Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 30, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXPONENTIAL STABILITY OF A DAMPED BEAM-STRING-BEAM TRANSMISSION PROBLEM BIENVENIDO BARRAZA MARTÍNEZ, JAIRO HERNÁNDEZ MONZÓN, GUSTAVO VERGARA ROLONG Abstract. We consider a beam-string-beam transmission problem, where two structurally damped or undamped beams are coupled with a frictionally damped string by transmission conditions. We show that for this structure, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution no matter how small is its size. For the exponential stability in the damped-damped-damped situation we use energy method. For the undamped-damped-undamped situation we use a frequency domain method from semigroups theory, which combines a contradiction ar- gument with the multiplier technique to carry out a special analysis for the resolvent. Additionally, we show that the solution first defined by the weak formulation has higher Sobolev space regularity. 1. Introduction Recent advances in material science have provided new means for suppressing vibrations from elastic multi-link structures, for instance, by applying some type of local or total damping. These structures consisting of connected flexible elements such as strings, beams, plates and shells have many applications in engineering areas such as in robot arms, frames, solar panels, aircrafts, satellite antennae, bridges and so on (see [6, 7, 15] and the references therein). In this context we consider a cou- pled beam-string-beam system, where we assume structural damping/no-damping for the beams and frictional damping for the string. More precisely, we consider an elastic structure composed by three parts. The first and the third parts are structurally damped or undamped beams in the open intervals I1 := (l0, l1) and I3 := (l2, l3), respectively, and the second is a frictionally damped string, occupying in equilibrium the open interval I2 := (l1, l2), where l0 < l1 < l2 < l3 as shown in Figure 1. We denote by u = u(x, t), w = w(x, t), and v = v(x, t) the vertical displacements of the points on the two beams and on the string with coordinates x at time t, respectively. 2020 Mathematics Subject Classification. 35M33, 35B35, 35B40, 93D23. Key words and phrases. Exponential stability; transmission problems; frictional damping; beam-string equations. ©2022. This work is licensed under a CC BY 4.0 license. Submitted September 30, 2021. Published April 13, 2022. 1 2 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 Figure 1. Beam-string-beam system The mathematical model for the structure is given by the equations utt + uxxxx − ρ1utxx = 0, in (l0, l1)× (0,∞), (1.1) vtt − vxx + βvt = 0, in (l1, l2)× (0,∞), (1.2) wtt + wxxxx − ρ2wtxx = 0, in (l2, l3)× (0,∞), (1.3) where ρi ≥ 0, i = 1, 2, and β ≥ 0 are fixed constants. The coefficients ρ1 ≥ 0 and ρ2 ≥ 0 describe the structural damping (or the absence of damping) for the beam equations (1.1) and (1.3), whereas β > 0 in (1.2) describes a frictional damping on the string. On the endpoints l0, l3 of the beams, we impose clamped (Dirichlet) boundary conditions u(l0, t) = ux(l0, t) = w(l3, t) = wx(l3, t) = 0, t ∈ (0,∞). (1.4) On the interface {l1, l2}, we have transmission conditions u(l1, t) = v(l1, t) and v(l2, t) = w(l2, t), t ∈ (0,∞), (1.5) uxxx(l1, t)− ρ1utx(l1, t) + vx(l1, t) = 0, t ∈ (0,∞), (1.6) wxxx(l2, t)− ρ2wtx(l2, t) + vx(l2, t) = 0, t ∈ (0,∞), (1.7) uxx(l1, t) = 0, t ∈ (0,∞), (1.8) wxx(l2, t) = 0, t ∈ (0,∞). (1.9) Condition (1.5) is known as the continuity transmission condition. (1.6) and (1.7) mean that the two forces which are the shear force of the beams and the stress of the string are such that one cancels the other. And (1.8) and (1.9) describe the fact that the beams present possible inflection point on l1 and l2 (compare with [14, p. 1934]). Finally, the boundary-transmission problem (1.1)–(1.9) is endowed the with ini- tial conditions: u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ (l0, l1), (1.10) v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ (l1, l2), (1.11) w(x, 0) = w0(x), wt(x, 0) = w1(x), x ∈ (l2, l3). (1.12) The aim of this article is to study the well-posedness, regularity, and exponential stability of the solution of (1.1)-(1.12). In recent years, the large-time behavior of structures consisting of elastic strings and beams with different damping has been studied. We refer the reader to [17, 19, 24], where structures formed by beams were studied. For instance, Shel in [24] showed, under certain conditions, the exponential stability of a network of elas- tic and thermoelastic Euler-Bernoulli beams. For transmission problems between strings, see for example, [1, 9, 13, 18, 22, 23]. Alves, Muñoz Rivera et al. consid- ered in [1] a transmission problem of a material composed of three components; EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 3 one of them is a Kelvin-Voigt viscoelastic material, the second is an elastic mate- rial (no dissipation) and the third is an elastic material with a frictional damping mechanism. They proved