Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 68, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS WITH DELAY ON THE BOUNDARY CONTROL HAIDAR BADAWI, MOHAMMAD AKIL, ZAYD HAJJEJ Abstract. In this article, we consider the Kirchhoff plate equation with delay terms on the boundary control. We give instability examples of systems for some choices of delays. Finally, we prove its well-posedness, strong stability, and exponential stability under a multiplier geometric control condition. 1. Introduction Let Ω ⊂ R2 be a bounded open set with boundary Γ of class C4 consisting of a clamped part Γ0 6= ∅ and a rimmed part Γ1 6= ∅ such that Γ0∩Γ1 = ∅. We consider the Kirchhoff plate equation with delay terms on the boundary controls, ϕtt(x, t) + ∆2ϕ(x, t) = 0 in Ω× (0,∞), ϕ(x, t) = ∂νϕ(x, t) = 0 on Γ0 × (0,∞), B1ϕ(x, t) = −β1∂νϕt(x, t)− β2∂νϕt(x, t− τ1) on Γ1 × (0,∞), B2ϕ(x, t) = γ1ϕt(x, t) + γ2ϕt(x, t− τ2) on Γ1 × (0,∞), ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x) in Ω, ϕt(x, t) = f0(x, t) on Γ1 × (−τ1, 0), ∂νϕt(x, t) = g0(x, t) on Γ1 × (−τ2, 0). (1.1) Here and below, β1, γ1, τ1, and τ2 are positive real numbers, β2 and γ2 are non- zero real numbers, ν = (ν1, ν2) is the unit outward normal vector along Γ, and τ = (−ν2, ν1) is the unit tangent vector along Γ. The constant 0 < µ < 1/2 is the Poisson coefficient and the boundary operators B1 and B2 are defined, respectively, by B1ϕ = ∆ϕ+ (1− µ)C1ϕ, B2ϕ = ∂ν∆ϕ+ (1− µ)∂τC2ϕ, where C1ϕ = 2ν1ν2ϕx1x2 − ν2 1ϕx2x2 − ν2 2ϕx1x1 , C2ϕ = (ν2 1 − ν2 2)ϕx1x2 − ν1ν2 (ϕx1x1 − ϕx2x2 ) . 2020 Mathematics Subject Classification. 74K20, 47D60, 90C31, 93D15, 93D20, 35B35, 35B40, 93C80, 93C43. Key words and phrases. Kirchhoff plate equation; boundary control; time delay; instability examples; strong stability; exponential stability. ©2023. This work is licensed under a CC BY 4.0 license. Submitted April 18, 2023. Published October 16, 2023. 1 2 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 Moreover, easy computations shows that C1ϕ = −∂2 τϕ− ∂τν2ϕx1 + ∂τν1ϕx2 , C2ϕ = ∂ντϕ− ∂τν1ϕϕx1 − ∂τν2ϕx2 . (1.2) To reformulate system (1.1), as in [15], we introduce the auxiliary variables η1(x, ρ, t) := ∂νut(x, t− ρτ1), x ∈ Γ1, ρ ∈ (0, 1), t > 0, η2(x, ρ, t) := ut(x, t− ρτ2), x ∈ Γ1, ρ ∈ (0, 1), t > 0. (1.3) Then, system (1.1) becomes ϕtt + ∆2ϕ = 0 in Ω× (0,∞), (1.4) ϕ = ∂νϕ = 0 on Γ0 × (0,∞), (1.5) B1ϕ+ β1∂νϕt + β2η 1(·, 1, t) = 0 on Γ1 × (0,∞), (1.6) B2ϕ− γ1ϕt − γ2η 2(·, 1, t) = 0 on Γ1 × (0,∞), (1.7) τ1η 1 t (·, ρ, t) + η1 ρ(·, ρ, t) = 0 on Γ1 × (0, 1)× (0,∞), (1.8) τ2η 2 t (·, ρ, t) + η2 ρ(·, ρ, t) = 0 on Γ1 × (0, 1)× (0,∞), (1.9) with the following initial conditions ϕ(·, 0) = ϕ0(·), ϕt(·, 0) = ϕ1(·) in Ω, η1(·, ρ, 0) = f0(·,−ρτ1) on Γ1 × (0, 1), η2(·, ρ, 0) = g0(·,−ρτ2) on Γ1 × (0, 1). (1.10) The energy of system (1.4)-(1.10) is E(t) = 1 2 { a(ϕ,ϕ) + ∫ Ω |ϕt|2dx+ τ1|β2| ∫ Γ1 ∫ 1 0 |η1(·, ρ, t)|2 dρ dΓ + τ2|γ2| ∫ Γ1 ∫ 1 0 |η2(·, ρ, t)|2 dρ dΓ } , (1.11) where the sesquilinear form a : H2(Ω)×H2(Ω) 7→ C is defined by a(ϕ,ψ) = ∫ Ω [ ϕx1x1 ψx1x1 + ϕx2x2 ψx2x2 + µ ( ϕx1x1 ψx2x2 + ϕx2x2 ψx1x1 ) + 2(1− µ)ϕx1x2 ψx1x2 ] dx. (1.12) We first recall the following Green’s formula (see [12]), a(ϕ,ψ) = ∫ Ω ∆2ϕψdx+ ∫ Γ ( B1ϕ∂νψ − B2ϕψ ) dΓ, (1.13) for all ϕ ∈ H4(Ω), ψ ∈ H2(Ω). For further purposes, we need a weaker version of it. As D(Ω) is dense in E(∆2, L2(Ω)) := {ϕ ∈ H2(Ω) : ∆2ϕ ∈ L2(Ω)} equipped with its natural norm, we deduce that ϕ ∈ E(∆2, L2(Ω)) (see [14, Theorem 5.6]) satisfies B1ϕ ∈ H−1/2(Γ) and B2ϕ ∈ H−3/2(Γ) with a(ϕ,ψ) = ∫ Ω ∆2ϕψdx+ 〈B1ϕ, ∂νψ〉H−1/2(Γ),H1/2(Γ) − 〈B2ϕ,ψ〉H−3/2(Γ),H3/2(Γ), ∀ψ ∈ H2(Ω). (1.14) EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 3 Similar to [1], for any regular solution Φ = (ϕ,ϕt, η 1, η2) of system (1.4)-(1.10), the energy E(t) satisfies the estimate d dt E(t) ≤ −(β1 − |β2|) ∫ Γ1 |∂νϕt|2 dΓ− (γ1 − |γ2|) ∫ Γ1 |ϕt|2 dΓ. (1.15) Let us recall some previous works that are related to our problem. Recently, in [6], the authors considered a