WELL-POSEDNESS AND EXPONENTIAL DECAY OF SOLUTIONS FOR A TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY

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1 Electronic Journal of Differential Equations, Vol. 7 (7), No. 74, pp. 3. ISSN: URL: or WELL-POSEDNESS AND EXPONENTIAL DECAY OF SOLUTIONS FOR A TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY GONGWEI LIU Abstract. In this article, we consider a transmission problem in a bounded domain with a distributed delay in the first equation. Using a semigroup theorem, we prove the existence and uniqueness of global solution under suitable assumptions on the weight of damping and the weight of distributed delay. Also we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.. Introduction In this article, we study the transmission problem with a distributed delay, τ µ (s)u t (t s)ds =, x, t >, u tt (x, t) au xx (x, t) + µ u t (x, t) + v tt (x, t) bv xx (x, t) =, x (, L ), t, (.) under the boundary and the transmission conditions u(, t) = u(l 3, t) =, u(l i, t) = v(l i, t), i =,, au x (L i, t) = bv x (L i, t), i =, and the initial conditions u(x, ) = u (x), u t (x, ) = u (x), x, v(x, ) = v (x), v t (x, ) = v (x), x (, L ), u t (x, t) = f (x, t), x, t (, τ ) (.) (.3) where < < L < L 3, = (, ) (L, L 3 ), a, b, µ are positive constants, and the initial data (u, u, v, v, f ) belongs to suitable space. Moreover, µ : [, τ ] R is a bounded function, where and τ are two real number satisfying < τ. It is known that transmission problems happen frequently in applications where the domain is occupied by two or several materials, whose elastic properties are different, joined together over the whole of a surface. From the mathematical point of view, a transmission problem for wave propagation consists on a hyperbolic Mathematics Subject Classification. 35B37, 35L55, 74D5, 93D5, 93D. Key words and phrases. Transmission problem; distributed delay; exponential decay. c 7 Texas State University. Submitted April, 7. Published July, 7.

2 G. LIU EJDE-7/74 equation for which the corresponding elliptic operator has discontinuous coefficients, see [, 6]. In absence of delay (µ (s) = ), the system (.)-(.3) has been investigated in [] by Bastaos and Raposo; for = [, ], they showed that the well-posedness and exponential stability of the total energy. Rivera and Oquendo [7] studied the transmission problem of viscoelastic waves and established that the dissipation produced by the viscoelastic part is strong enough to produce the exponential stability, no matter small its size is. Interested readers are referred to [, 3, 4, 6], for more results concerning other types of transmission problems. Introducing the delay term makes the problem different from those considered in the literatures. Delay effect arises in many applications depending not only on the present state but also on some past occurrences. It may turn a well-behaved system into a wild one. The presence of delay may be a source of instability. For example, it was shown in [5, 4, 8,,, 6] that an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used. Here we mention the some interesting results on the relation between the delay term and source term [,, 7, 3]. Nicaise and Pignotti [] considered the wave equation with liner frictional damping and internal distributed delay τ u tt u + µ u t + a(x) µ (s)u t (t s)ds = in (, ), with initial and mixed Dirichlet-Neumann boundary conditions and a is a suitable function. They obtained exponential decay of the solution under the assumption that a τ µ (s)ds < µ. The authors also obtained the same result when the distributed delay acted on the part of the boundary. Mustafa and Kafini [9] considered a thermoelastic system with internal distributed delay, they obtained exponential stability under suitable condition; for the boundary distributed delay, similar result was obtained by [8]. Here we also mention the work on Timoshenko system with second sound and internal distributed delay in [] by Apalara, and wave equation with strong distributed delay [5] by Messsaoudi et al. The effect of the delay term u t (x, t τ) in the transmission system has been investigated by Benseghir [3]. Recently, the well-posedness and the decay of solution for a transmission problem in a bounded domain with a viscoelastic term and a delay term u t (x, t τ) have been studied in [9, 5]. In this work we consider the transmission system (.)-(.3), and prove the wellposedness and the exponential stability. Our work extends the stability results in [, 3] to the transmission system with distributed delay. The plan of this paper is as follows. In section, we present some notations and assumptions needed for our work, and then establish the well-posedness of our problem by virtue of the semigroup methods. In section 3, we state and prove the stability result by introducing a suitable Lyapunov function.

