretical results.


Introduction
In the recent decades, the study of problems related to elastic solids with voids has attracted the attention of many researchers due to the extensive practical applications of such materials in different fields, such as petroleum industry, foundation engineering, soil mechanics, power technology, biology, material science and so on. Elastic solids with voids are one of the simple extensions of the theory of the classical elasticity. It allows the treatment of porous solids in which the matrix material is elastic and the interstices are void of material.
In 1972, Godman and Cowin [12] proposed an extension of the classical elasticity theory to porous media. They introduced the concept of a continuum theory of granular materials with interstitial voids into the theory of elastic solids with voids. In addition to their usual elastic effects, these materials have a microstructure with the property that the mass at each point is obtained as the product of the mass density of the material matrix by the volume fraction. This latter idea was introduced by Nunziato and Cowin [22] when they developed a nonlinear theory of elastic materials with voids. We refer the reader to [8,9,18,24] and the references therein for more details.
It is well known that the classical thermoelasticity using Fourier's Law of heat conduction, which states that the heat flux is proportional to the gradient of temperature (q = −δ∇θ ), leads to the physical paradox of infinite speed of heat propagation. In other words, any thermal disturbance at one point will be instantaneously transferred to the other parts of the body. This is practically unrealistic. To overcome this physical paradox but still keeping the essentials of heat conduction process, many theories have subsequently emerged. One of such theories was proposed by Green and Naghdi in 1990s [14][15][16]. They used an analogy between the concepts and equations of the purely thermal and the purely mechanical theories and arrived at three types of constitutive equations for heat flow in a stationary rigid solid labeled as type I, II, and III. Consequently, using these constitutive equations, they obtained three models, called thermoelasticity of type I, thermoelasticity of type II, and thermoelasticity of type III. The linear version of the first one coincides with the classical theory based on Fourier's law, the second one is known as thermoelasticity without energy dissipation because the heat equation is not a dissipative process and the third one is the most general and it contains the former two as limit cases. For a further historical review on these models, we refer the reader to [6,7,[13][14][15][16].
The basic evolution equations for one-dimensional theories of porous materials with temperature are given by: where T is the stress tensor, H is the equilibrated stress vector, G is the equilibrated body force, and q is the heat flux vector. The variables ω, ϕ, and θ are the displacement of the solid elastic material, the volume fraction, and the difference temperature, respectively. The positive parameters ρ, J , and β are the mass density, product of the mass density by the equilibrated inertia, and the coupling constant, respectively. Taking into account Green and Naghdi's theory, the constitutive equations are: where δ, k denote the thermal conductivity, is the so-called thermal displacement whose time derivative is the empirical temperature θ , that is, t = θ , and μ, ξ are constitutive constants which satisfy To keep the coupling, the constant b must be different from zero. We substitute (1.2) into (1.1) to obtain the following system For the asymptotic behaviors of the solutions for porous-elastic systems. Quintanilla [25] considered the one-dimensional porous dissipation elasticity: with initial and boundary conditions. He used Hurtwitz theorem to prove that the damping through porousviscosity (τ ϕ t ) is not strong enough to obtain an exponential decay but only a slow (nonexponential) decay.
However, Apalara [2,3] considered the same system and proved the exponential stability provided For various other damping mechanisms used and more results on porous elasticity, we refer the reader to [5,[26][27][28] and the references therein. Recently, Apalara [4] considered the following porous-elastic system with microtemperature: with Dirichlet-Neumann-Dirichlet boundary conditions. He showed that the unique dissipation given by microtemperatures is strong enough to produce exponential stability if and only if: and showed that the system is polynomially stable if χ = 0.
In the present work, we consider the system (1.4) which can be written as follows: with the following boundary conditions We study the well-posedness and the asymptotic stability of (1.8)-(1.10). Using the semigroup theory, we prove the existence and uniqueness of the solution. We then exploit the energy method to obtain the exponential decay result for the case of equal wave speeds. When (1.7) does not hold, we prove a polynomial decay result. The paper is organized as follows: in Sect. 2, we state the problem. In Sect. 3, we establish the wellposedness of the system. In Sect. 4, we show that the system is exponentially stable under condition (1.7). The polynomial stability when the wave-propagation speeds are different, is given in Sect. 5. In Sect. 6, we give some numerical illustrations.

Statement of the problem
To obtain the dissipative nature of System (1.8), we introduce the new variables: u = ω t and φ = ϕ t . So, with the following boundary conditions x ∈ (0, 1).

