Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 77, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.77 ASYMPTOTIC STABILITY FOR THERMODIFFUSION TIMOSHENKO SYSTEMS OF TYPE III JIALI QIN, JIANGHAO HAO Abstract. In this article, we study a Timoshenko model with thermal and mass diffusion effects. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam, where the heat conduction is given by Green and Naghdi, called thermoelasticity of type III. We obtain the stability of the system using the perturbed energy method and the system is exponentially stable when the wave speeds are equal. In the case of unequal wave speeds, we demonstrate that the system lacks exponential stability, and it is polynomially stable. These results indicate that the wave speed has a significant impact on the stability of the system, and the transmission performance of the system is better when the wave speeds are equal. 1. Introduction In this article, we study the thermodiffusion Timoshenko system ρ1φtt − k(φx + ψ)x = 0, (x, t) ∈ (0, 1)× (0,∞), ρ2ψtt − αψxx + k(φx + ψ)− r1θx − r2Px = 0, (x, t) ∈ (0, 1)× (0,∞), cθtt + dPtt − σ1θxx − σ2θxxt − r1ψxtt = 0, (x, t) ∈ (0, 1)× (0,∞), rPtt + dθtt − γ1Pxx − γ2Pxxt − r2ψxtt = 0, (x, t) ∈ (0, 1)× (0,∞), (1.1) with boundary conditions ψx(0, t) = φ(0, t) = θ(0, t) = P (0, t) = 0, t ∈ (0,∞), ψ(1, t) = φx(1, t) = θx(1, t) = Px(1, t) = 0, t ∈ (0,∞), (1.2) and initial conditions ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), x ∈ (0, 1), φ(x, 0) = φ0(x), φt(x, 0) = φ1(x), x ∈ (0, 1), P (x, 0) = P0(x), Pt(x, 0) = P1(x), x ∈ (0, 1), θ(x, 0) = θ0(x), θt(x, 0) = θ1(x), x ∈ (0, 1), (1.3) where φ denotes the transverse displacement of the beam and ψ is the rotation angle of the filament of the beam, P is the chemical potential and θ is the temperature difference, γ1, γ2, c, d, r, α, ρ1, ρ2 are physical positive constants. We assume that the symmetric matrix Λ = ( c d d r ) is positive definite, that is, δ := cr − d2 > 0. (1.4) We know that when the diffusion effect is added to the thermal effect, condition (1.4) is required to stabilize the system. 2020 Mathematics Subject Classification. 35B35, 35L55, 93D15. Key words and phrases. Timoshenko system; exponentially stable; polynomially stable; exponential stability; thermodiffusion of type III. ©2025. This work is licensed under a CC BY 4.0 license. Submitted February 1, 2025. Published July 23, 2025. 1 2 J. QIN, J. HAO EJDE-2025/77 It is widely known that the Timoshenko model is a specific model for vibration of elastic beam, arising from the coupling of shear force and bending moment within the system. Timoshenko [29] introduced the classic Timoshenko system ρφtt −K(φx + ψ)x = 0, (x, t) ∈ (0, 1)× (0,∞), Iρψtt − bψxx +K(φx + ψ) = 0, (x, t) ∈ (0, 1)× (0,∞). (1.5) The earliest study on the stability of system (1.5) was conducted by Soufyane [28], who considered adding a weak damping (βψt) to the second equation of system (1.5) and proved that the system is exponentially stable if and only if the wave speeds are equal, that is, K ρ = b Iρ . In addition, for the stability of system (1.5) under different damping mechanisms, the readers can also refer to the references [2, 20, 8, 14] and their references for more information and opinions. Thermoelastic damping is the source of material inherent damping, which is generated by the coupling between elastic field and temperature field in the structure caused by deformation. One of the thermoelastic dissipation is heat conduction through heat flux. There are different types of heat flux. When the heat flux is defined by Fourier’s law, Rivera and Racke [25] studied the thermoelastic Timoshenko system ρφtt −K(φx + ψ)x = 0, (x, t) ∈ (0, 1)× (0,∞), Iρψtt − bψxx +K(φx + ψ) + γθx = 0, (x, t) ∈ (0, 1)× (0,∞), ρ3θt − βθxx + γψxt = 0, (x, t) ∈ (0, 1)× (0,∞). (1.6) They proved that system (1.6) is exponentially stable when the wave speeds are equal. When the heat flux is defined by Cattaneo’s law, (1.6)3 can be replaced by the equations τ0qt + q + τθx = 0, θt + qx + γψxt = 0. Santosa, et al. [26] studied the stability of Timoshenko system with Cattaneo’s law. They in- troduced a new stability number χ0, and got that the system is exponentially with χ0 = 0, otherwise the system lacks exponential stability, and decays polynomially with rate t− 1 2 . They also proved that the decay rate is optimal. In addition, for type III heat dissipation, one can refer to [27, 12, 16, 17]. Dell’Oro and Pata [9] studied the stability for the heat dissipation of type Gurtin-Pipkin. Readers interested in the stability of the thermoelastic