Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 87, pp. 1–38. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.87 ASYMPTOTIC STABILIZATION FOR BRESSE TRANSMISSION SYSTEMS WITH FRACTIONAL DAMPING JIANGHAO HAO, DINGKUN WANG Abstract. In this article, we study the asymptotic stability of Bresse trans- mission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index θ ∈ [0, 1] and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with θ 6= 0, the system is not exponen- tial stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with θ = 0, the system is exponential stable. 1. Introduction In the previous decades, various types of equations models have been used to describe chemical, biological, physical, and engineering systems. In recent years, the mathematical model of arc-shaped elastic structures has been greatly promoted by more and more practical problems, and arc-shaped elastic structures are also widely studied in the fields of ocean, engineering, aviation, architecture and so on. Following the main idea of the deformation of elastic structures, we consider the circular arch problem given by the equations of motion, also known as the Bresse system (see [28] for details), ρ1ϕtt = Qx + lN, (1.1) ρ2ψtt = Mx −Q, (1.2) ρ1wtt = Nx − lQ, (1.3) where N = κ0l(wx − lϕ), (1.4) Q = κ(ϕx + ψ + lw), (1.5) M = bψx, (1.6) are the stress-strain relations for elastic behavior. Here ρ1 = ρA, ρ2 = ρI, κ = k′GA, κ0 = EA, b = EI, l = R−1. Here ρ is the material density, E is the elastic 2020 Mathematics Subject Classification. 35B37, 35L55, 74D05, 93D15. Key words and phrases. Bresse system; fractional damping; asymptotic stability; exponential decay; polynomial decay. ©2023. This work is licensed under a CC BY 4.0 license. Submitted July 2, 2023. Published December 28, 2023. 1 2 J. HAO, D. WANG EJDE-2023/87 modulus, G is the shear modulus, and k′ is the shear coefficient, A is the cross- sectional area, I is the area of the cross-sectional second moment, and R is the radius of curvature. These coefficients are normal numbers related to the physical properties of the beam. Functions ϕ, ψ and w denote vertical, shear angular and longitudinal displacement. In this article, we are interested in the asymptotic stability of Bresse systems (from coupled equations (1.1)-(1.6) whose dampings are given by fractional damping terms, and act on two equations respectively. The system is written as ρ1ϕtt − κ(ϕx + ψ + lw)x − κ0l(wx − lϕ) + γ1(−∂xx)θϕt = 0 in (0, L)× R+, ρ2ψtt − bψxx + κ(ϕx + ψ + lw) + γ2(−∂xx)θψt = 0 in (0, L)× R+, ρ1wtt − κ0(wx − lϕ)x + κl(ϕx + ψ + lw) + γ3(−∂xx)θwt = 0 in (0, L)× R+, (1.7) where θ is a parameter in the interval [0, 1] and damping coefficient γi ≥ 0, i = 1, 2, 3. We consider the Dirichlet-Neumann-Neumann boundary conditions ϕ(0, t) = ϕ(L, t) = ψx(0, t) = ψx(L, t) = wx(0, t) = wx(L, t) = 0 in R+, (1.8) and the initial conditions ϕ(x, 0) = ϕ0, ϕt(x, 0) = ϕ1 in (0, L), ψ(x, 0) = ψ0, ψt(x, 0) = ψ1 in (0, L), w(x, 0) = w0, wt(x, 0) = w1 in (0, L). (1.9) This fractional damping is an intermediate dissipation mechanism not previously considered in Bresse systems. In special cases, the mechanism includes friction damping (θ = 0) and Kelvin-Voigt damping (θ = 1). In books [13, 25] we find the following definition of fractional order operators: For α > 0, the bounded linear operator A−α is defined by A−α := 1 2πi ∫ γ λ−α(λI −A)−1dλ, where γ is a piecewise smooth path in Σ R+ going from ∞e−iδ to ∞eiδ for some δ > 0. We refer to [13, 25] for other relevant results on fractional powers. First, we introduce some relevant results that motivated this work. To stabilize the Bresse system, various kinds of damping are used and some decay results are established. From a large number of literature, three basic damping mechanisms can be distinguished, namely friction damping, Kelvin-Voigt damping and damping with memory. By comparison, the friction damping term is relatively simple, and the study of local Kelvin-Voigt damping is too much, while the damping with memory is more complex, because the damping term is represented by various forms of convolution products of the kernel. In the following content, we will briefly introduce the asymptotic stability of Bresse system under these three damping mechanisms. Guesmia [22] studied that when the friction damping only acts on a vertical displacement, under the Dirichlet-Neumann-Neumann boundary conditions, if l 6= mπ, ∀m ∈ Z, (1.10) the system is not exponentially stable. If (1.10) holds and l2 6= κ0ρ2 − bρ1 κ0ρ2 (mπ2)− κρ1 ρ2(κ+ κ0) , ∀m ∈ Z, (1.11) EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 3 the system is polynomial attenuated (the meaning of coefficient is consistent with that in this paper). Alabau Boussouira et al. studied in [5] that when the friction damping only acts on the shear angular displacement, under the complete Dirichlet boundary conditions, its stability is related to wave velocity κ ρ1 , b ρ2 and κ0 ρ1 . Denote the difference of wave velocity χ0 = κ ρ1 − b ρ2 and χ1 = κ− κ0. (1.12) when the wave velocity is equal, i.e. χ0, χ1 = 0, the system is decay exponentially. When the wave velocity is not equal, i.e. χ0 6= 0, χ1 6= 0 or χ0 6= 0, χ1 = 0, the system attenuates in polynomial form of t−1/6 or t−1/3. The optimality of poly- nomial decay is proved in [16]. Under the Dirichlet-Neumann-Neumann boundary conditions, if χ0 6= 0 or χ1 = 0, the system is not exponentially stable. Finally, numerical analysis is given to verify their conclusions. Afilal et al. [3] obtained that when the friction damping only acts on the longitudinal displacement, under the mixed boundary conditions ϕ(0, t) = ψx(0, t) = wx(0, t) = ϕx(L, t) = ψ(L, t) = w(L, t) = 0 in R+, if κ ρ1 = b ρ2 = κ0 ρ1 , l 6= π 2 +mπ, ∀m ∈ Z, (1.13) and l2 6= κ0ρ2 + bρ1 κ0ρ2 ( π 2 +mπ)− κρ1 ρ2(κ+ κ0) , ∀m ∈ Z, (1.14) the system is exponentially stable. If only (1.13) and (1.14) hold, the system is polynomial decay. The numerical analysis is also given. When there are two friction damping in the system, Alves et al. [2] proved that if there is no friction damping on the longitudinal displacement, the system is exponentially stable under the Dirichlet-Neumann-Neumann boundary conditions when χ1 = 0, and the system is non exponentially stable when χ1 6= 0. At the same time, they also proved that the decay is polynomial at the optimal rate t−1/2. Wehbe et al. [40] got that when there are two locally distributed feedbacks on the shear angular displacement and longitudinal displacement, ρ1ϕtt − κ(ϕx + ψ + lw)x − κ0l(wx − lϕ) = 0 in (0, L)× R+, ρ2ψtt − bψxx + κ(ϕx + ψ + lw) + a1(x)ψt = 0 in (0, L)× R+, ρ1wtt − κ0(wx − lϕ)x + κl(ϕx + ψ + lw) + a1(x)wt = 0 in (0, L)× R+, where the positive continuous functions aj(x), j = 1, 2 satisfy the conditions aj(x) ≥ a− > 0 for every x ∈ Θ := (0, c) ∪ (d, L), 0 < c < d < L. It turned out that under the Dirichlet-Neumann-Neumann boundary conditions, the system is exponentially stable when χ0 = 0. When χ0 6= 0, then for any positive integer m ≥ 1, there exists a constant Cm > 0 independent of initial value U0 ∈ D(Amj ), j = 1, 2 such that ‖Sj(t)U0‖2Hj ≤ Cm ( ln t t )m ln2 t‖U0‖2D(Amj ), ∀t > 0. 4 J. HAO, D. WANG EJDE-2023/87 For the Kelvin-Voigt damping system, Akil [1] studied the stability of Bresse system with only one discontinuous local Kelvin-Voigt damping on the axial force: ρ1ϕtt − κ(ϕx + ψ + lw)x − κ0l(wx − lϕ)− ld(x)(wtx − lϕt) = 0 in (0, L)× R+, ρ2ψtt − bψxx + κ(ϕx + ψ + lw) = 0 in (0, L)× R+, ρ1wtt − [κ0(wx − lϕ)x + d(x)(wtx − lϕt)]x + κl(ϕx + ψ + lw) = 0 in (0, L)× R+. Suppose that there exists 0 < α < β < L and a positive constant d0 such that d(x) = { d0 if x ∈ (α, β), 0, if x ∈ (0, α) ∪ (β, L), and under the complete Dirchlet boundary conditions, they proved that whether the wave velocities are equal or not, the system exhibits polynomial decay. When χ0 = 0, the decay rate is t−1. When χ0 6= 0, the decay rate is t−1/2. For other results on friction damping and Kelvin-Voigt damping, see [4, 14, 37] and their references. Recently some scholars have also studied the stability of Bresse systems whose damping term is dissipated through memory. When the memory terms of the three equations exist simultaneously, it has the following form∫ ∞ 0 g(s)ϕxx(x, t− s)ds, ∫ ∞ 0 g(s)ψxx(x, t− s)ds, ∫ ∞ 0 g(s)wxx(x, t− s)ds, where g : R+ → R+ is differentiable, non-increasing and integrable function on R+. Guesmia and Kafini [23] (three infinite memories), Guesmia and Kirane [24] (two infinite memories), Guesmia [19] (one infinite memory only acts on the lon- gitudinal displacement) and De Lima Santos et al. [36] (one infinite memory only acts on the shear angular displacement) obtained the asymptotic stability of the one-dimensional linear Bresse system under infinite memory, respectively. When the kernel function decays exponential at infinity, if the wave propagation velocity is the same, the exponential stability of the corresponding systems is obtained in these papers, otherwise it will lead to polynomial stability with decay rate t−1/2. Guesmia in [20] studied that an infinite memory only acts on the vertical displace- ment, they proved that even if the wave propagation velocity is the same and the kernel function has exponential decay at infinity, the exponential stability is not tenable, but it decays as a polynomial with t−1/4. In addition, the authors in [7] considered the stability of Bresse system with memory term acting on shear an- gular displacement under arbitrary growth of the relaxation function at infinity. They not only proved that the system is well-posedness, but also presented two general decay estimates: a uniform stability estimate under (1.12) and another general weak stability result. Some other authors have also considered the others dissipation mechanisms in Bresse systems, such as thermoelastic Bresse systems, (see [15, 21, 26, 30]). It is well known that the