exponential stability of the solution the if viscoelastic component is not in the middle of the material. Rissel and Wang established in [23] exponential stability for a coupled system composed of three parts: the first and third are purely elastic and the second thermo-elastic. For elastic structures composed of string and beam the reader is referred to [2, 3, 4, 5, 14, 16, 25, 26, 27]. Ammari et al. [2, 3, 4, 5] considered the nodal feedback stabilization for networks of strings and beams. They obtained that the decay rate of a closed-loop system depends on the positions of the nodal feedback controllers. Hassine in [14] studied a elastic transmission wave/beam systems with a local Kelvin-Voigt damping. He showed that the energy of this coupled system decays polynomially as the time variable goes to infinity if the damping (which is locally distributed) acts through one part of the structure. Li, Han and Xu in [16] obtained polynomial stability for a string-frictionally damped beam system and exponential stability for a frictionally damped string-beam system. Shel in [25] considered transmission problems for a coupling of a string and a beam with at least one of them being thermoelastic and established that the associated semigroup is exponentially stable when the string is thermoelastic and polynomial stables when only the beam is thermoelastic and sat- isfies certain additional condition. Wang in [26] obtained the strong stability of the semigroup associated to a frictionally damped string-beam system. F. Wang and J. M. Wang [27] established exponential stability for a beam-frictionally damped string system with some feedback at the interface point. In this article, we study the well-posedness of problem (1.1)-(1.12), higher regu- larity of the solution, and exponential stability of the energy of the system, depend- ing on the dampings. Using the energy method, we prove the exponential stability of (1.1)-(1.12) if the beams and the string are damped (i.e. if ρ1, ρ2 and β are positive), our proof does not need higher regularity of the solution. Moreover, by a frequency domain method from the semigroup theory, we show the exponential stability of (1.1)-(1.12) in the undamped-damped-undamped situation; i.e. ρ1 = ρ2 = 0 and β > 0. We use the higher Sobolev space regularity of the solutions, which implies that the transmission conditions, first defined by the weak formulation, hold in the classical sense. For this we follow ideas developed in Section 4 of [8]. This article is organized as follows: In Section 2, we define the basic spaces and operators. In Section 3 we show the generation of a C0s-emigroup of contrac- tions (and therefore the well-posedness of (1.1)-(1.12)). Exponential stability for the cases damped-damped-damped and undamped-damped-undamped, as well as higher regularity of the solutions, are shown in Section 4. Let us set some notation. Derivatives with respect to t of a function will be denoted by a “dot” over the name of the function. So φ̇ will denote the derivative of φ with respect to t. We also use ψ′, ψ′′, ψ′′′, or in general ψ(n) for n ≥ 4, for the derivatives of ψ with respect to the one-dimensional spatial variable x. 4 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 2. Base Spaces We let A be the matrix differential operator A :=  0 0 0 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I − ∂4 ∂x4 0 0 ρ1 ∂2 ∂x2 0 0 0 ∂2 ∂x2 0 0 −βI 0 0 0 − ∂4 ∂x4 0 0 ρ2 ∂2 ∂x2  . Then, if zt := ∂z ∂t , ztt := ∂2z ∂t2 , and U := (u, v, w, ut, vt, wt) >, we have that Ut = (ut, vt, wt, utt, vtt, wtt) > and (1.1)-(1.3) can be written as Ut =  ut vt wt utt vtt wtt  =  ut vt wt −∂ 4u ∂x4 + ρ1 ∂2ut ∂x2 ∂2v ∂x2 − βvt −∂ 4w ∂x4 + ρ2 ∂2wt ∂x2  = AU. Because of the initial condition (1.10)–(1.12), Ut(t) = AU(t) (t > 0), U(0) = U0, (2.1) where U0 := (u0, v0, w0, u1, v1, w1)>. On the intervals I1 = (l0, l1), I2 = (l1, l2), and I3 = (l2, l3), we define the following spaces H2 l0 := {u ∈ H2(I1) : u(l0) = u′(l0) = 0}, H2 l3 := {w ∈ H2(I3) : w(l3) = w′(l3) = 0}, with the inner products 〈u, ũ〉H2 l0 := 〈u′′, ũ′′〉L2(I1), (2.2) 〈w, w̃〉H2 l3 := 〈w′′, w̃′′〉L2(I3). (2.3) From the generalized Poincaré inequality, the induced norms ‖ · ‖H2 l0 and ‖ · ‖H2 l3 are equivalent to the standard norms ‖ · ‖H2(I1) and ‖ · ‖H2(I3) on H2 l0 and H2 l3 , respectively. Now, let us define the spaces H := { (u, v, w)> ∈ H2 l0 ×H 1(I2)×H2 l3 : u(l1) = v(l1) and v(l2) = w(l2) } , L := L2(I1)× L2(I2)× L2(I3), equipped with the inner products 〈(u, v, w)>, (ũ, ṽ, w̃)>〉H := 〈u, ũ〉H2 l0 + 〈v′, ṽ′〉L2(I2) + 〈w, w̃〉H2 l3 , (2.4) 〈(u, v, w)>, (ũ, ṽ, w̃)>〉L := 〈u, ũ〉L2(I1) + 〈v, ṽ〉L2(I2) + 〈w, w̃〉L2(I3). (2.5) Again, by Poincaré’s inequality, the norm in H, induced by the inner product (2.4), is equivalent to the standard norm in the product space H2(I1)×H1(I2)×H2(I3). From the continuity of the trace operator, H is a closed subspace of H2(I1) × H1(I2)×H2(I3) and therefore ( H, 〈·, ·〉H ) is a Hilbert space. EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 5 Also, we define the Hilbert space H := H× L, with the inner product 