Electronic Journal of Differential Equations, Vol. 2007(2007), No. 60, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) TRANSMISSION PROBLEM FOR WAVES WITH FRICTIONAL DAMPING WALDEMAR D. BASTOS, CARLOS A. RAPOSO Abstract. In this paper we consider the transmission problem, in one space dimension, for linear dissipative waves with frictional damping. We study the wave propagation in a medium with a component with attrition and another simply elastic. We show that for this type of material, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution, no matter how small is its size. 1. Introduction A number of authors have studied the wave equation with dissipation. We men- tion for example, the work of Zuazua [5] where it was obtained the uniform rate of decay of the solution for a large class of nonlinear wave equation with frictional damping acting in the whole domain. In this direction, the natural question that arises is about the rate of decay when the dissipation is effective only in a part of the domain. It is the purpose of this investigation, at least in part, to answer this question. We consider the wave propagation over a body consisting of two different type of materials. This is a transmission (or diffraction) problem. It happens fre- quently in applications where the domain is occupied by 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 equation for which the corresponding elliptic operator has discon- tinuous coefficients. Even though we consider a case of space dimension one and linear equations with constant coefficients, the problem studied here is interesting by its own. Existence, regularity, as well as the exact controllability for the transmission problem for the pure wave equation was studied in [2]. The transmission problem for viscoelastic waves was studied by Rivera and Oquendo [4] who proved the expo- nential decay of solution using regularity results of the Volterra’s integral equations and regularizing properties of the viscosity. The asymptotic behavior for a coupled system of equations of waves was studied by Raposo [3] by the same method used in this paper. 2000 Mathematics Subject Classification. 35B35, 35B40, 35L05, 35L20. Key words and phrases. Exponential stability; dissipative system; transmission problem. c©2007 Texas State University - San Marcos. Submitted June 23, 2006. Published April 22, 2007. 1 2 W. D. BASTOS, C. A. RAPOSO EJDE-2007/60 Let k1, k2 and α be positive real numbers and 0 < L0 < L. The system considered here is utt − k1uxx + αut = 0, x ∈ (0, L0), t > 0, (1.1) vtt − k2vxx = 0, x ∈ (L0, L), t > 0, (1.2) satisfying the boundary conditions u(0, t) = v(L, t) = 0, t > 0, (1.3) the transmission conditions u(L0, t) = v(L0, t), k1ux(L0, t) = k2vx(L0, t), t > 0, (1.4) and initial conditions u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ (0, L0), v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ (L0, L). (1.5) We are concerned with the asymptotic properties of the system above. The main result of this paper is Theorem 3.6 which shows that the solution of the transmission problem (1.1)–(1.5) decays exponentially to zero as time goes to infinity, no matter how large is the difference L − L0. The approach we use consists of choosing appropriate multipliers to build a functional of Lyapunov for the system. The notation used throughout this work is the standard one. For instance Hm, L2 = H0, Wm, p and Wm,∞ denote the usual Sobolev Spaces (see Adams [1]). By V we denote the space V := {(u, v) ∈ H1(0, L0)×H1(L0, L) : u(0) = v(L) = 0, u(L0) = v(L0)} which together with the inner product 〈(u1, v1) , (u2, v2)〉 := ∫ L0 0 u1 xu 2 x dx+ ∫ L L0 v1 xv 2 x dx is a Hilbert space. The energies associated to the equations (1.1) and (1.2) are: E1(t) = 1 2 ∫ L0 0 [|ut|2 + k1|ux|2] dx, E2(t) = 1 2 ∫ L L0 [|vt|2 + k2|vx|2] dx respectively. We denote E(t) = E1(t) + E2(t) the total energy associated to the system (1.1)–(1.5). The remainder of this paper is organized as follows. In Section 2 we show the existence of weak and strong solutions for the system (1.1)–(1.5), and in Section 3 we show the exponential decay of such solutions. 2. Existence of solutions We begin this section defining what is meant by weak solution to our transmission problem. Definition 2.1. The couple (u(x, t), v(x, t)) is a weak solution of the system (1.1)– (1.5) when (u, v) ∈ L∞(0, T ;V) ∩W 1,∞(0, T ;L2(0, L0)× L2(L0, L)), EJDE-2007/60 TRANSMISSION PROBLEM FOR WAVES 3 and satisfies − ∫ L0 0 u1φ(0) dx− ∫ L L0 v1ψ(0) dx− ∫ T 0 ∫ L0 0 utφt dx dt− ∫ T 0 ∫ L L0 vtψt dx dt + k1 ∫ T 0 ∫ L0 0 uxφx dx dt+ k2 ∫ T 0 ∫ L L0 vxψx dx dt+ α ∫ T 0 ∫ L0 0 utφdx dt = 0 for any (φ, ψ) ∈ L∞(0, T ; ) ∩W 1,∞(0, T ;L2(0, L0)× L2(L0, L)), such that (φ(T ), ψ(T )) = (0, 0). Theorem 2.2. Let us take (u0, v0) ∈ (H2(0, L0)×H2(L0, L))∩V and (u1, v1) ∈ V verifying the transmission conditions. Under this conditions the solution (u, v) of (1.1)–(1.5) satisfies (u, v) ∈ 2⋂ k=0 W k,∞(0, T ;H2−k(0, L0))×H2−k(L0, L). Proof. The existence is proved using Galerkin method. In order to do so we take a basis {(φ0, ψ0), (φ1, ψ1), (φ2, ψ2), · · ·} of V and let (u0 m, v 0 m), (u1 m, v 1 m) ∈ span{(φ0, ψ0), (φ1, ψ1) · · · (φm, ψm)} be a projection of the initial state on a finite dimensional subspace of V. Standard results on ordinary differential equations guarantee that there exists one and only one solution (um(t), vm(t)) := m∑ j=1 hj,m(t)(φj , ψj) of the approximated system,∫ L0 0 uttφ i dx+ ∫ L L0 vttψ i dx+k1 ∫ L0 0 uxφ i x dx+k2 ∫ L L0 vxψ i x dx+α ∫ L0 0 utφ i dx = 0 (2.1) i = 0, 1, 2, . . . ,m, with initial data (um(0), vm(0)) = (u0 m, v 0 m), (um t (0), vm t (0)) = (u1 m, v 1 m). We show next that the above solution remain bounded for any m ∈ N. In order to do so, we first multiply equation (2.1) by h′j,m(t) and then sum up in i, to obtain d dt Em(t) = −α ∫ L0 0 |um t |2 dx. Integrating the identity above from 0 to t, we get Em(t) ≤ Em(0) showing that the first order energy Em(t) is uniformly bounded for m ∈ N. Now we denote the second order energy by Em(t) = 1 2 ∫ L0 0 [|um tt |2 + k1|um xt|2] dx+ 1 2 ∫ L L0 [|vm tt |2 + k2|vm xt|2] dx. 4 W. D. BASTOS, C. A. RAPOSO EJDE-2007/60 Differentiating equation (2.1) with respect to t, we get∫ L0 0 utttφ i dx+ ∫ L L0 vtttψ i dx+ k1 ∫ L0 0 uxtφ i x dx + k2 ∫ L L0 vxtψ i x dx+ α ∫ L0 0 uttφ i dx = 0. (2.2) Multiplying equation (2.2) by h′′j,m(t) and