Bresse system evolved from the Timoshenko beam equation. If R → ∞ and g = 0, then l → 0 (see above for specific physical meanings), the model is simplified to Timoshenko beam equation (see [18]). If R → ∞ and g 6= 0, then l → 0, the model is simplified to Timoshenko beam equation with past history (see [31]). It is noteworthy that Higidio Portillo Oquendo et al. [32] dealt with the asymptotic behavior of the solution for a Timoshenko EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 5 system with a fractional damping, ρ1φtt − κ(φxx + ψx) = 0 in (0, L)× R+, ρ2ψtt − bψxx + κ(φx + ψ) + (−∂xx)θψt = 0 in (0, L)× R+, satisfying the boundary conditions φ(0, t) = φ(L, t) = ψx(0, t) = ψx(L, t) = 0 on R+. Here, the parameter θ ∈ [0, 1], the damping only acts on one equation of the system, and the exponentially decreasing kernel is considered. The authors obtained the exact decay rate, which depends on the difference of the propagation speeds of the two waves. To be precise, when the equations have different propagation speeds, if θ ≤ 1/2 then the system decays polynomially with rate t−1/(2−2θ), if θ ≥ 1/2 then the system decays polynomially with rate t−1/(2θ); when the equations have the same propagation speed and θ ∈ (0, 1], the system decays polynomially with rate t−1/(2θ), and these decay rates are optimal; when θ = 0 and the equations have the same propagation speed, the exponential decay of the system is obtained. Fur- thermore, Astudillo and Oquendo [6] studied the stability of the Timoshenko beam equation with fractional memory term under the exponentially decreasing kernels. The relationship between stability, the wave velocity and the fractional damping exponent is studied by using semi group method, and obtained the corresponding exponential stability and precise polynomial decay rates. The stability of some other Bresse systems with fractional derivatives have also been studied. In [9], Oquendo and Suárez introduced two internal damping terms expressed by the generalized Caputo fractional derivative and studied the asymp- totic stability of the following viscoelastic Bresse systems, ρ1ϕtt − κ(ϕx + ψ + lw)x − κ0l(wx − lϕ) + a1(x)∂α,ηt ϕ = 0, ρ2ψtt − bψxx + ∫ ∞ 0 g(s)ψxx(t− s)ds+ κ(ϕx + ψ + lw) + a2(x)∂β,ηt ψ = 0, ρ1wtt − κ0(wx − lϕ)x + κl(ϕx + ψ + lw) = 0, in (0, L) × R+, where the symbol ∂α,ηt (or ∂β,ηt ) refers to the generalized Caputo fractional derivative corresponding to the time variable t of order α (or β) and it is expressed for the order α by ∂α,ηt f(t) = 1 Γ(1− α) ∫ t 0 (t− s)−αe−η(t−s) df ds (s)ds. The authors not only proved the strong stability, lack of exponential stability and polynomial stability of the system, but also gave an accurate decay rate (see Theo- rem 3.8 of [9] for details), and also used numerical simulation to verify their results. Earlier, Benaissa and Kasmi [8] considered the Bresse system with three control boundary conditions of fractional derivative type, and they obtained the polynomial decay result. There are many studies on fractional damping. For other types of references, the readers can see [11, 12, 27, 33, 38, 39] and the references therein. Inspired by these works, we study the asymptotic behavior of the Bresse system (1.7)-(1.9). Firstly we introduce some notation. For 1 ≤ p ≤ ∞, Lp := Lp(0, L) denotes the usual Lebesgue space with the norm ‖ · ‖Lp . For the convenience of notation, we will use ‖ · ‖ instead of ‖ · ‖L2 and 〈·, ·〉 instead of 〈·, ·〉L2 . Let s be 6 J. HAO, D. WANG EJDE-2023/87 a nonnegative number, Hs := Hs(0, L) denotes the usual Sobolev space, equipped with the norm ‖ · ‖Hs . In the following, C denotes a generic positive constant. The set L2 ∗(0, L) := { h ∈ L2(0, L) : ∫ L 0 h(x)dx = 0 } is a closed subspace with the L2-norm, therefore it is a Hilbert space. As we know, the operators E := −∂xx : D(E) ⊂ L2(0, L)→ L2(0, L), (1.15) E∗ := −∂xx : D(E∗) ⊂ L2 ∗(0, L)→ L2 ∗(0, L), (1.16) with respective domains D(E) =: H2(0, L) ∩H1 0 (0, L), D(E∗) := { ψ,w ∈ H2(0, L) ∩ L2 ∗(0, L) : ψx(0) = ψx(L) = 0, wx(0) = wx(L) = 0 } , are positive, self-adjoint and have compact inverse. Therefore, the operators Eσ , Eσ∗ are bounded for σ ≤ 0, and positive self-adjoint for σ ∈ R. Furthermore, the embeddings D(Eσ1) ↪→ D(Eσ2), D(Eσ1 ∗ ) ↪→ D(Eσ2 ∗ ) are continuous for σ1 > σ2. The norms in D(Eσ) and D(Eσ∗ ) for σ ≥ 0 are given by ‖ϕ‖D(Eσ) := ‖Eσϕ‖, ‖ψ‖D(Eσ∗ ) := ‖Eσ∗ψ‖, and ‖w‖D(Eσ∗ ) := ‖Eσ∗w‖ respectively. Because the operators E and E∗ are positive, self-adjoint and they have compact inverse, the spectrum of these operators is constituted only by positive eigenvalues. The eigenvalues for both operators are given by ξ2n, where ξn = nπ L , n ∈ N, and the corresponding unitary eigenfunctions associated to these eigenvalues are en(x) = √ 2 L sin(ξnx), e∗n(x) = √ 2 L cos(ξnx). (1.17) The sequences {en} and {e∗n} form the bases of the spaces L2(0, L) and L2 ∗(0, L) respectively, then for ϕ ∈ L2(0, L) and ψ,w ∈ L2 ∗(0, L) we have ϕ = ∞∑ n=1 〈ϕ, en〉en, ψ = ∞∑ n=1 〈ψ, e∗n〉e∗n, w = ∞∑ n=1 〈w, e∗n〉e∗n. Note that, for ϕ ∈ D(Eσ+1/2), we have the following identities Eσ+1/2ϕ = ∞∑ n=1 ξ2σ+1 n 〈ϕ, en〉en, Eσ∗ ∂xϕ = ∞∑ n=1 ξ2σ+1 n 〈ϕ, en〉e∗n, by Parseval’s identity, we obtain ‖Eσ+1/2ϕ‖ = ‖Eσ∗ ∂xϕ‖. (1.18) In particular, for σ = 0 we have ‖E1/2ϕ‖ = ‖∂xϕ‖. In a similar way, for ψ,w ∈ D(E σ+1/2 ∗ ) it follows that ‖Eσ+1/2 ∗ ψ‖ = ‖Eσ∂xψ‖, ‖Eσ+1/2 ∗ w‖ = ‖Eσ∂xw‖. (1.19) At the same time, for ϕ ∈ D(Eσ0) and ψ,w ∈ D(Eσ0 ∗ ), with σ0 = max{σ, 1/2}, we easily verify that 〈Eσ∗ψ,ϕx〉 = −〈ψx, Eσϕ〉, 〈Eσ∗w,ϕx〉 = −〈wx, Eσϕ〉. (1.20) EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 7 Our main results deal with the asymptotic behavior of the solution of this sys- tem. The innovation of this paper is to extend the dissipation mechanism of control in some literatures to the case of fractional damping, and is to study the asymptotic stability of the system (1.7)-(1.9) when there are only two fractional damping re- spectively. It is found that the stability is related to the wave speed (1.12) and the value of θ ∈ [0, 1]. These results are clarified in Theorems 4.11, which are mainly stated as follows: (i) When fractional damping acts on vertical displacement and shear angular displacement, that is, γ1 > 0, γ2 > 0, γ3 = 0, if χ1 = 0 and θ ∈ (0, 1], then the semigroup etA decays polynomially with rate t−1/(2θ), when θ = 0, then the semigroup etA decay exponentially; if χ1 6= 0 and θ ≤ 1/2, then the semigroup etA decays polynomially with rate t−1/(2−2θ), when θ ≥ 1/2, then the semigroup etA decays polynomially with rate t−1/(2θ). (ii) When fractional damping acts on vertical displacement and longitudinal displacements, that is, γ1 > 0, γ3 > 0, γ2 = 0, if χ0 = 0 and θ ∈ (0, 1], then the semigroup etA decays polynomially with rate t−1/(2θ), when θ = 0, then the semigroup etA decay exponentially; if χ0 6= 0 and θ ≤ 1/2, then the semigroup etA decays polynomially with rate t−1/(2−2θ), when θ ≥ 1/2, then the semigroup etA decays polynomially with rate t−1/(2θ). (iii) When fractional damping acts on longitudinal displacements and shear an- gular displacement, that is, γ2 > 0, γ3 > 0, γ1 = 0, if χ0 = 0 and θ ∈ (0, 1], then the semigroup etA decays polynomially with rate t−1/(2θ), when θ = 0, then the semigroup etA decay exponentially; if χ0 6= 0 and θ ≤ 1/2, then the semigroup etA decays polynomially with rate t−1/(2−2θ), when θ ≥ 1/2, then the semigroup etA decays polynomially with rate t−1/(2θ). The outline of this article is the followings. In section 2 we study the well- posedness result of solution to system (1.7)-(1.9). In section 3, we prove the case of lack of exponential stability. In section 4, we give the asymptotic behavior of the corresponding semigroups, including exponential stability and polynomial stability, and precise decay rates are obtained. 2. Well-posedness of solution In this section, we use the semigroup theory to obtain the existence and unique- ness of solution for system (1.7)-(1.9). We denote the state space by H := H1 0 (0, L)× L2(0, L)×H1 ∗ (0, L)× L2 ∗(0, L)×H1 ∗ (0, L)× L2 ∗(0, L), (2.1) where H1 ∗ (0, L) := H1(0, L) ∩ L2 ∗(0, L). Note that H is an Hilbert space with the inner product 〈U1, U2〉H = ρ1〈ϕ̃1, ϕ̃2〉+ ρ2〈ψ̃1, ψ̃2〉+ ρ1〈w̃1, w̃2〉+ κ0〈∂xw1 − lϕ1, ∂xw2 − lϕ2〉 + κ〈∂xϕ1 + ψ1 + lw1, ∂xϕ2 + ψ2 + lw2〉+ b〈∂xψ1, ∂xψ2〉, (2.2) and the norm ‖U‖2H = ρ1‖ϕ̃‖2 + ρ2‖ψ̃‖2 + ρ1‖w̃‖2 + κ‖∂xϕ+ ψ + lw‖2 + κ0‖∂xw − lϕ‖2 + b‖∂xψ‖2, (2.3) where Ui = (ϕi, ϕ̃i, ψi, ψ̃i, wi, w̃i) T , i = 1, 2. 