Kirchhoff-type parabolic problem on a geodesic ball of hyperbolic space, they derived the growth rate of the blow-up solution and the decay rate of the global solution. The stabilization of the Kirchhoff plate equation with non-linear boundary controls was addressed by Rao in [18] (in the linear case, it corresponds to the system (1.1) with β2 = γ2 = 0). He proved that the energy of solutions decays exponentially if the multiplier geometric control condition is met. Time delays can appea in a variety of applications, including physics, chemistry, biology, and thermal phenomena, and they might depend both on the current state and on past occurrences (see [8, 11]). Since time delays frequently cause instabil- ities, scientists have recently become interested in controlling partial differential equations with time delays (see [3, 4, 5, 7]). Nicaise and Pignotti [15] examined the multidimensional wave equation with boundary feedback and a delay term at the boundary, by considering the system ztt(x, t)−∆z(x, t) = 0 in Ω× (0,∞), z(x, t) = 0 on ΓD × (0,∞), ∂z ∂ν (x, t) = −µ1zt(x, t)− µ2zt(x, t− τ) on ΓN × (0,∞), z(x, 0) = z0(x), zt(x, 0) = z1(x) in Ω, zt(x, t) = f0(x, t) on ΓN × (−τ, 0), (1.16) where µ1 and µ2 are positive real numbers, and Ω is an open bounded domain of Rn with a boundary Γ of class C2 with Γ = ΓD ∪ ΓN , such that ΓD ∩ ΓN = ∅. An exponential decay is established when µ2 < µ1. If this later is false, they found a sequence of delays {τk}k, τk → 0, for which the corresponding solutions have increased energy. To the best of our knowledge, there are no results concerning the case of the Kirchhoff plate equation with boundary controls and time delay. We fill this gap, by examining both instability and stability of system (1.1). The outline of this article is as follows. In section 2, we give some instability examples of system (1.1) for some particular choices of delays, when |β2| ≥ β1 and |γ2| ≥ γ1. In subsection 3.1 , we prove the well-posedness of our system. The subsection 3.2 is devoted to establish the strong stability of our system by following a general criteria of Arendt and Batty. Finally, in subsection 3.3, under condition (MGC), we show that system (1.1) is exponentially stable. We complete the introduction by introducing some notation. The usual norm and semi-norm of the Sobolev space Hs(Ω) (s > 0) are denoted by ‖ · ‖Hs(Ω) and | · |Hs(Ω), respectively. By A . B, we mean that there exists a constant C > 0 independent of A and B such that A ≤ CB. 2. Instability results In this section, we give some instability examples of system (1.1) in the cases |β2| ≥ β1 and |γ2| ≥ γ1. This is achieved by distinguishing between the following 4 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 cases: |β2| = β1 and |γ2| = γ1, (2.1) |β2| ≥ β1 and |γ2| ≥ γ1 and |β2| − β1 + |γ2| − γ1 > 0. (2.2) Theorem 2.1. If (2.1) or (2.2) hold, then there exist sequences of delays and solutions of (1.1) corresponding to these delays such that their standard energy is constant. Proof. We seek for a solution of system (1.1) of the form ϕ(x, t) = eiλtu(x), with λ 6= 0. (2.3) Inserting (2.3) in (1.1), we obtain −λ2u+ ∆2u = 0 in Ω, u = ∂νu = 0 on Γ0, B1u = −iλ(β1 + β2e −iλτ1)∂νu on Γ1, B2u = iλ(γ1 + γ2e −iλτ2)u on Γ1. (2.4) Let g ∈ H2 Γ0 (Ω). Multiplying the first equation in (2.4) by g, then using Green’s formula, we obtain − λ2 ∫ Ω ug dx+ a(u, g) + iλ(β1 + β2e −iλτ1) ∫ Γ1 ∂νu∂νg dΓ + iλ(γ1 + γ2e −iλτ2) ∫ Γ1 ug dΓ = 0, (2.5) for all g ∈ H2 Γ0 (Ω). Now, since |β2| ≥ β1 and |γ2| ≥ γ1, then we assume that cos(λτ1) = −β1 β2 and cos(λτ2) = −γ1 γ2 . (2.6) Thus, we choose β2 sin(λτ1) = √ β2 2 − β2 1 and γ2 sin(λτ2) = √ γ2 2 − γ2 1 . (2.7) Inserting (2.6) and (2.7) in (2.5), we obtain − λ2 ∫ Ω ugdx+ a(u, g) + λ √ β2 2 − β2 1 ∫ Γ1 ∂νu∂νg dΓ + λ √ γ2 2 − γ2 1 ∫ Γ1 ug dΓ = 0, (2.8) for all g ∈ H2 Γ0 (Ω). Now, taking g = u in (2.8), we obtain − λ2 ∫ Ω |u|2dx+ a(u, u) + λ √ β2 2 − β2 1 ∫ Γ1 |∂νu|2 dΓ + λ √ γ2 2 − γ2 1 ∫ Γ1 |u|2 dΓ = 