Timoshenko system can refer to [18, 7, 19, 1]. Thermodiffusion, known as the Soret effect, refers to the mass diffusion of components in a mixture due to a temperature gradient. In solid materials, this phenomenon is less common than in liquids or gases but can still occur under certain conditions, particularly in multicomponent solids, alloys, or doped semiconductors. High-tech research has revealed that diffusion is a common physical phenomenon occurring not only in fluids but also distinctly in solids. In solids, thermal diffusion arises from the interaction among strain, temperature, and mass diffusion fields, driving processes crucial to materials science and engineering. For thermodiffusion coupling mechanisms (strain-temperature-concentration) in solids and their engineering implementations, see Olesiak’s review [21] and the references therein. Because of their widespread use in modern engineering structures, Timoshenko beams have become a key research focus. Their stability, particularly under thermodiffusion effects, has at- tracted significant scholarly attention. Aouadi et al. [3] considered the effect of mass diffusion effect in a thermo-Timoshenko beam, ρ1φtt − κ(φx + ψ)x = 0, (x, t) ∈ (0, 1)× (0,∞), ρ2ψtt − αψxx + κ(φx + ψ)− γ1θx − γ2Px = 0, (x, t) ∈ (0, 1)× (0,∞), cθt + dPt + qx − γ1ψtx = 0, (x, t) ∈ (0, 1)× (0,∞), dθt + rPt + ηx − γ2ψtx = 0, (x, t) ∈ (0, 1)× (0,∞), (1.7) EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 3 where q is the heat flux, η is the mass diffusion flux. When heat and mass diffusion follow Fourier’s law and Fick’s law respectively q = −Kθx, η = −ℏPx, (1.8) system (1.7) can be written as ρ1φtt − κ(φx + ψ)x = 0, (x, t) ∈ (0, 1)× (0,∞), ρ2ψtt − αψxx + κ(φx + ψ)− γ1θx − γ2Px = 0, (x, t) ∈ (0, 1)× (0,∞), cθt + dPt −Kθxx − γ1ψtx = 0, (x, t) ∈ (0, 1)× (0,∞), dθt + rPt − ℏPxx − γ2ψtx = 0, (x, t) ∈ (0, 1)× (0,∞). (1.9) Aouadi et al. [3] proved the well-posedness and stability of system (1.9) with Dirichlet or Neumann boundary conditions. Without assuming equal wave velocities, they proved that the system lacks exponential stability under Neumann boundary conditions. By adding a linear frictional damping to the first equation, the system with Dirichlet boundary can be exponentially stable. This result was extended by Feng [11] to the same exponential stability results in the conditions of equal wave velocities and dropping of the linear frictional damping. Djellali et al. [10] considered the stability of system (1.9) when the thermal and mass diffusion coupling was on the shear force, and proved that the system is exponentially stable if and only if the wave speeds are equal, otherwise the system is polynomial stable. Ramos et al. [24] studied the qualitative and numerical behavior of (1.9) with Kelvin-Voigt damping. The flaw of Fourier’s law lies in the physical paradox of infinite heat propagation speed, which is a typical side effect of parabolic behavior. Cattaneo’s law eliminates this paradox, based on which (1.8) can be written as τ0qt + q = −Kθx, τ1ηt + η = −ℏPx. (1.10) By substituting (1.10) into (1.7), Aouadi et al. [4] studied the model ρ1φtt − κ(φx + ψ)x = 0, (x, t) ∈ (0, 1)× (0,∞), ρ2ψtt − αψxx + κ(φx + ψ)− γ1θx − γ2Px = 0, (x, t) ∈ (0, 1)× (0,∞), cθt + dPt + qx − γ1ψtx = 0, (x, t) ∈ (0, 1)× (0,∞), τ0qt + q +Kθx = 0, (x, t) ∈ (0, 1)× (0,∞), dθt + rPt + ηx − γ2ψtx = 0, (x, t) ∈ (0, 1)× (0,∞), τ1ηt + η + ℏPx = 0, (x, t) ∈ (0, 1)× (0,∞). (1.11) They proved the well-posedness of the system using semigroup theory. Then they introduced two numbers χ0 and χ1, the system is exponentially stable if and only if χ0 = 0 and χ1 = 0, otherwise the system lacks exponential stability, and the semigroup associated with the system decays to zero polynomially as t− 1 2 . Recently, Zhang [30] extended the research on thermodiffusion to the Type III model and studied the system ρvtt − (λ+ 2µ)vxx + γ1θ1xt + γ2θ2xt = 0, (x, t) ∈ (0, 1)× (0,∞), cθ1tt − kθ1xx − δ1θ1txx + γ1vtx + dθ2tt = 0, (x, t) ∈ (0, 1)× (0,∞), nθ2tt −Dθ2xx − δ2θ2txx + γ2vtx + dθ1tt = 0, (x, t) ∈ (0, 1)× (0,∞). (1.12) He used the semigroup method and the energy perturbation method to prove the global existence and exponential stability of the system under Dirichlet boundary conditions. Aouadi et al. [6] considered a thermoelastic diffusion problem of type III in one space dimension with boundary constant delays and proved that the system is asymptotically stable by constructing an appropriate Lyapunov functional. For similar results we can refer to [5] and its references. Taking inspiration from the above literature, in this article