8 J. HAO, D. WANG EJDE-2023/87 If we consider the vector U(t) = (ϕ(t), ϕ̃(t), ψ(t), ψ̃(t), w(t), w̃(t))T , then system (1.7)-(1.9) can be written as the Cauchy problem d dt U(t) = AU(t), U(0) = U0, (2.4) where U0 = (ϕ0, ϕ1, ψ0, ψ1, w0, w1)T is the vector of initial data and the operator A is given by AU =  ϕ̃ κ ρ1 (ϕxx + ψx + lwx) + κ0l ρ1 (wx − lϕ)− γ1 ρ1 Eθϕ̃ ψ̃ b ρ2 ψxx − κ ρ2 (ϕx + ψ + lw)− γ2 ρ2 Eθ∗ ψ̃ w̃ κ0 ρ1 (wxx − lϕx)− kl ρ1 (ϕx + ψ + lw)− γ3 ρ1 Eθ∗w̃  , (2.5) with D(A) = { U ∈ H : ϕ̃ ∈ H1 0 (0, L), ψ̃ ∈ H1 ∗ (0, L), w̃ ∈ H1 ∗ (0, L), ϕ ∈ H1 0 (0, L) ∩H2(0, L), ψ, w ∈ H1 ∗ (0, L) ∩H2(0, L), κEϕ+ γ1E θϕ̃ ∈ L2(0, L), bE∗ψ + γ2E θψ̃ ∈ L2 ∗(0, L), κ0E∗w + γ3E θ ∗w̃ ∈ L2 ∗(0, L) } . (2.6) We use the following Lumer-Phillips theorem [34] to prove the existence of solu- tion of Cauchy problem (2.4). Theorem 2.1 ([34]). Let A a linear operator with dense domain D(A) in a Hilbert space H. If A is dissipative and 0 ∈ ρ(A), the resolvent set of A, then the operator A is the generator of a C0-semigroup of contractions on H. We give the well-posedness result of solution as the following theorem. Theorem 2.2. For U0 = (ϕ0, ϕ1, ψ0, ψ1, w0, w1)T ∈ H, there exists a unique solu- tion of Cauchy problem (2.4) U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T ∈ C([0,∞);H). Moreover, if U0 ∈ D(A), then the solution is more regular, i.e. U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T ∈ C([0,∞);D(A)) ∩ C1([0,∞);H). Proof. We prove that the operator A in (2.5) satisfies the conditions of Theorem 2.1. Firstly, from (2.6) we can obtain that the domain of the operator A is dense in H. In addition, for any U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T ∈ D(A) we obtain Re〈AU,U〉 = −γ1‖Eθ/2ϕ̃‖2 − γ2‖Eθ/2∗ ψ̃‖2 − γ3‖Eθ/2∗ w̃‖2 ≤ 0. (2.7) Therefore, the operator A is dissipative. Secondly, we need to check that 0 ∈ ρ(A). To do this, for any F = (f1, f2, f3, f4, f5, f6)T ∈ H, let us prove that the problem AU = F has a unique solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T in D(A). According to the definition of the operator A, the system can be written as ϕ̃ = f1, (2.8a) κ(ϕxx + ψx + lwx) + κ0l(wx − lϕ)− γ1Eθϕ̃ = ρ1f2, (2.8b) EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 9 ψ̃ = f3, (2.8c) bψxx − κ(ϕx + ψ + lw)− γ2Eθ∗ ψ̃ = ρ2f4, (2.8d) w̃ = f5, (2.8e) κ0(wxx − lϕx)− kl(ϕx + ψ + lw)− γ3Eθ∗w̃ = ρ1f6, (2.8f) then from (2.8b), (2.8d) and (2.8e), we obtain κ(ϕxx + ψx + lwx) + κ0l(wx − lϕ) = h1, (2.9a) bψxx − κ(ϕx + ψ + lw) = h2, (2.9b) κ0(wxx − lϕx)− kl(ϕx + ψ + lw) = h3, (2.9c) where h1 = ρ1f2+γ1E θf1, h2 = ρ2f4+γ2E θ ∗f3 and h3 = ρ1f6+γ3E θ ∗f5. Multiplying (2.9a) by Φ ∈ H1 0 (0, L) , (2.9b) by Ψ ∈ H1 ∗ (0, L), and (2.9c) by W ∈ H1 ∗ (0, L), summing them, then system (2.9a)-(2.9c) can be studied as a variational problem B((ϕ,ψ,w), (Φ,Ψ,W )) = L(Φ,Ψ,W ), (2.10) where B((ϕ,ψ,w), (Φ,Ψ,W )) = κ〈ϕx + ψ + lw,Φx + Ψ + lW 〉+ b〈ψx,Ψx〉+ κ0〈wx − lϕ,Wx − lΦ〉, L(Φ,Ψ,W ) = −〈h1,Φ〉 − 〈h2,Ψ〉 − 〈h3,W 〉. We can verify that B is a continuous sesquilinear form on (H1 0 (0, L) ×H1 ∗ (0, L) × H1 ∗ (0, L))2 and L is a continuous linear form on H−1(0, L)×H−1∗ (0, L)×H−1∗ (0, L). At the same time, taking (Φ,Ψ,W ) = (ϕ,ψ,w), we have B((ϕ,ψ,w), (ϕ,ψ,w)) = κ‖ϕx + ψ + lw‖2 + b‖ψx‖2 + κ0‖wx − lϕ‖2. (2.11) Thus, we obtain the coercivity of this sesquilinear form. Now, applying Lax- Milgram theorem and considering (2.8a), (2.8c) and (2.8e), we have a unique so- lution U ∈ H. Since the solution satisfies system (2.8a)-(2.8f) in a weak sense, by these equations, we can obtain that U ∈ D(A). Finally, from (2.10) and (2.11), we deduce κ‖ϕx + ψ + lw‖2 + b‖ψx‖2 + κ0‖wx − lϕ‖2 = −ρ1〈f2, ϕ〉 − ρ2〈f4, ψ〉 − ρ1〈f6, w〉 − γ1〈Eθf1, ϕ〉 − γ2〈Eθ∗f3, ψ〉 − γ3〈Eθ∗f5, w〉. (2.12) From (2.7), we obtain − γ1‖Eθ/2ϕ̃‖2− γ2‖Eθ/2∗ ψ̃‖2− γ3‖Eθ/2∗ w̃‖2 ≤ −CRe〈AU,U〉 ≤ C‖F‖‖U‖. (2.13) Substituting (2.13) to (2.12), and using the Cauchy-Schwarz, Young’s and poincaré inequalities, we obtain κ‖ϕx + ψ + lw‖2 + b‖ψx‖2 + κ0‖wx − lϕ‖2 ≤ ε(‖ϕx‖2 + ‖ψx‖2 + ‖wx‖2 + C(‖F‖‖U‖+ ‖F‖2) for any constant ε > 0. Using this inequality gives ‖ϕx‖2 ≤ C(‖ϕx + ψ + lw‖2 + ‖ψ‖2 + ‖w‖2) ≤ C(‖ϕx + ψ + lw‖2 + ‖ψx‖2 + ‖wx‖2), and ‖wx‖2 ≤ C(‖wx − lϕ‖2 + ‖ϕ‖2) 10 J. HAO, D. WANG EJDE-2023/87 ≤ C(‖wx − lϕ‖2 + ‖ϕx‖2) ≤ C(‖wx − lϕ‖2 + ‖ϕx + ψ + lw‖2 + ‖ψx‖2 + ε‖wx‖2), for fixing constant ε enough small. By using poincaré inequality we obtain κ‖ϕx + ψ + lw‖2 + ‖ψx‖2 + κ0‖wx − lϕ‖2 ≤ C(‖F‖‖U‖+ ‖F‖2). Furthermore, from (2.8a), (2.8c), (2.8e), it follows that ρ1‖ϕ‖2 + ρ2‖ψ‖2 + ρ1‖w‖2 ≤ C‖F‖2. Therefore, from the above inequalities we conclude ‖U‖2 ≤ C‖F‖2, that is, 0 ∈ ρ(A). From Theorem 2.1, we obtain that A is the generator of a C0-semigroup of contractions in H, and the well-posedness of the Cauchy problem (2.4) is a result of the semigroup theory. The proof of Theorem 2.2 is complete. � 3. Lack of exponential stability In this section, we show that the semigroup associated with the Bresse system is not exponentially stable. We will use Pruss’s theorem [35] to prove the lack of exponential stability. That is, we will show that there exists a sequence of values λn such that ‖(λnI −A)−1‖L(H) →∞. It is equivalent to prove the existence of a sequence {Fn} ⊂ H and a sequence of complex numbers {λn} ⊂ iR, with Fn is bounded in H such that ‖(λnI −A)−1Fn‖H →∞, where (λn −A)Un = Fn with Un not bounded. Taking Fn = (f1, f2, f3, f4, f5, f6)T , we write firstly the spectral equation in terms of its components as follows, λnϕ− ϕ̃ = f1, (3.1a) λnϕ̃− κ ρ1 (ϕx + ψ + lw)x − κ0l ρ1 (wx − lϕ) + γ1 ρ1 Eθϕ̃ = f2, (3.1b) λnψ − ψ̃ = f3, (3.1c) λnψ̃ − b ρ2 ψxx + κ ρ2 (ϕx + ψ + lw) + γ2 ρ2 Eθ∗ ψ̃ = f4, (3.1d) λnw − w̃ = f5, (3.1e) λnw̃ − κ0 ρ1 (wx − lϕ)x + kl ρ1 (ϕx + ψ + lw) + γ3 ρ1 Eθ∗w̃ = f6. (3.1f) The main result of this section is stated as follows. Theorem 3.1. (i) When γ1, γ2 > 0, γ3 = 0, if χ1 6= 0, or χ1 = 0 and θ ∈ (0, 1], then the semigroup associated to system (1.7)-(1.9) is not ex- ponentially stable. (ii) When γ1, γ3 > 0, γ2 = 0, if χ0 6= 0, or χ0 = 0 and θ ∈ (0, 1], then the semigroup associated to system (1.7)-(1.9) is not exponentially stable. (iii) When γ2, γ3 > 0, γ1 = 0, if χ0 6= 0, or χ0 = 0 and θ ∈ (0, 1], then the semigroup associated to system (1.7)-(1.9) is not exponentially stable. EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 11 Proof. Using Pruss’s theorem [35], and taking f1 = f3 = f5 = 0 in (3.1a)-(3.1f) we obtain λnϕ = ϕ̃, (3.2a) λnϕ̃− κ ρ1 (ϕx + ψ + lw)x − κ0l ρ1 (wx − lϕ) + γ1 ρ1 Eθϕ̃ = f2, (3.2b) λnψ = ψ̃, (3.2c) λnψ̃ − b ρ2 ψxx + γ2 ρ2 Eθ∗ ψ̃ + κ ρ2 (ϕx + ψ + lw) = f4, (3.2d) λnw = w̃, (3.2e) λnw̃ − κ0 ρ1 (wx − lϕ)x + kl ρ1 (ϕx + ψ + lw) + γ3 ρ1 Eθ∗w̃ = f6. (3.2f) Substituting (3.2a), (3.2c), and (3.2e) into (3.2b), (3.2d) and (3.2f) respectively, we obtain λ2nϕ− κ ρ1 (ϕx + ψ + lw)x − κ0l ρ1 (wx − lϕ) + γ1 ρ1 λnE θϕ = f2, λ2nψ − b ρ2 ψxx + κ ρ2 (ϕx + ψ + lw) + γ2 ρ2 λnE θ ∗ψ = f4, λ2nw − κ0 ρ1 (wx − lϕ)x + kl ρ1 (ϕx + ψ + lw) + γ3 ρ1 λnE θ ∗w = f6. (3.3) Because of the Dirichlet-Neumann-Neumann boundary conditions (1.8), we take ϕ, ψ, w are of the form ϕ = An sin(nπL x), ψ = Bn cos(nπL x) and w = Cn cos(nπL x) with n ∈ N, where An, Bn and Cn depend on λn and will be explicitly determined below. After performing some simplifications, we will obtain a system of the form ΛŪ = Ξ, with Ū := (An, Bn, Cn)T , Ξ := ( f2 sin(nπL x) , f4 cos(nπL x) , f6 cos(nπL x) )T , Λ :=  P1(λn) κ ρ1 (nπL ) l(κ+κ0) ρ1 (nπL ) κ ρ2 (nπL ) P2(λn) lκ ρ2 l(κ+κ0) ρ1 (nπL ) lκ ρ1 P3(λn)  , where P1(λn) := λ2n + κ ρ1 ( nπ L )2 + κ0l 2 ρ1 + γ1 ρ1 λn( nπ L )2θ, P2(λn) := λ2n + b ρ2 ( nπ L )2 + κ ρ2 + γ2 ρ2 λn( nπ L )2θ, P3(λn) := λ2n + κ0 ρ1 ( nπ L )2 + κl2 ρ1 + γ3 ρ1 λn( nπ L )2θ. By solving this system, the expressions of An, Bn, Cn are obtained. And to do that, we set det(Λ) := ∣∣∣∣∣∣∣ P1(λn) κ ρ1 (nπL ) l(κ+κ0) ρ1 (nπL ) κ ρ2 (nπL ) P2(λn) lκ ρ2 l(κ+κ0) ρ1 (nπL ) lκ ρ1 P3(λn) ∣∣∣∣∣∣∣ . 12 J. HAO, D. WANG EJDE-2023/87 Next, we will discuss the non-exponential stability of the system in the following three cases. Case (i) When γ1, γ2 > 0, γ3 = 0. We consider the expression Cn = ∣∣∣∣∣∣∣ P1(λn) κ ρ1 (nπL ) f2 sin(nπL x) κ ρ2 (nπL ) P2(λn) f4 cos(nπL x) l(κ+κ0) ρ1 (nπL ) lκ ρ1 f6 cos(nπL x) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ P1(λn) κ ρ1 (nπL ) l(κ+κ0) ρ1 (nπL ) κ ρ2 (nπL ) P2(λn) lκ ρ2 l(κ+κ0) ρ1 (nπL ) lκ ρ1 P3(λn) ∣∣∣∣∣∣∣ . Taking f2 = f4 = 0, f6 = cos(nπL x), and P3(λn) = λ2n + κ0 ρ1 (nπL )2 + κl2 ρ1 = c0 ∈ R. Here, if χ1 6= 0, then c0 := l2(κ+κ0) 2 ρ1(κ−κ0) , while if χ1 = 0, then c0 is a given constant. Thus we have λ2n = c0 − κ0 ρ1 ( nπ L )2 − κl2 ρ1 . For large values of n, we have λn ∈ iR and |λn| ∼ O(n). So we obtain Cn = P1(λn)P2(λn)− κ2 ρ1ρ2 (nπL )2 det1(λn) , where det 1 (λn) = P1(λn)P2(λn)c0 + 2κ2l2(κ+ κ0) ρ21ρ2 ( nπ L )2 − P2(λn) l2(κ+ κ0)2 ρ21 ( nπ L )2 − P1(λn) κ2l2 ρ1ρ2 − c0 κ2 ρ1ρ2 ( nπ L )2, with the polynomials reduced to P1(λn) = c0 − κ0 ρ1 ( nπ L )2 − κl2 ρ1 + κ ρ1 ( nπ L )2 + κ0l 2 ρ1 + γ1 ρ1 λn( nπ L )2θ = c0 + |χ1| ρ1 ( nπ L )2 + |χ1|l2 ρ1 + γ1 ρ1 λn( nπ L )2θ, and P2(λn) = c0 − κ0 ρ1 ( nπ L )2 − κl2 ρ1 + b ρ2 ( nπ L )2 + κ ρ2 + γ2 ρ2 λn( nπ L )2θ = c0 + ( b ρ2 − κ0 ρ1 )( nπ L )2 − κl2 ρ1 + κ ρ2 + γ2 ρ2 λn( nπ L )2θ. Next, we discuss the classifications. For the subcase χ1 = 0, we obtain |P1(λn)| = c0 + γ1 ρ1 λn( nπ L )2θ ∼ O(n1+2θ), EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 13 |P2(λn)| = c0 + ( b ρ2 − κ ρ1 )( nπ L )2 − κl2 ρ1 + κ ρ2 + γ2 ρ2 λn( nπ L )2θ ∼  O(n1+2θ), if χ0 = 0, O(n1+2θ), if χ0 6= 0 and θ ≥ 1/2, O(n2), if χ0 6= 0 and θ ≤ 1/2. Thus, we have |P1(λn)P2(λn)− κ2 ρ1ρ2 ( nπ L )2| ∼ P1(λn)P2(λn) ∼  O(n2+4θ), if χ0 = 0, O(n2+4θ), if χ0 6= 0, θ ≥ 1/2, O(n3+2θ), if χ0 6= 0, θ ≤ 1/2, |det 1 (λn)| ∼ { O(n2+4θ), if θ ≥ 1/2, χ0 = 0, O(n3+2θ), if θ ≤ 1/2, χ0 = 0, |det 1 (λn)| ∼ { O(n2+4θ), if θ ≥ 1/2, χ0 6= 0, O(n4), if θ ≤ 1/2, χ0 6= 0, According to the ratio, it is found that whether χ0 is 0 or not, the asymptotic behavior of Cn can be estimated as |Cn| ∼ { O(1), if θ ≥ 1/2, O(n2θ−1), if θ ≤ 1/2. For the subcase χ1 6= 0, we have |P1(λn)|, |P2(λn)| ∼ { O(n1+2θ), if θ ≥ 1/2, O(n2), if θ ≤ 1/2, then |P1(λn)P2(λn)− κ2 ρ1ρ2 ( nπ L )2| ∼ { O(n2+4θ), if θ ≥ 1/2, O(n4), if θ ≤ 1/2, Substituting c0 = l2(κ+κ0) 2 ρ1(κ−κ0) and P1(λn) into det1(λn), we obtain |det 1 (λn)| = c0P2(λn) [ P1(λn)− l2(κ+ κ0)2 c0ρ21 ( nπ L )2 ] + 2κ2l2(κ+ κ0) ρ21ρ2 ( nπ L )2 − P1(λn) κ2l2 ρ1ρ2 − c0 κ2 ρ1ρ2 ( nπ L )2 = c0P2(λn) [ c0 + |χ1| ρ1 ( nπ L )2 + |χ1|l2 ρ1 + γ1 ρ1 λn( nπ L )2θ − |χ1| ρ1 ( nπ L )2 ] + 2κ2l2(κ+ κ0) ρ21ρ2 ( nπ L )2 − P1(λn) κ2l2 ρ1ρ2 − c0 κ2 ρ1ρ2 ( nπ L )2 = c0P2(λn) [ c0 + |χ1|l2 ρ1 + γ1 ρ1 λn( nπ L )2θ ] + 2κ2l2(κ+ κ0) ρ21ρ2 ( nπ L )2 − P1(λn) κ2l2 ρ1ρ2 − c0 κ2 ρ1ρ2 ( nπ L )2 14 J. HAO, D. WANG EJDE-2023/87 ∼ { O(n2+4θ), if θ ≥ 1/2, O(n3+2θ), if θ ≤ 1/2. So, the asymptotic behavior of Cn can be estimated as |Cn| ∼ { O(1), if θ ≥ 1/2, O(n1−2θ), if θ ≤ 1/2. Hence, ‖Un‖2H ≥ ρ1‖w̃‖2L2(0,L) = ρ1|λnCn|2 ∫ L 0 cos2( nπ L x)dx = ρ1L 2 |λnCn|2. So, when χ1 6= 0, we have ‖Un‖H ≥ √ ρ1L 2 |λn| |Cn| ∼ { O(n), if θ ≥ 1/2, O(n2−2θ), if θ ≤ 1/2, lim n→∞ ‖Un‖H = +∞, while when χ1 = 0, we have ‖Un‖H ≥ √ ρ1L 2 |λn| |Cn| ∼ { O(n), if θ ≥ 1/2, O(n2θ), if θ ≤ 1/2. If θ ∈ (0, 1], then limn→∞ ‖Un‖H = +∞. This means that the corresponding semigroup is not exponentially stable when χ1 6= 0 or χ1 = 0 and θ ∈ (0, 1]. The first result of this theorem is proved. Case (ii) When γ1, γ3 > 0, and γ2 = 0, we consider the expression Bn = ∣∣∣∣∣∣∣∣ P1(λn) f2 