0. (2.9) Without loss of generality, we can assume that ‖u‖L2(Ω) = 1. (2.10) Thus, from (2.9) and (2.10), we obtain λ2 − a(u, u)− λ √ β2 2 − β2 1qν(u)− λ √ γ2 2 − γ2 1q(u) = 0, (2.11) EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 5 where q(u) = ∫ Γ1 |u|2 dΓ and qν(u) = ∫ Γ1 |∂νu|2 dΓ. (2.12) We define Z := {z ∈ H2 Γ0 (Ω) : ‖z‖L2(Ω) = 1}. Now, we distinguish two cases. Case 1: If (2.1), then from (2.11), we have a(u, u) = λ2. (2.13) Let us define λ2 := min z∈Z a(z, z). (2.14) Now, if u satisfies a(u, u) = minz∈Z a(z, z). Then it easy to see that u is a solution of (2.5) and consequently (2.3) is a solution of (1.1). Moreover, from (2.3) and (1.11), we obtain E(t) = E(0) ≥ a(u, u) + λ2 ∫ Ω |u|2dx = 2λ2 > 0, ∀t ≥ 0. Thus, the energy of (1.1) is constant and positive. Further from our assumptions cos(λτ1) = −1, sin(λτ1) = 0, cos(λτ2) = −1, sin(λτ2) = 0, system (2.4) becomes −λ2u+ ∆2u = 0 in Ω, u = ∂νu = 0 on Γ0, B1u = 0 on Γ1, B2u = 0 on Γ1. (2.15) So, we can take a sequence (λn)n of positive real numbers defined by λ2 n = Λ2 n, n ∈ N, where Λ2 n, n ∈ N, are the eigenvalues for the bi-Laplacian operator with the bound- ary conditions (2.15)2-(2.15)4. Then, setting λnτ1 = (2k + 1)π, k ∈ N and λnτ2 = (2l + 1)π, l ∈ N, we obtain the sequences of delays τ1,n,k = (2k + 1)π λn , k, n ∈ N and τ2,n,l = (2l + 1)π λn , l, n ∈ N, which becomes arbitrarily small (or large) for suitable choices of the indices n, k, l ∈ N. Therefore, we have found sets of time delays for which system (1.1) is not asymptotically stable. Case 2: If (2.2) holds, then from (2.11), we have λ = 1 2 [√ β2 2 − β2 1qν(u) + √ γ2 2 − γ2 1q(u) ± √(√ β2 2 − β2 1qν(u) + √ γ2 2 − γ2 1q(u) )2 + 4a(u, u) ] . (2.16) 6 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 Let us define λ := 1 2 min z∈Z {√ β2 2 − β2 1qν(z) + √ γ2 2 − γ2 1q(z) + √(√ β2 2 − β2 1qν(z) + √ γ2 2 − γ2 1q(z) )2 + 4a(z, z) } . (2.17) Let us prove that if the minimum in the right-hand side of (2.17) is attained at u, that is √ β2 2 − β2 1qν(u) + √ γ2 2 − γ2 1q(u) + √(√ β2 2 − β2 1qν(u) + √ γ2 2 − γ2 1q(u) )2 + 4a(u, u) := min z∈Z {√ β2 2 − β2 1qν(z) + √ γ2 2 − γ2 1q(z) + √(√ β2 2 − β2 1qν(z) + √ γ2 2 − γ2 1q(z) )2 + 4a(z, z) } , (2.18) then u is a solution of (2.8). For this aim, take for ε ∈ R as z = u+ εg (2.19) with g ∈ H2 Γ0 (Ω) such that ∫ Ω ugdx = 0. Thus, we have ‖z‖2L2(Ω) = ‖u‖2L2(Ω) + ε2‖g‖2L2(Ω) = 1 + ε2‖g‖2L2(Ω). (2.20) Now, if we define f(ε) := 1 1 + ε2‖g‖2L2(Ω) (√ β2 2 − β2 1qν(u+ εg) + √ γ2 2 − γ2 1q(u+ εg) + √(√ β2 2 − β2 1qν(u+ εg) + √ γ2 2 − γ2 1q(u+ εg) )2 + 4a(u+ εg, u+ εg) ) ; thus, from (2.18), we obtain f(ε) ≥ f(0) = √ β2 2 − β2 1qν(u) + √ γ2 2 − γ2 1q(u) + √(√ β2 2 − β2 1qν(u) + √ γ2 2 − γ2 1q(u) )2 + 4a(u, u), which gives f ′(0) = 0. Consequently, after an easy computation, we obtain a(u, g) + λ √ β2 2 − β2 1 ∫ Γ1 ∂νu∂νg dΓ + λ √ γ2 2 − γ2 1 ∫ Γ1 ug dΓ = 0. (2.21) Since any function g̃ ∈ H2 Γ0 (Ω) can be decomposed as g̃ = αu+ g with α ∈ R and g ∈ H2 Γ0 (Ω) such that ∫ Ω ug dx = 0, from (2.21) and (2.9), we obtain that u satisfies (2.8). Thus, for such λ > 0, λτ1 = arccos ( − β1 β2 ) + 2kπ, k ∈ N and λτ2 = arccos ( − γ1 γ2 ) + 2lπ, l ∈ N, define a sequences of time delays for which (1.1) is not asymptotically stable. � EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 7 3. Stability results In this section, we will prove the wellposedness, strong stability and exponential stability of system (1.4)-(1.10). For this aim, we make the following assumptions β1, γ1 > 0, β2, γ2 ∈ R∗, |β2| < β1, |γ2| < γ1. (3.1) 3.1. Wellposedness of system (1.4)-(1.10). Under hypothesis (3.1), (1.15), sys- tem (1.4)-(1.10) is dissipative in the sense that its energy is non-increasing with respect to time (i.e. E′(t) ≤ 0). Let us define the Hilbert space H = H2 Γ0 (Ω)× L2(Ω)× ( L2(Γ1 × (0, 1)) )2 , where H2 Γ0 (Ω) = {f ∈ H2(Ω) : f = ∂ν f = 0 on Γ0}. This Hilbert space equipped with the inner product (Φ,Φ1)H = a(ϕ,ϕ1) + ∫ Ω ψψ1 dx+ τ1|β2| ∫ Γ1 ∫ 1 0 η1η1 1 dρ dΓ + τ2|γ2| ∫ Γ1 ∫ 1 0 η2η2 1 dρ dΓ, (3.2) where Φ = (ϕ,ψ, η1, η2)>, Φ1 = (ϕ1, ψ1, η 1 1 , η 2 1)> ∈ H. We define the linear un- bounded operator A : D(A) ⊂ H 7→ H by D(A) = { Φ = (ϕ,ψ, η1, η2)> ∈ DΓ0 (∆2)×H2 Γ0 (Ω)× (L2(Γ1;H1(0, 1)))2 : B1ϕ = −β1∂νψ − β2η 1(·, 1), B2ϕ = γ1ψ + γ2η 2(·, 1), η1(·, 0) = ∂νψ, η 2(·, 0) = ψ on Γ1 } where DΓ0 (∆2) = { ϕ ∈ H2 Γ0 (Ω) : ∆2ϕ ∈ L2(Ω), B1ϕ ∈ L2(Γ1), B2ϕ ∈ L2(Γ1) } and A  ϕ ψ η1 η2  =  ψ −∆2ϕ − 1 τ1 η1 ρ − 1 τ2 η2 ρ  , ∀Φ = (ϕ,ψ, η1, η2)> ∈ D(A). (3.3) Remark 3.1. From the fact that 2< ( ϕx1x1ϕx2x2 ) = |ϕx1x1 + ϕx2x2 |2 − |ϕx1x1 |2 − |ϕx2x2 |2, we have |ϕx1x1 |2 + |ϕx2x2 |2 + 2µ< ( ϕx1x1 ϕx2x2 ) + 2(1− µ)|ϕx1x2 |2 = (1− µ)|ϕx1x1 |2 + (1− µ)|ϕx2x2 |2 + µ|ϕx1x1 + ϕx2x2 |2 + 2(1− µ)|ϕx1x2 |2 ≥ 0 ; (3.4) consequently, from (1.12), we obtain a(ϕ,ϕ) ≥ (1− µ)|ϕ|H2(Ω). Hence the sesquilinear form a is coercive on H2 Γ0 (Ω), since Γ0 is non empty. On the other hand, from (1.14) (see also [18, Lemma 3.1 and Remark 3.1]), we have a(ϕ,ψ) = ∫ Ω ∆2ϕψdx+ ∫ Γ1 (B1ϕ∂νψ − B2ϕψ) dΓ, (3.5) for all ϕ ∈ DΓ0 (∆2) and all ψ ∈ H2 Γ0 (Ω). 8 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 Now, if Φ = (ϕ,ϕt, η 1, η2)> is solution of (1.4)-(1.10) and is sufficiently regular, then system (1.4)-(1.10) can be written as the first order evolution equation Φt = AΦ, Φ(0) = Φ0, (3.6) where Φ0 = (ϕ0, ϕ1, f0(·,−ρτ1), g0(·,−ρτ2))> ∈ H. Proposition 3.2. Under hypothesis (3.1), the unbounded linear operator A is m- dissipative in the energy space H. Proof. For all Φ = (ϕ,ψ, η1, η2)> ∈ D(A), from (3.2) and (3.3), we have < (AΦ,Φ)H = < { a(ψ,ϕ)− ∫ Ω ∆2ϕψdx− |β2| ∫ Γ1 ∫ 1 0 η1 ρη 1 dρ dΓ− |γ2| ∫ Γ1 ∫ 1 0 η2 ρη 2 dρ dΓ } . Using (3.5) and that Φ ∈ D(A), we obtain < (AΦ,Φ)H = −β1 ∫ Γ1 |∂νψ|2 dΓ−< { β2 ∫ Γ1 η1(·, 1)∂νψ dΓ } − γ1 ∫ Γ1 |ψ|2 dΓ −< { γ2 ∫ Γ1 η2(·, 1)ψ dΓ } − |β2| 2 ∫ Γ1 |η1(·, 1)|2 dΓ + |β2| 2 ∫ Γ1 |∂νψ|2 dΓ − |γ2| 2 ∫ Γ1 |η2(·, 1)|2 dΓ + |γ2| 2 ∫ Γ1 |ψ|2 dΓ. (3.7) Now, by using Young’s inequality, we obtain −< { β2 ∫ Γ1 η1(·, 1)∂νψ dΓ } ≤ |β2| 2 ∫ Γ1 |η1(·, 1)|2 dΓ + |β2| 2 ∫ Γ1 |∂νψ|2 dΓ, −< { γ2 ∫ Γ1 η2(·, 1)ψ dΓ } ≤ |γ2| 2 ∫ Γ1 |η2(·, 1)|2 dΓ + |γ2| 2 ∫ Γ1 |ψ|2 dΓ. Inserting the above inequalities into (3.7) and using hypothesis (3.1), we obtain <(AΦ,Φ)H ≤ −(β1 − |β2|) ∫ Γ1 |∂νψ|2 dΓ− (γ1 − |γ2|) ∫ Γ1 |ψ|2 dΓ ≤ 0, (3.8) which implies that A is dissipative. Now, let us prove that A is maximal. For this aim, if F = (f1, f2, f3, f4)> ∈ H, we look for Φ = (ϕ,ψ, η1, η2)> ∈ D(A) unique solution of − AΦ = F. (3.9) Equivalently, we have the system −ψ = f1, (3.10) ∆2ϕ = f2, (3.11) 1 τ1 η1 ρ = f3, (3.12) 1 τ2 η2 ρ = f4, (3.13) EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 9 with the boundary conditions ϕ = ∂νϕ = 0 on Γ0, B1ϕ = −β1∂νψ − β2η 1(·, 1), B2ϕ = γ1ψ + γ2η 2(·, 1) on Γ1, η1(·, 0) = ∂νψ, η2(·, 0) = ψ on Γ1. (3.14) From (3.10) and that F ∈ H, we obtain ψ = −f1 ∈ H2 Γ0 (Ω). (3.15) In light of (3.12), (3.13), (3.14) and since F ∈ H, we obtain η1 ρ ∈ L2(Γ1 × (0, 1)), η1(·, ρ) = τ1 ∫ ρ 0 f3(·, s)ds+ ∂νψ, (3.16) η2 ρ ∈ L2(Γ1 × (0, 1)), η2(·, ρ) = τ2 ∫ ρ 0 f4(·, s)ds+ ψ. (3.17) Consequently, owing to (3.15),(3.16), (3.17), taking into account that f3, f4 ∈ L2(Γ1 × (0, 1)), we deduce that η1, η2 ∈ L2(Γ1;H1(0, 1)). It follows from (3.11), (3.14), (3.16) and (3.17) that ∆2ϕ = f2 in Ω, ϕ = ∂νϕ = 0 on Γ0, B1ϕ = (β1 + β2)∂νf1 − τ1β2 ∫ 1 0 f3(·, s)ds on Γ1, B2ϕ = −(γ1 + γ2)f1 + τ2γ2 ∫ 1 0 f4(·, s)ds on Γ1. (3.18) Let u ∈ H2 Γ0 (Ω). Multiplying the first equation in (3.18) by u and integrating over Ω, then using Green’s formula, we obtain a(ϕ, u) = l(u), ∀u ∈ H2 Γ0 (Ω), (3.19) where l(u) = ∫ Ω f2u dx+ ∫ Γ1 ( (β1 + β2)∂νf1 − τ1β2 ∫ 1 0 f3(·, s)ds ) ∂νu dΓ + ∫ Γ1 ( (γ1 + γ2)f1 − τ2γ2 ∫ 1 0 f4(·, s)ds ) u dΓ. It is easy to see that, a is a sesquilinear, continuous and coercive form on H2 Γ0 (Ω)× H2 Γ0 (Ω) and l is an antilinear and continuous form on H2 Γ0 (Ω). Then, thanks to Lax-Milgram theorem, (3.19) admits a unique solution u ∈ H2 Γ0 (Ω). By taking the test function ϕ ∈ D(Ω), we see that the first identity of (3.18) holds in the distributional sense, hence ∆2ϕ ∈ L2(Ω). Going back to (3.19), and again applying Greens’s formula (1.14), we find that B1ϕ = (β1 + β2)∂νf1 − τ1β2 ∫ 1 0 f3(·, s)ds on Γ1 , B2ϕ = −(γ1 + γ2)f1 + τ2γ2 ∫ 1 0 f4(·, s)ds on Γ1 . 