(2.3)
Since the boundary conditions on φ are of Newmann type, we introduce some transformation that allows the use of Poincaré's inequality on φ. From the second equation in (2.1) and the boundary conditions (2.2), it follows that So, by solving (2.4) and using the initial data of φ, we obtain Consequently, if we let we get 1 0φ (x, t)dx = 0, ∀t ≥ 0, which allows the use of Poincaré's inequality onφ. So, (u,φ, θ) satisfies (2.1), (2.2) and similar initial conditions (2.3). Therefore, we work with (u,φ, θ) but we write (u, φ, θ) for simplicity.

The well-posedness of the problem
In this section, we prove the existence, uniqueness and smoothness of solutions for the system (2.1)-(2.3) using the semigroup theory. Introducing the vector function 3) can be written as We consider the energy space H is a Hilbert space with respect to the following inner product Remark 3.1 Under the hypothesis μξ > b 2 , it is easy to see that (3.3) defines an inner product. In fact, from (3.3), we have Hence, since μξ > b 2 we conclude that (U,Ũ ) H defines an inner product on H and the associated norm · H is equivalent to the usual one.
The domain of A is given by We have the following well-posedness result: Proof The result follows from Lumer-Phillips theorem (see [19,23]) provided we prove that A is a maximal dissipative operator, that is A is dissipative and that (I − A) is surjective. Thus, for any U ∈ D(A), we have To solve (3.6), we consider the following variational formulation and L : W −→ R is the linear form given by It is clear that W is a Hilbert space with the usual norm and we can easily show, using Cauchy-Schwarz inequality, that B and L are continuous. On the other hand, using Young's inequality and the fact μξ > b 2 , we have for some c > 0. Hence, B is coercive. Consequently, Lax-Milgram lemma guarantees the existence of a unique Thus, the elliptic regularity theory implies that and, moreover, we obtain Here, we can't use the regularity theorem, becauseφ ∈ H 1 * (0, 1). So, we take˜ ∈ H 1 0 (0, 1) and set It is clear thatφ ∈ H 1 * (0, 1). Then, a substitution in (3.8) leads to This gives (3.5) 4 . Since −δφ x x = r (x), then and the other hand, we have (3.8). Thus Sinceφ ∈ H 1 * is arbitrary. Then, and, hence, we get, for anyθ ∈ H 1 0 (0, 1), This, in turns, yields Then, Since f 5 = θ − q, then (δθ + kq) ∈ H 2 (0, 1) and we have which solves (3.5) 6 . Hence, there exists a unique U ∈ D(A) satisfies (3.4). Finally, using Lumer-Phillips theorem we deduce that A is an infinitesimal generator of a contraction semigroup in H and this complete the proof.

Exponential stability
In this section, we use the energy method to prove that system (2.1)-(2.3) is exponentially stable in the case of equal wave-speed propagation (1.7). To achieve this goal, we first establish some technical lemmas needed in the proof of exponential stability result. We also use c to be a positive generic constant. (u, φ, θ) be the solution of (2.1)-(2.3). Then the energy functional E, defined by

Lemma 4.1 Let
Proof Multiplying (2.1) by u t , φ t and θ t respectively, integrating over (0, 1) and using integration by parts and the boundary conditions, we obtain The second equation The third equation Adding up the above identities we arrive at This is exactly (4.2).

