Research Article Exponential and Polynomial Decay Rates of a Porous Elastic System with Thermal Damping

This paper concerns a linear porous thermoelastic system, where the heat conduction is given by Cattaneo ’ s law. By using the theory of the semigroup of the linear operator, the well-posedness is proven. By constructing some suitable multipliers, we establish exponential and polynomial decay rates of the system depending on a stability number π . In addition, we remove the assumption that the constant b is positive in the previous works, which extends some existing results in the literature.


Introduction
In the past decades, the elasticity problems of materials have attracted the attention of many mathematical researchers since these materials are widely used in many practical problems. Cowin and Nunziato first proposed the theory of porous elastic materials and established a nonlinear theory of elastic materials with voids in [1]. See also [2,3]. The one-dimensional model of porous elasticity is formulated as follows: Here uðx, tÞ is the displacement of the solid elastic material, and ϕðx, tÞ denotes the volume fraction. The functions G, T , and S are the equilibrated body force, the stress, and the equilibrated stress, respectively, and are given as follows: Here ρ, J, μ, ν, b, α, ξ, and τ are constitutive coefficients. In 1D, they satisfy investigated a porous elastic system with viscoelastic damping γu xxt in the first equation. It is proven that the damping is not enough to obtain exponential decay. One can refer to [10][11][12][13][14][15][16][17][18][19][20] and so on, for more stability results of the porous elastic system. The porous thermoelastic model can be obtained from [21,22]. The porous thermoelasticity with Fourier's law was first studied in [23]: cθ t = kθ xx − βu xt − mϕ t : The authors established the exponential decay of the system. If we neglect the porous damping, i.e., τ = 0, in [24], the authors proved that the energy decay is slow. In other words, the heat effect alone is not strong enough to get the exponential decay unless condition (4) holds; see [25]. Han and Xu [26] considered a nonuniform Lord-Shulman-type porous thermoelastic model. By using spectral theory, they obtained the exponential decay of energy assuming the coefficients are C 1 functions. Instead of the use of the spectral theory, by using the energy method, Messaoudi and Fareh [27] obtained the same stability result as in [26] but assuming coefficients are constants. In addition, in [27], the authors also investigated a porous thermoelastic system with elastic damping: They proved the system decays exponentially whether condition (4) holds or not. But the authors assumed that the constant b is positive. Feng et al. [28] studied the stability of an inhomogeneous porous-elastic system with temperature and microtemperature.
It is to be noted that system (6) decays exponentially whether condition (4) holds or not, since the elastic damping plays an important role. We may ask the following: Is the system exponentially stable without the assumption of equal wave speeds (4) if the elastic damping is absent in system (6)? We will address this issue in this paper.
In this paper, we investigate the following porous thermoelastic system in ð0, 1Þ × ð0,∞Þ, subject to the following boundary conditions and initial conditions Here qðx, tÞ denotes the heat flux, and θðx, tÞ is the difference temperature. Besides (3), the constitutive coefficients ρ 1 , β, γ, and τ satisfy The main goal in the present work is to prove that systems (7)-(13) are globally well posed and establish the stability of the system.
The main contributions of this paper are as follows: (i) The global well-posedness of systems (7)-(13) is proven by using the semigroup theory of the linear operator. See Theorem 1 (ii) We introduce a stability number π given by It will be proven that the stability depends on the number π. In addition, we will prove that the system decays exponentially under the condition π = 0 and decays polynomially if π ≠ 0. See Theorems 2 and 3.
(iii) As coupling is considered, the constant b must be different from 0. But we remove the assumption that the constant b is positive, which extends the stability result in [27] In Section 2, the well-posedness of the system is proven. The study of stability is given in Section 3. The constant c > 0 is a generic constant in the whole paper. Journal of Function Spaces
Proof. We use the approach of the semigroup of the linear operator to prove the theorem.