10 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 Furthermore, since F ∈ H, we deduce that ϕ ∈ DΓ0 (∆2). Consequently, if we define Φ = (ϕ,ψ, η1, η2)> with ϕ ∈ H2 Γ0 (Ω) the unique solution of (3.19), ψ = −f1, and η1 (resp. η2) defined by (3.16) (resp. (3.17)), Φ belongs to D(A) is the unique solution of (3.9). Then, A is an isomorphism and since ρ (A) is open set of C (see [10, Theorem 6.7 (Chapter III)]), we easily obtain R(λI −A) = H for a sufficiently small λ > 0. This, together with the dissipativeness of A, imply that D (A) is dense in H and that A is m-dissipative in H (see [16, Theorems 4.5, 4.6]). � Based on Lumer-Phillips theorem (see [16]), Proposition 3.2 implies that the operator A generates a C0-semigroup of contractions etA in H which gives the well- posedness of (3.6). Then, we have the following result. Theorem 3.3. For all Φ0 ∈ H, system (3.6) admits a unique weak solution Φ(t) = etAΦ0 ∈ C0(R+,H). Moreover, if Φ0 ∈ D(A), then system (3.6) admits a unique strong solution Φ(t) = etAΦ0 ∈ C0(R+, D(A)) ∩ C1(R+,H). 3.2. Strong stability of system (1.4)-(1.10). The following theorem is the main result of this subsection. Theorem 3.4. Under hypotheses (3.1), the C0-semigroup of contraction (etA)t≥0 is strongly stable in H; i.e., for all Φ0 ∈ H, the solution of (3.6) satisfies lim t→∞ ‖etAΦ0‖H = 0. According to Arendt-Batty [2], to prove Theorem 3.4, we need to prove that the operator A has no pure imaginary eigenvalues and σ(A) ∩ iR is countable. The proof of these results is not reduced to the analysis of the point spectrum of A on the imaginary axis since its resolvent is not compact. Hence the proof of Theorem 3.4 has been divided into the following two Lemmas. Lemma 3.5. For all λ ∈ R, iλI − A is injective i.e., ker(iλI − A) = {0} Proof. In accordance with Proposition 3.2, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R∗. For the sake of this, suppose that λ 6= 0 and let Φ = (ϕ,ψ, η1, η2)> ∈ D(A) be such that AΦ = iλΦ. (3.20) Equivalently, we have the system ψ = iλϕ, (3.21) −∆2ϕ = iλψ, (3.22) − 1 τ1 η1 ρ = iλη1, (3.23) − 1 τ2 η2 ρ = iλη2. (3.24) From (3.8), (3.20) and (3.1), we obtain 0 = < (AΦ,Φ)H ≤ −(β1 − |β2|) ∫ Γ1 |∂νψ|2 dΓ− (γ1 − |γ2|) ∫ Γ1 |ψ|2 dΓ ≤ 0. Thus, we have ∂νψ = ψ = 0 on Γ1, (3.25) EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 11 which gives, from (3.21) and since λ 6= 0, that ϕ = ∂νϕ = 0 on Γ1. (3.26) Using (3.23), (3.24), (3.25) and that Φ ∈ D(A), we obtain η1(·, ρ) = ∂νψe −iλτ1ρ = 0 on Γ1 × (0, 1), (3.27) η2(·, ρ) = ψe−iλτ2ρ = 0 on Γ1 × (0, 1). (3.28) Now, from equations (3.25), (3.27), (3.28) and seeing that Φ ∈ D(A), we obtain B1ϕ = ∆ϕ+ (1− µ)C1ϕ = 0 on Γ1, (3.29) B2ϕ = ∂ν∆ϕ+ (1− µ)∂τC2ϕ = 0 on Γ1. (3.30) Using (3.26) and considering ∇ϕ = ∂τϕτ + ∂νϕν on Γ1, we obtain ϕx1 = ϕx2 = 0 on Γ1. (3.31) Now, (1.2), (3.26) and (3.31), yield C1ϕ = C2ϕ = 0 on Γ1, (3.32) consequently, from (3.29) and (3.30), we infer ∆ϕ = ∂ν∆ϕ = 0 on Γ1. (3.33) Inserting (3.21) in (3.22), we obtain λ2ϕ−∆2ϕ = 0 in Ω, ϕ = ∂νϕ = 0 on Γ0, ϕ = ∂νϕ = ∆ϕ = ∂ν∆ϕ = 0 on Γ1. (3.34) Holmgren uniqueness theorem (see [13]) yields ϕ = 0 in Ω. (3.35) Finally, from (3.21), (3.27), (3.28), and (3.35), we obtain Φ = 0. � Lemma 3.6. Under hypothesis (3.1), we have R(iλI − A) = H for all λ ∈ R. Proof. From Proposition 3.2, we have 0 ∈ ρ(A). We still need to show the result for λ ∈ R?. For this aim, for F = (f1, f2, f3, f4)> ∈ H, we look for Φ = (ϕ,ψ, η1, η2)> ∈ D(A) solution of (iλI − A)Φ = F. (3.36) Correspondingly, we have the system iλϕ− ψ = f1, (3.37) iλψ + ∆2ϕ = f2, (3.38) iλη1 + 1 τ1 η1 ρ = f3, (3.39) iλη2 + 1 τ2 η2 ρ = f4, (3.40) with the boundary conditions ϕ = ∂νϕ = 0 on Γ0, B1ϕ = −β1∂νψ − β2η 1(·, 1), B2ϕ = γ1ψ + γ2η 2(·, 1) on Γ1, η1(·, 0) = ∂νψ, η2(·, 