Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 44, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.44 EXPONENTIAL STABILITY FOR POROUS THERMOELASTIC SYSTEMS WITH GURTIN-PIPKIN FLUX JIANGHAO HAO, JING YANG Abstract. In this article, we study the stability of a porous thermoelastic system with Gurtin-Pipkin flux. Under suitable assumptions for the derivative of the heat flux relaxation kernel, we establish the existence and uniqueness of solution by applying the semigroup theory, and prove the exponential stability of system without considering the wave velocity by the means of estimates of the resolvent operator norm. 1. Introduction In this work, we consider the porous thermoelastic transmission system with Gurtin-Pipkin flux, ρutt − µuxx − bϕx − γuxxt = 0 in (0, 1)× R+, Jϕtt − δϕxx + bux + ξϕ+ βθx = 0 in (0, 1)× R+, cθt + qx + βϕxt = 0 in (0, 1)× R+, (1.1) where R+ = [0,∞) and q = − ∫ t −∞ g(t− s)θx(x, s) ds. (1.2) This system of equations was firstly derived by Gurtin and Pipkin [1]. The initial and boundary conditions for system (1.1) are as follows, u(x, 0) = u0(x), ut(x, 0) = u1(x) in (0, 1), ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x) in (0, 1), θ(x, t) = θ0(x,−t) in (0, 1)× (−∞, 0], u(0, t) = u(1, t) = ϕx(0, t) = ϕx(1, t) = 0 in R+, θ(0, t) = θ(1, t) = 0 in R. (1.3) Here, u is transversal displacement, ϕ is the volume fraction, θ temperature, and q is the heat flux. We assume the coefficients ρ, J, c, µ, b, δ, γ, ξ are positive constants such that µξ ≥ b2. The heat conductivity relaxation kernel g > 0 and the parameter 2020 Mathematics Subject Classification. 35L70, 35B35. Key words and phrases. Gurtin-Pipkin flux; porous thermoelastic system; semigroup theory; exponential stability. ©2023. This work is licensed under a CC BY 4.0 license. Submitted January 16, 2023. Published June 28, 2023. 1 2 J. HAO, J. YANG EJDE-2023/44 β denotes a non-zero coupling coefficient. Note that the coupling coefficient does not play an important role in the analysis. The system we are studying highlights the heat flux q, which can be used in some materials to describe how memory effects can dominate. As far as we know, q comes in many forms and is used in a variety of systems, such as thermoelastic systems, Timoshenko systems, Bresse systems and so on. When heat flux q is expressed by Fourier’s law or Cattaneo’s law, a large number of scholars have studied the existence and asymptotic behavior of solutions for related systems. When the heat flux q is in terms of Fourier’s law, we have q = −kθx. (1.4) Casas and Quintanilla [2] studied the thermoelastic system ρutt − µuxx − bϕx + βθx = 0 in (0, π)× R+, Jϕtt − αϕxx + bux + ξϕ−mθ + τϕt = 0 in (0, π)× R+, cθt − kθxx + βutx +mϕt = 0 in (0, π)× R+. (1.5) Using the semigroup method, they demonstrated that the system was exponen- tially stable under a combination of porous dissipation and thermal effect. Apalara [3] considered the porous thermoelastic system with memory terms, mainly, the memory term was used to replace the porous dissipation term in (1.5). He used the energy method to obtain stable results of various forms of solution through different memory effects. Al-Mahdi et al. [4] considered the new kernel g′(t) ≤ −γ(t)G(g(t)) and established new general decay results in the case of infinite memory. Magaña and Quintanilla [5] introduced a strong damping mechanism, ρutt − µuxx − bϕx + βθx − γuxxt = 0 in (0, π)× R+, Jϕtt − δϕxx + bux + ξϕ−mθ = 0 in (0, π)× R+, cθt − kθxx + βutx +mϕt = 0 in (0, π)× R+. (1.6) They used the same method as in [2] to prove that the system decays slowly in the presence thermal effect. In addition, they introduced microtemperature and found out the exponential decay of this system. Also, we can see [6]. Pamplona et al. [7] obtained the conclusion of (1.6) using higher-order energy methods. Djebabla et al. [8] studied porous thermoelastic system with time delay, they used the energy method combined with multiplicative technique and showed the polynomial decay estimate. We can also refer to [9, 10] to study more thermoelastic systems. In Timoshenko systems, Rivera and Racke [11] considered the system ϕtt − k(ϕx + ψ)x = 0 in (0, L)× R+, ρ2ψtt − bψxx + k(ϕx + ψ) + γθx = 0 in (0, L)× R+, ρ3θt − κθxx + γψtx = 0 in (0, L)× R+. (1.7) EJDE-2023/44 STABILITY FOR POROUS THERMOELASTIC SYSTEMS 3 They demonstrated exponential stability through the damping effect of heat con- duction. For the angle of rotation system with memory term ρ1ϕtt − k(ϕx + ψ)x + βθx = 0 in (0, L)× R+, ρ2ψtt − αψxx + k(ϕx + ψ)− βθ + ∫ t 0 g(t− s)ψxx(s) ds = 0 in (0, L)× R+, ρ3θt − κθxx + β(ϕxt + ψt) = 0 in (0, L)× R+. (1.8) When β = 1, Messaoudi and Fareh [12] established the general decay result by constructing energy functional for equal wave velocities, that is, X = k ρ1 − α ρ2 = 0. (1.9) Later, they used the same method to consider the case of X 6= 0 in [13] and also obtained the general decay result. When β > 0 and β 6= 1, Almeida Júnior et al. [14] studied (1.8) at g = 0. Considering the case of X 6= 0, they found that related semigroups had different polynomial decay rates under different boundary conditions. The semigroup decays optimally at the rate of 1/ √ t for fully Dirich- let boundary conditions and at the rate of 1/ 4 √ t for Dirichlet-Neumann-Dirichlet boundary conditions. In the presence of memory term, Apalara [15] extended the above system for any β > 0, and obtained a general stability result independent of wave velocity