A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective
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Archiv der Mathematik
A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective ´nior, A.J.A. Ramos , M. Aouadi, D.S. Almeida Ju ´jo M.M. Freitas, and M.L. Arau
Abstract. In this work, we analyze a truncated version for the Timoshenko beam model with thermal and mass diffusion effects derived by Aouadi et al. (Z Angew Math Phys 70:117, 2019). In particular, we study some issues related to the second spectrum of frequency according to a procedure due to Elishakoff (in: Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, 2010). In Aouadi et al. (2019), the lack of exponential stability for the classical Timoshenko beam with thermodiffusion effects without assuming the nonphysical condition of equal wave speeds has be proved. By using the classical Faedo–Galerkin method combined with the a priori estimates, we prove the existence and uniqueness of a global solution of the truncated version of this problem. Then we prove that this solution is exponentially stable without assuming the condition of equal wave speeds. Mathematics Subject Classification. Primary 74K10; Secondary 37N15, 74F05. Keywords. Timoshenko, Thermo-diffusion, Well-posedness, Exponential stability.
1. Introduction. Recently, Aouadi et al. [5] introduced a new Timoshenko beam model with thermal and mass diffusion effects given by ρ1 ϕtt − κ(ϕx + ψ)x = 0
in
]0, L[×]0, ∞[, (1.1)
ρ2 ψtt − αψxx + κ(ϕx + ψ) − γ1 θx − γ2 Px = 0 cθt + dPt − Kθxx − γ1 ψxt = 0
in in
]0, L[×]0, ∞[, (1.2) ]0, L[×]0, ∞[, (1.3)
dθt + rPt − Pxx − γ2 ψxt = 0
in
]0, L[×]0, ∞[, (1.4)
Arch. Math.
A.J.A. Ramos et al.
where ϕ is the transverse displacement, ψ is the rotation of the neutral axis due to bending, θ is the temperature, and P is the chemical potential. The constants ρ1 , ρ2 , κ, α, γ1 , γ2 , c, r, d, , and K are physical positive parameters. They showed, without assuming the well-known equal wave speeds condition χ := κ/ρ1 − b/ρ2 = 0, the lack of exponential stability for the problem. Based on [5] and the recent studies due to Almeida J´ unior et al. [1–4], we consider the truncated version given by ρ1 ϕtt − κ(ϕx + ψ)x = 0 in
]0, L[×]0, ∞[,
(1.5)
−ρ2 ϕxtt − αψxx + κ(ϕx + ψ) − γ1 θx − γ2 Px = 0 in cθt + dPt − Kθxx − γ1 ψxt = 0 in
]0, L[×]0, ∞[, ]0, L[×]0, ∞[,
(1.6) (1.7)
dθt + rPt − Pxx − γ2 ψxt = 0 in
]0, L[×]0, ∞[,
(1.8)
with the initial conditions ϕ(x, 0) = ϕ0 (x), ϕt (x, 0) = ϕ1 (x), ϕtt (x, 0) = ϕ2 (x),
x ∈ (0, L), x ∈ (0, L), (1.9)
ψ(x, 0) = ψ0 (x), θ(x, 0) = θ0 (x), P (x, 0) = P0 (x), and boundary conditions of Dirichlet-Neumann-type
ϕ(0, t) = ϕ(L, t) = ψx (0, t) = ψx (L, t) = 0, t ≥ 0 θ(0, t) = θ(L, t) = P (0, t) = P (L, t) = 0, t ≥ 0.
(1.10)
The truncated version (1.5)–(1.10) is obtained by following the procedure of Elishakoff [7] which involves replacing the term ψtt in (1.2) by −ϕxtt based on d’Alembert’s principle for dynamic equilibrium. This eliminates the second spectrum of frequency and its dama
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