〈U, Ũ〉H := 〈(u1, v1, w1)>, (ũ1, ṽ1, w̃1)>〉H + 〈(u2, v2, w2)>, (ũ2, ṽ2, w̃2)>〉L = 〈u1, ũ1〉H2 l0 + 〈v′1, ṽ′1〉L2(I2) + 〈w1, w̃1〉H2 l3 + 〈u2, ũ2〉L2(I1) + 〈v2, ṽ2〉L2(I2) + 〈w2, w̃2〉L2(I3), for all U = (u1, v1, w1, u2, v2, w2)>, Ũ = (ũ1, ṽ1, w̃1, ũ2, ṽ2, w̃2)> ∈H . Since the functions in the spaces defined above are not regular enough to satisfy the transmission conditions (1.6)-(1.9) in the classic sense, or even in the trace sense, we interpret this transmission condition first in a “weak” sense. For this, we consider U = (u1, v1, w1, u2, v2, w2)> ∈H and Ũ = (ũ1, ṽ1, w̃1, ũ2, ṽ2, w̃2)> ∈ H×H sufficiently smooth, such that the following calculations make sense. Applying integration by parts we obtain 〈AU, Ũ〉H = a(U, Ũ) + b(U, (ũ2, ṽ2, w̃2)>), where a(U, Ũ) := 〈u2, ũ1〉H2 l0 + 〈v′2, ṽ′1〉L2(I2) + 〈w2, w̃1〉H2 l3 − 〈u′′1 ũ′′2〉L2(I1) − ρ1〈u′2, ũ′2〉L2(I1) − β〈v2, ṽ2〉L2(I2) − 〈v′1, ṽ′2〉L2(I2) − 〈w′′1 , w̃′′2 〉L2(I3) − ρ2〈w′2, w̃′2〉L2(I3) and b(U, (ũ2, ṽ2, w̃2)>) := −u(3) 1 (l1)ũ2(l1) + u (3) 1 (l0)ũ2(l0) + u′′1(l1)ũ′2(l1)− u′′1(l0)ũ′2(l0) + ρ1u ′ 2(l1)ũ2(l1)− ρ1u ′ 2(l0)ũ2(l0) + v′1(l2)ṽ2(l2)− v′1(l1)ṽ2(l1) − w(3) 1 (l3)w̃2(l3) + w (3) 1 (l2)w̃2(l2) + w′′1 (l3)w̃′2(l3)− w′′1 (l2)w̃′2(l2) + ρ2w ′ 2(l3)w̃2(l3)− ρ2w ′ 2(l2)w̃2(l2). Since Ũ ∈ H×H, we have that ũ2(l0) = ũ′2(l0) = w̃2(l3) = w̃′2(l3) = 0, ṽ2(l1) = ũ(l1), and ṽ2(l2) = w̃2(l2). Then, it follows that b(U, (ũ2, ṽ2, w̃2)>) = [ − u(3) 1 (l1) + ρ1u ′ 2(l1)− v′1(l1) ] ũ2(l1) + [ w (3) 1 (l2)− ρ2w ′ 2(l2) + v′1(l2) ] w̃2(l2) + u′′1(l1)ũ′2(l1)− w′′1 (l2)w̃′2(l2) = 0, if and only if u (3) 1 (l1)− ρ1u ′ 2(l1) + v′1(l1) = 0, w (3) 1 (l2)− ρ2w ′ 2(l2) + v′1(l2) = 0, u′′1(l1) = 0, w′′1 (l2) = 0, which are the transmission conditions (1.6)-(1.9) that we have in the description of the problem in Section 1 if u2 = ∂tu1 and w2 = ∂tw1. This motivates the following definition. 6 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 Definition 2.1. We say that U ∈ H satisfies the transmission conditions (1.6)- (1.9) in the weak sense if 〈AU, Ũ〉H = a(U, Ũ) for all Ũ ∈ H×H. (2.6) Now, we define the operator A : D(A ) ⊂H →H , A U := AU, (2.7) where D(A ) := { U ∈ H×H : u (4) 1 ∈ L2(I1), v′′1 ,∈ L2(I2), w (4) 1 ∈ L2(I3), U satisfies (1.6)-(1.9) in the weak sense } . In this way, problem (1.1)-(1.12) can be written in abstract form as the Cauchy problem dU dt (t) = A U(t) (t > 0), U(0) = U0. (2.8) 3. Well posedness Now, we show that problem (2.8) is well posed, which means, that for every U ∈ D(A ), (2.8) has one and only one classical solution which depends continu- ously on U0. For this, we prove that the operator A defined in (2.7) generates a C0-semigroup of contractions on H . To achieve that, we use the Lumer-Phillips theorem. Proposition 3.1. The following assertions hold: (a) A is dissipative. (b) I −A is surjective. (c) D(A ) is dense in H . Proof. Let U ∈ D(A ). From (2.6) we have 〈A U,U〉H = −ρ1‖u′2‖2L2(I1) − β‖v2‖2L2(I2) − ρ2‖w′2‖2L2(I3) ≤ 0, (3.1) from which (a) follows. For the surjectivity we will use the Lax-Milgram theorem. Let F = (f1, g1, h1, f2, g2, h2)> ∈ H . We need to show that there exists a U = (u1, v1, w1, u2, v2, w2)> ∈ D(A ) such that (I −A )U = F , i.e. u1 − u2 = f1 ∈ H2 l0 , (3.2) v1 − v2 = g1 ∈ H1(I2), (3.3) w1 − w2 = h1 ∈ H2 l3 , (3.4) u (4) 1 + u2 − ρ1u ′′ 2 = f2 ∈ L2(I1), (3.5) −v′′1 + (1 + β)v2 = g2 ∈ L2(I2), (3.6) w (4) 1 + w2 − ρ2w ′′ 2 = h2 ∈ L2(I3). (3.7) Plugging (3.2)–(3.4) in (3.5)–(3.7) respectively, we have to solve u (4) 1 + u1 − ρ1u ′′ 1 = f1 + f2 − ρ1f ′′ 1 , (3.8) −v′′1 + (1 + β)v1 = g2 + (1 + β)g1, (3.9) w (4) 1 + w1 − ρ2w ′′ 1 = h1 + h2 − ρ2h ′′ 1 . (3.10) EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 7 We define the sesquilinear form B : H×H→ C by B(Y,Φ) := 〈y1, φ1〉H2 l0 + ρ1〈y′1, φ′1〉L2(I1) + 〈y1, φ1〉L2(I1) + 〈y′2, φ′2〉L2(I2) + (1 + β)〈y2, φ2〉L2(I2) + 〈y3, φ3〉H2 l3 + ρ2〈y′3, φ′3〉L2(I3) + 〈y3, φ3〉L2(I3), for Y := (y1, y2, y3)>, Φ := (φ1, φ2, φ3)> ∈ H. It is easy to see that B : H×H→ C is continuous and coercive. Now, for (f1, g1, h1, f2, g2, h2)> ∈H , we define Λ : H→ C by Λ(Φ) := 〈f1 + f2, φ1〉L2(I1) + ρ1〈f ′1, φ′1〉L2(I1) + 〈g2 + (1 + β)g1, φ2〉L2(I2) + 〈h1 + h2, φ3〉L2(I3) + ρ2〈h′1, φ′3〉L2(I3), for all Φ ∈ H. It is also easy to see that Λ : H → C is an antilinear continuous functional. By the Lax-Milgram theorem exists a unique Y = (y1, y2, y3)> ∈ H such that B(Y,Φ) = Λ(Φ) for all Φ ∈ H. (3.11) In particular, if φ1 ∈ C∞c (I1) and φ2 = φ3 = 0, we have in (3.11) that 〈y1, φ1〉H2 l0 + ρ1〈y′1, φ′1〉L2(I1) + 〈y1, φ1〉L2(I1) = 〈f1 + f2, φ1〉L2(I1) + ρ1〈f ′1, φ′1〉L2(I1), which can be written, in distributional sense, as 〈y(4) 1 − ρ1y ′′ 1 + y1, φ1〉 = 〈f1 + f2 − ρ1f ′′ 1 , φ1〉, for all φ1 ∈ C∞c (I1). This implies that y (4) 1 − ρ1y ′′ 1 + y1 = f1 + f2 − ρ1f ′′ 1 (3.12) in sense of distributions. Since y1, y ′′ 1 , f1, f2, f ′′ 1 ∈ L2(I1), we conclude that y (4) 1 ∈ L2(I1). Similarly, taking φ1 = φ2 = 0 and φ3 ∈ C∞c (I3), and then φ1 = φ3 = 0 and φ2 ∈ C∞c (I2), we obtain y (4) 3 − ρ2y ′′ 3 + y3 = h1 + h2 − ρ2h ′′ 1 (3.13) and − y′′2 + (1 + β)y2 = g2 + (1 + β)g1 (3.14) in distributional sense, respectively. These equations imply that y (4) 3 ∈ L2(I3) and y′′2 ∈ L2(I2). Let (u1, v1, w1) := (y1, y2, y3) and (u2, v2, w2) := (u1−f1, v1−g1, w1−h1). Hence U := (u1, v1, w1, u2, v2, w2)> ∈ H×H, u (4) 1 ∈ L2(I1), v′′1 ∈ L2(I2) and w (4) 1 ∈ L2(I3). From (3.11), for Ũ = (ũ1, ṽ1, w̃1, ũ2, ṽ2, w̃2)> ∈ H×H arbitrary we have 〈u1, ũ2〉H2 l0 + ρ1〈u′1, ũ′2〉L2(I1) + 〈u1, ũ2〉L2(I1) + 〈v′1, ṽ′2〉L2(I2) + (1 + β)〈v1, ṽ2〉L2(I2) + 〈w1, w̃2〉H2 l3 + ρ2〈w′1, w̃′2〉L2(I3) + 〈w1, w̃2〉L2(I3) = 〈f1 + f2, ũ2〉L2(I1) + ρ1〈f ′1, ũ′2〉L2(I1) + 〈g2 + (1 + β)g1, ṽ2〉L2(I2) + 〈h1 + h2, w̃2〉L2(I3) + ρ2〈h′1, w̃′2〉L2(I3), 8 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 which is equivalent to 〈u1 − (f1 + f2), ũ2〉L2(I1) + 〈v1 − (g1 + g2), ṽ2〉L2(I2) + 〈w1 − (h1 + h2), w̃2〉L2(I3) = −〈u1, ũ2〉H2 l0 − ρ1〈(u1 − f1)′, ũ′2〉L2(I1) − 〈v′1, ṽ′2〉L2(I2) − β〈v1 − g1, ṽ ′ 2〉L2(I2) − 〈w1, w̃2〉H2 l3 − ρ2〈(w1 − h1)′, w̃′2〉L2(I3). (3.15) From (3.12)–(3.14) and (u2, v2, w2) = (u1 − f1, v1 − g1, w1 − h1) it follows that 〈−u(4) 1 + ρ1u ′′ 2 , ũ2〉L2(I1) = 〈u1 − (f1 + f2), ũ2〉L2(I1), 〈v′′1 − βv2, ṽ2〉L2(I2) = 〈v1 − (g1 + g2), ṽ2〉L2(I2), 〈−w(4) 1 + ρ2w ′′ 2 , w̃2〉L2(I3) = 〈w1 − (h1 + h2), w̃2〉L2(I3). Adding the three equations above and using (3.15) we obtain 〈−u(4) 1 + ρ1u ′′ 2 , ũ2〉L2(I1) + 〈v′′1 − βv2, ṽ2〉L2(I2) + 〈−w(4) 1 + ρ2w ′′ 2 , w̃2〉L2(I3) = 〈u1 − (f1 + f2), ũ2〉L2(I1) + 〈v1 − (g1 + g2), ṽ2〉L2(I2) + 〈w1 − (h1 + h2), w̃2〉L2(I3) = −〈u1, ũ2〉H2 l0 − ρ1〈(u1 − f1)′, ũ′2〉L2(I1) − 〈v′1, ṽ′2〉L2(I2) − β〈(v1 − g1), ṽ2〉L2(I2) − 〈w1, w̃2〉H2 l3 − ρ2〈(w1 − h1)′, w̃′2〉L2(I3) = −〈u1, ũ2〉H2 l0 − ρ1〈u′2, ũ′2〉L2(I1) − 〈v′1, ṽ′2〉L2(I2) − β〈v2, ṽ2〉L2(I2) − 〈w1, w̃2〉H2 l3 − ρ2〈w′2, w̃′2〉L2(I3). Therefore, 〈AU, Ũ〉H = 〈u2, ũ1〉H2 l0 + 〈v′2, ṽ′1〉L2(I2) + 〈w2, w̃1〉H2 l3 +〈−u(4) 1 + ρ1u ′′ 2 , ũ2〉L2(I1) + 〈v′′1 − βv2, ṽ2〉L2(I2) + 〈−w(4) 1 + ρ2w ′′ 2 , w̃2〉L2(I3) = 〈u2, ũ1〉H2 l0 + 〈v′2, ṽ′1〉L2(I2) + 〈w2, w̃1〉H2 l3 − 〈u1, ũ2〉H2 l0 − ρ1〈u′2, ũ′2〉L2(I1) − 〈v′1, ṽ′2〉L2(I2) − β〈v2, ṽ2〉L2(I2) − 〈w1, w̃2〉H2 l3 − ρ2〈w′2, w̃′2〉L2(I3). Hence U satisfies the transmission conditions (1.6)-(1.9) in the weak sense. From this we conclude that U ∈ D(A ), i.e. (b) holds. Since H is a Hilbert space, from (a) and (b) it follows that D(A ) is dense in H , see [20, Thm. 4.6, Chapter 1]. � Theorem 3.2. The operator A is the generator of a C0-semigroup (S(t))t≥0 of contractions on the Hilbert space H . In consequence, for each U0 ∈ D(A ) the Cauchy problem (2.8) has a unique classical solution U ∈ C1([0,∞),H ) which depends continuously on the initial data. Proof. From Proposition 3.1 and the Lumer-Phillips theorem we have that A is the generator of a contraction C0-semigroup over H . Hence, for each U0 ∈ D(A ), the Cauchy problem (2.8) has a unique classical solution U ∈ C1([0,∞),H ) which depends continuously on the initial data, i.e. the problem is well posed. � EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 9 4. Exponential stability In this section, we prove the main results of this article. First we prove the exponential stability of the semigroup (S(t))t≥0 generated by A if we have damping in the three subdomains, i.e., if ρ1, ρ2, and β are positive. We recall that we use the convention z(t)′ to indicate the derivative of the function z(t) := z(·, t) with respect to the spatial variable x, where z = z(x, t), whereas that the derivative of z(t) with respect to t will be denoted by ż(t) (see last paragraph in the introduction). For U0 ∈ D(A ), we have from Theorem 3.2 that U(t) := S(t)U0 (t ≥ 0) is the classical solution of (2.8). In this case the energy E(t) of the system is defined by E(t) := 1 2 ‖U(t)‖2H = 1 2 ( ‖u1(t)‖2H2 l0 + ‖v1(t)′‖2L2(I2) + ‖w1(t)‖2H2 l3 + ‖u2(t)‖2L2(I1) + ‖v2(t)‖2L2(I2) + ‖w2(t)‖2L2(I3) ) . Note that d dt E(t) = Re〈A U(t), U(t)〉H = −ρ1‖u2(t)′‖2L2(I1) − β‖v2(t)‖2L2(I2) − ρ2‖w2(t)′‖2L2(I3), (4.1) which shows that the system is dissipative if at least one damping is active (ρ1 + ρ2 + β > 0) and conservative if there is no damping at all (ρ1 = ρ2 = β = 0). Now, we prove the first main result of this article. Theorem 4.1. Let ρ1 > 0 , ρ2 > 0 and β > 0. Then, the semigroup (S(t))t≥0 is exponentially stable, i.e., for any U0 ∈ D(A ) and U(t) := S(t)U0 (t ≥ 0) we have E(t) ≤ Ce−αtE(0) with positive constants C and α. Proof. For U0 ∈ D(A ) and t ≥ 0, let U(t) := (u1(t), v1(t), w1(t), u2(t), v2(t), w2(t))> := S(t)U0, F (t) := 〈u1(t), u2(t)〉L2(I1) + 〈v1(t), v2(t)〉L2(I2) + 〈w1(t), w2(t)〉L2(I3). Then, using Cauchy-Schwarz and Young’s inequalities we have |F (t)| ≤ 1 2 ( ‖u1(t)‖2L2(I1) + ‖u2(t)‖2L2(I1) + ‖v1(t)‖2L2(I2) + ‖v2(t)‖2L2(I2) + ‖w1(t)‖2L2(I3) + ‖w2(t)‖2L2(I3) ) ≤ 1 2 ‖U(t)‖2X , with X := H2(I1)×H1(I2)×H2(I3)× L. Because of the equivalence between the standard norm in X and the norm ‖ · ‖H , there exists c1 > 0 such that |F (t)| ≤ c1E(t). 