by using the energy method under Neumann-Dirichlet-Dirichlet boundary conditions. General forms of Bresse system can also be coupled to ther- mal effect, ρ1ϕtt = k(ϕx + ψ + lω)x + lk0(ωx − lϕ) = 0 in (0, π)× R+, ρ2ψtt = bψxx − k(ϕx + ψ + lω)− γθx in (0, π)× R+, ρ1ωtt = k0(ωx − lϕ)x − lk(ϕx + ψ + lω) in (0, π)× R+, ρ3θt = qx − γψtx in (0, π)× R+. (1.10) Fatori and Rivera [16] showed the Bresse-Fourier system was exponentially stable if and only if k = k0 and (1.9) holds. For a discussion of the type III thermoelastic Bresse system readers may refer to [17] which considered the effect of memory item. When the heat flux q is in terms of Cattaneo’s law, q satisfies τ0qt + q + κθx = 0. (1.11) Fareh and Messaoudi [18] investigated the porous thermoelastic system with un- necessary positive definite energy ρutt − µuxx − bϕx = 0 in (0, 1)× R+, Jϕtt − αϕxx + bux + ξϕ+ βθx = 0 in (0, 1)× R+, cθt + qx + βϕtx + δθ = 0 in (0, 1)× R+, τ0qt + q + κθx = 0 in (0, 1)× R+. (1.12) Under Dirichlet-Neumann-Dirichlet boundary conditions, they assumed that µξ = b2 and introduced the stability number X = β2 − (cαµ ρ − ακ ρ0 ) − (J α − ρ µ ) . (1.13) 4 J. HAO, J. YANG EJDE-2023/44 When X = 0, they got exponential stability of this system, and when X 6= 0, this system was polynomially stable. Fernàndez-Sare and Racke [19] considered the Timoshenko system ρ1ϕtt − k(ϕx + ψ)x = 0 in (0, L)× R+, ρ2ψtt − αψxx + k(ϕx + ψ) + γθx = 0 in (0, L)× R+, ρ3θt + κqx + γψtx = 0 in (0, L)× R+, τ0qt + q + κθx = 0 in (0, L)× R+. (1.14) They showed that the solution of the above system was not exponentially stable, even if the condition (1.9) be satisfied. Santos et al. [20] considered (1.14), they introduced a new stability number X = ( τ − κρ1 ρ3 ) − ( ρ2 − αρ1 κ ) − (τδ2ρ1 κρ3 ) , (1.15) and established an exponential stability for X = 0. Also, they discussed the case of X 6= 0, obtained the optimal polynomial decay. For the related Timoshenko system with frictional damping, we can also refer to [21]. The new Bresse system established by coupling with (1.11) through (1.10) was studied by Keddi et al. [22], they used the same method as [18] to get the exponential decay result of system. For heat flux q of Gurtin-Pipkin type, we refer the readers to [23]. Here, we briefly describe a few systems. Pata and Vuk [24] considered the linear thermoelastic system utt − uxx + θx = 0 in (0, l)× R+, θt − ∫ t −∞ g(t− s)θxx(x, s) ds+ utx = 0 in (0, l)× R+. (1.16) They used the semigroup method to achieve that the solution of system had an exponential decay result. And in the latest literature, Fareh [25] studied the porous thermoelastic system with porous damping ρutt = µuxx + bϕx − βθx in (0, π)× R+, Jϕtt = αϕxx − bux − ξϕ+ δθ − τϕt in (0, π)× R+, cθt = ∫ t −∞ g(t− s)θxx(x, s) ds− βuxt − δϕt in (0, π)× R+, (1.17) and showed that the exponential decay of solution in the presence of the more general convolution integral form and the porous dissipation coefficient τ . Dell’Oro and Pata [26] considered the coupled Timoshenko system ρ1ϕtt − k(ϕx + ψ)x = 0 in (0, l)× R+, ρ2ψtt − bψxx + k(ϕx + ψ) + δθx = 0 in (0, l)× R+, ρ3θt − 1 β ∫ ∞ 0 g(s)θxx(t− s) ds+ δψtx = 0 in (0, l)× R+. (1.18) When (1.2) was applied in (1.10), Dell’Oro [27] studied the asymptotic stability of the system. They defined the stability number Xg = ( ρ1 ρ3k − 1 g(0)k1 )(ρ1 k − ρ2 b ) − 1 g(0)k1 ρ1γ 2 ρ3bk . (1.19) EJDE-2023/44 STABILITY FOR POROUS THERMOELASTIC SYSTEMS 5 and showed that this system was exponentially stable if and only if Xg = 0 and k = k0. In this article, we study the asymptotic behavior of a porous thermoelastic sys- tem with Gurtin-Pipkin flux under strong damping which improves the conclusions of [7, 28]. Thus, we know that in some materials, memory items can dominate. The rest of this article is structured as follows: In section 2, we give some preliminaries and reset the system (1.1)-(1.3) to an abstract Cauchy problem. In section 3, we give the well-posedness of the system. In in section 4, we give the main conclusion that the solution of the system is exponentially stable. 2. Preliminaries In this section, we give the definitions and assumptions for proving the conclusion of this article. Here (·, ·) and ‖ · ‖ denote the usual scalar product and the norm in L2(0, 1), respectively. ‖ · ‖−1 denotes the norm of the space H−1(0, 1) which is the conjugate space of H1 0 (0, 1) and 〈·, ·〉 denotes the conjugate pairs. We set the spaces L2 ∗(0, 1) = { ψ ∈ L2(0, 1) : ∫ 1 0 ψ(x) dx = 0 } , H1 ∗ (0, 1) = H1(0, 1) ∩ L2 ∗(0, 1), M = L2 k((0,∞);H1 0 (0, 1)) = {ζ(x, s) ∈ L2 ( (0,∞);H1 0 (0, 1) ) : ∫ ∞ 0 k(s) ∫ 1 0 ζ2x(x, s) dx ds < +∞}. The space M is endowed with the inner product and norm: 〈ζ, ξ〉M = ∫ ∞ 0 k(s) ( ζx(s), ξx(s) ) ds, ‖ζ‖2M = ∫ ∞ 0 k(s)‖ζx(s)‖2 ds. Meanwhile, we define the space K = {ζ|ζs ∈M : lim s→0 ‖ζx(s)‖ = 0}. Now, we define the state space H = H1 0 (0, 1)× L2(0, 1)×H1 ∗ (0, 1)× L2 ∗(0, 1)× L2(0, 1)×M endowed with the inner product 〈Z,Z∗〉H = µ ∫ 1 0 uxu ∗ x dx+ ξ ∫ 1 0 ww∗ dx+ b ∫ 1 0 wu∗x dx+ b ∫ 1 0 w∗ux dx + ρ ∫ 1 0 vv∗ dx+ J ∫ 1 0 zz∗ dx+ c ∫ 1 0 θθ∗ dx + δ ∫ 1 0 wxw ∗ x dx+ ∫ ∞ 0 ∫ 1 0 k(s)ζx(s)ζ∗x(s) dx ds, (2.1) for any Z = (u, v, w, z, θ, ζ)T ∈ H, Z∗ = (u∗, v∗, w∗, z∗, θ∗, ζ∗)T ∈ H. As in [24, 25], we introduce some new variables θt(x, s) = θ(x, t− s), s ≥ 0, ηt(x, s) = ∫ s 0 θt(x, τ) dτ, s ≥ 0, 6 J. HAO, J. YANG EJDE-2023/44 which denote the past history and the summed past history of θ up to t, respectively. We denote η0(x, s) = ∫ s 