we consider to extend the thermod- iffusion of the Type III model to the Timoshenko system. According to the constitutive equation of heat flux in type III theory, (1.8) can be written as qt = −σ1θx − σ2θxt, ηt = −γ1Px − γ2Pxt. (1.13) 4 J. QIN, J. HAO EJDE-2025/77 By substituting (1.13) into (1.7), we can obtain the system (1.1). This paper’s core achievement is to construct a more universal thermodiffusion Timoshenko model. It breaks through the limits of previous models, achieving theoretical expansion. Compared with the classical Fourier and Cattaneo models, it has two key advantages: resolving Fourier’s infinite speed paradox and overcoming Cattaneo’s single relaxation time limit. Expanding its applicable boundaries helps accurately describe practical engineering heat conduction, providing a reliable basis for applied research. The outline of this article is follows. In Section 2, we give the well-posedness of the system (1.1)-(1.3) by using the semigroup method. In section 3, we use the energy perturbation method and multiplier technique to obtain that the system is exponentially stable when k ρ1 = α ρ2 . By using the Gearhart-Herbst-Prüss-Huang [13, 23, 15] theorem we show that the system (1.1)-(1.3) lacks exponential stability. In Section 5, we obtain that the system is polynomially stable when k ρ1 ̸= α ρ2 by using the method of higher order energy. In Section 6, we give a summary of the content of this paper. 2. Well-posedness of the system In this section, we use the semigroup theory to give the existence and uniqueness of system (1.1)-(1.3). Let ϕ = φt, w = ψt. Then, the system (1.1)-(1.3) takes the form ρ1ϕtt − k(ϕx + w)x = 0, (x, t) ∈ (0, 1)× (0,∞), ρ2wtt − αwxx + k(ϕx + w)− r1θxt − r2Pxt = 0, (x, t) ∈ (0, 1)× (0,∞), cθtt + dPtt − σ1θxx − σ2θxxt − r1wxt = 0, (x, t) ∈ (0, 1)× (0,∞), rPtt + dθtt − γ1Pxx − γ2Pxxt − r2wxt = 0, (x, t) ∈ (0, 1)× (0,∞), (2.1) with the boundary conditions ϕ(0, t) = wx(0, t) = θ(0, t) = P (0, t) = 0, t ∈ (0,∞), ϕx(1, t) = w(1, t) = θx(1, t) = Px(1, t) = 0, t ∈ (0,∞), (2.2) and the initial conditions ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x), x ∈ (0, 1), w(x, 0) = w0(x), wt(x, 0) = w1(x), x ∈ (0, 1), P (x, 0) = P0(x), Pt(x, 0) = P1(x), x ∈ (0, 1), θ(x, 0) = θ0(x), θt(x, 0) = θ1(x), x ∈ (0, 1), (2.3) where ϕ0(x) = φ1(x), w0(x) = ψ1(x), x ∈ (0, 1), ϕ1(x) = k ρ1 φ0xx + k ρ1 ψ0x, x ∈ (0, 1), w1(x) = α ρ2 ψ0xx − k ρ2 ψ0 − k ρ2 φ0x + r1 ρ2 θ0x + r2 ρ2 P0x, x ∈ (0, 1). Next we construct an abstract Cauchy problem. Let H1 a(0, 1) = {v : v ∈ H1(0, 1) : v(0) = 0}, H1 b (0, 1) = {v : v ∈ H1(0, 1) : v(1) = 0}. We define the state space H = H1 a(0, 1)×H1 b (0, 1)× [ H2(0, 1) ∩H1 a(0, 1) ]2 × [ L2(0, 1) ]4 , EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 5 with the inner product ⟨U, Ũ⟩H = ρ1 ∫ 1 0 ss̃dx+ ρ2 ∫ 1 0 zz̃dx+ α ∫ 1 0 wxw̃xdx+ d ∫ 1 0 ϖũdx + c ∫ 1 0 uũdx+ r ∫ 1 0 ϖϖ̃dx+ σ1 ∫ 1 0 θxθ̃xdx+ γ1 ∫ 1 0 PxP̃xdx + k ∫ 1 0 (ϕx + w) ( ϕ̃x + w̃ ) dx+ d ∫ 1 0 uϖ̃dx, (2.4) for U = (ϕ,w, θ, P, s, z, u,ϖ)T , Ũ = (ϕ̃, w̃, θ̃, P̃ , s̃, z̃, ũ, ϖ̃)T . From (1.4), we have ∥U∥2H is nonneg- ative. We set the operator A : D(A) ⊂ H → H defined by A =  s z u ϖ k ρ1 (ϕx + w)x 1 ρ2 [αwxx − k (ϕx + w) + r1ux + r2ϖx] 1 δ [rσ1θxx + rσ2uxx − dγ1Pxx − dγ2ϖxx + (rr1 − dr2) zx] 1 δ [cγ1Pxx + cγ2ϖxx − dσ1θxx − dσ2uxx + (cr2 − dr1) zx]  , with domain D(A) = { U ∈ H : w ∈ H2(0, 1) ∩H1 b (0, 1), ϕ, θ, P ∈ H2(0, 1) ∩H1 a(0, 1), s ∈ H1 a(0, 1), z ∈ H1 b (0, 1), u,ϖ ∈ H2(0, 1) ∩H1 a(0, 1), wx(0) = ϕx(1) = θx(1) = Px(1) = 0 } for U = (ϕ,w, θ, P, s, z, u,ϖ)T ∈ D(A). Now, we set a vector function U = (ϕ,w, θ, P, ϕt, wt, θt, Pt, ) T , then the system (2.1)-(2.3) can be written as the Cauchy problem d dt U(t) = AU(t), U(0) = U0 = (ϕ0, w0, θ0, P0, ϕ1, w1, θ1, P1) T . (2.5) The well-posedness result is stated as follows. Theorem 2.1. For U0 ∈ H, then there exists a unique solution U ∈ C(R+,H) of problem (2.5). Moreover, if U0 ∈ D(A), then U ∈ C(R+, D(A)) ∩ C1(R+,H). Proof. For U = (ϕ,w, θ, P, s, z, u,ϖ)T ∈ D(A), we have ⟨AU,U⟩H = k ∫ 1 0 (ϕx + w)xs dx+ ∫ 1 0 [αwxx − k (ϕx + w) + r1ux + r2ϖx] zdx + α ∫ 1 0 zxwx dx+ k ∫ 1 0 (sx + z) (ϕx + w) dx + c ∫ 1 0 1 δ [rσ1θxx + rσ2uxx − dγ1Pxx − dγ2ϖxx + (rr1 − dr2)zx]u dx +r ∫ 1 0 1 δ [cγ1Pxx + cγ2ϖxx − dσ1θxx − dσ2uxx + (cr2 − dr1)zx]ϖdx + σ1 ∫ 1 0 uxθx dx+ γ1 ∫ 1 0 ϖxPx dx 6 J. QIN, J. HAO EJDE-2025/77 + d ∫ 1 0 1 δ [cγ1Pxx + cγ2ϖxx − dσ1θxx − dσ2uxx + (cr2 − dr1)zx]u dx + d ∫ 1 0 1 δ [rσ1θxx + rσ2uxx − dγ1Pxx − dγ2ϖxx + (rr1 − dr2)zx]ϖdx. Integrating by parts and using the boundary conditions, we obtain ⟨AU,U⟩H = −σ2 ∫ 1 0 u2xdx− γ2 ∫ 1 0 ϖ2 xdx ≤ 0, (2.6) which gives