ON THE DECAY RATES FOR A ONE-DIMENSIONAL POROUS ELASTICITY SYSTEM WITH PAST HISTORY

. This paper studies a porous elasticity system with past history 0 . By introducing a new variable, we establish an explicit and a general decay of energy for the case of equal-speed wave propagation as well as for the nonequal-speed case. To establish our results, we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [Electron. J. Diﬀer. Equa. 2012(2012), 1-45] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [C. R. Acad. Sci. Paris Ser. I, 347(2009), 867-872], Lasiecka and Tataru [Diﬀer. Inte. Equa., 6(1993), 507-533]. In addition we remove the assumption that b is positive constant in [J. Math. Anal. Appl., 469(2019), 457-471] and hence improve

1. Introduction. In this paper we are concerned with the following porous elasticity system with past history u(x, 0) = u 0 , u t (x, 0) = u 1 , φ(x, −t) = φ 0 , φ t (x, 0) = φ 1 , x ∈ (0, 1), (3) u x (0, t) = u x (1, t) = φ(0, t) = φ(1, t) = 0, t > 0, (4) where (x, t) ∈ (0, 1)×R + . The viscoelastic materials exhibit a natural weak damping which is related to their special property of retaining a long time range memory of their past histories. The function g(t) is generally called relaxation function. The materials often arise in many practical problems, for instance, soil mechanics, engineering, power technology, biology, material science and so on. The theory of porous elastic materials has been proposed by Cowin and Nunziato [24], where the authors established a nonlinear theory of elastic materials with voids. See also [7,8]. The equations for one-dimensional theories of porous materials are given by where T is the stress, H is the equilibrated stress and G is the equilibrated body force. The function u(x, t) represents the displacement of the solid elastic material, φ(x, t) is the volume fraction. The constitutive equations are where the constitutive coefficients ρ, J, µ, γ, b, δ, ξ and τ , in one-dimensional case, satisfy ξ > 0, δ > 0, µ > 0, ρ > 0, J > 0, and µξ ≥ b 2 .
In recent years, there are so many mathematical researchers paid their attentions to study asymptotic behavior of solutions to the equations proposed to study elastic materials. From a mathematical point of view, some of this interest comes from the need to establish general results, which are useful to clarify the empirical observations of engineers. First we mention the work [28]. In this paper, Quintanilla considered (5) only with porous dissipation, i.e., γ = 0, and proved that the weak damping τ φ t is not strong enough to obtain an exponential decay ofthe solutions. In addition the author established a slow decay. Recently Apalara [3] proved that the dissipation is strong enough to exponentially stabilize the system in the case of equal-speed wave propagations, i.e., Furthermore, Apalara [4] considered the system when the weak damping τ φ t is of nonlinear term, i.e., α(t)g(φ t ), and obtained an explicit and general decay rate provided (7) holds. Santos, Campelo and Almeida Júnior [32] studied two cases of (5): the porous elastic system with porous dissipation (γ = 0, τ > 0) and the porous elastic system with elastic dissipation (γ > 0, τ = 0). They proved that the corresponding semigroup is exponentially stable if and only if the wave speeds of the system are equal. In the case of lack of exponential stability they showed that the solution decays polynomially and proved that the rate of decay is optimal. Magańa and Quintanilla [19] studied a porous elastic solids with viscoelastic damping They proved that the viscoelasticity damping (−γu xxt ) is not enough to get exponential stability. With respect to porous elastic solids with memory term, the only one we find was due to Apalara [2]. In [2], the author considered By assuming g (t) ≤ −η(t)g(t), the author established a general decay result in the case of the equal-speed wave propagation case under the assumption that the constant b is positive. Recently, the present author and his co-author, Feng and Yin [11], extended the result to the case of the non-equal wave speeds, and the result also holds for b < 0. For more results concerning the stability to porous elastic solids, one can refer to [12,23,26,29,30,31] and so on.
The model of porous thermoelasticity can be obtained from the classical work of Ieşan [15,16]. In [5], the authors considered the following porous thermoelasticity of the form where the function θ is the temperature difference. They established the exponential decay of the system based on the methods developed by Liu and Zheng [18]. If the porous dissipation is absence, i.e., τ = 0, the heat effect alone is not strong enough to exponentially stabilize the system but only a slow decay is established, see [6]. However, Santos et. al. [33], proved that the system is exponentially stable if and only if (7) holds. In [34], Soufyane investigated a thermoelasticity system with finite memory term acting on porous equation Under the assumption g (t) ≤ −κg p (t), the author proved the system is exponential decay if p = 1 and polynomial decay if 1 < p < 3/2. Messaoudi and Fareh [20,21] considered a more general system than the one in [34]: Under the assumption g (t) ≤ −ξ(t)g(t), they established a general decay for the energy in the case of equal wave speeds in [20], and in the case of nonequal speeds in [21]. The author of the present work, Feng [10], studied the general model (8) with past history, i.e., the memory term in the second equation is replaced by ∞ 0 g(s)ψ xx (t − s)ds. Under the assumption g (t) ≤ −κg p (t), he extended the results in [34] to the case of nonequal speeds. For more results concerning porous thermoelasticity, one can refer to [25,35,36] and so on.
In this paper, we established an explicit and a general decay result of energy by introducing suitable energy and perturbed Lyapunov functionals for system (1)-(4) under equal wave speeds and non-equal wave speeds cases. And the case of nonequal wave speeds is more realistic from the physics point of view. Here we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [14], Guesmia [13] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [1], Lasiecka and Tataru [17]. In the present work we remove the assumption that b is positive constant in Apalara [2]. As coupling is considered, the constant b must be different from 0. Hence we extend and improve the result in Apalara [2].
Following the same arguments as in Dafermos [9], we introduce a new variable η to deal with the infinite memory, defined by It is easy to verify η t (x, t, s) + η s (x, t, s) = φ t (x, t).