0) = ψ on Γ1. (3.41) 12 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 From (3.39), (3.40) and (3.41), we deduce that η1(·, ρ) = ∂νψe −iλτ1ρ + τ1 ∫ ρ 0 f3(x, s)eiλτ1(s−ρ) ds on Γ1 × (0, 1), (3.42) η2(·, ρ) = ψe−iλτ2ρ + τ2 ∫ ρ 0 f4(x, s)eiλτ2(s−ρ) ds on Γ1 × (0, 1). (3.43) It follows from (3.37), (3.38), (3.41), (3.42) and (3.43) that −λ2ϕ+ ∆2ϕ = iλf1 + f2 in Ω, ϕ = ∂νϕ = 0 on Γ0, B1ϕ = −Ciλ(∂νϕ+ iλ−1∂νf1)− Fiλ on Γ1, B2ϕ = Diλ(ϕ+ iλ−1f1) +Giλ on Γ1, (3.44) where Ciλ = iλ(β1 + β2e −iλτ1), Fiλ = β2τ1 ∫ 1 0 f3(x, s)eiλτ1(s−1) ds, Diλ = iλ(γ1 + γ2e −iλτ2), Giλ = γ2τ2 ∫ 1 0 f4(x, s)eiλτ2(s−1) ds. Let u ∈ H2 Γ0 (Ω). Multiplying the first equation in (3.44) by u, integrating over Ω, then using Green’s formula, we obtain b(ϕ, u) = l(u), ∀u ∈ V := H2 Γ0 (Ω), (3.45) where b(ϕ, u) = b1(ϕ, u) + b2(ϕ, u), with b1(ϕ, u) = a(ϕ, u), b2(ϕ, u) = −λ2 ∫ Ω ϕudx+ Ciλ ∫ Γ1 ∂νϕ∂νu dΓ +Diλ ∫ Γ1 ϕudΓ (3.46) and l(u) = ∫ Ω (iλf1 + f2)u dx− ∫ Γ1 (iλ−1Ciλ∂νf1 + Fiλ)∂νu dΓ − ∫ Γ1 (iλ−1Diλ +Giλ)u dΓ. (3.47) Let V′ be the dual space of V. We define the operators B : V→ V′ as ϕ 7→ Bϕ and the operators Bi : V→ V′ as ϕ 7→ Biϕ for i = 1, 2, such that (Bϕ)(u) = b(ϕ, u), ∀u ∈ V, (Biϕ)(u) = bi(ϕ, u), ∀u ∈ V, i ∈ {1, 2}. (3.48) We need to prove that the operator B is an isomorphism. So, we divide the proof into two steps: Step 1. In this step, we prove that the operator B2 is compact. For this purpose, let us define the Hilbert space Hs Γ0 (Ω) := {u ∈ Hs(Ω) : u = ∂νu = 0 on Γ0} with s ∈ ( 3 2 , 2). Now, from (3.46) and a trace theorem, we obtain |b2(ϕ, u)| . ‖ϕ‖L2(Ω)‖u‖H2(Ω) + ‖∂νϕ‖L2(Γ1)‖∂νu‖L2(Γ1) + ‖ϕ‖L2(Γ1)‖u‖L2(Γ1) . ‖ϕ‖Hs(Ω)‖u‖H2(Ω), EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 13 for all s ∈ ( 3 2 , 2 ) . As V is compactly embedded in Hs Γ0 (Ω) for any s ∈ ( 3 2 , 2 ) , B2 is indeed a compact operator. This compactness property and the fact that B1 is an isomorphism imply that the operator B = B1 + B2 is a Fredholm operator of index zero. Now, following Fredholm alternative, we simply need to prove that the operator B is injective to obtain that it is an isomorphism. Step 2. In this step, we prove that the operator B is injective (i.e. ker(B) = {0}). To this end, let φ ∈ ker(B) which gives b(φ, u) = 0, ∀u ∈ V. Likewise, we have a(φ, u)− λ2 ∫ Ω φu dx+ Ciλ ∫ Γ1 ∂νφ∂νu dΓ +Diλ ∫ Γ1 φu dΓ = 0, ∀u ∈ V. Thus, we find that −λ2φ+ ∆2φ = 0 in D′(Ω), φ = ∂νφ = 0 on Γ0 B1φ = −Ciλ∂νφ on Γ1, B2φ = Diλφ on Γ1. Therefore, the vector Φ defined by Φ = (φ, iλφ, iλe−iλτ1ρ∂νφ, iλe −iλτ2ρφ)> belongs to D(A) and satisfies iλΦ− AΦ = 0, and consequently Φ ∈ ker(iλI − A). Hence Lemma 3.5 yields Φ = 0 and conse- quently φ = 0 and ker(B) = {0}. Steps 1 and 2 guarantee that the operator B is isomorphism. Furthermore it is easy to see that the operator l is an antilinear and continuous form on V. As a consequence, (3.45) admits a unique solution φ ∈ V. In (3.45), by taking test functions u ∈ D(Ω), we see that the first identity of (3.44) holds in the distributional sense, hence ∆2ϕ ∈ L2(Ω). Coming back to (3.45), and again applying Green’s formula (1.14), we find that B1ϕ = −Ciλ(∂νϕ+ iλ−1∂νf1)− Fiλ on Γ1, B2ϕ = Diλ(ϕ+ iλ−1f1) +Giλ on Γ1. Further, since ϕ, ∂νϕ, f1, ∂νf1, Fiλ and Giλ belong to L2(Γ1), we deduce that ϕ ∈ DΓ0 (∆2). As a result, if ϕ ∈ V is the unique solution of (3.45) and if we define η1 (resp. η2) by (3.42) (resp. (3.43)), we deduce that Φ = (ϕ, iλϕ− f1, η 1, η2)> belongs to D(A) and is the unique solution of (3.36). � Proof of Theorem 3.4. From Lemma 3.5, the operator A has no pure imaginary eigenvalues (i.e. σp(A) ∩ iR = ∅). Moreover, from Lemma 3.5 and Lemma 3.6, iλI − A is bijective for all λ ∈ R and since A is closed, we conclude, with the help of the closed graph theorem, that iλI − A is an isomorphism for all λ ∈ R, hence that σ(A) ∩ iR = ∅. � 14 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68 3.3. Exponential stability. In this subsection, we will prove the strong stability of system (1.4)-(1.10). We start this