10 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 because d dtU(t) = A U(t), i.e., u̇1(t) v̇1(t) ẇ1(t) u̇2(t) v̇2(t) ẇ2(t)  =  u2(t) v2(t) w2(t) −u1(t)(4) + ρ1u2(t)′′ v1(t)′′ − βv2(t) −w1(t)(4) + ρ2w2(t)′′  , we have d dt F (t) = 〈u̇1(t), u2(t)〉L2(I1) + 〈u1(t), u̇2(t)〉L2(I1) + 〈v̇1(t), v2(t)〉L2(I2) + 〈v1(t), v̇2(t)〉L2(I2) + 〈ẇ1(t), w2(t)〉L2(I3) + 〈w1(t), ẇ2(t)〉L2(I3) = ‖u2(t)‖2L2(I1) + ‖v2(t)‖2L2(I2) + ‖w2(t)‖2L2(I3) + 〈Φ(t),A U(t)〉H = ‖u2(t)‖2L2(I1) + ‖v2(t)‖2L2(I2) + ‖w2(t)‖2L2(I3) − ‖u1(t)‖2H2 l0 − ρ1〈u1(t)′, u2(t)′〉L2(I1) − ‖v1(t)′‖2L2(I2) − β〈v1(t), v2(t)〉L2(I2) − ‖w1(t)‖2H2 l3 − ρ2〈w1(t)′, w2(t)′〉L2(I3) = ‖(u2(t), v2(t), w2(t))‖2L − ‖(u1(t), v1(t), w1(t))‖2H − 〈(ρ1u1(t)′, βv1(t), ρ2w1(t)′), (u2(t)′, v2(t), w2(t)′)〉L. Where the weak transmission conditions were used with Φ(t) := (0, 0, 0, u1(t), v1(t), w1(t))>. Let δ > 0. By Young’s inequality and Poincaré’s inequality, there exists Cδ > 0 and c2 > 0 such that − 〈(ρ1u1(t)′, βv1(t), ρ2w1(t)′), (u2(t)′, v2(t), w2(t)′)〉L ≤ δ‖(ρ1u1(t)′, βv1(t), ρ2w1(t)′)‖2L + Cδ‖(u2(t)′, v2(t), w2(t)′)‖2L ≤ c2δ‖(u1(t), v1(t), w1(t))‖2H + Cδ‖(u2(t)′, v2(t), w2(t)′)‖2L. Since U(t) ∈ D(A ) we have u2(t) ∈ H2 l0 and w2(t) ∈ H2 l3 which implies u2(t)(l0) = 0 and w2(t)(l3) = 0. Therefore from Poincaré’s inequality we have ‖u2(t)‖L2(I1) ≤ const.‖u2(t)′‖L2(I1), ‖w2(t)‖L2(I3) ≤ const.‖w2(t)′‖L2(I3). From the three inequalities above, taking δ small enough such that c2δ ≤ 1/2, we obtain d dt F (t) ≤ ‖(u2(t), v2(t), w2(t))‖2L − 1 2 ‖(u1(t), v1(t), w1(t))‖2H + Cδ‖(u2(t)′, v2(t), w2(t)′)‖2L ≤ c3‖(u2(t)′, v2(t), w2(t)′)‖2L − 1 2 ‖(u1(t), v1(t), w1(t))‖2H (4.2) for a positive constant c3. Now, let L(t) := c4E(t)+F (t) with c4 a positive constant. Then, for c4 large enough such that 2c1 ≤ c4 and −c4 max{ρ1, β, ρ2}+c3 ≤ −1/2, it EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 11 follows from (4.1), (4.2) and Poincaré’s inequality applied to u2 and w2 that there are constants c5 and c6 such that d dt L(t) ≤ −1 2 ‖(u2(t)′, v2(t), w2(t)′)‖2L − 1 2 ‖(u1(t), v1(t), w1(t))‖2H ≤ −c5 2 ‖(u2(t), v2(t), w2(t))‖2L − 1 2 ‖(u1(t), v1(t), w1(t))‖2H ≤ −min{c5, 1} 1 2 ‖U(t)‖2H×L = −c6E(t). (4.3) Since |F (t)| ≤ c1E(t) ≤ c4E(t)/2, we have c4 2 E(t) ≤ L(t) ≤ 3c4 2 E(t). Therefore, (4.3) implies that d dtL(t) ≤ −αL(t) for some positive constant α. By Gronwall’s lemma, L(t) ≤ Ce−αtL(0), which implies E(t) ≤ 2 c4 L(t) ≤ 2 c4 Ce−αtL(0) ≤ 3Ce−αtE(0). � Now, we consider the case in which both beams are undamped. For this we need to show some regularity results. Lemma 4.2. Let I an open interval in the real line. For each g ∈ L2(I) there exists a unique v ∈ H1 0 (I) such that∫ I v′ ϕ′ = − ∫ I g ϕ ∀ϕ ∈ H1 0 (I). (4.4) Furthermore, v ∈ H2(I). Proof. It is easy to see that the bilinear form (v, ϕ) 7→ ∫ I v′ ϕ′ is continuous in H1 0 (I)×H1 0 (I) and that, due to Poincaré’s inequality, it is also coercive in H1 0 (I). Then, Lax-Milgram theorem give the existence and uniqueness of the solution v of (4.4), since ϕ 7→ ∫ I g ϕ is a continuous linear functional on H1 0 (I), whenever g ∈ L2(I). Now, (4.4) implies that (v′)′ = g in distributional sense. Since g ∈ L2(I) we have that v′ ∈ H1(I) and therefore v ∈ H2(I). � We will use the following lemma whose proof is similar to that of [8, Corollary 4.3]. Lemma 4.3. Let a < b, f ∈ L2((a, b)) and z ∈ C. For sufficiently large λ > 0 there exists a unique u ∈ H4((a, b)) such that u(4) + λu = f in (a, b), u(a) = u′(a) = 0, u′′(b) = 0, u′′′(b) = z. Theorem 4.4. Let U = (u1, v1, w1, u2, v2, w2)> ∈ D(A ). Then u1 ∈ H4(I1), v1 ∈ H2(I2) and w1 ∈ H4(I3). In particular, the transmission conditions (1.6)– (1.9) hold in the classical sense. 12 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 Proof. If F = (f1, g1, h1, f2, g2, h2)> := A U , then u2 = f1 ∈ H2 l0 , (4.5) v2 = g1 ∈ H1(I2), (4.6) w2 = h1 ∈ H2 l3 , (4.7) −u(4) 1 + ρ1u ′′ 2 = f2 ∈ L2(I1), (4.8) v′′1 − βv2 = g2 ∈ L2(I2), (4.9) −w(4) 1 + ρ2w ′′ 2 = h2 ∈ L2(I3). (4.10) From (4.6) and (4.9) we have v′′1 = βg1 + g2. (4.11) Now, let v0 be a smooth function on I2 such that v0(l1) = u1(l1) and v0(l2) = w1(l2) (we can take v0 as an affine function for example) and set v̂ := v1 − v0. Then v̂ ∈ H1 0 (I2) and, by (4.11), we have v̂′′ = βg1 + g2 − v′′0 in the distributional sense. Then v̂ satisfies∫ I2 v̂′ φ′ = − ∫ I2 (βg1 + g2 − v′′0 )φ ∀φ ∈ H1 0 (I2). By Lemma 4.2 and βg1 + g2 − v′′0 ∈ L2(I2), v̂ ∈ H2(I2). Therefore, v1 = v0 + v̂ ∈ H2(I2) and, in particular, Sobolev embedding theorem implies that v1 ∈ C1(I2). On the other hand, we consider Φ := (0, 0, 0, φ, ψ, 0)> with φ ∈ H2 l0 arbitrary and ψ ∈ H1(I2) such that ψ(l1) = φ(l1) and ψ(l2) = 0. From the definition of D(A ) we obtain〈 A U,Φ 〉 H = 〈−u(4) 1 + ρ1u ′′ 2 , φ〉L2(I1) + 〈v′′1 − βv2, ψ〉L2(I2) = −〈u′′1 , φ′′〉L2(I1) − ρ1〈u′2, φ′〉L2(I1) − β〈v2, ψ〉L2(I2) − 〈v′1, ψ′〉L2(I2). Also, integrating by parts the terms 〈u′2, φ′〉L2(I1) and 〈v′1, ψ′〉L2(I2) in the above equality, we have 〈u(4) 1 , φ〉L2(I1) = 〈u′′1 , φ′′〉L2(I1) + [ ρ1f ′ 1(l1)− v′1(l1) ] φ(l1). (4.12) By Lemma 4.3, for sufficiently large λ > 0, there exists a unique ũ1 ∈ H4(I1) such that λũ1 + ũ (4) 1 = λu1 + ρf ′′1 − f2 in I1, ũ1(l0) = ũ′1(l0) = 0, ũ′′1(l1) = 0, ũ′′′1 (l1) = ρ1f ′ 1(l1)− v′1(l1). Then, for all φ ∈ H2 l0 , integration by parts twice and the boundary conditions above yield 〈λũ1 + ũ (4) 1 , φ〉L2(I1) = λ〈ũ1, φ〉L2(I1) + 〈ũ′′1 , φ′′〉L2(I1) + [ ρ1f ′ 1(l1)− v′1(l1) ] φ(l1). (4.13) EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 13 For the same λ, adding up the term 