0 θ0(x, τ) dτ, s ≥ 0. We can easily show that ηtt(x, s) = θ(x, t)− ηts(x, s). (2.2) Further, we assume that g(∞) = 0 and ηt(x, 0) = lims→0 η t(x, s) = 0, then − ∫ t −∞ g(t− s)θxx(x, s) ds = ∫ ∞ 0 g′(s)ηtxx(x, s) ds. (2.3) Setting k(s) = −g′(s), combining (2.2) and (2.3), system (1.1)-(1.3) can be written as ρutt − µuxx − bϕx − γuxxt = 0 in (0, 1)× R+, Jϕtt − δϕxx + bux + ξϕ+ βθx = 0 in (0, 1)× R+, cθt − ∫ ∞ 0 k(s)ηtxx(x, s) ds+ βϕxt = 0 in (0, 1)× R+, ηtt(x, s) = θ(x, t)− ηts(x, s) in (0, 1)× R+ × R+, (2.4) supplemented with the initial and boundary conditions u(x, 0) = u0(x), ut(x, 0) = u1(x), ϕ(x, 0) = ϕ0(x) in (0, 1), ϕt(x, 0) = ϕ1(x), θ(x, 0) = θ0(x) in (0, 1), η0(x, s) = η0(x, s) in (0, 1)× R+, u(0, t) = u(1, t) = ϕx(0, t) = ϕx(1, t) = 0 in R+, θ(0, t) = θ(1, t) = ηt(0, s) = ηt(1, s) = 0 in R+ × R+. (2.5) Remark 2.1 ([29]). From (2.4)2 and the boundary conditions, we easily verify that d2 dt2 ∫ 1 0 ϕ(x, t) dx+ ξ J ∫ 1 0 ϕ(x, t) dx = 0. By solving this ordinary differential equation and using the initial data of ϕ, we obtain∫ 1 0 ϕ(x, t) dx = (∫ 1 0 ϕ0(x) dx ) cos (√ ξ J t ) + √ J ξ (∫ 1 0 ϕ1(x) dx ) sin (√ ξ J t ) . We introduce ϕ̄(x, t) = ϕ(x, t)− (∫ 1 0 ϕ0(x) dx ) cos (√ ξ J t ) − √ J ξ (∫ 1 0 ϕ1(x) dx ) sin (√ ξ J t ) , then ϕ̄x(x, t) = ϕx(x, t) in (0, 1)× R+, ϕ̄xx(x, t) = ϕxx(x, t) in (0, 1)× R+, and ϕ̄x(0, t) = ϕx(0, t) = 0 in R+, ϕ̄x(1, t) = ϕx(1, t) = 0 in R+. EJDE-2023/44 STABILITY FOR POROUS THERMOELASTIC SYSTEMS 7 Furthermore, we find that (u, ϕ̄, θ, η) satisfies the same boundary conditions as (2.5)4, (2.5)5 and ∫ 1 0 ϕ̄(x, t) dx = 0. Hence, the Poincaré inequality is applicable for ϕ̄ provided that ϕ̄ ∈ H1(0, 1). For the rest of the paper, we will use ϕ̄ instead of ϕ. For convenience, we still denote ϕ in the followings. The relevant Poincaré inequality is∫ 1 0 ψ2 dx ≤ Cp ∫ 1 0 ψ2 x dx, ∀ψ ∈ H1 ∗ (0, 1). To prove our results more easily, we make some hypotheses: (H1) The relaxation function k : R+ → R+ is non-increasing of class C1(R+) ∩ L1(R+) such that k(s) ≥ 0, k′(s) ≤ 0, s ≥ 0, (H2) k is summable on R+, we have∫ ∞ 0 k(s) ds = k0 > 0, ∫ ∞ 0 sk(s) ds = k1 > 0, (H3) There exists a positive constant ν such that k′(s) ≤ −νk(s), s ≥ 0. Let U = (u, v, ϕ, w, θ, ηt)T , where v = ut and w = ϕt, then system (2.4)-(2.5) is equivalent to the abstract Cauchy problem d dt U(t) = AU(t), U(0) = (u0, u1, ϕ0, ϕ1, θ0, η0)T , (2.6) where the operator A is defined as A  u v ϕ w θ ηt  =  v µ ρuxx + b ρϕx + γ ρvxx w δ Jϕxx − b J ux − ξ Jϕ− β J θx 1 c ∫∞ 0 k(s)ηtxx(s) ds− β cwx θ − ηts  , (2.7) with domain D(A) = { U ∈ H : u ∈ H2(0, 1) ∩H1 0 (0, 1), v ∈ H1 0 (0, 1), ϕ ∈ H2(0, 1) ∩H1 ∗ (0, 1) w ∈ H1 ∗ (0, 1), θ ∈ H1 0 (0, 1), ∫ ∞ 0 k(s)ηt(s) ds ∈ H2(0, 1), ηt ∈ K, ηt(0) = 0 } . We introduce the