us that A is dissipative. Given F = (f1, f2, f3, f4, f5, f6, f7, f8) ∈ H, we need to find a solution U for (I −A)U = F, (2.7) that is ϕ− s = f1, w − z = f2, θ − u = f3, P −ϖ = f4, ρ1s− k(ϕx + w)x = ρ1f5, ρ2z − αwxx + k (ϕx + w)− r1ux − r2ϖx = ρ2f6, δu− rσ1θxx − rσ2uxx + dγ1Pxx + dγ2ϖxx − (rr1 − dr2) zx = δf7, δϖ − cγ1Pxx − cγ2ϖxx + dσ1θxx + dσ2uxx − (cr2 − dr1) zx = δf8. (2.8) From (2.8)1 to (2.8)4, we have s = ϕ− f1, z = w − f2, u = θ − f3, ϖ = P − f4. (2.9) Combining (2.8) and (2.9), we have ρ1ϕ− k(ϕx + w)x = h1, ρ2w − αwxx + k (ϕx + w)− r1θx − r2Px = h2, δθ − rσ1θxx − rσ2θxx + dγ1Pxx + dγ2Pxx − (rr1 − dr2)wx = h3 δP − cγ1Pxx − cγ2Pxx + dσ1θxx + dσ2θxx − (cr2 − dr1)wx = h4, (2.10) in which h1 = ρ1(f5 + f1), h2 = ρ2f6 + ρ2f2 − r1f3x − r2f4x, h3 = δ(f7 + f4)− rσ2f3xx + dγ2f4xx − (rr1 − dr2)f2x, h4 = δ(f8 + f4)− cγ2f4xx + dσ2f3xx − (cr2 − dr1)f2x. Multiplying (2.10)1 by ϕ̃, (2.10)2 by w̃, (2.10)3 by c δ θ̃, (2.10)4 by r δ P̃ , (2.10)3 by d δ P̃ and (2.10)4 by d δ θ̃, and integrating over (0, 1). By adding them up, we can obtain the variational formulation B((ϕ,w, θ, P ), (ϕ̃, w̃, θ̃, P̃ )) = L(ϕ̃, w̃, θ̃, P̃ ), (2.11) where the bilinear form B : [ H1 a(0, 1)×H1 b (0, 1)×H2(0, 1) ∩H1 a(0, 1)×H2(0, 1) ∩H1 a(0, 1) ] → R is defined by B((ϕ,w, θ, P ), (ϕ̃, w̃, θ̃, P̃ )) = ρ1 ∫ 1 0 ϕϕ̃dx+ ρ2 ∫ 1 0 ww̃dx+ k ∫ 1 0 (ϕx + w)(ϕ̃x + w̃)dx +α ∫ 1 0 wxw̃x dx+ c ∫ 1 0 θθ̃dx+ (σ1 + σ2) ∫ 1 0 θxθ̃x dx+ d ∫ 1 0 θP̃ dx EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 7 +(γ1 + γ2) ∫ 1 0 PxP̃x dx+ r ∫ 1 0 PP̃dx+ d ∫ 1 0 P θ̃dx+ r1 ∫ 1 0 θxw̃dx −r1 ∫ 1 0 wθ̃x dx− r2 ∫ 1 0 Pxw̃dx+ r2 ∫ 1 0 wP̃x dx, and L : [ H1 a(0, 1)×H1 b (0, 1)×H2(0, 1) ∩H1 b (0, 1)×H2(0, 1) ∩H1 b (0, 1) ] → R is linear form defined by L(ϕ̃, w̃, θ̃, P̃ ) = ∫ 1 0 h1ϕ̃dx+ ∫ 1 0 h2w̃dx+ c δ ∫ 1 0 h3θ̃dx+ d δ ∫ 1 0 h3P̃ dx+ r δ ∫ 1 0 h4P̃ dx+ d δ ∫ 1 0 h4θ̃dx. It is easy to verify that B(·, ·) and L(·) are continuous. Furthermore we have B((ϕ,w, θ, P ), (ϕ,w, θ, P )) = ρ1 ∫ 1 0 ϕ2dx+ ρ2 ∫ 1 0 w2dx+ k ∫ 1 0 (ϕx + w)2dx+ (γ1 + γ2) ∫ 1 0 P 2 xdx +α ∫ 1 0 w2 xdx+ c ∫ 1 0 θ2dx+ (σ1 + σ2) ∫ 1 0 θ2xdx+ 2d ∫ 1 0 θPdx+ r ∫ 1 0 P 2dx ≥ ∥(ϕ,w, θ, P )∥2H; thus we can derive B is coercive. By using the Lax-Milgram theorem, we infer that (2.11) has a unique solution ϕ ∈ H1 a(0, 1), w ∈ H1 b (0, 1), θ, P ∈ H2(0, 1) ∩H1 a(0, 1). By substituting ϕ, w, θ, P into (2.9), we obtain s ∈ H1 a(0, 1), z ∈ H1 b (0, 1), u,ϖ ∈ H2(0, 1) ∩H1 a(0, 1). For each θ̃ ∈ H2(0, 1) ∩H1 a(0, 1) and by taking (ϕ̃, w̃, P̃ ) = (0, 0, 0) ∈ H1 a(0, 1)×H1 b (0, 1)×H2(0, 1) ∩H1 a(0, 1), we can rewrite (2.11) as c ∫ 1 0 θθ̃dx+ (σ1 + σ2) ∫ 1 0 θxθ̃x dx+ d ∫ 1 0 P θ̃dx− r1 ∫ 1 0 wθ̃x dx = c δ ∫ 1 0 h3θ̃dx+ d δ ∫ 1 0 h4θ̃dx, (2.12) that is (σ1 + σ2)θxx = cθ + dP + r1wx − c δ h3 − d δ h4. (2.13) Moreover, (2.12) is also true for any ϑ ∈ C1([0, 1]), ϑ(0) = 0 instead of θ̃ ∈ H2(0, 1) ∩ H1 a(0, 1). Using integration by parts and (2.13) we have θx(1)ϑ̃(1) = 0. Noting that ϑ is arbitrary, we can obtain θx(1) = 0. Using the same method, we can obtain Px(1) = 0. Next, by taking (ϕ̃, θ̃, P̃ ) = (0, 0, 0) ∈ H1 a(0, 1)×H2(0, 1) ∩H1 a(0, 1)×H2(0, 1) ∩H1 a(0, 1), ∀w̃ ∈ H1 b (0, 1), we can rewrite (2.11) as ρ2 ∫ 1 0 ww̃dx+ k ∫ 1 0 (ϕx + w)w̃dx− α ∫ 1 0 wxw̃x dx+ r1 ∫ 1 0 θxw̃dx− r2 ∫ 1 0 Pxw̃dx = ∫ 1 0 h2w̃dx, (2.14) that is −αwxx = ρ2w + k(ϕx + w) + r1θx − r2Px − h2 ∈ L2(0, 1). 8 J. QIN, J. HAO EJDE-2025/77 Then we have w ∈ H2(0, 1). On the other hand, for each ξ ∈ C1[0, 1] with ξ(1) = 0, (2.12) is also true. Thus we obtain ρ2 ∫ 1 0 wξdx+ k ∫ 1 0 (ϕx + w)ξdx− α ∫ 1 0 wxxξdx+ r1 ∫ 1 0 θxξdx− r2 ∫ 1 0 Pxξdx = ∫ 1 0 h2ξdx. Integrating by parts, we have αwx(0)ξ(0) = 0, ∀ξ ∈ C1[0, 1]. Combining this with the arbitrariness of ξ, we can obtain wx(0) = 0; then w ∈ H2(0, 1) ∩H1 b (0, 1). Using the same method, we can obtain ϕ ∈ H2(0, 1) ∩H1 a(0, 1), ϕx(0) = 0. Hence, there exists a unique U ∈ D(A) such that (2.7) is satisfied, the operator I−A is surjective. And obviously, D(A) is dense on H. Therefore, according to the Lumer-Philips theorem[22], we can obtain the well-posedness result of the problem (2.5). □ 3. Exponential stability for ρ1 k = ρ2 α In this section, we assume that ρ1 k = ρ2 α holds, and use the energy perturbation method to prove that the solution of the system is exponential stable. To do this, we need to give the following estimations. Lemma 3.1. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). Then the energy functional E(t) := 1 2 ∫ 1 0 [ρ1ϕ 2 t + ρ2w 2 t + αw2 x + k(ϕx + w)2 + cθ2t + rP 2 t + σ1θ 2 x]dx + 1 2 ∫ 1 0 [γ1P 2 x + 2dθtPt]dx, (3.1) satisfies E′(t) = −σ2 ∫ 1 0 θ2xtdx− γ2 ∫ 1 0 P 2 xtdx ≤ 0. (3.2) Proof. Multiplying the equations in system (2.1) by ϕt, wt, θt, Pt, respectively, and integrating over (0, 1), using integration by parts and the boundary conditions, we obtain (3.1) and (3.2). □ Remark 3.2. Using (1.4), we have the estimate cθ2t + 2dPtθt + rP 2 t = 1 2 [ c(θt + d c Pt) 2 + r(Pt + d r θt) 2 + (c− d2 r )θ2t + (r − d2 c )P 2 t ] ≥ 1 2 ( c− d2 r ) θ2t + 1 2 ( r − d2 c ) P 2 t . (3.3) Then, we have E(t) ≥ 1 2 ∫ 1 0 [ ρ1ϕ 2 t + ρ2w 2 t + αw2 x + k(ϕx + w)2 + c1θ 2 t ] dx + 1 2 ∫ 1 0 [ c2P 2 t + σ1θ 2 x + γ1P 2 x ] dx ≥ 0, (3.4) where c1 = c− d2 r and c2 = c− d2 r . On the other hand, we have E(t) ≤ 1 2 ∫ 1 0 ρ1ϕ 2 t + αw2 x + ρ2w 2 t + k(ϕx + w)2 + (c+ d)θ2t + (r + d)P 2 t dx+ 1 2 ∫ 1 0 σ1θ 2 x + r1P 2 xdx. So, we obtain E(t) ∼ 1 2 ∫ 1 0 ρ1ϕ 2 t + αw2 x + ρ2w 2 t + k(ϕx + w)2 + θ2t + P 2 t + σ1θ 2 x + r1P 2 xdx. EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 9 Lemma 3.3. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). For any ε1 > 0, the functional F1(t) := ρ2 ∫ 1 0 wtw dx+ ρ1 ∫ 1 0 w ∫ x 0 ϕt(y, t) dy dx, satisfies F ′ 1(t) ≤ −α 2 ∫ 1 0 w2 xdx+ c ∫ 1 0 θ2xtdx+ c ∫ 1 0 P 2 xtdx+ c ( 1 + 1 ε1 ) ∫ 1 0 w2 t dx+ ε1 ∫ 1 0 ϕ2tdx. (3.5) Proof. By taking the derivative of F1(t), and using integration by parts and the first and second equations of (2.1), we can obtain F ′ 1(t) = ρ2 ∫ 1 0 wttw dx+ ρ2 ∫ 1 0 w2 t dx+ ρ1 ∫ 1 0 w ∫ x 0 ϕtt(y, t) dy dx+ ρ1 ∫ 1 0 wt ∫ x 0 ϕt(y, t) dy dx = −α ∫ 1 0 w2 xdx+ r1 ∫ 1 0 θxtw dx+ r2 ∫ 1 0 Pxtw dx+ ρ2 ∫ 1 0 w2 t dx + ρ1 ∫ 1 0 wt ∫ x 0 ϕt(y, t) dy dx. (3.6) Using Young’s inequality and Poincaré’s inequality, for any ε1 > 0 we have the estimates r1 ∫ 1 0 θxtw dx ≤ α 4 ∫ 1 0 w2 xdx+ c ∫ 1 0 θ2xtdx, r2 ∫ 1 0 Pxtw dx ≤ α 4 ∫ 1 0 w2 xdx+ c ∫ 1 0 P 2 xtdx, ρ1 ∫ 1 0 wt ∫ x 0 ϕt(y, t) dy dx ≤ ε1 ∫ 1 0 ϕ2tdx+ c ε1 ∫ 1 0 w2 t dx. Substituting the above estimates into (3.6), we obtain (3.5). □ Lemma 3.4. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). Then the functional F2(t) := −ρ2 ∫ 1 0 ϕtϕdx satisfies F ′ 2(t) ≤ −ρ1 ∫ 1 0 ϕ2tdx+ c ∫ 1 0 w2 xdx+ 2k ∫ 1 0 (ϕx + w)2dx. (3.7) Proof. We take the derivative of F2(t), and using (2.1)1 and integration by parts, we obtain F ′ 2(t) = −ρ1 ∫ 1 0 ϕ2tdx− k ∫ 1 0 (ϕx + w)x ϕdx = −ρ1 ∫ 1 0 ϕ2tdx+ k ∫ 1 0 (ϕx + w) 2 dx− k ∫ 1 0 (ϕx + w)w dx. (3.8) Using Young’s inequality and Poincaré’s inequality, for any ε2 > 0, we obtain k ∫ 1 0 (ϕx + w)w dx ≤ c ∫ 1 0 w2 xdx+ k ∫ 1 0 (ϕx + w)2dx. (3.9) Combining (3.8) and (3.9), we obtain (3.7). □ Lemma 3.5. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). The functional F3(t) := ρ1 k ∫ 1 0 ϕtwx dx+ ρ2 α ∫ 1 0 wt(ϕx + w) dx 10 J. QIN, J. HAO EJDE-2025/77 satisfies F ′ 3(t) ≤ − k 2α ∫ 1 0 (ϕx + w)2dx+ ρ2 α ∫ 1 0 w2 t dx+ c ∫ 1 0 θ2xtdx +c ∫ 1 0 P 2 xtdx+ (ρ2 α − ρ1 k )∫ 1 0 ϕtwxtdx. (3.10) Proof. Differentiating F3(t) with respect to t, using integration by parts and (2.1)1 and (2.1)2, we arrive at F ′ 3(t) = ∫ 1 0 (ϕx + w)xwx dx+ ρ1 k ∫ 1 0 ϕtwxtdx+ ∫ 1 0 (ϕx + w)wx dx + r1 α ∫ 1 0 θxt(ϕx + w)dx+ r2 α ∫ 1 0 Pxt(ϕx + w)dx − k α ∫ 1 0 (ϕx + w)2dx+ ρ2 α ∫ 1 0 wt(ϕx + w)tdx = − k α ∫ 1 0 (ϕx + w)2dx+ r1 α ∫ 1 0 θxt(ϕx + w)dx+ ρ2 α ∫ 1 0 w2 t dx + r2 α ∫ 1 0 Pxt(ϕx + w)dx+ (ρ1 k − ρ2 α ) ∫ 1 0 ϕtwxtdx. (3.11) Using Young’s inequality, we have r1 α ∫ 1 0 θxt(ϕx + w)dx ≤ k 4α ∫ 1 0 (ϕx + w)2dx+ c ∫ 1 0 θ2xtdx, (3.12) r2 α ∫ 1 0 Pxt(ϕx + w)dx ≤ k 4α ∫ 1 0 (ϕx + w)2dx+ c ∫ 1 0 P 2 xtdx. (3.13) Combining estimates (3.12) and (3.13) in (3.11), we obtain (3.10). □ Lemma 3.6. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). For any ε4 > 0, ε5 > 0, the functional F4(t) := ρ2c ∫ 1 0 wt ∫ 1 x θt(y, t) dy dx+ dρ2 ∫ 1 0 wt ∫ 1 x Pt(y, t) dy dx, satisfies F ′ 4(t) ≤ −r1ρ1 2 ∫ 1 0 w2 t + ε4 ∫ 1 0 w2 xdx+ ε5 ∫ 1 0 (ϕx + w)2dx+ c ∫ 1 0 θ2xdx +c ( 1 + 1 ε4 + 1 ε5 ) ∫ 1 0 θ2xtdx+ c ( 1 + 1 ε4 + 1 ε5 ) ∫ 1 0 P 2 xtdx. (3.14) EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 11 Proof. We take the derivative of F4(t) with respect to t, use (2.1)2 and (2.1)3, and integration by parts, thus we obtain F ′ 4(t) = cα ∫ 1 0 wxx ∫ 1 x θt(y, t) dy dx− ck ∫ 1 0 (ϕx + w) ∫ 1 x θt(y, t) dy dx +r1c ∫ 1 0 θxt ∫ 1 x θt(y, t) dy dx+ r2c ∫ 1 0 Pxt ∫ 1 x θt(y, t) dy dx −dρ2 ∫ 1 0 wt ∫ 1 x Ptt(y, t) dy dx+ σ1ρ2 ∫ 1 0 wt ∫ 1 x θxx(y, t) dy dx +σ2ρ2 ∫ 1 0 wt ∫ 1 x θxxt(y, t) dy dx+ r1ρ2 ∫ 1 0 wt ∫ 1 x wxt(y, t) dy dx +dα ∫ 1 0 wxx ∫ 1 x Pt(y, t) dy dx− dk ∫ 1 0 (ϕx + w) ∫ 1 x Pt(y, t) dy dx +dr1 ∫ 1 0 θxt ∫ 1 x Pt(y, t) dy dx+ dr2 ∫ 1 0 Pxt ∫ 1 x Pt(y, t) dy dx +dρ2 ∫ 1 0 wt ∫ 1 x Ptt(y, t) dy dx = −r1ρ2 ∫ 1 0 w2 t dx+ cα ∫ 1 0 θtwx dx− kc ∫ 1 0 (ϕx + w) ∫ 1 x θt(y, t) dy dx +cr1 ∫ 1 0 θ2t dx+ cr2 ∫ 1 0 θtPtdx− σ1ρ2 ∫ 1 0 θxwtdx− σ2ρ2 ∫ 1 0 θxtwtdx +dα ∫ 1 0 Ptwx dx− dk ∫ 1 0 (ϕx + w) ∫ 1 x Pt(y, t) dy dx +dr1 ∫ 1 0 θtPtdx+ dr2 ∫ 1 0 P 2 t dx. (3.15) Using Young’s inequality and Poincaré’s inequality, we have cα ∫ 1 0 θtwx dx ≤ ε4 2 ∫ 1 0 w2 xdx+ c ε4 ∫ 1 0 θ2xtdx, dα ∫ 1 0 Ptwx dx ≤ ε4 2 ∫ 1 0 w2 xdx+ c ε4 ∫ 1 0 P 2 xtdx, −kc ∫ 1 0 (ϕx + w) ∫ 1 x θt(y, t) dy dx ≤ ε5 2 ∫ 1 0 (ϕx + w) 2 dx+ c ε5 ∫ 1 0 θ2xtdx, −dk ∫ 1 0 (ϕx + w) ∫ 1 x Pt(y, t) dy dx ≤ ε5 2 ∫ 1 0 (ϕx + w) 2 dx+ c ε5 ∫ 1 0 P 2 xtdx, (dr1 + cr2) ∫ 1 0 Ptθtdx ≤ (dr1 + cr2)Cp 2 ∫ 1 0 θ2xtdx+ (dr1 + cr2)Cp 2 ∫ 1 0 P 2 xtdx, −σ1ρ2 ∫ 1 0 θxwtdx ≤ r1ρ2 4 ∫ 1 0 w2 t dx+ c ∫ 1 0 θ2xdx, −σ2ρ2 ∫ 1 0 θxtwtdx ≤ r1ρ2 4 ∫ 1 0 w2 t dx+ c ∫ 1 0 θ2xtdx. Based on the above estimates and equation (3.15), we can derive (3.14). □ Lemma 3.7. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). For any ε6 > 0, the functional F5(t) := c ∫ 1 0 θtθdx+ r1 ∫ 1 0 θxw dx+ σ2 2 ∫ 1 0 θ2xdx+ d ∫ 1 0 Ptθdx, 12 J. QIN, J. HAO EJDE-2025/77 satisfies F ′ 5(t) ≤ −σ1 ∫ 1 0 θ2xdx+ ε6 ∫ 1 0 w2 xdx+ c ( 1 + 1 ε6 ) ∫ 1 0 θ2xtdx+ c ∫ 1 0 P 2 xtdx. (3.16) Proof. Differentiating F5(t), using (2.1)3 and integrating by parts, we obtain F ′ 5(t) = c ∫ 1 0 θ2t dx+ c ∫ 1 0 θttθdx+ r1 ∫ 1 0 θxtw dx+ r1 ∫ 1 0 θxwtdx +σ2 ∫ 1 0 θxθxtdx+ d ∫ 1 0 Ptθtdx+ d ∫ 1 0 Pttθdx = −σ1 ∫ 1 0 θ2xdx+ c ∫ 1 0 θ2t dx+ r1 ∫ 1 0 θxtw dx+ d ∫ 1 0 Ptθtdx. (3.17) Using Young’s inequality and Poincaré’s inequality, we have r1 ∫ 1 0 θxtw dx ≤ ε6 ∫ 1 0 w2 xdx+ c ε6 ∫ 1 0 θ2xtdx, (3.18) d ∫ 1 0 θtPtdx ≤ cCp ∫ 1 0 θ2xtdx+ d2Cp 4c ∫ 1 0 P 2 xtdx. (3.19) Combining (3.18), (3.19) and (3.17), we obtain (3.16). □ Lemma 3.8. Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). For any ε7 > 0, the functional F6(t) := r ∫ 1 0 PtPdx+ r2 ∫ 1 0 Pxw dx+ γ2 2 ∫ 1 0 P 2 xdx+ d ∫ 1 0 θtPdx, satisfies F ′ 6(t)≤ −γ1 ∫ 1 0 P 2 xdx+ ε7 ∫ 1 0 w2 xdx+ c ( 1 + 1 ε7 )∫ 1 0 P 2 xtdx+ c ∫ 1 0 θ2xtdx. (3.20) Proof. Differentiating F6(t), using (2.1)4 and integrating by parts, we obtain F ′ 6(t) = r ∫ 1 0 P 2 t dx+ d ∫ 1 0 PttPdx+ γ2 ∫ 1 0 Pxtw dx+ r2 ∫ 1 0 Pxwtdx +γ2 ∫ 1 0 PxPxtdx+ d ∫ 1 0 Ptθtdx+ d ∫ 1 0 θttPdx = −γ1 ∫ 1 0 P 2 xdx+ r ∫ 1 0 P 2 t dx+ r2 ∫ 1 0 Pxtw dx+ d ∫ 1 0 Ptθtdx. (3.21) Using Young’s inequality and Poincaré’s inequality, we have r2 ∫ 1 0 Pxtw dx ≤ ε7 ∫ 1 0 w2 xdx+ c ε7 ∫ 1 0 P 2 xtdx, (3.22) d ∫ 1 0 θtPtdx ≤ rCp ∫ 1 0 P 2 xtdx+ d2Cp 4r ∫ 1 0 θ2xtdx. (3.23) Combining (3.22), (3.23) and (3.21), we obtain (3.20). □ Now we define a Lyapunov functional L(t) by L(t) := NE(t) +N1F1(t) + F2(t) + F3(t) +N4F4(t) +N5F5(t) + F6(t), where Ni, i = 1, 2, 4, 5 are positive constants to be chosen later. Lemma 3.9. Assume ρ1 k = ρ2 α . Let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). Then there exists positive constant c1, c2 and λ such that c1E(t) ≤ L(t) ≤ c2E(t), t > 0, (3.24) L′(t) ≤ −λE(t). (3.25) EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 13 Proof. According to the definition of L(t), and using Young’s, Poincaré’s and Cauchy-Schwarz’s inequalities and Remark 3.2, we obtain |L(t)−NE(t)| = |N1F1(t) +N2F2(t) +N3F3(t) +N4F4(t) +N5F5(t) + F6(t)| ≤ ρ2N1 ∫ 1 0 |wtw|dx+ ρ1N1 ∫ 1 0 |w| (∫ 1 0 |ϕt(y, t)|dy ) dx +ρ2 ∫ 1 0 |ϕϕt|dx+ N3ρ2 α ∫ 1 0 |wt||ϕx + w|dx+ d ∫ 1 0 |θPt|dx +N4ρ2c ∫ 1 0 |θt| (∫ 1 0 |wt(y, t)|dy ) dx+N4σ1ρ2 ∫ 1 0 |θxw|dx +N4ρ2d ∫ 1 0 |Pt| (∫ 1 0 |wt(y, t)|dy ) dx+ cρ2 ∫ 1 0 |θtθ|dx +r1 ∫ 1 0 |θxw|dx+ σ2 2 ∫ 1 0 θ2xdx+ d ∫ 1 0 |θPt|dx+ r ∫ 1 0 |PtP |dx +r2 ∫ 1 0 |Pxw|dx+ γ2 2 ∫ 1 0 P 2 xdx+ N3ρ1 k ∫ 1 0 |wx||ϕt(y, t)|dx ≤ m ∫ 1 0 [ ϕ2t + w2 t + w2 x + (ϕx + w)2 + θ2t + P 2 t + θ2x + P 2 x ] dx ≤ cE(t), (3.26) where m and c are different positive constants. Thus we have (N − c)E(t) ≤ L(t) ≤ (N + c)E(t). Let c1 = N − c, c2 = N + c, we obtain (3.24). Taking the derivative of L(t) with respect to t, using (3.2), (3.5), (3.7), (3.10), (3.14), (3.16) and (3.20), we have L′(t) ≤ − [ σ2N − cN1 − c ( 1 + 1 ε4 + 1 ε5 ) N4 − c ( 1 + 1 ε6 ) N5 − c ] ∫ 1 0 θ2xtdx − [ γ2N − cN1 − c ( 1 + 1 ε4 + 1 ε5 ) N4 − cN5 − c ] ∫ 1 0 P 2 xtdx − [α 2 N1 − ε4N4 − ε6N5 − c ] ∫ 1 0 w2 xdx − [ρ1 − ε1N1] ∫ 1 0 ϕ2tdx− [ k 2α N3 − ε5N4 − c] ∫ 1 0 (ϕx − w)2dx − [ r1ρ1 2 N4 − c(1 + 1 ε1 )N1 − cN3] ∫ 1 0 w2 t dx − [σ1N5 − cN4] ∫ 1 0 θ2xdx− σ1 ∫ 1 0 P 2 xdx. 14 J. QIN, J. HAO EJDE-2025/77 Choosing ε1 = ρ1 2N1 , ε4 = ε5 = c N4 , ε6 = c N5 , we have L′(t) ≤ − [ σ2N − cN1 − cN4 − cN2 4 − cN5 − cN2 5 − c ] ∫ 1 0 θ2xtdx − [γ2N − cN1 − cN4 − cN2 4 − cN5 − c] ∫ 1 0 P 2 xtdx− [ α 2 N1 − c] ∫ 1 0 w2 xdx − ρ1 2 ∫ 1 0 ϕ2tdx− [ k 2α N3 − c ] ∫ 1 0 (ϕx − w)2dx − [r1ρ1 2 N4 − cN1 − 2cN2 1 ρ1 − cN3 ] ∫ 1 0 w2 t dx − [σ1N5 − cN4] ∫ 1 0 θ2xdx− σ1 ∫ 1 0 P 2 xdx. (3.27) Taking N1 large enough such that