3 EJDE-7/74 TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY 3. Well-posedness of the problem Throughout this paper, c and c i are used to denote the generic positive constant. From now on, we shall omit x and t in all functions of x and t if there is no ambiguity. As in [], we introduce the new variable z(x, ρ, t, s) = u t (x, t ρs), x, ρ (, ), t >, s (, τ ). Then the above variable z satisfies sz t (x, ρ, t, s) + z ρ (x, ρ, t, s) =, x, ρ (, ), t >, s (, τ ). (.) Consequently, system (.) is equivalent to τ u tt (x, t) au xx (x, t) + µ u t (x, t) + µ (s)z(x,, t, s)ds =, x, t >, v tt (x, t) bv xx (x, t) =, x (, L ), t, sz t (x, ρ, t, s) + z ρ (x, ρ, t, s) =, x, ρ (, ), t >, s (, τ ). Defining U = (u, v, ϕ, ψ, z) T, we formally get that U satisfies U = AU, U() = U = (u, v, u, v, f ), where the operator A is defined as u ϕ v A ϕ ψ = ψ au xx µ ϕ τ µ (s)z(x,, t, s)ds bv xx. z s z ρ(x, ρ, t, s) Introducing the space X = { (u, v) = H () H (, L ) : u(, t) = u(l 3, t) =, we define the energy space as u(l i, t) = v(l i, t), au x (L i, t) = bv x (L i, t), i =, }, H = X L () L (, L ) L ( (, ) (, τ ) ) equipped with the inner product U, Ũ H = (ϕ ϕ + au x ũ x )dx + The domain of A is + τ L (ψ ψ + bv x ṽ x )dx s µ (s) z(x, ρ, s) z(x, ρ, s) ds dρ dx. D(A) = { (u, v, ϕ, ψ, z) T H : (u, v) (H () H (, L )) X, Clearly, D(A) is dense in H. ϕ H (), ψ H (, L ), z(x,, s) = ϕ, z, z ρ L ( (, ) (, τ ) )}. (.) (.3)

4 4 G. LIU EJDE-7/74 Concerning the weight of the distributed delay, we assume that τ µ (s) ds µ. (.4) The well-posedness of the system (.), (.)and (.3) is ensured by the following theorem. Theorem.. Under the assumption (.4), for any U H, there exists a unique weak solution U C(R +, H) of problem (.3). Moreover, if U D(A), then U C(R +, D(A)) C(R +, H). Proof. We use the semigroup approach and the Hille-Yosida theorem to prove the well-posedness of the problem. First, we prove that the operator A is dissipative. Indeed, for U = (u, v, ϕ, ψ, z) D(A), where ϕ(l i ) = ψ(l i ), i =,, we have AU, U H ( τ = auxx µ ϕ + L bv xx ψ dx + L µ (s)z(x,, t, s)ds ) ϕ dx + a bv x ψ x dx + τ For the last term of the right hand side of (.5), we have u x ϕ x dx µ (s) zz ρ dρ ds dx. (.5) = τ + µ (s) zz ρ dρ ds dx τ τ µ (s) d dρ z(x, ρ, t, s) dρ ds dx µ (s) z (x,, s)dsx τ µ (s) ds z (x,, s)dx. (.6) Integrating by parts in (.5), and noticing the fact z(x,, t, s) = ϕ(x, t), from (.6), we have ( AU, U H = [au x ϕ] + [bv x ψ] L µ τ ) µ (s) ds ϕ dx + τ τ µ (s) z (x,, s) ds dx µ (s)z(x,, s)ϕ ds dx. Using Young s inequality, and the equality ϕ(l i ) = ψ(l i ), i =,, from (.) and (.6) we have τ AU, U H (µ µ (s) ) ϕ dx τ µ (s) z (x,, s) ds dx + τ µ (s) z (x,, s) ds dx ( τ ) µ µ (s) ϕ dx, by (.4). Hence, the operator A is dissipative.