Lemma 4.2 Let (u, φ, θ) be the solution of (2.1)-(2.3). Then the functional
satisfies, for any ε 1 > 0, the estimate Proof By taking the derivative of F 1 , using (2.1) and integrating by parts, we get We use integration by parts and So, Using Young's and Cauchy-Schwarz inequalities, we have, and, for any ε 1 > 0, Then, by substituting the above inequalities into (4.7), we get satisfies, for any ε 2 > 0, the estimate Proof The differentiation of F 2 , using (2.1), integration by parts, and the boundary conditions (2.2), gives Now, we estimate the terms in the right-hand side of the above identity. Using Young's and Cauchy-Schwarz inequalities, (2.1), and calculations as in (4.8), we find, and, for any So, by Poincaré's inequality and the above estimate, we arrive at satisfies, for some positive constant m 0 , the estimate Proof Direct computations, exploiting χ = 0 (χ = 0 defined by (2.7)) and using (2.1) and integration by parts, yield Using Young's inequality, we get, for any ε 3 > 0, Thus, by taking ε 3 small enough such that we obtain (4.12). satisfies Proof A differentiation of F 4 , using (2.1) and integrating by parts and (2.2), gives Then use of Young's and Poincaré's inequalities leads to satisfies, for ε 2 > 0, Proof A simple differentiation of F 5 , using (2.1), (2.2) and integrating by parts, leads to Next, by Young's inequality, we arrive at satisfies, for N sufficiently large, L ∼ E (4.19) and the estimate (4.20) where λ is a positive constant.
Proof The equivalence (4.19) is a matter of a routine calculations. See, for instance [5].
To prove (4.20), we differentiate L(t), and recall (4.2), (4.5), (4.10), (4.12), (4.15) and (4.17). So, we have We apply Poincaré's inequality for θ t and take N 5 = 2δ β N 2 , to get At this point, we choose the constants carefully. First, let us take ε 1 = ρ 2N 1 , and choose N 3 large enough such that We then choose N 1 large enough so that Next, we select N 2 so large that then pick ε 2 small enough so that Finally, we choose N large enough so that (4.19) remains valid and, further, Therefore, we arrive at We finally use Poincaré's inequality to substitute − Proof First, using Young's inequality, (4.1) becomes The combination of (4.20) and (4.22) gives Using L ∼ E, we get A simple integration over (0, t) yields Consequently, (4.21) is established by recalling L ∼ E.

Polynomial stability
In this section, we prove the polynomial decay result for the non-equal speed of propagation case, that is (1.7) does not holds. To establish our result, we work with the strong solution of (2.1)-(2.3) and define the second-order energy functional satisfies, for any ε 7 > 0 and for some positive constant m 1 , the estimatẽ Proof A simple differentiation of (5.3) gives Using integration by parts for the second term in the right-hand of (5.5) and exploiting (2.1) 3 , we get Substituting (5.6) and (4.13) into (5.5), we obtaiñ Using Young's and Poincaré's inequalities, we find Finally, we choose ε 6 small enough such that to obtain (5.4).

Numerical tests
To illustrate the theoretical results of this work, we present in this section two numerical tests. We solve the system (2.1) under the initial and boundary conditions (2.2), (2.3). The system is discritized using a second order finite difference method in time and space. For more stability, we implement the conservative scheme of Lax-Wendroff. for more details, we refer to our previous works [1,11,17]. We examine the following two tests: • TEST 1: Based on the result (4.21) of our Theorem, we examine the exponential decay of the energy (4.1) using the equality condition of the parameters χ = 0, given by (1.7). Here, we take all parameters of the system (2.1) equal to 1. • TEST 2: In Test 2, we examine the polynomial decay of the energy (4.1) using the parameters condition χ = 0, where the parameters of the system (2.1) are taken as follows μ = 5; ρ = 1; δ = 0.05; J = 1 and the remaining parameters are equal to 1.
To ensure the numerical stability of the implemented method and the executed code, we use t << 0.5dx satisfying the stability condition according to the Courant-Friedrichs-Lewy (CFL) inequality, where dt represents the time step and dx the spatial step. The spatial interval [0, 1] is subdivided into 200 subintervals and the temporal interval [0, T e ] = [0, 1] is deduced from the stability condition above. We run our code for 10,000 time steps using the following initial conditions: u(x, 0) = 2 sin (π x) ; φ(x, 0) = 2x sin (π x) ; θ(x, 0) = 1 4 x(1 − x) in [0, 1]. (6.1) Under the same initial and boundary conditions mentioned above, we show in Fig. 1 the numerical results of the exponential decay case. Whereas we present in Fig. 2 the results obtained for the polynomial case. We show three cross section cuts for the numerical solution (u, φ, θ) at x = 0.25, x = 0.5 and at x = 0.75. For all components of the solution, the decay behavior is clearly demonstrated for both experiments, the exponential and the polynomial decays. Moreover, it should be stressed that the graphical presentations are normalized to ensure a clear comparisons. Therefore, we can clearly compare the energy decay obtained in Test 1 and in Test 2. For this, see Fig. 3.
Finally, we noticed that the case χ = 0 ensures an exponential energy decay and, therefore, the decay of all components of the solution (u, φ, θ). While the case χ = 0 ensures the polynomial decay. But for some special choices of the system parameters generating the damping speed, we could obtain an exponential-like decay of the energy and a damped waves similar to the exponential case. u u u φ φ φ