Stability
The energy functional of systems (7)-(13) is defined by Next, we give the stability results of systems (7)-(13). More precisely, the exponential and polynomial decay rates depend on π given by The exponential stability result is as follows: Theorem 2. Let π = 0. For U 0 ∈ H , the energy functional (50) satisfies where a 1 > 0 and a 2 > 0 are independent in the initial data.
The polynomial stability result is as follows: where ϖ is a positive constant given in (112) below.
In this section, we will give detail proofs of Theorems 2 and 3. To proceed, we need some technical lemmas to construct some suitable multipliers.
Proof. Differentiating F 1 ðtÞ with respect to t and using Equations (9) and (10), we can obtain 5 Journal of Function Spaces which, together with integration by parts, implies that Applying the inequality of Young, we have for any ε 1 > 0, −τβ which, along with (59), implies (57).

Lemma 6. Define functional F 2 ðtÞ by
Then we get that for any ε 2 > 0 and ε 3 > 0, Proof. We take the derivative of F 2 ðtÞ with respect to t and notice (8) and (9) to obtain We integrate by parts to obtain which, together with the inequalities of Young and Hölder, implies (62).

Lemma 8. The functional F 4 ðtÞ defined by
satisfies that Proof. Taking the derivative of F 4 and using (7) and (10) and integrating by parts, we arrive at By the Young inequality, (76) follows.

Lemma 9.
Define the functional F 5 ðtÞ by Then we obtain for any ε 4 > 0, Proof. We differentiate F 5 with respect to t and apply (7)- (8) to conclude that Noting (11), we get for any x ∈ ð0, 1Þ, Performing the inequality of Young, we infer that for any ε 4 > 0. The proof is done.
Next, we will prove Theorems 2 and 3.

Proof of Theorem 2.
In this subsection, we prove the exponential energy decay.
Proof. First the functional LðtÞ is defined by where M and M i ði = 1, 2, 3, 5Þ are positive constants. The following estimate can be obtained: where λ 1 > 0 and λ 2 > 0. In fact, the application of the 7 Journal of Function Spaces Young inequality implies that there exists C > 0 such that i.e., Taking N > 0 to be large if needed such that λ 1 ≔ M − C > 0 and λ 2 ≔ M + C > 0, then (84) follows.
Combining (54)-(57), (62)-(66), and (76), we see that In (87), we take Then we can get At this point, we take M 3 > 0 to be so large that We take M 5 > 0 to be large such that Then we pick the constant M 2 > 0 to be large such that And then, we choose M 1 to be large enough such that We pick M > 0 to be large enough so that (84) holds, and in addition, In view of (57), there is a positive constant λ 3 > 0 such that for any t > 0, Noting (84), the estimate is obtained: which, recalling (84) again, yields (52).

Proof of Theorem 3.
In this subsection, we prove the polynomial energy decay.
Proof. We differentiate (7)-(10) with respect to t to get Jϕ ttt − αϕ xxt + bu xt + ξϕ t + βθ xt = 0, together with the following boundary conditions: By using the same arguments as (54), we know that the energyẼðtÞ satisfiesẼ In (66), we have proven that We use (10) and Young's inequality to obtain Then for any ε 5 > 0, Taking ε 5 = τjbj/4α, we have The functionalLðtÞ is defined bỹ Combining (54)-(57), (62), (76), and (101)-(105), we obtainL ′ t ð Þ ≤ − γM − cM 1 With the same choice of constants as above, we further take M > 0 to be larger (if necessary) so that αM − cM 3 > 0: Thanks to (57), we infer that for any t > 0, where δ 1 is a positive constant. Recalling that the energy functional EðtÞ is nonincreasing and positive, thus (109) gives us, for any t > 0, And consequently, Here, Remark 10. It is inferred from Theorem 3 that if condition (51) does not hold, the energy decays polynomially. In fact, if the condition fails, we can get the lack of exponential stability. One can prove the claim by using the Gearhart-Herbst-Prüss-Huang theorem [41,42]. One can find some detailed proofs of related porous elastic systems about the lack of exponential stability in [10, 19, 43] among others.