sin(nπL x) l(κ+κ0) ρ1 (nπL ) κ ρ2 (nπL ) f4 cos(nπL x) lκ ρ2 l(κ+κ0) ρ1 (nπL ) f6 cos(nπL x) P3(λn) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ P1(λn) κ ρ1 (nπL ) l(κ+κ0) ρ1 (nπL ) κ ρ2 (nπL ) P2(λn) lκ ρ2 l(κ+κ0) ρ1 (nπL ) lκ ρ1 P3(λn) ∣∣∣∣∣∣∣ . We take f2 = f6 = 0, f4 = cos(nπL x), and P2(λn) = λ2n + b ρ2 (nπL )2 + κ ρ2 = b0 ∈ R. Here, if χ0 6= 0, then b0 := κ2 κρ2−bρ1 , while if χ0 = 0, then b0 is a given constant. Then λ2n = b0 − b ρ2 ( nπ L )2 − κ ρ2 . For large values of n, we have λn ∈ iR and |λn| ∼ O(n). So, we obtain Bn = P1(λn)P3(λn)− l2(κ+κ0) 2 ρ21 (nπL )2 det2(λn) , where det 2 (λn) = P1(λn)P2(λn)b0 + 2κ2l2(κ+ κ0) ρ21ρ2 ( nπ L )2 − b0 l2(κ+ κ0)2 ρ21 ( nπ L )2 − P1(λn) κ2l2 ρ1ρ2 − P3(λn) κ2 ρ1ρ2 ( nπ L )2, EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 15 with the polynomials reduced to P1(λn) = b0 − b ρ2 ( nπ L )2 − κ ρ2 + κ ρ1 ( nπ L )2 + κ0l 2 ρ1 + γ1 ρ1 λn( nπ L )2θ, = b0 + |χ0|( nπ L )2 − κ ρ2 + κ0l 2 ρ1 + γ1 ρ1 λn( nπ L )2θ, and P3(λn) = b0 − b ρ2 ( nπ L )2 − κ ρ2 + κ0 ρ1 ( nπ L )2 + κl2 ρ1 + γ3 ρ1 λn( nπ L )2θ, = b0 + ( κ0 ρ1 − b ρ2 )( nπ L )2 − κ ρ2 + κl2 ρ1 + γ3 ρ1 λn( nπ L )2θ. Through calculation, we find that the estimate of the asymptotic behavior of Bn is consistent with the estimate of Cn in the item (i) of this Theorem, and only replace χ1 and χ0 in Cn with χ0 and χ1 in Bn respectively. So we will not go into details here. Case (iii) When γ2, γ3 > 0, and γ1 = 0, we consider the expression An = ∣∣∣∣∣∣∣ f2 sin(nπL x) κ ρ1 (nπL ) l(κ+κ0) ρ1 (nπL ) f4 cos(nπL x) P2(λn) lκ ρ2 f6 cos(nπL x) lκ ρ1 P3(λn) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ P1(λn) κ ρ1 (nπL ) l(κ+κ0) ρ1 (nπL ) κ ρ2 (nπL ) P2(λn) lκ ρ2 l(κ+κ0) ρ1 (nπL ) lκ ρ1 P3(λn) ∣∣∣∣∣∣∣ . We Take f2 = sin(nπL x), f4 = cos(nπL x), f6 = 0, and P1(λn) = λ2n+ κ ρ1 (nπL )2+ κ0l 2 ρ1 = a0, where c0 ∈ R is a given constant. Then λ2n = a0 − κ ρ1 ( nπ L )2 − κ0l 2 ρ1 . For large values of n, we have λn ∈ iR and |λn| ∼ O(n). So, we obtain An = P2(λn)P3(λn)− κl2(κ+κ0) ρ21 (nπL )2 − κ2l2 ρ1ρ2 − P3(λn) κρ1 (nπL )2 det3(λn) , where det 3 (λn) = P2(λn)P3(λn)a0 + 2κ2l2(κ+ κ0) ρ21ρ2 ( nπ L )2 − P2(λn) l2(κ+ κ0)2 ρ21 ( nπ L )2 − a0(λn) κ2l2 ρ1ρ2 − P3(λn) κ2 ρ1ρ2 ( nπ L )2, with the polynomials reduced to P2(λn) = a0 − κ ρ1 ( nπ L )2 − κ0l 2 ρ1 + b ρ2 ( nπ L )2 + κ ρ2 + γ2 ρ2 λn( nπ L )2θ, = a0 + |χ0|( nπ L )2 + κ ρ2 − κ0l 2 ρ1 + γ2 ρ2 λn( nπ L )2θ, 16 J. HAO, D. WANG EJDE-2023/87 and P3(λn) = a0 − κ ρ1 ( nπ L )2 − κ0l 2 ρ1 + κ0 ρ1 ( nπ L )2 + κl2 ρ1 + γ3 ρ1 λn( nπ L )2θ, = a0 + |χ1| ρ1 ( nπ L )2 − |χ1|l2 ρ1 + γ3 ρ1 λn( nπ L )2θ. Next, we discuss the classification. For the subcase χ0 = 0, we have |P2(λn)| = a0 + κ ρ2 − κ0l 2 ρ1 + γ2 ρ2 λn( nπ L )2θ ∼ O(n1+2θ), |P3(λn)| ∼  O(n1+2θ), if χ1 = 0, O(n1+2θ), if χ1 6= 0, θ ≥ 1/2, O(n2), if χ1 6= 0. θ ≤ 1/2. Thus, we have |P2(λn)P3(λn)− κl2(κ+ κ0) ρ21 ( nπ L )2 − κ2l2 ρ1ρ2 − P3(λn) κ ρ1 ( nπ L )2| ∼  O(n2+4θ), if χ1 = 0, O(n2+4θ), if χ1 6= 0, θ ≥ 1/2, O(n3+2θ), if χ1 6= 0, θ ≤ 1/2, |det 1 (λn)| ∼ { O(n2+4θ), if θ ≥ 1/2, χ1 = 0, O(n3+2θ), if θ ≤ 1/2, χ1 = 0. |det 1 (λn)| ∼ { O(n2+4θ), if θ ≥ 1/2, χ1 6= 0, O(n4), if θ ≤ 1/2, χ1 6= 0. According to the ratio, it is found that whether χ1 is 0 or not, the asymptotic behavior of An can be estimated as |An| ∼ { O(1), if θ ≥ 1/2, O(n2θ−1), if θ ≤ 1/2. For the subcase χ0 6= 0, we have P3(λn) is unchanged and |P2(λn)| ∼ { O(n1+2θ), if θ ≥ 1/2, O(n2), if θ ≤ 1/2. Then |P2(λn)P3(λn)| ∼  O(n2+4θ), if θ ≥ 1/2, χ1 = 0, O(n3+2θ), if θ ≤ 1/2, χ1 = 0, O(n2+4θ), if θ ≥ 1/2, χ1 6= 0, O(n4), if θ ≤ 1/2, χ1 6= 0, |P3(λn) κ ρ1 ( nπ L )2| ∼  O(n3+2θ), if χ1 = 0, O(n3+2θ), if θ ≥ 1/2, χ1 6= 0, O(n4), if θ ≤ 1/2, χ1 6= 0. Thus we obtain |P2(λn)P3(λn)− κl2(κ+ κ0) ρ21 ( nπ L )2 − κ2l2 ρ1ρ2 − P3(λn) κ ρ1 ( nπ L )2| EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 17 ∼ { O(n2+4θ), if θ ≥ 1/2, O(n4), , if θ ≤ 1/2, when χ1 = 0 or χ1 6= 0; and |det 3 (λn)| ∼ { O(n2+4θ), if θ ≥ 1/2, O(n4), if θ ≤ 1/2, when χ1 = 0 or χ1 6= 0. Therefore regardless of the value of χ1 and θ, the asymptotic behavior of An can be estimated as |An| ∼ O(1). It follows that ‖Un‖2H ≥ ρ1‖ϕ̃‖2L2(0,L) = ρ1|λnAn|2 ∫ L 0 sin2( nπ L x)dx = ρ1L 2 |λnAn|2. So when χ0 6= 0, we have ‖Un‖H ≥ √ ρ1L 2 |λn| |An| ∼ O(n) and lim n→∞ ‖Un‖H = +∞, while when χ0 = 0, we have ‖Un‖H ≥ √ ρ1L 2 |λn| |An| ∼ { O(n), if θ ≥ 1/2, O(n2θ), if θ ≤ 1/2, and if θ ∈ (0, 1], then limn→∞ ‖Un‖H = +∞. This means that the corresponding semigroup is not exponentially stable when χ1 6= 0 or χ1 = 0 and θ ∈ (0, 1]. This shows the third result of Theorem 3.1, which completes the proof. � 4. Stability results In this section we study the asymptotic behavior of the semigroup etA associ- ated to the system (1.7)-(1.9) when fractional damping is applied to two equations separately. The following spectral characteristics of exponential and polynomial stability of semigroups will be used to obtain the stability results. Firstly we give the following useful theorems. Theorem 4.1 ([17]). Let A be the generator of a C0-semigroup of contractions on a Hilbert space. Then, the semigroup etA is exponentially stable if and only if iR ⊂ ρ(A) and lim sup |λ|→∞ ‖(iλI −A)−1‖ <∞. Theorem 4.2 ([10, 29]). Let A the generator of a C0-semigroup of bounded oper- ators on a Hilbert space with iR ⊂ ρ(A). Then we have ‖etAU0‖ ≤ Ct−1/θ‖U0‖D(A), ∀t > 0, U0 ∈ D(A), if and only if lim sup |λ|→∞ |λ|−θ‖(iλI −A)−1‖ <∞. In the remainder of this article, C and Cδ represent positive constants that assume different values at different locations. In most cases, it may be Cδ → ∞ when δ → 0+. Using the above theorems, we obtain some estimates for the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T 18 J. HAO, D. WANG EJDE-2023/87 of the equation (iλI −A)U = F , where λ ∈ R and F = (f1, f2, f3, f4, f5, f6)T ∈ H. Then the system can be decomposed into the following forms iλϕ− ϕ̃ = f1, (4.1a) iλρ1ϕ̃− κ(ϕx + ψ + lw)x − κ0l(wx − lϕ) + γ1E θϕ̃ = ρ1f2, (4.1b) iλψ − ψ̃ = f3, (4.1c) iλρ2ψ̃ − bψxx + κ(ϕx + ψ + lw) + γ2E θ ∗ ψ̃ = ρ2f4, (4.1d) iλw − w̃ = f5, (4.1e) iλρ1w̃ − κ0(wx − lϕ)x + kl(ϕx + ψ + lw) + γ3E θ ∗w̃ = ρ1f6. (4.1f) Before stating the main stability results we need to introduce some lemmas. In all the lemmas below, we assume that θ ∈ [0, 1], U ∈ D(A) is the solution of the equation (iλI − A)U = F for real number λ > 0. Note that similar to (2.13) and using (2.7), we obtain the first estimate γ1‖Eθ/2ϕ̃‖2+γ2‖Eθ/2∗ ψ̃‖2+γ3‖Eθ/2w̃‖2 ≤ CRe〈(iλI−A)U,U〉 ≤ C‖F‖‖U‖; (4.2) that is, ‖Eθ/2ϕ̃‖2 ≤ C‖F‖‖U‖, (4.3) ‖Eθ/2∗ ψ̃‖2 ≤ C‖F‖‖U‖, (4.4) ‖Eθ/2w̃‖2 ≤ C‖F‖‖U‖. (4.5) Using (4.1a) and taking into account estimates (4.3), we have λ2‖Eθ/2ϕ‖2 ≤ ‖Eθ/2ϕ̃‖2 + ‖Eθ/2f1‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.6) in the same way, we obtain λ2‖Eθ/2ψ‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.7) λ2‖Eθ/2w‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) . (4.8) Lemma 4.3. Let δ > 0 and α ≤ 0. There exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies the following estimates: (i) λ2‖E α 2 ϕ̃‖2 ≤ Cδ ( ‖E α 2 + 1 2 ∗ ϕ̃‖2 + ‖E α 2 ∗ ψ̃‖2 + ‖E α 2 ∗ w̃‖2 ) + ε1λ 2‖Eα+ θ 2 ϕ̃‖2 + Cε1 ( ‖F‖‖U‖+ ‖F‖2 ) , and (ii) ‖E α 2 + 1 2 ϕ̃‖2 ≤ Cδ ( λ2‖E α 2 ϕ̃‖2 + ‖E α 2 ∗ ψ̃‖2 + ‖E α 2 ∗ w̃‖2 ) + ε1λ 2‖Eα+ θ 2 ϕ̃‖2 + Cε1 ( ‖F‖‖U‖+ ‖F‖2 ) , where ε1 is positive or zero if the the damping coefficient γ1 is present or not present respectively. Proof. Multiplying (4.1b) by iλ, taking the inner product with Eαϕ̃ and using the definition of the operator E, applying the self-adjointness of Eσ for σ ∈ R, we obtain ρ1λ 2‖E α 2 ϕ̃‖2 = iλκ〈Eϕ,Eαϕ̃〉 − iλκ〈ψx, Eαϕ̃〉 − iλκl〈wx, Eαϕ̃〉 − iλκ0l〈wx, Eαϕ̃〉 + iλκ0l 2〈ϕ,Eαϕ̃〉+ iλγ1〈Eθϕ̃, Eαϕ̃〉 − iλρ1〈f2, Eαϕ̃〉. EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 19 Substituting (4.1a) in the previous equality, we obtain ρ1λ 2‖E α 2 ϕ̃‖2 = κ‖E α 2 + 1 2 ϕ̃‖2 + κ〈E α 2 + 1 2 f1, E α 2 + 1 2 ϕ̃〉 − iλκ〈ψx, Eαϕ̃〉 − iλ(κ+ κ0)l〈wx, Eαϕ̃〉+ κ0l 2‖E α 2 ϕ̃‖2 + κ0l 2〈E α 2 f1, E α 2 ϕ̃〉 + iλγ1〈Eθ/2ϕ̃, Eα+ θ 2 ϕ̃〉 − iλρ1〈f2, Eαϕ̃〉. (4.9) Next, let’s estimate some items in (4.9). First, we estimate the items with f in (4.9). Applying Cauchy-Schwarz and Young’s inequalities, for ε small enough to be chosen later, we have |κ〈E α 2 + 1 2 f1, E α 2 + 1 2 ϕ̃〉| ≤ C‖E α 2 + 1 2 f1‖2 + C‖E α 2 + 1 2 ϕ̃‖2 ≤ C‖F‖2 + C‖E α 2 + 1 2 ϕ̃‖2, |κ0l2〈E α 2 f1, E α 2 ϕ̃〉| ≤ C‖E α 2 f1‖2 + C‖E α 2 ϕ̃‖2 ≤ C‖F‖2 + C‖E α 2 ϕ̃‖2, |iλρ1〈f2, Eαϕ̃〉| ≤ C‖E α 2 f2‖2 + ελ2‖E α 2 ϕ̃‖2 ≤ C‖F‖2 + ελ2‖E α 2 ϕ̃‖2. Then, using integration by parts, the self-adjointness of Eσ and (1.19), applying Cauchy-Schwarz, Young’s inequalities and (4.3), for ε small enough to be chosen later, we have |iλκ〈ψx, Eαϕ̃〉| = |iλκ〈E α 2 ∗ ψ,E α 2 ∗ ϕ̃x〉| = |iλκ〈E α 2 ∗ ψ,E α 2 + 1 2 ϕ̃〉| = |κ〈E α 2 ∗ (ψ̃ + f3), E α 2 + 1 2 ϕ̃〉| ≤ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 + 1 2 ϕ̃‖2 + C‖F‖2, |iλ(κ+ κ0)l〈wx, Eαϕ̃〉| = |iλ(κ+ κ0)l〈E α 2 ∗ w,E α 2 ∗ ϕ̃x〉| = |(κ+ κ0)l〈E α 2 ∗ (w̃ + f5), E α 2 + 1 2 ϕ̃〉| ≤ C‖E α 2 ∗ w̃‖2 + C‖E α 2 + 1 2 ϕ̃‖2 + ‖F‖2, |iλγ1〈Eθ/2ϕ̃, Eα+ θ 2 ϕ̃〉| ≤ C‖Eθ/2ϕ̃‖2 + ε1‖Eα+ θ 2 ϕ̃‖2 ≤ ε1‖Eα+ θ 2 ϕ̃‖2 + Cε1‖F‖‖U‖. Substituting the above estimates into (4.9) and considering α ≤ 0, we obtain the first result of Lemma 4.3. Similarly, rewriting (4.9) introduces the item (ii) of this Lemma. The proof is complete. � Lemma 4.4. Let δ > 0 and α ≤ 0. There exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies the following estimates: (i) λ2‖E α 2 ψ̃‖2 ≤ Cδ ( ‖E α 2 + 1 2 ∗ ψ̃‖2 + ‖E α 2 ϕ̃‖2 + ‖E α 2 ∗ w̃‖2 ) + ε2λ 2‖Eα+ θ 2 ψ̃‖2 + Cε2 ( ‖F‖‖U‖+ ‖F‖2 ) , (ii) ‖E α 2 + 1 2 ψ̃‖2 ≤ Cδ ( λ2‖E α 2 ψ̃‖2 + ‖E α 2 ϕ̃‖2 + ‖E α 2 ∗ w̃‖2 ) + ε2λ 2‖Eα+ θ 2 ϕ̃‖2 + Cε2 ( ‖F‖‖U‖+ ‖F‖2 ) , (iii) λ2‖E α 2 w̃‖2 ≤ Cδ ( ‖E α 2 + 1 2 ∗ w̃‖2 + ‖E α 2 ϕ̃‖2 + ‖E α 2 ∗ ψ̃‖2 ) + ε3λ 2‖Eα+ θ 2 w̃‖2 20 J. HAO, D. WANG EJDE-2023/87 + Cε3 ( ‖F‖‖U‖+ ‖F‖2 ) , and (iv) ‖E α 2 + 1 2 w̃‖2 ≤ Cδ ( λ2‖E α 2 w̃‖2 + ‖E α 2 ϕ̃‖2 + ‖E α 2 ∗ ψ̃‖2 ) + ε3λ 2‖Eα+ θ 2 w̃‖2 + Cε3 ( ‖F‖‖U‖+ ‖F‖2 ) , where ε2 and ε3 are positive or zero if the the damping