related energy functional E(t) = 1 2 ∫ 1 0 ( µ|ux|2 + ξ|ϕ|2 + 2buxϕ+ ρ|v|2 + J |w|2 + c|θ|2 + δ|ϕx|2 ) dx + 1 2 ‖ηt‖2M. (2.8) 8 J. HAO, J. YANG EJDE-2023/44 Note that from the assumption µξ ≥ b2,∫ 1 0 ( µ|ux|2 + ξ|ϕ|2 + 2buxϕ ) dx = µ 2 ‖ux + b µ ϕ‖2 + ξ 2 ‖ϕ+ b ξ ux‖2 + 1 2 (µ− b2 ξ )‖ux‖2 + 1 2 (ξ − b2 µ )‖ϕ‖2 ≥ 0. (2.9) 3. Well-posedness In this section, we give the existence and the uniqueness of solution for sys- tem (2.6). We use the semigroup method to prove this conclusion, which involves the Lax-Milgram theorem, the Lumer-Phillips theorem and Hille-Yosida theorem. Among them, the content of the Lumer-Phillips theorem is as follows. Lemma 3.1 ([30]). A densely defined linear operator A : D(A) ⊂ H → H generates a C0-semigroup of contractions on H if and only if A is m-dissipative, i.e., it satisfies (i) 0, λI −A is surjective. In the reflexive Banach space H, we know that operator A is densely defined from (ii) of Lemma 3.1. Theorem 3.2. Assume (H1)–(H3) and that for each U(0) = (u0, u1, ϕ0, ϕ1, θ0, η0)T in H, system (2.6) has a unique solution U ∈ C(R+;H). Moreover, if U(0) = (u0, u1, ϕ0, ϕ1, θ0, η0)T ∈ D(A) then the solution U satisfies U ∈ C ( R+;D(A) ) ∩ C1 ( R+;H ) . Proof. We first prove that the operator A generates a C0-semigroup of contractions on H. Firstly, for all U = (u, v, ϕ, w, θ, ηt)T ∈ D(A), we have 0. Indeed, let ψ ∈ H1 0 (0, 1) such that ‖ψx‖ ≤ 1, and by applying some formulas, we have |〈γf1xx, ψ〉| = |〈γf1x, ψx〉| ≤ γ‖f1x‖ <∞, and ∣∣〈 ∫ ∞ 0 k(s) (∫ s 0 eτ−sf6xx(τ) dτ ) ds, ψ 〉∣∣ = ∣∣〈 ∫ ∞ 0 k(s) (∫ s 0 eτ−sf6x(τ) dτ ) ds, ψx 〉∣∣ ≤ ∫ ∞ 0 k(s)e−s (∫ s 0 eτ‖f6x(τ)‖ dτ ) ds ≤ ∫ ∞ 0 eτ‖f6x(τ)‖ ∫ ∞ τ k(s)e−s ds dτ 10 J. HAO, J. YANG EJDE-2023/44 ≤ ∫ ∞ 0 k(τ)eτ‖f6x(τ)‖ ∫ ∞ τ e−s ds dτ = ∫ ∞ 0 k(τ)‖f6x(τ)‖ dτ <∞. The first, second, and third equations of (3.12) are multiplied by u1 ∈ H1 0 (0, 1), ϕ1 ∈ H1 ∗ (0, 1), and θ1 ∈ H1 0 (0, 1) respectively, integrate over (0, 1) and add them, we have the variational formulation B ( (u, ϕ, θ), (u1, ϕ1, θ1) ) = L(u1, ϕ1, θ1), (3.13) where B : [ H1 0 (0, 1)×H1 ∗ (0, 1)×H1 0 (0, 1) ]2 → R is a bounded bilinear form defined by B ((u, ϕ, θ), (u1, ϕ1, θ1)) = ρ ∫ 1 0 uu1 dx+ (µ+ γ) ∫ 1 0 uxu1x dx− b ∫ 1 0 ϕxu1 dx+ (J + ξ) ∫ 1 0 ϕϕ1 dx + δ ∫ 1 0 ϕxϕ1x dx+ b ∫ 1 0 uxϕ1 dx+ β ∫ 1 0 θxϕ1 dx+ c ∫ 1 0 θθ1 dx + ∫ ∞ 0 k(s)(1− e−s) ds ∫ 1 0 θxθ1x dx+ β ∫ 1 0 ϕxθ1 dx, and L : H1 0 (0, 1)×H1 ∗ (0, 1)×H1 0 (0, 1)→ R is the linear functional L (u1, ϕ1, θ1) = ρ ∫ 1 0 (f1 + f2)u1 dx+ γ ∫ 1 0 f1xu1x dx+ J ∫ 1 0 (f3 + f4)ϕ1 dx + β ∫ 1 0 f3xθ1 dx+ c ∫ 1 0 f5θ1 dx + ∫ 1 0 θ1 ∫ ∞ 0 k(s) (∫ s 0 eτ−sf6xx(τ) dτ ) ds. Utilizing Poincaré inequality, we obtain B ((u, ϕ, θ), (u, ϕ, θ)) = ρ ∫ 1 0 u2 dx+ (µ+ γ) ∫ 1 0 u2x dx+ 2b ∫ 1 0 uxϕdx+ (J + ξ) ∫ 1 0 ϕ2 dx + δ ∫ 1 0 ϕ2 x