5 EJDE-7/74 TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY 5 Next, we prove the operator A is maximal. It is sufficient to show that the operator λi A is surjective for a fixed λ >. Indeed, given F = (f, f, f 3, f 4, f 5 ) H, we prove that there exists U = (u, v, ϕ, ψ, z) D(A) satisfying that is (λi A)U = F, (.7) λu ϕ = f, λv ψ = f, τ λϕ au xx + µ ϕ + µ (s) z(x,, t, s)ds = f 3, λψ bv xx = f 4, λsz + z ρ = sf 5. Suppose we have obtained (u, v) with the suitable regularity, then ϕ = λu f, ψ = λv f, (.8) (.9) so we have ϕ H () and ψ H (, L ). Moreover, using the approach as in Nicaise and Pignotti [], we obtain that the last equation in (.8) with z(x,, s) has a unique solution It follows from (.9) that ρ z(x, ρ, s) = ϕ(x)e λρs + se λρs e λσs f 5 (x, σ, s)dσ. ρ z(x, ρ, s) = λue λρs f e λρs + se λρs e λσs f 5 (x, σ, s)dσ, (.) in particular, z(x,, s) = λue λs + z (x, s) with z L ( (, τ )) defined by z (x, s) = f e λs + se λs e λσs f 5 (x, σ, s)dσ. By (.8) and (.9), the functions (u, v) satisfy the equations where ku au xx = f, λ v bv xx = f + λf 4, τ k = λ + λµ + λ µ (s) e λs ds >, f = f 3 + (λ + λµ )f τ which can be reformulated as ( ku au xx )w dx = L for any (w, w ) X. (λ v bv xx )w dx = µ (s) z (x, s)ds L (), L fw dx, (f + λf 4 )w dx, (.) (.)

6 6 G. LIU EJDE-7/74 Integrating by parts in (.), we obtain that the variational formulation corresponding to (.) takes the form Φ ( (u, v), (w, w ) ) = l(w, w ), (.3) where the bilinear form Φ : (X, X ) R and the linear form l : X R are defined by Φ ( (u, v), (w, w ) ) = kuw dx + and + L l(w, w ) = L au x w x [au x w ] + λ vw dx v x w x dx [bv x w ] L, fw dx + L (f + λf 4 )w dx. By the properties of the space X, it is easy to see that Φ is continuous and coercive, and l is continuous. Applying the Lax-Milgram theorem, we deduce that problem l (.3) admits a unique solution (u, v) X for all (w, w ) X. It follows from (.) that (u, v) ( (H () H (, L )) ) X. Thus, the operator λi A is surjective for any λ >. Hence the Hille-Yosida theorem guarantees the existence of a unique solution to the problem (.7). This completes the proof. 3. Exponential stability In this section, we state and prove the stability result for the energy of the system (.)-(.3). For the regular solution of the system (.)-(.3), we define the energy as (see [3]) E (t) = u t (x, t)dx + a u x(x, t)dx, (3.) L L E (t) = vt (x, t)dx + b vx(x, t)dx. (3.) And the total energy is defined as E(t) = E (t) + E (t) + τ s µ (s) z (x, ρ, t, s) ds dρ dx. (3.3) For the energy decay result, we assume a restriction on the weight of the distribute delay and the damping as τ µ (s) ds < µ. (3.4) The stability result reads as follows. Theorem 3.. Let (u, v, z) be the solution of the system (.), (.) and (.3). Assume (3.4) and a b < + L 3 L (L ), L 3 > 3(L ). (3.5) Then there exist two positive constants K and κ, such that E(t) Ke κt, t. (3.6) The proof will be established through the following Lemmas.