coefficient γ2 and γ3 are present or not present, respectively. Proof. Multiplying (4.1d) and (4.1f) by iλ, taking the inner product with Eα∗ ψ̃ and Eα∗ w̃, respectively. Using the definition of the operator E∗, applying the self- adjointness of Eσ for σ ∈ R, by integration by parts, we obtain the results of this Lemma. Since the corresponding steps and estimates are similar to Lemma 4.3, we will not repeat them here. � Lemma 4.5. Let δ > 0 and α ≤ 0. There exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies the following estimates, ‖E α 2 ∗ (ϕx + ψ + lw)‖2 ≤ ρ1ρ2 κ |χ0|λ2〈ψ,Eα∗ ϕx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ∗ w̃‖2 + C‖E α 2 ϕ̃‖2 − γ2〈Eθ∗ ψ̃, Eα∗ (ϕx + ψ + lw)〉 − b κ γ1〈ψx, Eα+θ∗ ϕ̃〉 − bl κ γ3〈ψ,Eα+θ∗ w̃〉+ C‖F‖2. Proof. Taking the inner product of (4.1d) and Eα∗ (ϕx+ψ+ lw), using the fact that Eσ is self-adjoint for any σ ∈ R, and applying (4.1c), we deduce that κ‖E α 2 ∗ (ϕx + ψ + lw)‖2 = ρ2λ 2〈ψ,Eα∗ (ϕx + ψ + lw)〉+ ρ2〈(iλf3 + f4), Eα∗ (ϕx + ψ + lw)〉 + b〈ψxx, Eα∗ (ϕx + ψ + lw)〉 − γ2〈Eθ∗ ψ̃, Eα∗ (ϕx + ψ + lw)〉. (4.10) Next, we estimate the third term on the right of the equality. Using the definition of f∗ and E, integration by parts, from (4.1b), (4.1a), (4.1e), and (4.1f) we have b〈ψxx, Eα∗ (ϕx + ψ + lw)〉 = −b〈ψx, Eα∗ (ϕx + ψ + lw)x〉 = − b κ 〈ψx, Eα∗ (iλρ1ϕ̃− κ0l(wx − lϕ) + γ1E θϕ̃− ρ1f2)〉 = ρ1b κ 〈ψx, Eα∗ (iλf1 + f2)〉 − ρ1b κ λ2〈ψ,Eα∗ ϕx〉 − κ0bl κ 〈ψ,Eα∗ (wx − lϕ)x〉 − bγ1 κ 〈ψx, Eα+θ∗ ϕ̃〉 = ρ1b κ 〈ψx, Eα∗ (iλf1 + f2)〉 − ρ1b κ λ2〈ψ,Eα∗ ϕx〉 − bγ1 κ 〈ψx, Eα+θ∗ ϕ̃〉 − bl κ 〈ψ,Eα∗ (iλρ1w̃ + κl(ϕx + ψ + lw) + γ3E θ ∗w̃ − ρ1f6)〉 = ρ1b κ 〈ψx, Eα∗ (iλf1 + f2)〉 − ρ1b κ λ2〈ψ,Eα∗ ϕx〉 − bγ1 κ 〈ψx, Eα+θ∗ ϕ̃〉 + blρ1 κ λ2〈ψ,Eα∗ w〉+ blρ1 κ 〈ψ,Eα∗ (iλf5 + f6)〉 − blγ3 κ 〈ψ,Eα+θ∗ w̃〉 EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 21 − bl2〈ψ,Eα∗ (ϕx + ψ + lw)〉. Substituting the above expression in (4.10), from the definition of (1.12) we obtain κ‖E α 2 ∗ (ϕx + ψ + lw)‖2 = |χ0| ρ1ρ2 κ λ2〈ψ,Eα∗ ϕx〉+ ρ2λ 2‖E α 2 ∗ ψ‖2 + (lρ2 + ρ1bl κ )λ2〈ψ,Eα∗ w〉 + ρ1b κ 〈ψx, Eα∗ (iλf1 + f2)〉+ ρ2〈(iλf3 + f4), Eα∗ (ϕx + ψ + lw)〉 + ρ1bl κ 〈ψ,Eα∗ (iλf5 + f6)〉 − bl2〈ψ,Eα∗ (ϕx + ψ + lw)〉 − bγ1 κ 〈ψx, Eα+θ∗ ϕ̃〉 − γ2〈Eθ∗ ψ̃, Eα∗ (ϕx + ψ + lw)〉 − blγ3 κ 〈ψ,Eα+θ∗ w̃〉. (4.11) Now, we estimate some items on the right side of formula (4.11). Using the self-adjointness of Eσ∗ and (1.19), Cauchy-Schwarz and Young’s inequalities, we have ρ1b κ 〈ψx, Eα∗ (iλf1 + f2)〉 = ρ1b κ 〈Eα∗ ψx, f2〉 − iλ ρ1b κ 〈Eα∗ ψx, f1〉 = ρ1b κ 〈Eα∗ ψx, f2〉+ iλ ρ1b κ 〈Eα∗ ψ, (f1)x〉 ≤ C‖Eα+ 1 2 ∗ ψ‖‖F‖+ Cλ2‖Eα∗ ψ‖2 + C‖F‖2 ≤ Cλ2‖E α 2 ∗ ψ‖2 + C‖F‖‖U‖+ C‖F‖2, since α < 0, then α + 1 2 ≤ 1 2 and α ≤ α 2 . Considering the continuous embedding D(Eσ1 ∗ ) ↪→ D(Eσ2 ∗ ), for σ1 ≥ σ2, using equivalent norm and ‖E1/2 ∗ ψ‖ ≤ ‖U‖ to obtain the above estimate. Similarly, ρ1bl κ 〈ψ,Eα∗ (iλf5 + f6)〉 ≤ Cλ2‖E α 2 ∗ ψ‖2 + C‖F‖2. Using Cauchy-Schwarz and Young’s inequalities, by (1.18)-(1.19) and the continu- ous embedding α− 1 2 ≤ α 2 , we have ρ2〈(iλf3 + f4), Eα∗ (ϕx + ψ + lw)〉 = ρ2〈(f4), Eα∗ (ϕx + ψ + lw)〉+ iλρ2〈f3, Eα∗ (ϕx + ψ + lw)〉 ≤ C‖Eα∗ (ϕx + ψ + lw)‖‖F‖+ C|λ|‖E1/2 ∗ f3‖‖E α− 1 2 ∗ (ϕx + ψ + lw)‖ ≤ C‖Eα∗ (ϕx + ψ + lw)‖2 + Cλ2‖Eαϕ‖2 + Cλ2‖Eα− 1 2 ∗ ψ‖2 + Cλ2‖Eα− 1 2 ∗ w‖2 + C‖F‖2 ≤ C‖E α 2 ∗ (ϕx + ψ + lw)‖2 + Cλ2‖E α 2 ϕ‖2 + Cλ2‖E α 2 ∗ ψ‖2 + Cλ2‖E α 2 ∗ w‖2 + C‖F‖2. We also have from Cauchy-Schwarz and Young’s inequalities (lρ2 + ρ1bl κ )λ2〈ψ,Eα∗ w〉 = (lρ2 + ρ1bl κ )λ2〈E α 2 ∗ ψ,E α 2 ∗ w〉 ≤ Cλ2‖E α 2 ∗ ψ‖2 + Cλ2‖E α 2 ∗ w‖2, 22 J. HAO, D. WANG EJDE-2023/87 |bl2〈ψ,Eα∗ (ϕx + ψ + lw)〉| = |bl2〈E α 2 ∗ ψ,E α 2 ∗ (ϕx + ψ + lw)〉| ≤ C‖E α 2 ∗ ψ‖2 + C‖E α 2 ∗ (ϕx + ψ + lw)‖2. Substituting the above estimates into (4.11), using the continuous embedding and (4.1a), (4.1c) and (4.1e), we conclude that ‖E α 2 ∗ (ϕx + ψ + lw)‖2 ≤ ρ1ρ2 κ |χ0|λ2〈ψ,Eα∗ ϕx〉+ Cλ2‖E α 2 ∗ ψ‖2 + Cλ2‖E α 2 ∗ w‖2 + Cλ2‖E α 2 ϕ‖2 − b κ γ1〈ψx, Eα+θ∗ ϕ̃〉 − γ2〈Eθ∗ ψ̃, Eα∗ (ϕx + ψ + lw)〉 − bl κ γ3〈ψ,Eα+θ∗ w̃〉+ C‖F‖2 ≤ ρ1ρ2 κ |χ0|λ2〈ψ,Eα∗ ϕx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ∗ w̃‖2 + C‖E α 2 ϕ̃‖2 − b κ γ1〈ψx, Eα+θ∗ ϕ̃〉 − γ2〈Eθ∗ ψ̃, Eα∗ (ϕx + ψ + lw)〉 − bl κ γ3〈ψ,Eα+θ∗ w̃〉+ C‖F‖2. The proof is complete. � Lemma 4.6. Let δ > 0 and α ≤ 0. There exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies the following estimates: ‖E α 2 ∗ w̃‖2 + ‖E α 2 ∗ (wx − lϕ)‖2 ≤ |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C‖E α 2 ∗ (ϕx + ψ + lw)‖2 + γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉+ κγ3 κ0 〈ϕx + ψ + lw,Eα+θ∗ w̃〉+ C‖F‖2. Proof. Performing the duality product between (4.1b) and Eα∗ (wx − lϕ), using (4.1a), and integrating by parts, we obtain κ0l‖E α 2 ∗ (wx − lϕ)‖2 = 〈iλρ1ϕ̃− κ(ϕx + ψ + lw)x + γ1E θϕ̃− ρ1f2, Eα∗ (wx − lϕ)〉 = ρ1〈−λ2ϕ− iλf1, Eα∗ (wx − lϕ)〉 − ρ1〈f2, Eα∗ (wx − lϕ)〉 + κ〈(ϕx + ψ + lw), Eα∗ (wx − lϕ)x〉+ γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉 = −ρ1λ2〈ϕ,Eα∗ wx〉 − ρ1〈iλf1 + f2, E α ∗ (wx − lϕ)〉+ ρ1lλ 2‖E α 2 ϕ‖2 + γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉+ κ〈(ϕx + ψ + lw), Eα∗ (wx − lϕ)x〉. (4.12) EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 23 Substituting (4.1f) into the last item of the above estimate, and using (4.1e) yield κ〈(ϕx + ψ + lw), Eα∗ (wx − lϕ)x〉 = κ〈(ϕx + ψ + lw), 1 κ0 Eα∗ (iλρ1w̃ + κl(ϕx + ψ + lw) + γ3E θ ∗w̃ − ρ1f6)〉 = ρ1κ κ0 〈(ϕx + ψ + lw), Eα∗ (−λ2w − iλf5)〉+ κ2l κ0 ‖E α 2 ∗ (ϕx + ψ + lw)‖2 + κγ3 κ0 〈(ϕx + ψ + lw), Eα+θ∗ w̃〉 − κρ1 κ0 〈(ϕx + ψ + lw), Eα∗ f6〉 = ρ1κ κ0 λ2〈ϕ,Eα∗ wx〉 − ρ1κ κ0 λ2〈ψ,Eα∗ w〉+ κγ3 κ0 〈(ϕx + ψ + lw), Eα+θ∗ w̃〉 + κ2l κ0 ‖E α 2 ∗ (ϕx + ψ + lw)‖2 − κρ1 κ0 〈(ϕx + ψ + lw), Eα∗ (iλf5 + f6)〉 − ρ1κl κ0 λ2〈w,Eα∗ w〉. (4.13) Substituting (4.13) into (4.12), from the definition of (1.12) we have κ0l‖E α 2 ∗ (wx − lϕ)‖2 = |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ ρ1lλ 2‖E α 2 ϕ‖2 + κ2l κ0 ‖E α 2 ∗ (ϕx + ψ + lw)‖2 − ρ1κl κ0 λ2‖E α 2 ∗ w‖2 − ρ1κ κ0 λ2〈ψ,Eα∗ w〉 − ρ1〈iλf1 + f2, E α ∗ (wx − lϕ)〉 − κρ1 κ0 〈(ϕx + ψ + lw), Eα∗ (iλf5 + f6)〉+ κγ3 κ0 〈(ϕx + ψ + lw), Eα+θ∗ w̃〉 + γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉. (4.14) Now, we estimate some terms on the right-hand side of (4.14). Using the self- adjointness of Eσ∗ and (1.19), Cauchy-Schwarz and Young’s inequalities, we have |ρ1〈iλf1 + f2, E α ∗ (wx − lϕ)〉| = |ρ1〈f2, Eα∗ (wx − lϕ)〉|+ |ρ1〈iλf1, Eα∗ (wx − lϕ)〉| = |ρ1〈f2, Eα∗ (wx − lϕ)〉|+ |ρ1〈iλE1/2 ∗ f1, E α− 1 2 ∗ wx〉 − |lρ1〈iλf1, Eαϕ〉‖ ≤ C‖F‖‖Eα∗ (wx − lϕ)‖+ C|λ|‖F‖‖Eα∗ w‖+ C|λ|‖F‖‖Eαϕ‖ ≤ C‖F‖2 + C‖Eα∗ (wx − lϕ)‖2 + Cλ2‖Eα∗ w‖2 + Cλ2‖Eαϕ‖2, and |ρ1κ κ0 λ2〈(ψ,Eα∗ w〉| = | ρ1κ κ0 〈λE α 2 ∗ ψ, λE α 2 ∗ w〉| ≤ Cλ2‖E α 2 ∗ ψ‖2 + Cλ2‖E α 2 ∗ w‖2. Considering the continuous embedding D(Eσ1 ∗ ) ↪→ D(Eσ2 ∗ ) for σ1 ≥ σ2, using (1.19), Cauchy-Schwarz and Young’s inequalities, we obtain |κρ1 κ0 〈(ϕx + ψ + lw), Eα∗ (iλf5 + f6)〉| = |κρ1 κ0 〈(ϕx + ψ + lw), Eα∗ f6〉|+ | κρ1 κ0 〈iλ(ϕx + ψ + lw), Eα∗ f5〉| = |κρ1 κ0 〈(ϕx + ψ + lw), Eα∗ f6〉| − κρ1 κ0 iλ〈Eαϕ, (f5)x〉+ κρ1 κ0 iλ〈Eα∗ ψ, f5〉 + κρ1 κ0 iλl〈Eα∗ w, f5〉 24 J. HAO, D. WANG EJDE-2023/87 ≤ C‖F‖2 + C‖Eα∗ (ϕx + ψ + lw)‖2 + Cλ2‖Eα∗ w‖2 + Cλ2‖Eαϕ‖2 + Cλ2‖Eα∗ ψ‖2. Substituting the above estimates in (4.14), using the continuous embedding, α ≤ α/2, and (4.1a), (4.1c), (4.1e), we conclude that ‖E α 2 ∗ w̃‖2 + ‖E α 2 ∗ (wx − lϕ)‖2 ≤ |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ Cλ2‖E α 2 ∗ ψ‖2 + Cλ2‖E α 2 ϕ‖2 + C‖E α 2 ∗ (ϕx + ψ + lw)‖2 + γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉+ κγ3 κ0 〈ϕx + ψ + lw,Eα+θ∗ w̃〉+ ‖F‖2 ≤ |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C‖E α 2 ∗ (ϕx + ψ + lw)‖2 + γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉+ κγ3 κ0 〈ϕx + ψ + lw,Eα+θ∗ w̃〉+ ‖F‖2. The proof is complete. � Lemma 4.7. The solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies ‖E1/2 ∗ ψ‖2 = ‖ψx‖2 ≤ Cδ ( ‖ψ̃‖2 + ‖ϕ‖2 + ‖w‖2 + ‖F‖‖U‖+ ‖F‖2 ) . Proof. Taking the inner product of (4.1d) with ψ, using (1.19) and integrating by parts, we obtain b‖E1/2 ∗ ψ‖2 = b‖ψx‖2 = −iλρ2〈ψ̃, ψ〉 − κ〈(ϕx + ψ + lw), ψ〉 − γ2〈Eθ∗ ψ̃, ψ〉+ ρ2〈f4, ψ〉 = ρ2‖ψ̃‖2 + ρ2〈ψ̃, f3〉 − κ〈ϕ,ψx〉+ κ‖ψ‖2 + lκ〈w,ψ〉 − γ2〈Eθ/2∗ ψ̃, E θ/2 ∗ ψ〉+ ρ2〈f4, ψ〉. Applying Cauchy-Schwarz and Young’s inequalities, by (4.4) and (4.1c) we have b‖E1/2 ∗ ψ‖2 = b‖ψx‖2 ≤ C ( ‖ψ̃‖2 + ‖ϕ‖2 + ‖w‖2 + ‖ψ‖2 + ‖Eθ/2∗ ψ̃‖2 + ‖Eθ/2∗ ψ‖2 ) ≤ C ( ‖ψ̃‖2 + ‖ϕ‖2 + ‖w‖2 + ‖F‖‖U‖+ ‖F‖2 ) , which completes the proof. � Next, we prove the asymptotic stability of system (1.7)-(1.9) when the fractional damping acts on two of the equations. Firstly we consider the case γ1, γ2 > 0 and γ3 = 0. Lemma 4.8. Let δ > 0 and γ1, γ2 > 0, γ3 = 0. There exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies the following: (i) ‖w̃‖2 ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ1 = 0, (ii) ‖w̃‖2 ≤ Cδλ2−2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ1 6= 0 and θ ≤ 1/2, (iii) ‖w̃‖2 ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ1 6= 0 and θ ≥ 1/2. Proof. Adding (4.1a), (4.1c) and (4.1e), we obtain iλ(ϕx + ψ + lw) = (ϕ̃x + ψ̃ + lw̃) + (f1)x + f3 + f5. EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 25 The inner product of the above formula and κEα∗ (ϕx +ψ+ lw), by partial integral and (4.1b) to deduce κ‖E α 2 ∗ (ϕx + ψ + lw)‖2 = κ iλ 〈(ϕ̃x + ψ̃ + lw̃) + (f1)x + f3 + f5, E α ∗ (ϕx + ψ + lw)〉 = 1 iλ 〈ϕ̃, Eα∗ (−iλρ1ϕ̃+ κ0l(wx − lϕ)− γ1Eθϕ̃+ ρ1f2)〉 + κ iλ 〈(ψ̃ + lw̃) + (f1)x + f3 + f5, E α ∗ (ϕx + ψ + lw)〉 = ρ1‖E α 2 ∗ ϕ̃‖2 + γ1 iλ ‖E α 2 + θ 2 ∗ ϕ̃‖2 + κ0l iλ 〈ϕ̃, Eα∗ (wx − lϕ)〉+ ρ1 iλ 〈ϕ̃, Eα∗ f2〉 + κ iλ 〈(ψ̃ + lw̃) + (f1)x + f3 + f5, E α ∗ (ϕx + ψ + lw)〉 ≤ C‖E α 2 ∗ ϕ̃‖2 + C λ ‖Eθ/2∗ ϕ̃‖2 + C‖E α 2 ∗ ψ‖2 + C‖E α 2 ∗ w‖2 + C λ2 ‖E α 2 ∗ (wx − lϕ)‖2 + ‖F‖2. (4.15) The last step follows from the self-adjointness of Eσ∗ , Cauchy-Schwarz and Young’s inequalities, the continuous embedding and (4.1c), (4.1e). It can be seen from (4.15) that the estimate here is different from Lemma 4.5, which is independent of χ0. Substituting (4.15) into the result of Lemma 4.6, note that γ3 = 0, using contin- uous embedding and (4.1c), (4.1e), (4.3), we obtain ‖E α 2 ∗ w̃‖2 + ‖E α 2 ∗ (wx − lϕ)‖2 ≤ |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C‖E α 2 ∗ (ϕx + ψ + lw)‖2 + γ1〈Eθϕ̃, Eα∗ (wx − lϕ)〉+ C‖F‖2 ≤ |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C λ ‖Eθ/2∗ ϕ̃‖2 + C‖E α 2 ∗ ψ‖2 + C‖E α 2 ∗ w‖2 + C‖Eθ/2ϕ̃‖2 + C‖Eα+ θ 2 (wx − lϕ)‖2 + ‖F‖2 ≤ |χ1|ρ1λ2〈ϕ,Eα∗ wx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C‖Eα+ θ 2 (wx − lϕ)‖2 + ‖F‖ ‖U‖+ ‖F‖2. (4.16) When χ1 = 0, taking α = −θ in (4.16), then from the continuous embedding we have ‖E−θ/2∗ w̃‖2 + ‖E−θ/2∗ (wx − lϕ)‖2 ≤ C‖E−θ/2∗ ψ̃‖2 + C‖E−θ/2ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , since −θ/2 < θ/2, we have (4.3) and (4.4). So, we obtain ‖E−θ/2∗ w̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.17) ‖E−θ/2∗ (wx − lϕ)‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) . (4.18) On the other hand, we have ‖E−θ/2∗ wx‖2 ≤ C ( ‖E−θ/2∗ (wx − lϕ)‖2 + ‖E−θ/2∗ ϕ‖2 ) . 