dx+ c ∫ 1 0 θ2 dx+ ∫ ∞ 0 k(s)(1− e−s) ds ∫ 1 0 θ2x dx ≥ α‖(u, ϕ, θ)‖2, for some constant α > 0. Thus, B(·, ·) is coercive. According to the Lax-Milgram theorem, (3.13) has a unique solution (u, ϕ, θ) ∈ H1 0 (0, 1)×H1 ∗ (0, 1)×H1 0 (0, 1). If we take (u1, ϕ1, θ1) = (u1, 0, 0) in (3.13), we have (µ+ γ) ∫ 1 0 uxu1x dx = b ∫ 1 0 (ϕx − u)u1 dx+ ρ ∫ 1 0 (f1 + f2)u1 dx+ γ ∫ 1 0 f1xu1x dx, which means that u ∈ H2(0, 1) ∩H1 0 (0, 1). EJDE-2023/44 STABILITY FOR POROUS THERMOELASTIC SYSTEMS 11 Similarly, if we take (u1, ϕ1, θ1) = (0, ϕ1, 0) in (3.13), we have δ ∫ 1 0 ϕxϕ1x = J ∫ 1 0 (f3 +f4−ϕ)ϕ1 dx−ξ ∫ 1 0 ϕϕ1 dx−b ∫ 1 0 uxϕ1 dx−β ∫ 1 0 θxϕ1 dx which means that ϕ ∈ H2(0, 1) ∩H1 0 (0, 1). Moveover, from (3.5), (3.7) and (3.9), we observe that v ∈ H1 0 (0, 1), w ∈ H1 ∗ (0, 1), ∫ ∞ 0 k(s)ηt(s) ds ∈ H2(0, 1). Inserting (??) in (3.10), we obtain ηts(s) = e−sθ + f6(s)− ∫ s 0 ey−sf6(y)dy, thus, we have ηt ∈ K and ηt(0) = 0. Hence, there exists a unique solution U ∈ D(A). Consequently, A is a maximal monotone operator, i.e., the operator A generates a C0-semigroup of contractions on H. Finally, the conclusion of Theorem 3.2 can be obtained by applying the Hille-Yosida theorem. � 4. Stability In this section, we give the stability result of system (2.6) by means of estimates of the resolvent operator norm. Lemma 4.1 ([27]). Let A be the infinitesimal generator of a contraction semigroup S(t) acting on space H. Then, the following statements are equivalent: (i) S(t) is exponentially stable; (ii) There exists ε > 0, such that inf λ∈R ‖(iλ−A)U‖H ≥ ε‖U‖H, ∀U ∈ D(A); (iii) The imaginary axis iR is contained in the resolvent set ρ(A) of the operator A and sup λ∈R ‖(iλ−A)−1‖L(H) <∞. Theorem 4.2. Assume that (H1)–(H3) are satisfied and U(0) ∈ D(A). Then the energy of system (2.6) is exponentially stable. Proof. We prove (ii) by a contradiction argument. Suppose that the claim is false, then there exist two sequences {λn} ⊂ R and {Un} ⊂ D(A), with ‖Un‖H = 1, (4.1) such that ‖iλnUn −AUn‖H → 0. (4.2) Equivalently, we have iλnun − vn → 0 in H1 0 (0, 1), (4.3) iρλnvn − µD2un − bDϕn − γD2vn → 0 in L2(0, 1), (4.4) iλnϕn − wn → 0 in H1 ∗ (0, 1), (4.5) iJλnwn − δD2ϕn + bDun + ξϕn + βDθn → 0 in L2 ∗(0, 1), (4.6) icλnθn − ∫ ∞ 0 k(s)D2ηtn(s) ds+ βDwn → 0 in L2(0, 1), (4.7) 12 J. HAO, J. YANG EJDE-2023/44 iλnη t n − θn +Dsη t n(s)→ 0 in M, (4.8) in which we denote D = ∂ ∂x and Ds = ∂ ∂s . We just have to show that each component of Un goes to 0 in the norm of H. We will prove it in two cases. Case 1. Assuming λn 6→ 0, that is sequence λn satisfies inf n∈N |λn| > 0. Now, taking the inner product of both sides of (4.2) with Un, and then taking the real part, we obtain 0; therefore, |bn| ≤ |λn|‖θn‖−1 ∫ ∞ 0 √ k(s) ∫ s 0 e− ν(s−r) 2 √ k(r)‖Dζn(r)‖ dr ds ≤ C ∫ ∞ 0 √ k(s)e−νr/2 ∫ s 0 eνr/2 √ k(r)‖Dζn(r)‖dr ds ≤ C ∫ ∞ 0 eνr/2 √ k(r)‖Dζn(r)‖ ∫ ∞ r e−νr/2 √ k(s) ds dr ≤ C ∫ ∞ 0 eνr/2k(r)‖Dζn(r)‖ ∫ ∞ r e−νr/2 ds dr ≤ C ∫ ∞ 0 k(r)‖Dζn(r)‖dr ≤ C‖ζn‖M → 0. On the other hand, an → ∫ ∞ 0 k(s) ds > 0. Back to (4.13), we obtain ‖θn‖ → 0. (4.14) Next, we will prove that ‖wn‖ → 0. Setting Wn = ∫ x 0 wn(y)dy ∈ H1 0 (0, 1). Integrating both sides of (4.6) over (0, x), we have sup n∈N |λn|‖Wn‖ <∞. Now, taking the inner product of both sides of (4.7) with Wn, namely icλn(θn,Wn)− ∫ ∞ 0 k(s)〈D2ηtn(s),Wn〉 ds+ β(Dwn,Wn)→ 0. (4.15) Using the Cauchy-Schwartz inequality and (4.14), we obtain an estimate of the first term of (4.15), |icλn(θn,Wn)| ≤ c|λn|‖Wn‖‖θn‖ → 0. 14 J. HAO, J. YANG EJDE-2023/44 By calculations, the second term of (4.15) satisfies∣∣ ∫ ∞ 0 k(s)〈D2ηtn(s),Wn〉 ds ∣∣ ≤ ‖wn‖ ∫ ∞ 0 k(s)‖Dηtn(s)‖2 ds→ 0. Therefore, |β(Dwn,Wn)| = β‖wn‖2 → 0; this means that ‖wn‖ → 0. (4.16) From (4.5), we can easily check that ‖ϕn‖ → 0. (4.17) Moreover, using Cauchy-Schwartz inequality, we obtain 2b ∫ 1 0 Dunϕn dx→ 0. (4.18) Taking the inner product of both sides of (4.4) and (4.6) with un and ϕn, respec- tively, we find that ‖Dun‖ → 0, (4.19) ‖Dϕn‖ → 0. (4.20) According to (4.10)-(4.11), (4.14), (4.16)-(4.20), we obtain that ‖Un‖H → 0 which contradicts (4.1). Case 2. Assuming λn → 0, from (4.2), we have ‖vn‖ → 0, ‖wn‖ → 0, (4.21) and µD2un + bDϕn + γD2vn → 0 in L2(0, 1), (4.22) δD2ϕn − bDun − ξϕn − βDθn → 0 in H1 ∗ (0, 1), (4.23) θn −Dsη t n(s)→ 0 in M. (4.24) Taking the inner product of both sides of (4.24) with sθ̂n, we have 〈θn, sθ̂n〉M − 〈Dsη t n(s), sθ̂n〉M → 0. Since |〈Dsη t n(s), sθ̂n〉M| = ∣∣ ∫ ∞ 0 sk(s) d ds ∫ 1 0 DηtnDθ̂n dx ds ∣∣ = ∣∣ ∫ ∞ 0 sk(s) d ds ∫ 1 0 ηtnθn dx ds ∣∣ = ∣∣ ∫ ∞ 0 k(s) ∫ 1 0 ηtnθn dx ds+ ∫ ∞ 0 sk′(s) ∫ 1 0 ηtnθn dx ds ∣∣ ≤ ‖θn‖ [ ∫ ∞ 0 k(s)‖ηtn(s)‖ ds+ ∫ ∞ 0 sk′(s)‖ηtn(s)‖ ds ] ≤ ∫ ∞ 0 k(s)‖ηtn(s)‖ ds+ ∫ ∞ 0 sk′(s)‖ηtn(s)‖ ds, (4.25) and − ∫ ∞ 0 s2k′(s) ds = 2 ∫ ∞ 0 sk(s) ds = C2 <∞. EJDE-2023/44 STABILITY FOR POROUS THERMOELASTIC SYSTEMS 15 Applying Hölder and Poincaré inequalities, we obtain∫ ∞ 0 k(s)‖ηtn(s)‖ ds = ∫ ∞ 0 √ k(s) √ k(s)‖ηtn(s)‖ ds ≤ √∫ ∞ 0 k(s) ds √∫ ∞ 0 k(s)‖ηtn(s)‖2 ds ≤ Cp √ k0 √∫ ∞ 0 k(s)‖Dηtn(s)‖2 ds ≤ C‖ηtn‖M → 0, and ∫ ∞ 0 sk′(s)‖ηtn(s)‖ ds = ∫ ∞ 0 s √ −k′(s) √ −k′(s)‖ηtn(s)‖ ds ≤ ( − ∫ ∞ 0 s2k′(s) ds ∫ ∞ 0 −k′(s)‖ηtn(s)‖2 ds )1/2 ≤ (−C2 CP ∫ ∞ 0 −k′(s)‖Dηtn(s)‖2 ds )1/2 → 0. Therefore, |〈Dsη t n(s), sθ̂n〉M| → 0. In addition, 〈θn, sθ̂n〉M = ∫ ∞ 0 sk(s)(Dθn, Dθ̂n) ds = k1‖θn‖2 → 0, this means ‖θn‖ → 0. (4.26) Taking the inner product of both sides of (4.22) and (4.23) with un and ϕn, we have µ‖Dun‖2 + b(ϕn, Dun)→ 0, (4.27) δ‖Dϕn‖2 + b(Dun, ϕn) + ξ‖ϕn‖2 → 0. (4.28) Summing (4.27) and (4.28), we have µ‖Dun‖2 + 2b