〈λu1, φ〉L2(I1) in (4.12) we obtain 〈λu1 + u (4) 1 , φ〉L2(I1) = λ〈u1, φ〉L2(I1) + 〈u′′1 , φ′′〉L2(I1) + [ ρ1f ′ 1(l1)− v′1(l1) ] φ(l1). (4.14) From (4.6) and (4.8) we have λu1 + u (4) 1 = λu1 + ρ1f ′′ 1 − f2 = λũ1 + ũ4 1. Subtracting (4.14) from (4.13), with û1 := ũ1 − u1, we obtain 0 = λ〈û1, φ〉L2(I1) + 〈û′′1 , φ′′〉L2(I1) for all φ ∈ H2 l0 . Since û1 ∈ H2 l0 we can set φ = û1 in the last equality and we obtain û1 = 0, which implies u1 = ũ1 ∈ H4(I1). In particular, the Sobolev embedding theorem guarantees that u1 ∈ C3(I1) and therefore the transmission conditions hold in the classical sense. The proof of w1 ∈ H4(I3) is similar, and therefore w1 ∈ C3(I3). � Now, we use the following frequency domain result, which gives us a necessary and sufficient condition for the exponential stability of a C0-semigroup of contrac- tions. For its proof see [11, 12, 21]. Proposition 4.5. Let (T (t))t≥0 be a C0-semigroup of contractions in a Hilbert space H, generated by an operator A. Then the semigroup is exponentially stable if and only if iR ⊂ ρ(A) and ‖(iλI −A)−1‖L (H) ≤ C ∀λ ∈ R. (4.15) The second main result of this article is the following. Theorem 4.6. If ρ1 = ρ2 = 0 and β > 0, then the semigroup (S(t))t≥0 generated by A is exponentially stable. Proof. By Proposition 4.5 it is sufficient to show that A satisfies (4.15). First, we will show that 0 ∈ ρ(A ). Let F = (f1, g1, h1, f2, g2, h2)> ∈ H , then we have to find a U = (u1, v1, w1, u2, v2, w2)> ∈ D(A ) such that −A U = F , which is equivalent to equations (4.5)–(4.10) with −F replacing F . Then, plugging the first three equations in the last three, we obtain u (4) 1 = f2, (4.16) −v′′1 = g2 + βg1, (4.17) w (4) 1 = h2. (4.18) Now, we define the sesquilinear form B0 : H×H→ C by B0(Y,Φ) := 〈y1, φ1〉H2 l0 + 〈y′2, φ′2〉L2(I2) + 〈y3, φ3〉H2 l3 , for Y := (y1, y2, y3)>, Φ := (φ1, φ2, φ3)> ∈ H, and the antilinear functional Λ : H→ C by Λ(Φ) := 〈f2, φ1〉L2(I1) + 〈g2 + βg1, φ2〉L2(I2) + 〈h2, φ3〉L2(I3), for all Φ ∈ H. Here antilinear means linear up to conjugated scalars, i.e. Λ(Φ1 + Φ2) = Λ(Φ1) + Λ(Φ2) and Λ(αΦ) = αΛ(Φ). It is easy to see that B0 : H×H→ C 14 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 is continuous and coercive, and that Λ : H → C is continuous. Then, by the Lax-Milgram theorem, there exists a unique Y := (y1, y2, y3)> ∈ H such that B0(Y,Φ) = Λ(Φ) for all Φ ∈ H. (4.19) As in the proof of Proposition 3.1, we obtain that U := (y1, y2, y3,−f1,−g1,−h1)> ∈ D(A ) and satisfies−A U = F . On the other hand, if Ũ := (ũ1, ṽ1, w̃1, ũ2, ṽ2, w̃2)> ∈ D(A ) solves −A Ũ = F , then B0((ũ1, ṽ1, w̃1)>,Φ) = Λ(Φ) holds for all Φ ∈ H by the definition of D(A ) and the weak transmission conditions. Therefore U = Ũ and A is a bijection. Since A is the generator of C0-semigroup of contractions by Theorem 3.2, A is closed and hence 0 ∈ ρ(A ). By the Sobolev embedding theorem, we obtain that A −1 is a compact operator on H , and therefore, the spectrum of A consists of eigenvalues only. Thus, we have to establish that there are no purely imaginary eigenvalues. Let λ ∈ R, λ 6= 0, and U ∈ D(A ) with A U = iλU , i.e. iλu1 = u2, (4.20) iλv1 = v2, (4.21) iλw1 = w2, (4.22) iλu2 + u (4) 1 = 0, (4.23) iλv2 − v′′1 + βv2 = 0, (4.24) iλw2 + w (4) 1 = 0. (4.25) From the dissipativity of A , 0 = Re〈(iλI −A )U,U〉H = Re(iλ‖U‖H )− Re〈A U,U〉H = β‖v2‖2L2(I2). Then, v1 = v2 = 0, i.e. U = (u1, 0, w1, u2, 0, w2)>. Since v1 ∈ H1(I2) the Sobolev inequality implies that v1(x) = 0 for all x ∈ I2 = [l1, l2]. Multiplying (4.20) by iλ and substituting in (4.23), we obtain u (4) 1 − λ2u1 = 0 in (l0, l1). (4.26) Moreover u1 satisfies the boundary conditions u1(l0) = u′1(l0) = 0 and u′′1(l1) = u′′′1 (l1) = 0. (4.27) Let x := l1 + (l0 − l1)η, η ∈ [0, 1]. Then u1(x) = u1(l1 + (l0 − l1)η) =: z(η) and dku1 dxk = 1 (l0 − l1)k dkz dηk . Therefore, problem (4.26)–(4.27) can be transformed into z(4) − a2z = 0 in (0, 1), z(0) = z′′(0) = z′′′(0) = 0, z(1) = z′(1) = 0, (4.28) where a := (l0 − l1)2|λ| 6= 0. The general solution of the ordinary differential equation in (4.28) is z(η) = c1 cosh( √ aη) + c2 sinh( √ aη) + c3 cos( √ aη) + c4 sin( √ aη). Now, we see that the boundary conditions in (4.28) imply that c1 = c2 = c3 = c4 = 0 and therefore z ≡ 0, i.e. u1 ≡ 0. In similar way we obtain w1 ≡ 0. From (4.20) EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 15 and (4.22) we have also u2 ≡ 0 and w2 ≡ 0. Thus, U = 0 and we conclude that iR ⊂ ρ(A ). Now, we will show that sup λ∈R ‖(iλI −A )−1‖L (H ) <∞. (4.29) If this inequality is false, then there are sequences (λn)n∈N ⊂ R and (Un)n∈N ⊂ D(A ) such that |λn| −−−−→ n→∞ ∞, ‖Un‖H = 1 for all n ∈ N and ‖(iλnI −A )Un‖H → 0 n→∞. (4.30) Let Fn := (iλnI−A )Un. Because the standard norm on H2(I1)×H1(I2)×H2(I3)× L2(I1)× L2(I2)× L2(I3) is equivalent to the norm ‖ · ‖H on H , (4.30) implies iλnu1,n − u2,n = f1,n → 0 in H2(I1), (4.31) iλnv1,n − v2,n = g1,n → 0 in H1(I2), (4.32) iλnw1,n − w2,n = h1,n → 0 in H2(I3), (4.33) iλnu2,n + u (4) 1,n = f2,n → 0 in L2(I1), (4.34) iλnv2,n − v′′1,n + βv2,n = g2,n → 0 in L2(I2), (4.35) iλnw2,n + w (4) 1,n = h2,n → 0 in L2(I3), (4.36) where ρ1 = ρ2 = 