26 J. HAO, D. WANG EJDE-2023/87 Then using (1.19) and (4.18) we have ‖E 1 2− θ 2 ∗ w‖2 = ‖E−θ/2∗ wx‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) . (4.19) From (4.1e) we have w̃ = iλw − f5, then by (4.19) we obtain ‖E 1 2− θ 2 ∗ w̃‖2 = ‖E 1 2− θ 2 ∗ (iλw − f5)‖2 ≤ Cλ2‖E 1 2− θ 2 ∗ w‖2 + C‖F‖2 ≤ Cδλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.20) Then, by applying the interpolation inequalities, from (4.18) and (4.20) we conclude that ‖w̃‖ ≤ ‖E−θ/2∗ w̃‖1−θ‖E 1 2− θ 2 ∗ w̃‖θ ≤ Cδ (√ ‖F‖‖U‖+ ‖F‖2 )1−θ(|λ|√‖F‖‖U‖+ ‖F‖2 )θ ≤ Cδ|λ|θ √ ‖F‖‖U‖+ ‖F‖2. (4.21) This yields ‖w̃‖ ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) . This leads to the first result of this Lemma. When χ1 6= 0, we have ‖E α 2 ∗ w̃‖2 + ‖E α 2 ∗ (wx − lϕ)‖2 ≤ Cλ2〈ϕ,Eα∗ wx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C‖Eα+ θ 2 (wx − lϕ)‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2〈ϕ,Eα∗ (wx − lϕ)〉+ Cλ2〈ϕ,Eα∗ ϕ)〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ϕ̃‖2 + C‖Eα+ θ 2 (wx − lϕ)‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2‖E α 2 ϕ̃‖2 + C‖E α 2 ∗ ψ̃‖2 + C‖Eα+ θ 2 (wx − lϕ)‖2 + ‖F‖‖U‖+ ‖F‖2. (4.22) Considering α = θ− 1 and θ ≤ 1/2 in (4.22), then θ 2 + α ≤ α 2 , from the continuous embedding we obtain ‖E θ 2− 1 2 ∗ w̃‖2 + ‖E θ 2− 1 2 ∗ (wx − lϕ)‖2 ≤ Cλ2‖E θ 2− 1 2 ∗ ϕ̃‖2 + C‖E θ 2− 1 2 ∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2. Taking into account α = θ − 1 in the item (i) of Lemma 4.3, we obtain λ2‖E θ 2− 1 2 ϕ̃‖2 ≤ C ( ‖Eθ/2ϕ̃‖2 + ‖E θ 2− 1 2 ∗ ψ̃‖2 + ‖E θ 2− 1 2 ∗ w̃‖2 ) + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E θ 2− 1 2 ∗ w̃‖2 + C ( ‖F‖‖U‖+ ‖F‖2 ) , since θ ≤ 1/2, then α+ θ 2 ≤ α 2 , θ2 − 1 2 < θ 2 and (4.4). Substituting the last formula to obtain ‖E θ 2− 1 2 ∗ w̃‖2 + ‖E θ 2− 1 2 ∗ (wx − lϕ)‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) ; that is, ‖E θ 2− 1 2 ∗ w̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.23) ‖E θ 2− 1 2 ∗ (wx − lϕ)‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.24) EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 27 Similar to the estimates of (4.19) and (4.6), we deduce that ‖Eθ/2∗ w̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) , (4.25) Then, by applying the interpolation inequalities, from (4.23) and (4.25) we conclude that ‖w̃‖ ≤ ‖E θ 2− 1 2 ∗ w̃‖θ‖Eθ/2∗ w̃‖1−θ ≤ Cδ (√ ‖F‖‖U‖+ ‖F‖2 )θ(|λ|√‖F‖‖U‖+ ‖F‖2 )1−θ ≤ Cδ|λ|1−θ √ ‖F‖‖U‖+ ‖F‖2. (4.26) Thus it yields ‖w̃‖ ≤ Cδλ2−2θ ( ‖F‖‖U‖+ ‖F‖2 ) . This leads to the second result of this Lemma. For the third result of this lemma, that is, θ ≥ 1/2, considering α = −θ in (4.22), then θ 2 + α ≤ α 2 , because of the continuous embedding, we deduce that ‖E−θ/2∗ w̃‖2 + ‖E−θ/2∗ (wx − lϕ)‖2 ≤ Cλ2‖E−θ/2∗ ϕ̃‖2 + C‖E−θ/2∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2. Taking into account α = −θ in the item (i) of Lemma 4.3, we obtain λ2‖E−θ/2∗ ϕ̃‖2 ≤ C ( ‖E 1 2− θ 2 ∗ ϕ̃‖2 + ‖E−θ/2∗ ψ̃‖2 + ‖E−θ/2∗ w̃‖2 ) + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E−θ/2∗ w̃‖2 ) + C ( ‖F‖‖U‖+ ‖F‖2 ) , since θ ≥ 1/2, then α + θ 2 ≤ α 2 , 1 2 − θ 2 ≤ θ 2 and (4.4). furthermore, − θ2 < θ 2 is an identity. Substituting the last formula to obtain ‖E−θ/2∗ w̃‖2 + ‖E−θ/2∗ (wx − lϕ)‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) ; that is, ‖E−θ/2∗ w̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.27) ‖E−θ/2∗ (wx − lϕ)‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.28) Similar to the estimates in the previous two parts, we have ‖E 1 2− θ 2 ∗ w̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) , (4.29) Using the same interpolation inequalities as in (4.21), and applying (4.27), (4.29) we conclude that ‖w̃‖ ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) . The third conclusion of this Lemma has also been proved. So the proof is complete. � Secondly we consider the case γ1, γ3 > 0, and γ2 = 0. Lemma 4.9. Let δ > 0 and γ1, γ3 > 0, γ2 = 0. then there exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies (i) ‖ψ̃‖2 ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ0 = 0, (ii) ‖ψ̃‖2 ≤ Cδλ2−2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ0 6= 0 and θ ≤ 1/2, (iii) ‖ψ̃‖2 ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ0 6= 0 and θ ≥ 1/2. 28 J. HAO, D. WANG EJDE-2023/87 Proof. Taking the inner product of (4.1b) with Eα∗ ψx to deduce that κ‖E α 2 ∗ ψx‖2 = 〈iλρ1ϕ̃− κ(ϕx + lw)x − κ0l(wx − lϕ) + γ1E θϕ̃− ρ1f2, Eα∗ ψx〉 = −ρ1λ2〈ϕ,Eα∗ ψx〉 − iλρ1〈f1, Eα∗ ψx〉 − κ〈ϕxx, Eα∗ ψx〉 − l(κ+ κ0)〈wx, Eα∗ ψx〉+ κ0l 2〈ϕ,Eα∗ ψx〉+ γ1〈Eθ/2∗ ϕ̃, E α+ θ 2 ∗ ψx〉 − ρ1〈f2, Eα∗ ψx〉. (4.30) Next, we estimate the item −κ〈ϕxx, Eα∗ ψx〉 in (4.30). For this end, taking the inner product of (4.1d) with κEα∗ ϕx, noting that γ2 = 0, we have −κ〈ϕxx, Eα∗ ψx〉 = κ b 〈iλρ2ψ̃ + κ(ϕx + ψ + lw)− ρ2f4, Eα∗ ϕx〉 = −κρ2 b λ2〈Eα∗ ψ,ϕx〉 − iλ κρ2 b 〈f3, Eα∗ ϕx〉+ κ2 b ‖Eα∗ ϕx‖2 + κ2 b 〈ψ,Eα∗ ϕx〉+ κ2l b 〈w,Eα∗ ϕx〉 − κρ2 b 〈f4, Eα∗ ϕx〉. Substitute the last formula and use integration by parts to obtain κ‖E α 2 ∗ ψx‖2 = −ρ1λ2〈ϕ,Eα∗ ψx〉 − iλρ1〈f1, Eα∗ ψx〉 − l(κ+ κ0)〈wx, Eα∗ ψx〉 + κ0l 2〈ϕ,Eα∗ ψx〉+ γ1〈Eθ/2∗ ϕ̃, E α+ θ 2 ∗ ψx〉 − ρ1〈f2, Eα∗ ψx〉 + κρ2 b λ2〈ψx, Eα∗ ϕ〉 − iλ κρ2 b 〈f3, Eα∗ ϕx〉+ κ2 b ‖Eα∗ ϕx‖2 + κ2 b 〈ψ,Eα∗ ϕx〉+ κ2l b 〈w,Eα∗ ϕx〉 − κρ2 b 〈f4, Eα∗ ϕx〉, (4.31) where |iλρ1〈f1, Eα∗ ψx〉| = |iλρ1〈(f1)x, E α ∗ ψ〉| ≤ Cλ2‖E α 2 ∗ ψ‖2 + C‖F‖2, |l(κ+ κ0)〈wx, Eα∗ ψx〉| ≤ Cλ2‖E α 2 ∗ wx‖2 + Cλ2‖E α 2 ∗ ψx‖2, |κ0l2〈ϕ,Eα∗ ψx〉| ≤ Cλ2‖E α 2 ∗ ϕ‖2 + Cλ2‖E α 2 ∗ ψx‖2. The estimates for the remaining items are similar to the those above. Substituting these estimates into (4.31) and using Cauchy-Schwarz, Young’s inequalities, (1.12) and (4.3) yield κ‖E α 2 ∗ ψx‖2 ≤ |χ0|λ2〈ϕ,Eα∗ ψx〉+ Cλ2‖E α 2 ψ‖2 + Cλ2‖E α 2 ∗ ϕ‖2 + C‖E α 2 ∗ ϕx‖2 + C‖E α 2 ∗ wx‖2 + C‖Eα+ θ 2 ∗ ψx‖2 + ‖F‖‖U‖+ ‖F‖2. (4.32) For the case χ0 = 0, taking α = −θ in (4.32), because of the continuous embedding we have ‖E−θ/2∗ ψx‖2 ≤ C‖E−θ/2∗ ψ̃‖2 + C‖E−θ/2∗ ϕx‖2 + C‖E−θ/2∗ ϕ̃‖2 + C‖E−θ/2∗ wx‖2 + ‖F‖‖U‖+ ‖F‖2. (4.33) Taking into account α = −θ in the item (ii) of Lemma 4.3, using (1.18) we obtain ‖E−θ/2∗ ϕx‖2 = ‖E 1 2− θ 2 ∗ ϕ‖2 EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 29 ≤ C ( ‖E−θ/2∗ ϕ̃‖2 + ‖E−θ/2∗ ψ‖2 + ‖E−θ/2∗ w‖2 ) + ‖F‖‖U‖+ ‖F‖2. The estimate of ‖E−θ/2wx‖2 can be obtained in the same way. Substituting into (4.33) and using (1.19), we obtain ‖E 1 2− θ 2 ∗ ψ‖2 = ‖E−θ/2∗ ψx‖2 ≤ C‖E−θ/2∗ ψ̃‖2 + C‖E−θ/2∗ ϕ̃‖2 + C‖E−θ/2∗ w̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E−θ/2∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2, (4.34) since (4.1a) and (4.1c). Consider α = −θ in the item (i) of Lemma 4.4 and divide by λ2, by (4.1a), (4.1c) and (4.1e) to obtain ‖E−θ/2∗ ψ̃‖2 ≤ C ( ‖E 1 2− θ 2 ∗ ψ‖2 + ‖E−θ/2∗ ϕ‖2 + ‖E−θ/2∗ w‖2 ) + ‖F‖2. (4.35) From the γ2 = 0, we have ε2 = 0. Substituting (4.34) into (4.35) we have ‖E−θ/2∗ ψ̃‖2 ≤ C ( ‖E−θ/2∗ ψ̃‖2 + ‖E−θ/2∗ ϕ‖2 + ‖E−θ/2∗ w‖2 ) + ‖F‖‖U‖+ ‖F‖2 ≤ C ( ‖E−θ/2∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2 ) . (4.36) Thus, ‖E−θ/2∗ ψ̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) . (4.37) Taking into account α = −θ in the item (ii) of Lemma 4.4 and ε2 = 0, using − θ2 < θ 2 and (4.37), we deduce that ‖E 1 2− θ 2 ∗ ψ̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.38) Similar to the interpolation inequality as (4.21), use (4.37) and (4.38) to conclude ‖ψ̃‖ ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) . This leads to the first result of Lemma 4.9. For the case χ0 6= 0, we have ‖E α 2 ∗ ψx‖2 ≤ C‖E α 2 ∗ ψx‖2 + Cλ2‖E α 2 ∗ ϕ̃‖2 + Cλ2‖E α 2 ∗ ψ‖2 + C‖E α 2 ∗ ϕx‖2 + C‖E α 2 ∗ wx‖2 + C‖Eα+ θ 2 ∗ ψx‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2‖E α 2 ∗ ϕ̃‖2 + Cλ2‖E α 2 ∗ ψ‖2 + C‖E α 2 ∗ ϕx‖2 + C‖E α 2 ∗ wx‖2 + C‖Eα+ θ 2 ∗ ψx‖2 + ‖F‖‖U‖+ ‖F‖2. (4.39) Considering α = θ − 1 and θ ≤ 1/2 in (4.39), we have θ 2 + α ≤ α 2 , by continuous embedding, (1.18), (1.19), (4.3), and (4.5), we obtain ‖E θ 2− 1 2 ∗ ψx‖2 ≤ Cλ2‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖E θ 2− 1 2 ∗ ϕx‖2 + ‖E θ 2− 1 2 ∗ ψ̃‖2 + C‖E θ 2− 1 2 ∗ wx‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖Eθ/2∗ ϕ‖2 + ‖E θ 2− 1 2 ∗ ψ̃‖2 + C‖Eθ/2∗ w‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖E θ 2− 1 2 ∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2. 