7 EJDE-7/74 TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY 7 Lemma 3.. Let assumption (3.4) holds. Then the energy functional defined by (3.3), satisfies the estimate ( τ ) E (t) µ µ (s) ds u t (x, t)dx. (3.7) Proof. By differentiating (3.), using the first equation in (.), and integrating by parts, we obtain τ E (t) = [au x u t ] µ u t (x, t)dx µ (s)z(x,, t, s)u t (x, t) ds dx. Similarly, E (t) = [bv x v t ] L. Noticing that z(x,, t, s) = u t (x, t), from (.), we obtain d τ s µ (s) z (x, ρ, t, s) ds dρ dx dt = τ µ (s) z (x,, t, s) ds dx + τ µ (s) u t (x, t) ds dx. Meanwhile, using Young s inequality, we have τ µ (s)z(x,, t, s)u t (x, t) ds dx τ τ µ (s) ds u t (x, t)dx + τ µ (s) z (x,, t, s) ds dx. Combining the above equalities and using (3.4), we show that (3.7) holds, where we also use the fact [au x u t ] = [bv x v t ] L from (.). As in [5], we define the functional τ I(t) = se ρs µ (s) z (x, ρ, t, s) ds dρ dx, then we have the following estimate. Lemma 3.3. The functional I(t) satisfies the estimate τ I (t) e τ µ (s) z (x,, t, s) ds dx ( τ ) + µ (s) ds u t (x, t)dx τ e τ s µ (s) z (x, ρ, t, s) ds dρ dx. Proof. By differentiating I(t) and using the third equation in (.), we obtain τ I (t) = e ρs µ (s) d dρ z (x, ρ, t, s)dρ ds dx = τ µ (s) τ d dρ (e ρs z (x, ρ, t, s))dρ ds dx s µ (s) e ρs z (x, ρ, t, s) ds dρ dx. (3.8)

8 8 G. LIU EJDE-7/74 Hence I (t) = τ ( τ e s µ (s) z (x,, t, s) ds dx + τ s µ (s) e ρs z (x, rho, t, s) dρ ds dx. ) µ (s) ds u t (x, t)dx Recalling e s e ρs, for all ρ [, ], and e s e τ, for all s [, τ ], we obtain (3.8). Now we define the functional D(t) = uu t dx + µ Then we have the following estimate. L u dx + vv t dx. Lemma 3.4. The functional D(t) satisfies L L D (t) (a ε C) u xdx b vxdx + u t dx + vt dx + τ τ µ (s) ds µ (s) z (x,, t, s) ds dx. 4ε (3.9) Proof. Taking the derivative of D(t) with respect to t, using (.), we obtain L L D (t) = u t dx + vt dx a u xdx vxdx L τ (3.) µ (s)z(x,, t, s)u(x, t) ds dx + [au x u] + [bv x v] L. It follows from the boundary condition (.) that [au x u] + [bv x v] L =. Using the boundary condition (.), we obtain u (x, t) = ( x u x (x, t)dx ) L L u x(x, t)dx, x [, ], u (x, t) (L 3 L ) L3 L u x(x, t)dx, x [L, L 3 ], which imply the following Poincaré s inequality u (x, t)dx C u x(x, t)dx, x, (3.) where C = max{, L 3 L } is the Poincaré s constant. Using Young s inequality and (3.), we have τ µ (s)z(x,, t, s)u(x, t) ds dx ε C u x(x, t)dx + τ τ µ (s) ds µ (s) z (x,, t, s) ds dx, 4ε for any ε >. Inserting the above estimates in (3.), then (3.9) is fulfilled.

9 EJDE-7/74 TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY 9 Inspired by [3], we introduce the functional x L, x [, ], q(x) = x L+L3, x [L, L 3 ], + L L3 L (L (x L ) ), x [, L ]. (3.) We define the two functionals F (t) = q(x)u x u t dx, L F (t) = q(x)v x v t dx. Then, we have the following estimates. Lemma 3.5. For any ε >, the functionals F(t) and F (t) satisfy F (t) C(ε ) and + C(ε ) u t dx + ( a + ε ) τ µ (s) ds τ u xdx µ (s) z (x,, t, s) ds dx a 4 [(L 3 L )u x(l, t) + u x(, t)] 4 [u t (, t) + (L 3 L )u t (L, t)], (3.3) F (t) = + L 3 L ( L L vt dx + bv 4(L ) xdx ) + 4 v t (, t) + L 3 L vt (L, t) + b 4 4 [(L 3 L )vx(l, t) + vx(l, t)]. (3.4) Proof. Taking the derivative of F (t) with respect to t and using (.), we have F (t) = = q(x)u x u tt q(x)u xt u t dx q(x)u x ( auxx µ u t q(x)u xt u t dx. Integrating by parts, we have τ µ (s)z(x,, t, s)ds ) dx q(x)u xt u t dx = q (x)u t dx + [q(x)u t ], q(x)au x u xx dx = aq (x)u xdx + [aq(x)u x]. (3.5)