subsection with the definition of our multiplier geometric control condition. Definition 3.7. We say that the partition (Γ0,Γ1) of the boundary Γ satisfies the multiplier geometric control condition (MGC) if there exists a point x0 ∈ R2 and a positive constant δ such that h · ν ≥ δ−1 on Γ1 and h · ν ≤ 0 on Γ0, (3.49) where h(x) = x− x0. Theorem 3.8. Under hypotheses (3.1) and (3.49), the C0-semigroup etA is expo- nentially stable; i.e. there exists constants M ≥ 1 and ε > 0 independent of Φ0 ∈ H such that ‖etAΦ0‖H ≤Me−εt‖Φ0‖H, ∀t ≥ 0. Since iR ⊂ ρ(A) (see the previous subsection), according to [9] and [17], to prove Theorem 3.8, it remains to prove that lim sup λ∈R,|λ|→∞ ‖ (iλI − A) −1 ‖L(H) <∞. (3.50) We will prove condition (3.50) by a contradiction argument. For this purpose, suppose that (3.50) is false, then there exists {(λn,Φn := (ϕn, ψn, η 1 n, η 2 n)>)}n≥1 ⊂ R∗ ×D(A) with |λn| → ∞ as n→∞ and ‖Φn‖H = 1, ∀n ≥ 1, (3.51) such that (iλnI − A)Φn = Fn := (f1,n, f2,n, f3,n, f4,n)> → 0 in H, as n→∞. (3.52) For simplicity, we drop the index n. Equivalently, from (3.52), we have iλϕ− ψ = f1 → 0 in H2 Γ0 (Ω), (3.53) iλψ + ∆2ϕ = f2 → 0 in L2(Ω), (3.54) iλη1 + 1 τ1 η1 ρ = f3 → 0 in L2(Γ1 × (0, 1)), (3.55) iλzη2 + 1 τ2 η2 ρ = f4 → 0 in L2(Γ1 × (0, 1)). (3.56) Taking the inner product of (3.52) with Φ in H and using (3.8), we obtain (β1 − |β2|) ∫ Γ1 |∂νψ|2 dΓ + (γ1 − |γ2|) ∫ Γ1 |ψ|2 dΓ ≤ −<(AΦ,Φ)H = <(F,Φ)H ≤ ‖F‖H‖Φ‖H, From the above estimation, (3.1) and the fact that ‖F‖H = o(1) and ‖Φ‖H = 1, we obtain ∫ Γ1 |∂νψ|2 dΓ = o(1) and ∫ Γ1 |ψ|2 dΓ = o(1). (3.57) Lemma 3.9. Under hypothesis (3.1), the solution Φ = (ϕ,ψ, η1, η2)> ∈ D(A) of (3.53)-(3.56) satisfies the following estimates∫ Γ1 ∫ 1 0 |η1|2 dρ dΓ = o(1), ∫ Γ1 |η1(·, 1)|2 dΓ = o(1), (3.58) EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 15∫ Γ1 ∫ 1 0 |η2|2 dρ dΓ = o(1), ∫ Γ1 |η2(·, 1)|2 dΓ = o(1), (3.59)∫ Γ1 |B1ϕ|2 dΓ = o(1), ∫ Γ1 |B2ϕ|2 dΓ = o(1). (3.60) Proof. By (3.42), the Cauchy-Schwarz inequality, and that ρ ∈ (0, 1), we obtain∫ Γ1 ∫ 1 0 |η1|2 dρ dΓ ≤ 2 ∫ Γ1 |∂νψ|2 dΓ + 2τ2 1 ∫ Γ1 ∫ 1 0 (∫ ρ 0 |f3(·, s)|ds )2 dρ dΓ ≤ 2 ∫ Γ1 |∂νψ|2 dΓ + 2τ2 1 ∫ Γ1 ∫ 1 0 ρ ∫ ρ 0 |f3(·, s)|2ds dρ dΓ ≤ 2 ∫ Γ1 |∂νψ|2 dΓ + 2τ2 1 (∫ 1 0 ρ dρ )∫ Γ1 ∫ 1 0 |f3(·, s)|2ds dΓ = 2 ∫ Γ1 |∂νψ|2 dΓ + τ2 1 ∫ Γ1 ∫ 1 0 |f3(·, s)|2ds dΓ. The above inequality, (3.57) and since f3 → 0 in L2(Γ1 × (0, 1)) lead to the first estimation in (3.58). Now, from (3.42), we deduce that η1(·, 1) = ∂νψe −iλτ1 + τ1 ∫ 1 0 f3(·, s)eiλτ1(s−1)ds on Γ1, consequently, by using Cauchy-Schwarz inequality, we infer that∫ Γ1 |η1(·, 1)|2 dΓ ≤ 2 ∫ Γ1 |∂νψ|2 dΓ + 2τ2 1 ∫ Γ1 (∫ 1 0 |f3(·, s)|ds )2 dΓ ≤ 2 ∫ Γ1 |∂νψ|2 dΓ + 2τ2 1 ∫ Γ1 ∫ 1 0 |f3(·, s)|2ds dΓ. Therefore, from the above inequality, (3.57) and as f3 → 0 in L2(Γ1 × (0, 1)), we obtain the second estimation in (3.58). The same argument as before yields (3.59). Since Φ ∈ D(A), we have B1ϕ = −β1∂νψ − β2η 1(·, 1) on Γ1, B2ϕ = γ1ψ + γ2η 2(·, 1) on Γ1. Finally, from the above equations, (3.57), (3.58), (3.59), we deduce (3.60). � Lemma 3.10. Under hypothesis (3.1), the solution Φ = (ϕ,ψ, η1, η2)> ∈ D(A) of (3.53)-(3.56) satisfies the estimates∫ Γ1 |∂νϕ|2 dΓ = o(λ−2) and ∫ Γ1 |ϕ|2 dΓ = o(λ−2). (3.61) Proof. Equation (3.53) yields iλ∂νϕ = ∂νψ + ∂νf1 on Γ1, iλϕ = ψ + f1 on Γ1. From the above equations, we deduce that∫ Γ1 |λ∂νϕ|2 dΓ . ∫ Γ1 |∂νψ|2 dΓ + ∫ Γ1 |∂νf1|2 dΓ, (3.62) 16 H. BADAWI, M. AKIL, Z. HAJJEJ EJDE-2023/68∫ Γ1 |λϕ|2 dΓ . ∫ Γ1 |ψ|2 dΓ + ∫ Γ1 |f1|2 dΓ. (3.63) Using the trace theorem and that a(f1, f1) = o(1), we obtain∫ Γ1 |∂νf1|2 dΓ . ‖f1‖2H2(Ω) . a(f1, f1) = o(1),∫ Γ1 |f1|2 dΓ . ‖f1‖2H2(Ω) . a(f1, f1) = o(1). Inserting the above estimations into (3.62) and (3.63), then using (3.57), we obtain the desired result. � Lemma 3.11. Under hypotheses (3.1) and (3.49), the solution Φ = (ϕ,ψ, η1, η2)> in D(A) of (3.53)-(3.56) satisfies the following estimates∫ Ω |λϕ|2dx = o(1) and a(ϕ,ϕ) = o(1). (3.64) Proof. Inserting (3.53) in (3.54), we obtain −λ2ϕ+ ∆2ϕ = iλf1 + f2 in Ω. Multiplying the above equation by (h · ∇ϕ), integrating over Ω, then taking the real part, we obtain < {∫ Ω (−λ2ϕ+ ∆2ϕ)(h · ∇ϕ) dx } = < {∫ Ω (iλf1 + f2)(h · ∇ϕ) dx } (3.65) Now, by using Green’s formula and that ϕ = 0 on Γ0, then using (3.61), we have < { − λ2 ∫ Ω ϕ(h · ∇ϕ) dx } = 1 2 ∫ Ω |λϕ|2dx− 1 2 ∫ Γ1 (h · ν)|λϕ|2 dΓ = 1 2 ∫ Ω |λϕ|2dx+ o(1). (3.66) Now that a(ϕ,ϕ) = O(1) and a(f1, f1) = o(1), we obtain ‖∇ϕ‖L2(Ω) ≤ ‖ϕ‖H2(Ω) . √ a(ϕ,ϕ) = O(1), (3.67) ‖f1‖L2(Ω), ‖∇f1‖L2(Ω) ≤ ‖f1‖H2(Ω) . √ a(f1, f1) = o(1). (3.68) Consequently, by using Green’s formula, (3.61), (3.3), and f2 → 0 in L2(Ω), we deduce that < {∫ Ω (iλf1 + f2)(h · ∇ϕ) dx } = < { − iλ ∫ Ω (h · ∇f1)ϕdx− 2iλ ∫ Ω f1ϕdx− iλ ∫ Γ1 f1ϕ(h · ν) dΓ + ∫ Ω f2(h · ∇ϕ) dx } = o(1). Inserting (3.66) into (3.65) and using the above estimation, we acquire 1 2 ∫ Ω |λϕ|2dx = −< {∫ Ω ∆2ϕ(h · ∇ϕ) dx } + o(1). (3.69) EJDE-2023/68 STABILITY AND INSTABILITY OF KIRCHHOFF PLATE EQUATIONS 17 According to [1, Lemma 5.4], for all ϕ ∈ DΓ0 (∆2), we have −< {∫ Ω ∆2ϕ(h · ∇ϕ) dx } ≤ −1 2 a(ϕ,ϕ) + ε1R 2 2 ∫ Γ1 |B2ϕ|2 dΓ + (∫ Γ1 |B1ϕ|2 dΓ )1/2(∫ Γ1 |∂νϕ|2 dΓ )1/2 + R2ε2 2 ∫ Γ1 |B1ϕ|2 dΓ, (3.70) where R = ‖h‖L∞(Ω) and ε1, ε2 are positive constants. Consequently, using (3.61) and (3.60), we obtain −< {∫ Ω ∆2ϕ(h · ∇ϕ) dx } ≤ −1 2 a(ϕ,ϕ) + o(1). (3.71) Finally, inserting (3.71) into (3.69), we obtain 1 2 ∫ Ω |λϕ|2dx+ 1 2 a(ϕ,ϕ) = o(1). � Proof of Theorem 3.8. From Lemmas 3.9 and 3.11, we deduce that ‖Φ‖H = o(1), which contradicts (3.51). � 4. Conclusion In this article, we considered the Kirchhoff plate equation with delay terms on the boundary control. We gave some instability examples in the cases |β2| ≥ β1 and |γ2| ≥ γ1. Thanks to the general criteria of Arendt and Batty, which helped to establish the strong stability of our system without using any geometric control condition. Finally, under the multiplier geometric control condition (MGC), we showed that system (1.1) is exponentially stable. Acknowledgments. This work was supported by Researchers Supporting Project number (RSPD2023R736), King Saud University, Riyadh, Saudi Arabia. References [1] M. Akil, H. Badawi, S. Nicaise, A. Wehbe; Stability results on the kirchhoff plate equation with delay terms on the dynamical boundary controls. Revista Matemática Complutense, 36(3):749–777, Sep 2023. [2] W. Arendt, C. J. K. Batty; Tauberian theorems and stability of one-parameter semigroups. Transactions of the American Mathematical Society, 306(2):837–852, 1988. [3] R. Datko; Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 26(3):697–713, 1988. [4] R. 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SIAM Journal on Control and Optimization, 45(5):1561–1585, Jan. 2006. [16] A. Pazy; Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. [17] J. Prüss; On the spectrum of C0-semigroups. Transactions of the American Mathematical Society, 284(2):847–857, 1984. [18] B. Rao; Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear bound- ary feedback. Nonlinear Analysis: Theory, Methods & Applications, 20(6):605–626, 1993. Haidar Badawi (corresponding author) PDEs with applications in materials sciences and biology, Department of Mathematics and Physics, Lebanese International University LIU, Beirut, Lebanon. PDEs with applications in materials sciences and biology, Department of Mathematics and Physics, Lebanese International University of Beirut BIU, Beirut, Lebanon Email address: haidar.badawi@liu.edu.lb Mohammad Akil Univ. Polytechnique Hauts-de-France, INSA Hauts-de-France, CERAMATHS - Labora- toire de Materiaux Céramiques et de Mathématiques, F-59313 Valenciennes, France Email address: mohammad.akil@uphf.fr Zayd Hajjej Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Email address: zhajjej@ksu.edu.sa 1. Introduction 2. Instability results 3. Stability results 3.1. Wellposedness of system (??)-(??) 3.2. Strong stability of system (??)-(??) 3.3. Exponential stability 4. Conclusion Acknowledgments References