30 J. HAO, D. WANG EJDE-2023/87 Taking into account α = θ− 1 in the item (i) of Lemma 4.3, note that θ ≤ 1/2 and (4.3), substituting the last formula to obtain ‖E θ 2− 1 2 ∗ ψx‖2 ≤ C‖Eθ/2∗ ϕ̃‖2 + C‖E θ 2− 1 2 ∗ ψ̃‖2 + C‖E θ 2− 1 2 ∗ w̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E θ 2− 1 2 ∗ ψ̃‖2 + ( ‖F‖‖U‖+ ‖F‖2 ) . Similar to the process of (4.35)-(4.36), − θ2 is replaced by θ 2 − 1 2 , 1 2 − θ 2 is replaced by θ 2 , we have ‖E θ 2− 1 2 ∗ ψ̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) . (4.40) Taking α = θ − 1 in the item (ii) of Lemma 4.4 again, we obtain ‖Eθ/2∗ ψ̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.41) Using the same interpolation inequalities with (4.26), and applying (4.40)-(4.41), we conclude that ‖ψ̃‖ ≤ Cδλ2−2θ ( ‖F‖‖U‖+ ‖F‖2 ) . The second conclusion of Lemma 4.9 follows. For the third result of this Lemma, that is θ ≥ 1/2, then 1 2 − θ 2 ≤ θ 2 . Considering α = −θ in (4.39), then θ 2 + α ≤ α 2 , by the continuous embedding we deduce ‖E−θ/2∗ ψx‖2 ≤ Cλ2‖E−θ/2∗ ϕ̃‖2 + ‖E−θ/2∗ ϕx‖2 + ‖E−θ/2∗ ψ̃‖2 + C‖E−θ/2∗ wx‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E 1 2− θ 2 ∗ ϕ̃‖2 + ‖E−θ/2∗ ψ̃‖2 + ‖E−θ/2∗ w̃‖2 + ‖E 1 2− θ 2 ∗ ϕ‖2 + C‖E 1 2− θ 2 ∗ w‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ ‖E−θ/2∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2. Here, we use (1.18) and (1.19), and consider α = −θ in item (i) of lemma 4.4 and θ ≥ 1/2 to obtain the above inequality. Then similar to the process of (4.35)-(4.38), we obtain ‖E−θ/2∗ ψ̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.42) ‖E 1 2− θ 2 ∗ ψ̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.43) Similar to the interpolation inequality as (4.21), apply (4.42) and (4.43) to conclude that ‖ψ̃‖ ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) . The third conclusion of Lemma 4.9 can be obtained. The proof is complete. � Thirdly we consider the case γ2, γ3 > 0 and γ1 = 0. Lemma 4.10. Let δ > 0 and γ2, γ3 > 0, γ1 = 0. Then there exists positive constant Cδ, such that for |λ| ≥ δ the solution U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T of system (4.1a)-(4.1f) satisfies (i) ‖ϕ̃‖2 ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ0 = 0, (ii) ‖ϕ̃‖2 ≤ Cδλ2−2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ0 6= 0 and θ ≤ 1/2, (iii) ‖ϕ̃‖2 ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) when χ0 6= 0 and θ ≥ 1/2. EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 31 Proof. From (4.1a) and (4.1e) we obtain iλ(wx − lϕ) = (w̃x − lϕ̃) + (f5)x − lf1. Taking the inner product with κ0E α ∗ (wx − lϕ) we deduce that κ0‖E α 2 ∗ (wx − lϕ)‖2 = κ0 iλ 〈(w̃x − lϕ̃) + (f5)x − lf1, Eα∗ (wx − lϕ)〉 = 1 iλ 〈w̃, Eα∗ (−iλρ1w̃ − κl(ϕx + ψ + lw)− γ3Eθ∗w̃ + ρ1f6)〉 + κ0 iλ 〈−lϕ̃+ (f5)x − lf1, Eα∗ (wx − lϕ)〉 = ρ1‖E α 2 ∗ w̃‖2 + γ3 iλ ‖E α 2 + θ 2 ∗ w̃‖2 − κl iλ 〈w̃, Eα∗ (ϕx + ψ + lw)〉 + κ0 iλ 〈−lϕ̃+ (f5)x − lf1, Eα∗ (wx − lϕ)〉+ ρ1 iλ 〈w̃, Eα∗ f6〉 ≤ C‖E α 2 ∗ w̃‖2 + C λ ‖Eθ/2∗ w̃‖2 + C‖E α 2 ∗ (ϕx + ψ + lw)‖2 + C‖E α 2 ϕ‖2 + ‖F‖2, (4.44) here we used the facts of α ≤ 0, (4.1a), the self-adjointness of Eσ∗ , Cauchy-Schwarz and Young’s inequalities. It can be seen from (4.44) that the estimate here is different from Lemma 4.6, which is independent of χ1. From the result of Lemma 4.5, the fact γ1 = 0, and using (4.4), the self- adjointness of Eσ∗ , Cauchy-Schwarz and Young’s inequalities we deduce that ‖E α 2 ∗ (ϕx + ψ + lw)‖2 ≤ ρ1ρ2 κ |χ0|λ2〈ψ,Eα∗ ϕx〉+ C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ∗ w̃‖2 + C‖E α 2 ϕ̃‖2 + C‖Eθ/2∗ ψ̃‖2 + C‖E α 2 +θ ∗ ψ‖2 + C‖E θ 2+α ∗ (ϕx + ψ + lw)‖2 + C‖F‖2 ≤ C|χ0| ( λ2‖E α 2 ∗ ψ̃‖2 + ‖E α 2 ∗ ϕx‖2 ) + C‖E α 2 ∗ ψ̃‖2 + C‖E α 2 ∗ w̃‖2 + C‖E α 2 ϕ̃‖2 + C‖E α 2 +θ ∗ ψ‖2 + C‖E θ 2+α ∗ (ϕx + ψ + lw)‖2 + ‖F‖‖U‖+ ‖F‖2. (4.45) Case 1: χ0 = 0. Taking α = −θ in (4.45), we obtain α 2 + θ = θ 2 and θ 2 + α = α 2 , because of (4.4), (4.5), (4.1c), and the continuous embedding, we have ‖E−θ/2∗ (ϕx + ψ + lw)‖2 ≤ C‖E−θ/2ψ̃‖2 + C‖E−θ/2∗ ϕ̃‖2 + C‖E−θ/2∗ w̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E−θ/2∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. (4.46) On the other hand, ‖E−θ/2∗ ϕx‖2 ≤ C ( ‖E−θ/2∗ (ϕx + ψ + lw)‖2 + ‖E−θ/2∗ ψ‖2 + ‖E−θ/2∗ w‖2 ) . (4.47) Thus by − θ2 < θ 2 , (4.46), (4.1c), (4.1e), (4.4) and (4.5), we have ‖E−θ/2∗ ϕx‖2 ≤ C‖E−θ/2∗ ϕ̃‖2 + C‖E−θ/2∗ ψ‖2 + C‖E−θ/2∗ w‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E−θ/2∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. (4.48) 32 J. HAO, D. WANG EJDE-2023/87 Taking into account α = −θ in the item (i) of Lemma 4.3, dividing λ2, and from γ1 = 0 and (1.18) we obtain ‖E−θ/2ϕ̃‖2 ≤ C ( ‖E 1 2− θ 2ϕ‖2 + ‖E−θ/2∗ ψ‖2 + ‖E−θ/2∗ w‖2 ) + ‖F‖2 ≤ C‖E−θ/2ϕx‖2 + C ( ‖F‖‖U‖+ ‖F‖2 ) . The above estimate and (4.48) yield ‖E−θ/2∗ ϕ̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) . (4.49) Taking α = −θ in the item (ii) of Lemma 4.3, from (4.49) we obtain ‖E 1 2− θ 2 ∗ ϕ̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.50) Using the interpolation inequality similar to (4.21), by (4.49)-(4.50) we obtain ‖ϕ̃‖ ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) . This leads to the first result of Lemma 4.10. Case 2: χ0 6= 0. Taking α = θ− 1 in (4.45) and noting θ ≤ 1/2, from (4.4), (4.1c), (1.19) and continuous embedding we obtain ‖E θ 2− 1 2 ∗ (ϕx + ψ + lw)‖2 ≤ C|χ0| ( λ2‖E θ 2− 1 2 ∗ ψ̃‖2 + ‖E θ 2− 1 2 ∗ ϕx‖2 ) + C‖E θ 2− 1 2 ∗ w̃‖2 + C‖E θ 2− 1 2 ∗ ϕ̃‖2 + C‖E θ 2− 1 2 ∗ ψ̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2‖E θ 2− 1 2 ∗ ψ̃‖2 + C‖E θ 2− 1 2 ∗ ϕx‖2 + C‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. (4.51) Taking into account α = θ − 1 in the item (i) of Lemma 4.4, from (4.4) we obtain λ2‖E θ 2− 1 2 ∗ ψ̃‖2 ≤ C ( ‖Eθ/2ψ̃‖2 + ‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖E θ 2− 1 2 ∗ w̃‖2 ) + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. (4.52) Substituting (4.52) into (4.51) gives ‖E θ 2− 1 2 ∗ (ϕx + ψ + lw)‖2 ≤ C‖E θ 2− 1 2 ∗ ϕx‖2 + C‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. In addition, similar to (4.47)-(4.50), − θ2 is replaced by θ 2 − 1 2 , 1 2 − θ 2 is replaced by θ 2 , we have ‖Eθ/2∗ ϕ‖2 = ‖E θ 2− 1 2 ∗ ϕx‖2 ≤ C‖E θ 2− 1 2 ∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2, (4.53) ‖E θ 2− 1 2 ∗ ϕ̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.54) ‖Eθ/2∗ ϕ̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.55) Using the same interpolation inequalities with similar as (4.26), and applying (4.54)- (4.55), we conclude that ‖ϕ̃‖ ≤ Cδλ2−2θ ( ‖F‖‖U‖+ ‖F‖2 ) . The second conclusion of Lemma 4.10 follows. EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 33 Case 3: For the third result of this Lemma, that is, θ ≥ 1/2. Taking α = −θ in (4.45), and from the continuous embedding we obtain ‖E−θ/2∗ (ϕx + ψ + lw)‖2 ≤ Cλ2‖E−θ/2∗ ψ̃‖2 + C‖E−θ/2∗ ϕx‖2 + C‖E−θ/2∗ w̃‖2 + C‖E−θ/2∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ Cλ2‖E−θ/2∗ ψ̃‖2 + C‖E−θ/2∗ ϕx‖2 + C‖E−θ/2∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. (4.56) Next, we use Lemma 4.4 to estimate the first item in (4.54). Consider α = −θ and θ ≥ 1/2 in the item (i) of Lemma 4.3, by (4.4) we deduce that λ2‖E−θ/2∗ ψ̃‖2 ≤ C‖E 1 2− θ 2 ∗ ψ̃‖2 + C‖E−θ/2∗ ϕ̃‖2 + C‖E−θ/2∗ w̃‖2 + ‖F‖‖U‖+ ‖F‖2 ≤ C‖E−θ/2∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2. Substituting this into (4.56) gives ‖E−θ/2∗ (ϕx +ψ+ lw)‖2 ≤ C‖E−θ/2∗ ϕ̃‖2 +C‖E−θ/2∗ ϕx‖2 + ‖F‖‖U‖+ ‖F‖2. (4.57) In addition, similar to (4.47)-(4.50), we have ‖E 1 2− θ 2 ∗ ϕ‖2 = ‖E−θ/2∗ ϕx‖2 ≤ C‖E−θ/2∗ ϕ̃‖2 + ‖F‖‖U‖+ ‖F‖2, (4.58) ‖E−θ/2∗ ϕ̃‖2 ≤ C ( ‖F‖‖U‖+ ‖F‖2 ) , (4.59) ‖E 1 2− θ 2 ∗ ϕ̃‖2 ≤ Cλ2 ( ‖F‖‖U‖+ ‖F‖2 ) . (4.60) Using the same interpolation inequalities as for (4.21), and applying (4.59)-(4.60) we conclude that ‖ϕ̃‖ ≤ Cδλ2θ ( ‖F‖‖U‖+ ‖F‖2 ) . The third conclusion of this Lemma can be obtained. The proof is complete. � Finally, we present the stability results of system (1.7)-(1.9) (or problem (2.4)). Theorem 4.11. Assume that χ0 and χ1 defined by (1.12) and θ ∈ [0, 1]. Then the semigroup etA corresponding to problem (2.4) is stable as follows: (i) Assume γ1, γ2 > 0, γ3 = 0. • If χ1 = 0 and the exponents θ = 0, then the semigroup is stable exponen- tially, i.e., there exist positive constants C and δ0 such that ‖etAU0‖ ≤ Ce−δ0t, ∀t > 0. • If χ1 = 0 and the exponents θ1 ∈ (0, 1], then the semigroup is stable poly- nomially with the estimate ‖etAU0‖ ≤ Ct−1/(2θ)‖U0‖D(A), ∀t > 0, U0 ∈ D(A). • If χ1 6= 0, then the semigroup is stable polynomially with the estimates ‖etAU0‖ ≤ { Ct−1/(2−2θ)‖U0‖D(A), if θ ∈ [0, 1/2], Ct−1/(2θ)‖U0‖D(A), if θ ∈ [1/2, 1], for ∀t > 0, U0 ∈ D(A). (ii) Assume γ1, γ3 > 0, γ2 = 0. 34 J. HAO, D. WANG EJDE-2023/87 • If χ0 = 0 and the exponents θ = 0, then the semigroup is stable exponen- tially, i.e., there exist positive constants C and δ0 such that ‖etAU0‖ ≤ Ce−δ0t, ∀t > 0. • If χ0 = 0 and the exponents θ1 ∈ (0, 1], then the semigroup is stable poly- nomially with the estimate as following, ‖etAU0‖ ≤ Ct−1/(2θ)‖U0‖D(A), ∀t > 0, U0 ∈ D(A). • If χ0 6= 0, then the semigroup is stable polynomially with the estimate ‖etAU0‖ ≤ { Ct−1/(2−2θ)‖U0‖D(A), if θ ∈ [0, 1/2], Ct−1/(2θ)‖U0‖D(A), if θ ∈ [1/2, 1], for ∀t > 0, U0 ∈ D(A). (iii) Assume γ2, γ3 > 0, γ1 = 0. • If χ1 = 0 and the exponents θ = 0, then the semigroup is stable exponen- tially, i.e., there exist positive constants C and δ0 such that ‖etAU0‖ ≤ Ce−δ0t, ∀t > 0. • If χ1 = 0 and the exponents θ1 ∈ (0, 1], then the semigroup is stable poly- nomially with the estimate ‖etAU0‖ ≤ Ct−1/(2θ)‖U0‖D(A), ∀t > 0, U0 ∈ D(A). • If χ1 6= 0, then the semigroup is stable polynomially with the estimate ‖etAU0‖ ≤ { Ct−1/(2−2θ)‖U0‖D(A), if θ ∈ [0, 1/2], Ct−1/(2θ)‖U0‖D(A), if θ ∈ [1/2, 1], for ∀t > 0, U0 ∈ D(A). Proof. We use Theorems 4.1 and 4.2 to prove these stability results. So, we check the conditions of these two Theorems. Step 1. We prove iR ⊂ ρ(A) through a contradictory argument. Suppose that iR 6⊂ ρ(A). From Theorem 1, we know that 0 ∈ ρ(A), then we denote λ0 the maximum positive number such that (−iλ0, iλ0) ⊂ ρ(A), therefore −iλ0 or iλ0 is an element of the spectrum σ(A). Suppose that iλ0 ∈ σ(A) (if −iλ0 ∈ σ(A) the process is similar). Then, for δ ∈ (0, λ0), there exist a sequence of real numbers {λn}, such that λn ∈ [δ, λ0), λn → λ0, and a sequence of unit vectors {Un = (ϕn, ϕ̃n, ψn, ψ̃n, wn, w̃n)T } ⊂ D(A) satisfying ‖(iλn −A)Un‖ = ‖Fn‖ → 0 as n→∞. That is, if Fn = (f1n, f2n, f3n, f4n, f5n, f6n)T , then we have iλϕn − ϕ̃n = f1n → 0 in H1 0 (0, L), (4.61a) iλρ1ϕ̃n − κ(ϕnx + ψn + lwn)x − κ0l(wnx − lϕn) + γ1E θϕ̃n = ρ1f2n → 0, in L2(0, L) (4.61b) iλψn − ψ̃n = f3n → 0 in H1 ∗ (0, L), (4.61c) iλρ2ψ̃n − bψxx + κ(ϕnx + ψn + lwn) + γ2E θ ∗ ψ̃n = ρ2f4n → 0 in L2 ∗(0, L), (4.61d) iλwn − w̃n = f5n → 0 in H1 ∗ (0, L), (4.61e) EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 35 iλρ1w̃n − κ0(wnx − lϕn)x + kl(ϕnx + ψn + lwn) + γ3E θ ∗w̃n = ρ1f6n → 0 in L2 ∗(0, L). (4.61f) In the same way as we obtained in (4.2), we have