α 2 N1 − c > 0, and taking N3 large enough such that k 2α N3 − c > 0. For fixed N1 and N3, choosing N4 large enough so that r1ρ1 2 N4 − cN1 − 2cN2 1 ρ1 − cN3 > 0. For fixed N4, choosing N5 large enough so that σ1N5 − cN4 > 0. For fixed N1, N3, N4 and N5, choosing N large enough so that σ2N − cN1 − cN4 − cN2 4 − cN5 − cN2 5 − c > 0, γ2N − cN1 − cN4 − cN2 4 − cN5 − c > 0. Thus we have L′(t) ≤ −c1 ∫ 1 0 ( θ2xt + P 2 xt + w2 x + ϕ2t + (ϕx + w)2 + w2 t + P 2 x + θ2x ) dx. (3.28) Combining (3.28) and Remark 3.2, we have L′(t) ≤ −λE(t), where λ = c1 c2 . □ Theorem 3.10. Assume ρ1 k = ρ2 α , and let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). Then there exist positive constants λ1 and λ2 such that the energy functional given by (3.1) satisfies E(t) ≤ λ2e −λ1t, t ≥ 0. (3.29) Proof. From (3.24) and (3.25), we have L′(t) ≤ −λE(t) ≤ − λ c2 L(t). (3.30) By integrating both sides of (3.30), we obtain L(t) ≤ L(0)e−λt/c2 . (3.31) Combing this and (3.24), we have c1E(t) ≤ L(t) ≤ L(0)e−λ/c2t ≤ c2E(0)e−λt/c2 . Let λ1 = − λ c2 and λ2 = c2 c1 E(0), we obtain (3.29). This completes the proof of Theorem 3.1. □ EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 15 4. Lack exponential stability when ρ1 k ̸= ρ2 α To show that system (2.1)-(2.3) lacks exponential stability when ρ1 k ̸= ρ2 α , we use the Gearhart- Herbst-Prüss-Huang theorem. Theorem 4.1 ([13, 23, 15]). Let S(t) = eAt be a C0-semigroup of contraction on Hilbert space H. Then S(t) is exponentially stable if and only if ρ(A) ⊃ {iλ : λ ∈ R} ≡ iR and lim λ→∞ ∥(iλ−A)−1∥L(H) <∞. Theorem 4.2. Assume ρ1 k ̸= ρ2 α , and let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). Then the semigroup associated with (2.1)-(2.3) is not exponentially stable. Proof. To show the system lacks exponential stability, we need to show that there is a real number sequence {λµ}µ∈N and vector function sequence {Fµ}µ∈N with Fµ = (f1µ, f 2 µ, f 3 µ, f 4 µ, f 5 µ, f 6 µ, f 7 µ, f 8 µ) T ∈ H, with ∥Fµ∥H ≤ 1 such that ∥(iλµI −A)−1Fµ)∥L(H) → ∞, where iλµUµ −AUµ = Fµ, (4.1) with Uµ = (ϕµ, wµ, θµ, Pµ, sµ, zµ, uµ, ϖµ) is not bounded. For the sake of convenience, we omit the subscript µ of the components for Uµ and Fµ in the following of this section. Rewriting (4.1) in terms of its components, we obtain iλϕ− s = f1, iλw − z = f2, iλθ − u = f3, iλP −ϖ = f4, iλρ1s− kϕxx − kwx = ρ1f 5, iλρ2z − αwxx + kϕx + kw − r1ux − r2ϖx = ρ2f 6, iλδu− rσ1θxx − rσ2uxx + dγ1Pxx + dγ2ϖxx − (r1r − dr2)zx = δf7, iλδϖ − cγ1Pxx − cγ2ϖxx + dσ1θxx + dσ2uxx − (cr2 − dr1)zx = δf8. (4.2) Taking f1 = f2 = f3 = f4 = 0, system (4.2) can be rewritten as −λ2ρ1ϕ− kϕxx − kwx = ρ1f 5, −λ2ρ2w − αwxx + kw + kϕx − iλr1θx − iλr2Px = ρ2f 6, −λ2δθ − rσ1θxx − iλrσ2θxx + dγ1Pxx + iλdγ2Pxx − iλ(r1r − dr2)wx = δf7, −λ2δP − cγ1Pxx − iλcγ2Pxx + dσ1θxx + iλdσ2θxx − iλ(cr2 − dr1)wx = δf8. (4.3) Because of the boundary conditions is given by (2.2), we assume that w = Aµ cos (2µ+ 1)π 2 x, ϕ = Bµ sin (2µ+ 1)π 2 x, θ = Cµ sin (2µ+ 1)π 2 x, P = Dµ sin (2µ+ 1)π 2 x. Next, we choose f6 = f7 = f8 = 0, f5 = 1 ρ1 sin (2µ+ 1)π 2 x. 16 J. QIN, J. HAO EJDE-2025/77 Inserting w, ϕ, θ, P and f i, (i = 5, 6, 7, 8) into (4.3), we arrive at[ λ2ρ1 + k ( (2µ+ 1)π 2 )2] Bµ + (2µ+ 1)π 2 Aµ = 1,[ − λ2ρ2 + α ( (2µ+ 1)π 2 )2 + k ] Aµ + k (2µ+ 1)π 2 Bµ − iλr1 (2µ+ 1)π 2 Cµ − iλr2 (2µ+ 1)π 2 Dµ = 0, NµCµ − TµDµ + iλ (rr1 − dr2) (2µ+ 1)π 2 Aµ = 0, MµDµ − EµCµ + iλ (cr2 − dr1) (2µ+ 1)π 2 Aµ = 0, (4.4) where Nµ = −λ2δ + rσ1 ( (2µ+ 1)π 2 )2 + iλrσ2 ( (2µ+ 1)π 2 )2 , Tµ = iλdγ2 ( (2µ+ 1)π 2 )2 + dγ1 ( (2µ+ 1)π 2 )2 , Mµ = −λ2δ + cγ1 ( (2µ+ 1)π 2 )2 + iλcγ2 ( (2µ+ 1)π 2 )2 , Eµ = iλdσ2 ( (2µ+ 1)π 2 )2 + dσ1 ( (2µ+ 1)π 2 )2 . Let λ = √ k ρ1 ( (2µ+1)π 2 ) . Then from (4.4)1, we obtain Aµ = 2 (2µ+ 1)π . Combining (4.4)3 and (4.4)4, we obtain Dµ = − iλ (2µ+1)π 2 [(rr1 − dr2)Eµ + (cr2 − dr1)Nµ] MµNµ − TµEµ Aµ, Cµ = − iλ (2µ+1)π 2 [(rr1 − dr2)Mµ + (cr2 − dr1)Tµ] MµNµ − TµEµ Aµ. As µ→ ∞, we have MµNµ − TµEµ → −λ2[cr − d2]γ2σ2 ( (2µ+ 1)π 2 )2 , [(rr1 − dr2)Eµ + (cr2 − dr1)Nµ] → iλ [ cr − d2 ] r2σ2 ( (2µ+ 1)π 2 )2 , [(rr1 − dr2)Mµ + (cr2 − dr1)Tµ] → iλ [ cr − d2 ] r1γ2 ( (2µ+ 1)π 2 )2 . Then, we have Cµ → − r1 σ2 , Dµ → − r2 γ2 , µ→ ∞. Consequently, ∥U∥H≥ σ1 2 ∫ 1 0 θ2xdx = σ1 2 ∫ 1 0 ( (2µ+ 1)π 2 Cµ cos (2µ+ 1)π 2 x)2dx = σ1 2 (Cµ (2µ+ 1)π 2 )2 ∫ 1 0 (cos (2µ+ 1)π 2 x)2dx = σ1 4 ( (2µ+ 1)π 2 Cµ) 2 → ∞, µ→ ∞. From the above, it is easy to see that ∥U∥H → ∞ as µ → ∞, which is the desired conclusion. This completes the proof. □ EJDE-2025/77 THERMODIFFUSION TIMOSHENKO SYSTEMS 17 5. Polynomial stability when ρ1 k ̸= ρ2 α In this section, we construct a high order energy to show that system (2.1)-(2.3) is polynomial stable, with decay rate t−1. Theorem 5.1. Assume ρ1 k ̸= ρ2 α , and let (ϕ,w, θ, P ) be the solution of system (2.1)-(2.3). Then there exists positive constant λ3 such that the energy functional given by (3.1) satisfies E(t) ≤ λ3 t , ∀t > 0. (5.1) Proof. Firstly, we define a second order energy functional E2(t) = 1 2 ∫ 1 0 [ρ1ϕ 2 tt + ρ2w 2 tt + αw2 xt + k(ϕx + w)2t + cθ2tt + rP 2 t + σ1θ 2 xt]dx + 1 2 ∫ 1 0 [γ1P 2 xt + 2dθttPtt]dx, (5.2) so that E′ 2(t) = −σ2 ∫ 1 0 θ2xttdx− γ2 ∫ 1 0 P 2 xttdx. (5.3) Since ρ1 k ̸= ρ2 α , the last term in (3.11) can be handled as follows. From (2.1)3, we obtain∫ 1 0 ϕtwxtdx = 1 r1 ∫ 1 0 (cθtt + dPtt − σ1θxx − σ2θxxt)ϕtdx = r r1 ∫ 1 0 θttϕtdx+ d r1 ∫ 1 0 Pttϕtdx− σ1 r1 ∫ 1 0 θxtϕx dx− σ2 r1 ∫ 1 0 θxttϕx dx + d dt ∫ 1 0 (σ1 r1 θxϕx + σ2 r1 ϕxθxt ) dx. (5.4) By Young’s inequality and Poincaré’s inequality, we have(ρ1 k − ρ2 α )∫ 1 0 ϕtwxtdx ≤ (ρ1 k − ρ2 α ) d dt ∫ 1 0 (σ1 r1 θxϕx + σ2 r1 ϕxθxt ) dx +ε7 ∫ 1 0 ( ϕ2t + ϕ2x ) dx+ c ε7 ∫ 1 0 ( θ2xtt + θ2xt + P 2 xtt ) dx, (5.5) where we use that ϕ2x = (ϕx − w + w) 2 ≤ 2(ϕx + w)2 + 2w2. Now we define the Lyapunov functional F (t) := N(E(t) + E2(t)) +N1(t)I1(t) +N2I2(t) +N3I3(t) + I4(t) + I5(t) +N6I6(t) + (ρ2 α − ρ1 k )∫ 1 0 (σ1 r1 θxϕx + σ2 r1 ϕxθxt ) dx. Obviously, (N − c)E(t) +NE2(t) ≤ F (t) ≤ (N + c)E(t) +NE2(t). Thus, we can infer that there exist positive constants m1 and m2 such that m1(E(t) + E2(t)) ≤ F (t) ≤ m2(E(t) + E2(t)). (5.6) From (3.25) and (5.5), we obtain F ′(t) ≤ −cE(t) + ε7 ∫ 1 0 ( ϕ2t + ϕ2x ) dx− [N − c ε7 ] ∫ 1 0 θ2xttdx − [N − c ε7 ] ∫ 1 0 P 2 xttdx { − σ2N − cN1 − c ( 1 + 1 ε4 + 1 ε5 ) N4 − c ( 1 + 1 ε6 ) N5 − c ε7 − c }∫ 1 0 θ2xtdx. 18 J. QIN, J. HAO EJDE-2025/77 Then we choose ε7 small enough, take N large enough such that F (t) is positive and σ2N +N − cN1 − c ( 1 + 1 ε4 + 1 ε5 ) N4 − c ( 1 + 1 ε6 ) N5 − c ε7 − c > 0. Thus depending on the above constants, we have F ′(t) ≤ − c 2 E(t), in which c is some constant. Integrating over (0, t), we obtain∫ t 0 E(t)dt ≤ 1 c (F (0)− F (t)) ≤ 1 c F (0) ≤ m2 c (E(0) + E2(0)). (5.5) Since (tE(t))′ = E(t) + tE′(t) ≤ E(t), with (5.5), we derive that tE(t) ≤ ∫ t 0 E(t)dt ≤ m2 c1 (E(0) + E2(0)). Furthermore we have E(t) ≤ λ3 t , where λ3 = m2 c1 (E(0) + E2(0)). Thus the proof of Theorem 5.1 is complete. □ 6. Couclusions In this paper, we focus on the stability of a thermodiffusion Timoshenko system of type III. By using the energy perturbation method, we successfully prove that there is a close dependence between the stability of the system and the wave speed. Specifically, when the wave speed are equal, the system is exponentially stable. However, when the wave speed are not equal, the system only has polynomial stability. In addition, we also prove that the system does lack exponential stability when the wave speed are not equal. These results indicate that the wave speed has a significant impact on the stability of the system, and the transmission performance of the system is better when the wave speeds are equal. This finding has important implications for engineering applications, particularly in the design of high-performance aerospace and microelectronic structures, where precise control of thermal diffusion and stable system operation are critical. It is worth mentioning that our results extend and improve the previous relevant research results. At present, whether the decay rate of this polynomial has reached an optimal state remains an open problem yet to be solved. Future research could focus on investigating system stability under various boundary conditions. Additionally, the model could be extended to higher dimensions to better represent the complexity and diversity of real-world engineering scenarios. Acknowledgments. This work was supported by the National Natural Science Foundation of China (12271315), by the special fund for Science and Technology Innovation Teams of Shanxi Province (202204051002015), and by the Fundamental Research Program of Shanxi Province (202203021221018). 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Phys., 2023, 74(3): 14pp. Jiali Qin School of Mathematics and Statistics, Shanxi University, Taiyuan, Shanxi 030006, China Email address: 1953543775@qq.com Jianghao Hao Key Laboratory of Complex Systems and Data Science of Ministry of Education, Shanxi University, Taiyuan, Shanxi 030006, China Email address: hjhao@sxu.edu.cn 1. Introduction 2. Well-posedness of the system 3. Exponential stability for 1k=2 4. Lack exponential stability when 1k=2 5. Polynomial stability when 1k=2 6. Couclusions Acknowledgments References