BAOWEI FENG
Then we can obtain the following system, which is equivalent to problem (1)-(4), where ds. The outline of this paper is as follows: In Section 2, we give some assumptions and main results. In Section 3, we establish the decay rates of energy to the system (9)-(14).

Preliminaries and main results.
In this section, we give some assumptions for our consideration and our main results. In the following, c is used to denote a generic positive constant.
For the relaxation function g(t), we assume Assumption H2. There exists an increasing strictly convex function G : With respect to the new variable η, as in Pata and Zucchi [27], we define the operator T as which is the infinitesimal generator of a translation semigroup with domain The history space and norm
Theorem 2.1. For any initial data U 0 ∈ H, problem (9)- (14) and (6) has a unique weak solution Now we define the energy functional to problem (9)- (14) and (6) by The stability results of the present work are given in the following theorems.
Theorem 2.2. Suppose that Assumption H1 and Assumption H2 hold. Let µ ρ = δ J . For any initial data U 0 ∈ H such that there exists some positive constant m 0 , then there exist positive constants β 1 , β 2 and 0 depending on U 0 H such that for any t ≥ 0, where Theorem 2.3. Suppose that Assumption H1 and Assumption H2 hold. Let µ ρ = δ J . For any initial data U 0 ∈ H 1 such that there exists some positive constant m 0 , then there exist positive constants β 3 and 0 depending on U 0 H1 such that for any t ≥ 0, where Remark 1.
Remark 2. We can get (21) from (20) but it is weaker than which coincides with (20) when G = Id.
Remark 3. Generally speaking, the assumption which the constant b is positive is not required in the studies on porous-elasticity. Our results also hold that the constant b is negative, but the proposed analysis can not be applied for b = 0. In fact, the case b = 0 means the existence of a coupling meanwhile when b = 0 the system is uncoupled and we can not expect decay.

ONE-DIMENSIONAL POROUS ELASTICITY SYSTEM WITH PAST HISTORY 2911
Example 2.3. Let g(t) = µe −(1+t) p with p ∈ (0, 1) and µ > 0 small enough so that (15) holds. For when t is near zero, assumption H2 holds with with q ∈ (1, p 2 ). Then from (19) we can get there exist two constants ρ 1 > 0 and ρ 2 > 0 such that for any q ∈ (1, p 2 ), For any q > 1, assumption H2 also holds with G(t) = t q . Then (20) gives us for some ρ > 0, 3. General decay. In this section, we shall prove Theorems 2.2 and 2.3, which will be divided into the following three subsections.

Technical Lemmas.
Lemma 3.1. The energy functional E(t) is non-increasing and satisfies Proof. Multiplying (9) by u t and (10) by φ t , respectively, and using integration by parts, we can get Note that Inserting (25) into (24), we can get the desired estimate (23). The proof is done.
In the sequel we define the following functionals: and ). The functional I 1 (t) satisfies for any t ≥ 0, where c * > 0 is Poincaré's constant.
Proof. We take the derivative of I 2 (t) with respect to t and use (9)-(10) to get It follows from Hölder's inequality and Young's inequality that for any ε, ε 1 > 0, Then (27) follows from (28)- (30). This completes the proof.
Lemma 3.5. The functional I 4 (t) satisfies that for any ε 4 > 0, where c 3 and c 4 are two positive constants, and c 4 depends on ε 4 .   Then differentiating I 4 (t) with respect to t and using (9)-(10), we obtain that
In the sequel, we define the functional L(t) by where N, N 2 , N 3 and N 4 are positive constants will be chosen later. It is easy to show that for N > 0 large enough, the functional L(t) is equivalent to E(t).
Then we choose N 2 large such that At last we pick N 3 > 0 large enough so that In view of µξ > b 2 and (47), we know that which, together with (18), gives us (45). The proof is now complete.
Lemma 3.7. Under the assumptions of Theorem 2.2, there exists a positive constant γ 1 > 0 such that for any 0 > 0 Proof. This lemma can easily be proved by repeating the arguments of Guesmia [13].

BAOWEI FENG
For any τ > 0, we find that E(t), defined by is equivalent to E(t), and where we used the fact G ( 0 E(t)) is non-increasing.

3.3.
Proof of Theorem 2.3. In this subsection, we consider the case µ ρ = δ J , which is more realistic from the physics point of view.
Differentiating system (9)-(13) with respect to time, we get a new system For U 0 ∈ H 1 , system (53) is well posed. Next we introduce second-order energy functional to problem (9)- (13) bỹ By using the same arguments as in Lemma 3.1, we have the following lemma.
Now we employ the method in [22], see also [14], to estimate the last term in the right hand side of (45).   . (57) By using Young's inequality and Hölder's inequality, we infer that for any ν > 0, and