10 G. LIU EJDE-7/74 Inserting the above two equalities into (3.5), and noticing (3.) and Young s inequality, we obtain F (t) = u t dx + u xdx [aq(x)u x] ( τ [q(x)u t ] + q(x)u x µ u t + µ (s)z(x,, t, s)ds ) dx τ C(ε ) u t dx + ( a + ε ) u xdx [aq(x)u x] [q(x)u t ] τ τ + C(ε ) µ (s) ds µ (s) z (x,, t, s) ds dx, (3.6) for any ε >. On the other hand, by the boundary conditions (.), we have [q(x)u t ] = 4 [u t (, t) + (L 3 L )u t (L, t)], [aq(x)u x] = a 4 [(L 3 L )u x(l, t) + u x(, t)]. Inserting the above two equalities into (3.6), then (3.6) gives (3.3). By the same method, taking the derivative of F (t) with respect to t, we have L F (t) = = L = + L 3 L 4(L ) L q(x)v xt v t dx q(x)v x v tt dx q (x)v t dx [q(x)v t ] L + ( L v t dx + L L bq (x)v xdx [bq(x)v x] L bv xdx ) + 4 v t (, t) + L 3 L vt (L, t) + b 4 4 [(L 3 L )vx(l, t) + vx(l, t)]. Hence, the proof is complete. Proof of Theorem 3.. We define the Lyapunov functional L(t) = N E(t) + N I(t) + γ F (t) + γ F (t) + γ 3 D(t), (3.7) where N, N, γ, γ, γ 3 are positive constants that will be chosen later. It follows from the boundary conditions (.) that a u x(l i, t) = bv x(l i, t), i =,. (3.8)

11 EJDE-7/74 TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY Taking the derivative of (3.7) with respective to t, using the above lemmas and (3.8), we have L (t) { N (µ τ { N e τ γ C(ε ) µ (s) ds) N τ τ µ (s) ds γ 3 τ µ (s) z (x,, t, s) ds dx { (a ε C )γ 3 ( a + ε )γ } u xdx { + L 3 L 4(L ) γ } L + γ 3 { + L 3 L 4(L ) γ γ 3 } N e τ τ L µ (s) ds γ C(ε ) γ 3 bv xdx v t dx τ s µ (s) z (x, ρ, t, s) ds dρ dx µ (s) ds 4ε } { }( γ γ 4 u t (, t) + L 3 L u t (L, t) ) 4 { γ a b γ }(a 4 [u x(, t) + (L 3 L )u x(l, t)] ). } u t dx (3.9) At this point we will choose all the constants, carefully, such that all the coefficients in (3.9) will be negative. In fact, it follows from the assumption (3.5) that we can always choose γ, γ and γ 3 such that + L 3 L 4(L ) γ γ 3 >, γ > a b γ, γ > γ, γ 3 > γ. Once the above constants γ, γ, γ 3 are fixed, we may choose ε and ε sufficiently small such that γ 3 ε C + γ ε < a(γ 3 γ ). Then we can take N sufficiently large such that N e τ γ C(ε ) τ µ (s) ds γ 3 τ µ (s) ds 4ε >. Finally, noticing the assumption (3.4), we can always choose N sufficiently large such that the first coefficient in (3.9) is negative. Thus, we obtain that there exists a positive constant α such that (3.9) yields ( L L (t) α u t dx + au xxdx + vt dx L + bvxxdx + τ ) s µ (s) z (x, ρ, t, s) ds dρ dx,