0. From the dissipativity of A , it follows that 0← Re(〈(iλnI −A )Un, Un〉H ) = Re[iλn‖Un‖2H − 〈A Un, Un〉H ] = β‖v2,n‖2L2(I2), i.e. ‖v2,n‖L2(I2) → 0. (4.37) Now, (4.32), (4.35), and (4.37) imply ‖v1,n‖2L2(I2) → 0, |λn|‖v1,n‖L2(I2) → 0, ‖λ−1 n v′′1,n‖L2(I2) → 0. (4.38) From this and the Gagliardo-Nirenberg inequality, it follows that ‖v′1,n‖L2(I2) ≤ ‖λnv1,n‖1/2L2(I2)‖λ −1 n v′′1,n‖ 1/2 L2(I2) + ‖v1,n‖L2(I2) → 0. (4.39) Substituting v2,n = iλnv1,n − g1,n in (4.35), we have g2,n = −λ2 nv1,n − iλng1,n − v′′1,n + βv2,n. (4.40) Now, taking L2-product of (4.40) with (l2 − x)v′1,n for x ∈ I2, we obtain 〈g2,n, (l2 − x)v′1,n〉L2(I2) = λ2 n〈v1,n, (l2 − x)v′1,n〉L2(I2) − ‖λnv1,n‖2L2(I2) + (l2 − l1)|λnv1,n(l1)|2 − iλn〈g1,n, v1,n〉L2(I2) + 〈i (l2 − x)g′1,n, λnv1,n〉L2(I2) + i(l2 − l1)λng1,n(l1)v1,n(l1)− ‖v′1,n‖2L2(I2) + 〈v′′1,n, (l2 − x)v1,n〉L2(I2) + (l2 − l1)|v′1,n(l1)|2 + 〈βv2,n, (l2 − x)v′1,n〉L2(I2), 16 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 or equivalently − ‖λnv1,n‖2L2(I2) + (l2 − l1)|λnv1,n(l1)|2 − ‖v′1,n‖2L2(I2) + (l2 − l1)|v′1,n(l1)|2 + 2Re {〈βv2,n, (l2 − x)v′1,n〉L2(I2)} = 〈g2,n, (l2 − x)v′1,n〉L2(I2) + 〈βv2,n, (l2 − x)v′1,n〉L2(I2) − λ2 n〈v1,n, (l2 − x)v′1,n〉L2(I2) + 〈ig1,n, λnv1,n〉L2(I2) − 〈i(l2 − x)g′1,n, λnv1,n〉L2(I2) − i(l2 − l1)λng1,n(l1)v1,n(l1) − 〈v′′1,n, (l2 − x)v′1,n〉L2(I2). (4.41) From (4.40) we have that βv2,n = g2,n + λ2 nv1,n + iλng1,n + v′′1,n and therefore 〈βv2,n, (l2 − x)v′1,n〉L2(I2) = 〈g2,n, (l2 − x)v′1,n〉L2(I2) + λ2 n〈v1,n, (l2 − x)v′1,n〉L2(I2) + 〈ig1,n, λnv1,n〉L2(I2) − 〈i(l2 − x)g′1,n, λnv1,n〉L2(I2) − i(l2 − l1)λng1,n(l1)v1,n(l1) + 〈v′′1,n, (l2 − x)v′1,n〉L2(I2). Substituting this in (4.41), we obtain (l2 − l1)|λnv1,n(l1)|2 + (l2 − l1)|v′1,n(l1)|2 = 2 Re { 〈g2,n, (l2 − x)v′1,n〉L2(I2) − 〈i(l2 − x)g′1,n, λnv1,n〉L2(I2) − i(l2 − l1)g1,n(l1)λnv1(l1)− 〈βv2,n, (l2 − x)v′1,n〉L2(I2) + 〈ig1,n, λnv1,n〉L2(I2) } + ‖λnv1,n‖2L2(I2) + ‖v′1,n‖2L2(I2). Now, by the Cauchy-Schwarz inequality and Young’s inequality, for each ε > 0 there exists a Cε > 0 such that (l2 − l1)|λnv1,n(l1)|2 + (l2 − l1)|v′1,n(l1)|2 ≤ 2 [ (l2 − l1)‖g2,n‖L2(I2)‖v′1,n‖L2(I2) + (l2 − l1)‖g′1,n‖L2(I2)‖λnv1,n‖L2(I2) + (l2 − l1) ( ε|λnv1,n(l1)|2 + Cε|g1,n(l1)|2 ) + β(l2 − l1)‖v2,n‖L2(I2)‖v′1,n‖L2(I2) + ‖g1,n‖L2(I2)‖λnv1,n‖L2(I2) ] + ‖λnv1,n‖2L2(I2) + ‖v′1,n‖2L2(I2). From this inequality with ε = 1/4, (4.32), (4.35), (4.37)–(4.39), and the trace theorem, it follows that |λnv1,n(l1)| → 0 and |v′1,n(l1)| → 0 as n→∞. Therefore, |λnu1,n(l1)| → 0 and |u′′′1,n(l1)| → 0 as n→∞] (4.42) because the transmission conditions (1.5) and (1.6) with ρ1 = 0. Now, substituting u2,n = iλnu1,n − f1,n in (4.34), we obtain f2,n = −λ2 nu1,n − iλnf1,n + u (4) 1,n. (4.43) EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 17 Taking L2-product of (4.43) with (x− l0)u′1,n for x ∈ I1, we obtain with integration by parts that 〈f2,n, (x− l0)u′1,n〉L2(I1) = ‖λnu1,n‖2L2(I1) + λ2 n〈u1,n, (x− l0)u′1,n〉L2(I1) − (l1 − l0)|λnu1,n(l1)|2 + iλn〈f1,n, u1,n〉L2(I1) + 〈i(x− l0)f ′1,n, λnu1,n〉L2(I1) − i(l1 − l0)f1,n(l1)λnu1,n(l1) + ‖u′′1,n‖2L2(I1) − 〈u′′1,n, (x− l0)u′′′1,n〉L2(I1) + (l1 − l0)u′′′1,n(l1)u′1,n(l1). (4.44) From (4.43) we have λ2 nu1,n = −f2,n − iλnf1,n + u (4) 1,n and therefore λ2 n〈u1,n, (x− l0)u′1,n〉L2(I1) = −〈f2,n, (x− l0)u′1,n〉L2(I1) + 〈if1,n, λnu1,n〉L2(I1) + 〈i(x− l0)f ′1,n, λnu1,n〉L2(I1) − i(l1 − l0)f1,n(l1)λnu1,n(l1) + 2‖u′′1,n‖2L2(I1) + 〈u′′1,n, (x− l0)u′′′1,n〉L2(I1) + (l1 − l0)u′′′1,n(l1)u′1,n(l1). (4.45) Now, plugging (4.45) in (4.44), we obtain ‖λnu1,n‖2L2(I1) + 3‖u′′1,n‖2L2(I1) = 2 Re { 〈f2,n , (x− l0)u′1,n〉L2(I1) − 〈if1,n , λnu1,n〉L2(I1) − 〈i(x− l0)f ′1,n , λnu1,n〉L2(I1) + i(l1 − l0)f1,n(l1)λnu1,n(l1) − (l1 − l0)u′′′1,n(l1)u′1,n(l1) } + (l1 − l0)|λnu1,n(l1)|2. (4.46) Note that the Gagliardo-Nirenberg inequality implies ‖u′1,n‖L2(I1) ≤ ‖u1,n‖1/2L2(I1)‖u ′′ 1,n‖ 1/2 L2(I1) + ‖u1,n‖L2(I1) and thus ‖u′1,n‖2L2(I1) ≤ 3‖u1,n‖2L2(I1) + ‖u′′1,n‖L2(I1). (4.47) Moreover, from the trace theorem that there exists a positive constant C such that |u′1,n(l1)| ≤ C‖u1,n‖H2(I1) ≤ C‖Un‖H = C. (4.48) Let ε1, ε2, and ε3 be positive numbers. By Young’s inequality in (4.46), there are positive constants Cε1 , Cε2 , and Cε3 such that ‖λnu1,n‖2L2(I1) + 3‖u′′1,n‖2L2(I1) ≤ ε1‖u′1,n‖2L2(I1) + Cε1‖f2,n‖2L2(I1) + ε2‖λnu1,n‖2L2(I1) + Cε2‖f1,n‖2L2(I1) + ε3‖λnu1,n‖2L2(I1) + Cε3‖f ′1,n‖2L2(I1) + 2(l1 − l0) { |f1,n(l1)‖λnu1,n(l1)| + |u′′′1,n(l1)‖u′1,n(l1)|+ |λnu1,n(l1)|2 } ≤ 3ε1‖λnu1,n‖2L2(I1) + ε1‖u′′1,n‖2L2(I1) + Cε1‖f2,n‖2L2(I1) + ε2‖λnu1,n‖2L2(I1) + Cε2‖f1,n‖2L2(I1) + ε3‖λnu1,n‖2L2(I1) + Cε3‖f ′1,n‖2L2(I1) + 2(l1 − l0) { |f1,n(l1)‖λnu1,n(l1)|+ C|u′′′1,n(l1)| + |λnu1,n(l1)|2 } , (4.49) where we used (4.47) and (4.48). Choosing ε1, ε2, and ε3 small enough such that 3ε1 + ε2 + ε3 < 1/2, we obtain from (4.31), (4.34), (4.42), and (4.49) that ‖λnu1,n‖L2(I1) → 0, and ‖u′′1,n‖L2(I1) → 0 as n→∞. (4.50) 18 B. BARRAZA, J. HERNÁNDEZ, G. VERGARA EJDE-2022/30 Similarly we conclude that ‖λnw1,n‖L2(I3) → 0 and ‖w′′1,n‖L2(I3) → 0 as n→∞. (4.51) Therefore, ‖Un‖H → 0 as n → ∞, because (4.31), (4.32), (4.33), (4.37), (4.39), (4.50), and (4.51), which is a contradiction. Thus we have proved that the (S(t))t≥0 is exponentially stable. � Acknowledgments The authors would like to thank MINCIENCIAS (former COLCIENCIAS) for the financial support under Project 121571250194. The authors also want to thank the reviewers for their valuable comments and suggestions. References [1] M. Alves, J. E. Muñoz Rivera, M. Sepúlveda, O. Vera; The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt dissipation, SIAM J. Appl. Math. 74 (2) (2014), 345–365. [2] K. Ammari, D. Jellouli, M. Mehrenberger; Feedback stabilization of a coupled string-beam system, Netw. Heterog. Media 4 (1) (2009), 19–34. [3] K. Ammari, M. Mehrenberger; Study of the nodal feedback stabilization of a string-beams network, J Appl Math Comput 36 (1) (2011), 441–458. [4] K. Ammari, D. Mercier, V. Regnier, J. Valein; Spectral analysis and stabilization of a chain of serially Euler-Bernoulli beams and strings, Commun. Pure Appl. Anal 11 (2) (2012), 785–807. [5] K. Ammari, F. Shel; Stability of a tree-shaped network of strings and beams, Mathematical Methods in the Applied Sciences, vol. 41 (17), (2018) 7915–7935. [6] H. T. Banks, R. C. Smith, Y. Wang; Modeling aspects for piezoelectirc patch activation of shells, plates and beams, Quart. Appl. Math., LIII (1995), 353–381. [7] H. T. Banks, R. C. Smith, Y. Wang; Smart Materials Structures, Wiley, 1996. [8] B. Barraza Mart́ınez, R. Denk, J. Hernández Monzón, F. Kammerlander, M. Nendel; Regu- larity and asymptotic behavior for a damped plate-membrane transmission problem. Journal of Mathematical Analysis and Applications, 474(2) (2019),1082–1103. [9] W. D. Bastos, C. A. Raposo; Transmission problems for waves with frictional damping, Electronic Journal of Differential Equations, 2010 (2010) no. 60, pp. 1–10. [10] R. Dáger, E. Zuazua; Wave propagation, observation and control in 1-d flexible multi- structures, volume 50 of Mathématiques & Applications (Berlin), Springer-Verlag, 2006. [11] L. M. Gearhart; Spectral theory for contraction semigroups on Hilbert space, Trans. Am. Math. Soc. 236 (1978), 385–394. [12] F. L. Huang; Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ., 1 (1985) 43–56. [13] Z. J. Han, E. Zuazua; Decay rates for elastic-thermoelastic star-shaped networks, Networks & Heterogeneous Media, 12 (3) (2017), 461–488. [14] F. Hassine; Energy decay estimates of elastic transmission wave/beam systems with a local kelvin voigt damping. International Journal of control, 89(10) (2016), 1933–1950. [15] J . Lagnese, G. Leugering, E. J. P. G. Schmidt; Modeling, Analysis of Dynamic Elastic Multi-link Structures, Birkhäuser, Boston-Basel-Berlin, 1994. [16] Y. F. Li, Z. J. Han, G. Q. Xu; Explicit decay rate for coupled string-beam system with localized frictional damping, Applied Mathematics Letters, 78 (2018), 51–58. [17] K. Liu, Z. Liu; Exponential decay of energy of the euler-bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM J. Control Optim, 36 (3) (1998), 1086–1098. [18] T. K. Maryati, J. Muñoz Rivera, A. Rambaud, O. Vera; Stability of an n-component Tim- oshenko beam with localized Kelvin-Voigt and frictional dissipation, Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 136, 1-18. [19] J . Muñoz Rivera, H. Portillo; The transmission problem for thermoelastic beams. Journal of Thermal Stresses, 24 (12) (2001), 1137–1158. EJDE-2022/30 EXPONENTIAL STABILITY OF A TRANSMISSION PROBLEM 19 [20] A. Pazy; Semigroups of linear operators and applications to partial differential equations, volume 44. Springer Science & Business Media, 2012. [21] J. Prüss; On the spectrum of C0-semigroups, Trans. Am. Math. Soc. 284 (1984) 847–857. [22] C. Raposo, W. Bastos, J. Ávila; A transmission problem for Euler-Bernoulli beam with kelvin-voigt damping. Applied Mathematics and Information Sciences, 5 (1) (2011), 17–28. [23] M. Rissel, Y. G. Wang; Remarks on exponential stability for a coupled system of elasticity and thermoelasticity with second sound, J. Evol. Equ. (2020). https://doi.org/10.1007/s00028- 020-00636-4. [24] F. Shel; Exponential Stability of a Network of Beams, J. Dyn. Control Syst. 21 (2015), 443– 460. [25] F. Shel. Thermoelastic stability of a composite material, J. Differential Equations, 269 (2020), 9348–9383. [26] C. Wang; Spectral Analysis for a Wave/Plate Transmission System, Advances in Mathemat- ical Physics, vol. 2019, Article ID 7849561, 9 pages, 2019. [27] F. Wang, J. M. Wang; Stability of an interconnected system of Euler-Bernoulli beam and wave equation through boundary coupling, Systems & Control Letters 138 (2020), 104–664. Bienvenido Barraza Mart́ınez Universidad del Norte, Departamento de Matemáticas y Estad́ıstica, Barranquilla, Colombia Email address: bbarraza@uninorte.edu.co Jairo Hernández Monzón Universidad del Norte, Departamento de Matemáticas y Estad́ıstica, Barranquilla, Colombia Email address: jahernan@uninorte.edu.co Gustavo Vergara Rolong Universidad del Norte, Departamento de Matemáticas y Estad́ıstica, Barranquilla, Colombia Email address: gvergaraa@uninorte.edu.co 1. Introduction 2. Base Spaces 3. Well posedness 4. Exponential stability Acknowledgments References