γ1‖Eθ/2ϕ̃n‖2 + γ2‖Eθ/2∗ ψ̃n‖2 + γ3‖Eθ/2w̃n‖2 ≤ C‖F‖n‖U‖n → 0. (4.62) Then the following estimates hold ‖ϕ̃n‖2 ≤ ‖Eθ/2ϕ̃n‖2 ≤ C‖F‖n‖U‖n → 0, (4.63) ‖ψ̃n‖2 ≤ ‖Eθ/2ψ̃n‖2 ≤ C‖F‖n‖U‖n → 0, (4.64) ‖w̃n‖2 ≤ ‖Eθ/2w̃n‖2 ≤ C‖F‖n‖U‖n → 0. (4.65) From Lemma 4.7, (4.63), (4.65), (4.1c) and (4.1e) we find that ‖E1/2 ∗ ψn‖2 = ‖ψnx‖2 ≤ Cδ ( ‖ψ̃n‖2 + ‖ϕn‖2 + ‖wn‖2 + ‖F‖n‖U‖n + ‖F‖2n ) ≤ C ( ‖F‖n‖U‖n + ‖F‖2n ) → 0. (4.66) On the other hand, ‖ϕnx + ψn + lwn‖2 ≤ C(‖ϕnx‖2 + ‖ψn‖2 + ‖wn‖2) = C(‖E1/2 ∗ ϕn‖2 + ‖ψn‖2 + ‖wn‖2) ≤ C ( 1 λ2 ‖E1/2ϕ̃n‖2 + 1 λ2 ‖ψ̃n‖2 + 1 λ2 ‖w̃n‖2 + ‖F‖2 ) . Taking α = 0 in the item (ii) of Lemma 4.3 and dividing λ2, we obtain 1 λ2 ‖E1/2ϕ̃n‖2 ≤ C ( ‖ϕ̃n‖2 + ‖ψn‖2 + ‖wn‖2 ) + ε1‖Eθ/2ϕ̃n‖2 + Cε1 ( ‖F‖n‖U‖n + ‖F‖2n ) . Whether ε1 is zero or not, we can get from Lemmas 4.8, 4.9, and 4.10, 1 λ2 ‖E1/2ϕ̃n‖2 → 0. thus, from (4.63) and (4.65) we deduce that ‖ϕnx + ψn + lwn‖2 → 0. (4.67) Similarly, it follows that ‖wnx − lψn‖2 ≤ C(‖wnx‖2 + ‖ϕn‖2) = C(‖E1/2 ∗ wn‖2 + ‖ϕn‖2) ≤ C ( 1 λ2 ‖E1/2w̃n‖2 + 1 λ2 ‖ϕ̃n‖2 ) → 0. (4.68) The estimates (4.62)-(4.68) imply that ‖Un‖ → 0 which is absurd with ‖Un‖ = 1, for all n ∈ N. Consequently iR ⊂ ρ(A). Step 2. Let U = (ϕ, ϕ̃, ψ, ψ̃, w, w̃)T be the solution of the system (iλ−A)U = F . Once we have proven iR ⊂ ρ(A), and then according to Theorems 4.1 and 4.2, we need to prove the decay rate of the desired semigroup etA. Because the proving process is very similar, we only prove the item (i) of Theorem 4.11, and the other two results have similar proof. Assume γ1, γ2 > 0, γ3 = 0. From (4.2) we have ‖Eθ/2ϕ̃n‖2 + γ2‖Eθ/2∗ ψ̃n‖2 ≤ C‖F‖n‖U‖n; 36 J. HAO, D. WANG EJDE-2023/87 that is, ‖ϕ̃n‖2 ≤ ‖Eθ/2ϕ̃n‖2 ≤ C‖F‖n‖U‖n, (4.69) ‖ψ̃n‖2 ≤ ‖Eθ/2ψ̃n‖2 ≤ C‖F‖n‖U‖n. (4.70) From (4.66)-(4.68) we have ‖ψnx‖2 ≤ Cδ ( ‖ψ̃n‖2 + ‖ϕn‖2 + ‖wn‖2 + ‖F‖n‖U‖n + ‖F‖2n ) ≤ C ( ‖wn‖2 + ‖F‖n‖U‖n + ‖F‖2n ) , (4.71) ‖ϕnx + ψn + lwn‖2 ≤ C(‖E1/2 ∗ ϕn‖2 + ‖ψn‖2 + ‖wn‖2) ≤ C ( ‖ϕ̃n‖2 + ‖Eθ/2ϕ̃n‖2 + 1 λ2 ‖ψ̃n‖2 + 1 λ2 ‖w̃n‖2 ) + (‖F‖n‖U‖n + ‖F‖2n) ≤ C 1 λ2 ‖w̃n‖2 + ‖F‖n‖U‖n + ‖F‖2n. (4.72) ‖wnx − lψn‖2 ≤ C(‖E1/2 ∗ wn‖2 + ‖ϕn‖2) ≤ C ( ‖w̃n‖2 + ‖ϕn‖2 + ‖ψn‖2 + 1 λ2 ‖ϕ̃n‖2 + ‖F‖2n ) ≤ C‖w̃n‖2 + C‖F‖2n. (4.73) The above estimates imply that ‖Un‖2 ≤ ‖w̃n‖2 + ‖F‖n‖U‖n + ‖F‖2n. It can be seen from Lemma 4.8 that, when χ1 = 0 and θ = 0, we obtain ‖Un‖2 ≤ Cδ ( ‖F‖n‖U‖n + ‖F‖2n ) ; when χ1 = 0 or χ1 6= 0 and θ ≥ 1/2, we obtain ‖Un‖2 ≤ Cδλ2θ ( ‖F‖n‖U‖n + ‖F‖2n ) ; when χ1 6= 0 and θ ≤ 1/2, we obtain ‖Un‖2 ≤ Cδλ2−2θ ( ‖F‖n‖U‖n + ‖F‖2n ) . Hence, applying Young’s inequality, it follows that ‖Un‖2 ≤ Cδ‖F‖2n, or ‖Un‖2 ≤ Cδλ2θ‖F‖2n, or ‖Un‖2 ≤ Cδλ2−2θ‖F‖2n, which is the desired result. Consequently, when χ1 = 0 and θ = 0, by Theorem 4.1, we obtain that the semigroup is exponentially stable. When χ1 = 0 and θ ∈ (0, 1], by Theorem 4.2 the semigroup decays polynomially with the rate t−1/(2θ). When χ1 6= 0 and θ ∈ [0, 1/2], by Theorem 4.2 the semigroup decays polynomially with the rate t−1/(2−2θ). When χ1 6= 0 and θ ∈ [1/2, 1], by Theorem 4.2 the semigroup decays polynomially with the rate t−1/(2θ). Then the item (i) of Theorem 4.11 is be established. Similarly, we can get the other two results. Thus the proof is complete. � Remark 4.12. Alves et al. [2] studied the asymptotic behavior of the system when friction damping acts on the vertical displacement and the angle displacement at the same time (that is, θ = 0 and γ3 = 0 in system (1.7)-(1.9)). They obtained the exponential decay and polynomial decay of the system respectively. These results are consistent with our conclusions obtained in the item (i) of Theorem 4.11. Furthermore, Alves et al. showed that the polynomial decay rate t− 1 2 is EJDE-2023/87 ASYMPTOTIC STABILIZATION FOR BRESSE SYSTEMS 37 optimal. Thus we believe that our polynomial stability results are optimal, which will be our next research topic. Acknowledgments. This study was funded by the National Natural Science Foun- dation of China (12271315), by the special fund for Science and Technology In- novation Teams of Shanxi Province (202204051002015), and by the Fundamental Research Program of Shanxi Province (202203021221018). References [1] M. Akil, H. Badawi, S. Nicaise, A. Wehbe; On the stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force, Z. Angew. Math. Phys., 72(3) (2021), 1-27. [2] M. O. Alves, L. H. Fatori, M. A. Jorge Silva, R. N. Monteiro; Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38(5) (2015), 898-908. [3] M. Afilal, A. Guesmia, A. Soufyane, M. Zahri; On the exponential and polynomial stability for a linear Bresse system, Math. Methods Appl. Sci., 43(5) (2020), 2626-2645. [4] G. Aguilera Contreras, J. E. Muñoz Rivera; Bresse systems with localized Kelvin-Voigt dis- sipation, Electron. J. Differential Equations, (2021), 1-14. [5] F. Alabau Boussouira, J. E. Muñoz Rivera, D. da S. Almeida Júnior; Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374(2) (2011), 481-498. [6] M. Astudillo, H. P. Oquendo; Stability results for a Timoshenko system with a fractional operator in the memory, Appl. Math. Optim., 83(3) (2021), 1247-1275. [7] R. Bekhouche, A. Guesmia, S. Messaoudi; Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements, Arab. J. Math., 11(2) (2022), 155-1784. [8] A. Benaissa, A. Kasmi; Well-posedness and energy decay of solutions to a Bresse system with a boundary dissipation of fractional derivative type, Discrete Contin. Dyn. Syst. Ser. B, 23(10) (2018), 4361-4395. [9] T. Bentrcia, A. Mennouni; On the asymptotic stability of a Bresse system with two fractional damping terms: Theoretical and numerical analysis, Discrete Contin. Dyn. Syst. Ser. B, 28(1) (2023), 580-622. [10] A. Borichev, Y. Tomilov; Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347(2) (2010), 455-478. [11] V. R. Cabanillas, C. A. Raposo; Exponential stability for laminated beams with intermediate damping, Arch. Math. (Basel), 118(6) (2022), 625-635. [12] H. Dridi, A. Djebabla; Timoshenko system with fractional operator in the memory and spatial fractional thermal effect, Rend. Circ. Mat. Palermo (2), 70(1) (2021), 593-621. [13] K. J. Engel, R. Nagel; One-parameter semigroups for linear evolution equations, Grad. Texts in Math., 194 (2000). [14] T. El Arwadi, W. Youssef; On the stabilization of the Bresse beam with Kelvin-Voigt damp- ing, Appl. Math. Optim., 83(3) (2021), 1831-1857. [15] L. H. Fatori, J. E. Muñoz Rivera; Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75(6) (2010), 881-904. [16] L. H. Fatori, R. N. Monteiro; The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25(3) (2012), 600-604. [17] L. Gearhart; Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc., 236 (1978), 385-394. [18] K F. Graff; Wave motion in elastic solids, Dover Publications, New York (1991). [19] A. Guesmia; Asymptotic stability of Bresse system with one infinite memory in the longitu- dinal displacements, Mediterr. J. Math., 14(2) (2017), 1-19. [20] A. Guesmia; Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4(1) (2017), 78-97. [21] A. Guesmia; The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement, Arab. J. Math. (Springer), 8(1) (2019), 15-41. [22] A. Guesmia; Polynomial and non exponential stability of a weak dissipative Bresse system, preprint. 38 J. HAO, D. WANG EJDE-2023/87 [23] A. Guesmia, M. Kafini; Bresse system with infinite memories, Math. Methods Appl. Sci., 38(11) (2015), 2389-2402. [24] A. Guesmia, M. Kirane; Uniform and weak stability of Bresse system with two infinite mem- ories, Z. Angew. Math. Phys., 67(5) (2016), 1-39. [25] S. G. Kreui; Linear differential equations in Banach space, American Mathematical Soc, (2011). [26] A .A. Keddi, T. A. Apalara, S. A. Messaoudi; Exponential and polynomial decay in a thermoelastic-Bresse system with second sound, Appl. Math. Optim., 77(2) (2018), 315-341. [27] Z. B. Kuang, Z. Y. Liu, L. Tebou; Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping case, ESAIM Control Optim. Calc. Var., 28 (2022), 1-20. [28] J. E. Lagnese, G. Leugering, E. J. P. G. Schmidt; Modeling, analysis and control of dynamic elastic multi-link structures, Systems Control Found. Appl., (1994). [29] Z. Y. Liu, B. P. Rao; Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56(4) (2005), 630-644. [30] Z. Y. Liu, B. P. Rao; Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60(1) (2009), 54-69. [31] J. E. Muñoz Rivera, H. D. Fernández Sare; Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339(1) (2008), 482-502. [32] H. P. Oquendo, C. R. da Luz; Asymptotic behavior for Timoshenko systems with fractional damping, Asymptot. Anal., 118 (1-2) (2020), 123-142. [33] H. P. Oquendo, F. M. S. Suárez; Exact decay rates for coupled plates with partial fractional damping, Z. Angew. Math. Phys., 70(3) (2019), 1-18. [34] A. Pazy; Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44 (1983). [35] J. Prüss; On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284(2) (1984), 847- 857. [36] M. de L. Santos, A. Soufyane, D. da S. Almeida Júnior; Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73(1) (2015), 23-54. [37] A. Soufyane, B. Said-Houari; The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equ. Control Theory, 3(4) (2014), 713-738. [38] G. F. Tyszka, M. R. Astudillo, H. P. Oquendo; Stabilization by memory effects: Kirchhoff plate versus Euler-Bernoulli plate, Nonlinear Anal. Real World Appl., 68(2) (2022), 1-24. [39] L. Tebou; Regularity and stability for a plate model involving fractional rotational forces and damping, Z. Angew. Math. Phys., 72(4) (2021), 1-13. [40] A. Wehbe, W. Youssef; Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51(10) (2010), 1-17. Jianghao Hao School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, China Email address: hjhao@sxu.edu.cn Dingkun Wang School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi, 030006, China Email address: 1693229781@qq.com 1. Introduction 2. Well-posedness of solution 3. Lack of exponential stability 4. Stability results Acknowledgments References