12 G. LIU EJDE-7/74 recalling (3.3), which implies L (t) α E(t),. (3.) On the hand, it is not hard to see that L(t) E(t), i.e. there exist two positive constants β and β such that Combining (3.) and (3.), we obtain that β E(t) L(t) β E(t), t. (3.) L (t) κl(t), t for the positive constant κ = α/β. Integration over (, t) gives L(t) L()e κt, t, recall (3.) again, then (3.6) holds. Hence, the proof is complete. Acknowledgments. The authors would like to thank the referees for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper. This project is supported by Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (Grant No.5A7 )and the Basic Research Foundation of Henan University of Technology, China (No.3JCYJ). References [] T. A. Apalara; Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay. Electron. J. Differential Equations, 54 (4), -5. [] W. D. Bastos, C. A. Raposo; Transmission problem for waves with frictional damping. Electron. J. Differential Equations, 6 (7), -. [3] A. Benseghir; Existence and exponential decay of solutions for transmission problem with delay. Electron. J. Differential Equations, (4), -. [4] R. Datko; Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim. 6(3) (988), [5] R. Datko, J. Lagnese, M. P. Polis; An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 4() (986), [6] R. Dautray, J. L. Lions; Mathematical Analysis and Numerical Methods for Sciences and Technology. Vol., Springer-Verlag, Berlin-Heidelberg, 99. [7] B. W. Feng; Well-posedness and exponential stability for a plate equation with time-varying delay and past history. Z. Angew. Math. Phys., 68 (7), DOI.7/s [8] A. Guesmia; Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay, IMA J. Math. Control Inform., 3 (3), [9] G. Li, D. H. Wang, B. H. Zhu; well-posedness and decay of solutions for a transmission problem with history and delay, Electron. J. Differential Equations, 3 (4), -. [] G. W. Liu, H. Y. Yue, H. W. Zhang; Long time Behavior for a wave equation with time delay, Taiwan. J. Math., 7()(7), 7-9. [] G. W. Liu, H. W. Zhang; Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67()(6) -4. [] T. F. Ma, H. P. Oquendo; A transmission problem for beams on nonlinear supports. Bound. Value Probl., 4 (6), Art. ID 757. [3] A. Marzocchi, J. E. M. Rivera, M. G. Naso; Asymptotic behavior and exponential stability for a transmission problem in thermoelasticity. Math. Meth. Appl. Sci., 5 (), [4] A. Marzocchi, J. E. M. Rivera, M. G. Naso; Transmission problem in thermoelasticity with symmetry. IMA J. Appl. Math., 63(), [5] S. A. Messaoudi, A. Fareh, N. Doudi; Well posedness and exponential satbility in a wave equation with a strong damping and a strong delay. J. Math. Phys., 57(6), 5, 3pp.

13 EJDE-7/74 TRANSMISSION PROBLEM WITH DISTRIBUTED DELAY 3 [6] S. A. Messaoudi, B. Said-Houari; Energy decay in a transmission problem in thermoelasticity of type III. IMA. J. Appl. Math., 74 (9), [7] J. E. Muñoz Rivera, H. P. Oquendo; The transmission problem of viscoelastic waves. Acta Applicandae Mathematicae, 6()(), -. [8] M. I. Mustafa; A uniform stability result for thermoelasticity of type III with boundary distributed delay, J. Abstr. Diff. Equa. Appl.,() (4), -3. [9] M. I. Mustafa, M. Kafini; Exponential decay in thermoelastic systems with internal distributed delay, Palestine J. Math.() (3), [] S. Nicaise, C. Pignotti; Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45(5) (6), [] S. Nicaise, C. Pignotti; Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations (9-) (8), [] S. Nicaise, C. Pignotti; Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 4 (), -. [3] S. Nicaise, C. Pignotti; Exponential stability of abstract evolutions with time delay, J. Evol. Equ. 5 (5) 7-9. [4] S. Nicaise, J. Valein, E. Fridman; Stabilization of the heat and the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S (3) (9), [5] D. H. Wang, G. Li, B. Q. Zhu; Well-posedness and general decay of solution for a transimission problem with vicoelastic term and delay. J. Nonlinear, Sci. Appl., 9(3) (6), -5. [6] G. Q. Xu, S. P. Yung, L. K. Li; Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., (4) (6) Gongwei Liu College of Science, Henan University of Technology, Zhengzhou 45, China address: gongweiliu@6.com