summing up in i, we obtain d dt Em(t) = −α ∫ L0 0 |um tt |2 dx which integrated from 0 to t furnishes Em(t) ≤ Em(0). The next step is to estimate the second order energy. Letting t → 0+ in equation (2.1), multiplying the limit result by h′′j,m(t) we get∫ L0 0 |um tt (0)|2 dx+ ∫ L L0 |vm tt (0)|2 dx = −k1 ∫ L0 0 um x (0)um xtt(0) dx− k2 ∫ L L0 vm x (0)vm xtt(0) dx− α ∫ L0 0 um t (0)um tt (0) dx. Integrating by parts the equation above, we get∫ L0 0 |um tt (0)|2 dx+ ∫ L L0 |vm tt (0)|2 dx = k1 ∫ L0 0 um xx(0)um tt (0) dx+ k2 ∫ L L0 vm xx(0)vm tt (0) dx− α ∫ L0 0 um t (0)um tt (0) dx. (2.3) After application of Young’s inequality in equation (2.3) we find∫ L0 0 |um tt (0)|2 dx+ ∫ L L0 |vm tt (0)|2 dx ≤ c {∫ L0 0 |um xx(0)|2 dx+ ∫ L L0 |vm xx(0)|2 dx } + c ∫ L0 0 |um t (0)|2 dx. which implies that the initial data (um tt (0), vm tt (0)) is bounded in L2(0, L0)× L2(L0, L)), and so is Em(0). Whence we have Em(t) is bounded for every m ∈ N. The first and second order energy boundedness implies that there exists a subse- quence of (um, vm), which we still denote in the same way, such that (um, vm) ∗ ⇀ (u, v) in L∞(0.T ;V), (um t , v m t ) ∗ ⇀ (ut, vt) in L∞(0.T ;V), (um tt , v m tt ) ∗ ⇀ (utt, vtt) in L∞(0.T ;L2(0, L0)× L2(L0, L))). EJDE-2007/60 TRANSMISSION PROBLEM FOR WAVES 5 Therefore the couple (u, v) satisfies utt − k1uxx + αut = 0 vtt − k2vxx = 0. The rest of the proof is a matter of routine. � 3. Exponential stability With a view toward proving the main result of this paper we formulate and prove a series of five lemmas. They will provide some technical inequalities which play fundamental role in the proof of Theorem 3.6. Lemma 3.1. The total energy E(t) satisfies d dt E(t) = −α ∫ L0 0 |ut|2 dx. Proof. Multiplying equation (1.1) by ut and integrating in (0, L0) we have∫ L0 0 ututt dx− k1 ∫ L0 0 utuxx dx = −α ∫ L0 0 |ut|2 dx which integrated by parts leads to d dt 1 2 ∫ L0 0 [|ut|2 + k1|ux|2] dx = −α ∫ L0 0 |ut|2 dx+ k1ux(L0)ut(L0). (3.1) Multiplying equation (1.2) by vt and performing an integration in (L0, L) we get∫ L L0 vtvtt dx− k2 ∫ L L0 vtvxx dx = 0. After integrating by parts we arrive at d dt 1 2 ∫ L L0 [|vt|2 + k2|vx|2] dx = −k2vx(L0)vt(L0). (3.2) Adding (3.1) with (3.2) and using the transmission conditions (1.4) we conclude d dt E(t) = −α ∫ L0 0 |ut|2 dx. (3.3) � Lemma 3.2. There exist positive constants C0 and C1, independent of initial data, such that the functional defined by J1(t) = ∫ L0 0 (x− L0)utux dx satisfies d dt J1(t) ≤ −C1E1(t) + C0 ∫ LO 0 |ut|2 dx+ k1L0 2 |ux(0)|2. 6 W. D. BASTOS, C. A. RAPOSO EJDE-2007/60 Proof. Multiplying equation (1.1) by (x− L0)ux and performing an integration in (0, L0) we get∫ L0 0 (x−L0)uxutt dx− k1 ∫ L0 0 (x−L0)uxuxx dx = −α ∫ L0 0 (x−L0)uxut dx. (3.4) Note that d dt (x− L0)uxut = (x− L0)uxutt + (x− L0)uxtut. (3.5) Now using (3.5) in (3.4) we get d dt ∫ L0 0 (x− L0)uxut dx = ∫ L0 0 (x− L0) 1 2 [ d dx |ut|2] dx + k1 ∫ L0 0 (x− L0) 1 2 [ d dx |ux|2] dx− α ∫ L0 0 (x− L0)uxut dx and performing integration by parts we get d dt ∫ L0 0 (x− L0)uxut dx. = −1 2 ∫ L0 0 |ut|2 dx− k1 2 ∫ L0 0 |ux|2 dx − α ∫ L0 0 (x− L0)uxut dx+ k1L0 2 |ux(0)|2 from which it follows that d dt J1(t) ≤ −C1E1(t) + C0 ∫ L0 0 |ut|2 dx+ k1L0 2 |ux(0)|2. � Lemma 3.3. There exists a positive constant C2, independent of initial data, such that the functional defined by J2(t) = ∫ L L0 (x− L0)vtvx dx satisfies d dt J2(t) ≤ −C2E2(t) + k2(L− L0) 2 |vx(L)|2. Proof. Multiplying equation (1.2) by (x− L0)vx and performing an integration in (L0, L) we get ∫ L L0 (x− L0)vxvtt dx− k2 ∫ L L0 (x− L0)vxvxx dx = 0. (3.6) Notice that d dt (x− L0)vxvt = (x− L0)vxvtt + (x− L0)vxtvt. (3.7) Now using (3.7) in (3.6) we get d dt ∫ L L0 (x− L0)vxvt dx = ∫ L L0 (x− L0) 1 2 [ d dx |vt|2] dx+ k2 ∫ L L0 (x− L0) 1 2 [ d dx |vx|2] dx and performing integration by parts we get d dt ∫ L L0 (x− L0)vxvt dx. = −1 2 ∫ L L0 |vt|2 dx− k2 2 ∫ L L0 |vx|2 dx+ k2(L− L0) 2 |vx(L)|2 EJDE-2007/60 TRANSMISSION PROBLEM FOR WAVES 7 from which it follows that d dt J2(t) ≤ −C2E2(t) + k2(L− L0) 2 |vx(L)|2. � Now we must control the punctual terms |ux(0)|2 and |vx(L)|2 present in the inequalities given by the lemmas 3.2 and 3.3 respectively. In order to do so we introduce the two following lemmas. Lemma 3.4. Let us take p ∈ C1(0, L0) with p(0) > 0 and p(L0) = 0. Then, there exist positive constants C0, C4, N0 independent of initial data, such that the functional defined by J3(t) = N0J1(t) + ∫ L0 0 putux dx satisfies d dt J3(t) ≤ −C4E1(t) +N0C0 ∫ L0 0 |ut|2 dx. Proof. Multiplying equation (1.1) by p ux and performing an integration in (0, L0) we get ∫ L0 0 p uxutt dx− k1 ∫ L0 0 p uxuxx dx = −α ∫ L0 0 p uxut dx. (3.8) Notice that d dt p uxut = p uxutt + p uxtut. (3.9) Now using (3.9) in (3.8) we get d dt ∫ L0 0 p uxut dx = ∫ L0 0 p 1 2 [ d dx |ut|2] dx+ k1 ∫ L0 0 p 1 2 [ d dx |ux|2] dx− α ∫ L0 0 p uxut dx and performing integration by parts we get d dt ∫ L0 0 p uxut dx = −1 2 ∫ L0 0 p′ |ut|2 dx− k1 2 p(0) |ux(0)|2 − k1 2 ∫ L0 0 p′ |ux|2 dx− α ∫ L0 0 p uxut dx, from which it follows that d dt ∫ L0 0 p uxut dx ≤ −k1 2 p(0) |ux(0)|2 + C3E1(t). Denoting J3(t) = N0J1(t) + ∫ L0 0 putux dx, we have d dt J3(t) ≤ −N0C1E1(t) + C3E1(t) + N0k1L0 2 |ux(0)|2 − k1 2 p(0) |ux(0)|2 +N0 C0 ∫ L0 0 |ut|2 dx. 8 W. D. BASTOS, C. A. RAPOSO EJDE-2007/60 Now taking N0 such that N0 C1 > C3 and choosing p(0) = N0L0 we conclude that d dt J3(t) ≤ −C4E1(t) +N0C0 ∫ L0 0 |ut|2 dx. � Lemma 3.5. Let us take q ∈ C1(L0, L) with q(L0) = 0 and q(L) < 0. Then, there exist positive constants C5 and N1 independent of initial data such that the functional defined by J4(t) = N1J2(t) + ∫ L L0 qvtvx dx satisfies d dtJ4(t) ≤ −C5E2(t). Proof. Multiplying equation (1.2) by q vx and performing an integration in (L0, L) we get ∫ L L0 q vxvtt dx− k2 ∫ L L0 q vxvxx dx = 0. (3.10) Notice that d dt q vxvt = q vxvtt + q vxtvt. (3.11) Now using (3.11) in (3.10) we get d dt ∫ L L0 q vxvt dx = ∫ L L0 q 1 2 [ d dx |vt|2] dx+ k2 ∫ L L0 q 1 2 [ d dx |vx|2] dx and performing integration by parts we arrive at d dt ∫ L L0 q vxvt dx = −1 2 ∫ L L0 q′ |vt|2 dx+ k2 2 q(L) |vx(L)|2 − k2 2 ∫ L L0 q′ |vx|2 dx, from which it follows that d dt ∫ L L0 q vxvt dx ≤ k2 2 q(L) |vx(L)|2 + C4E2(t). Denoting J4(t) = N1J2(t) + ∫ L L0 qvtvx dx, we have d dt J4(t) ≤ −N1C2E2(t) + C4E2(t) + N1k2(L− L0) 2 |vx(L)|2 + k2 2 q(L) |vx(L)|2. Now taking N1 such that N1 C2 > C4 and choosing q(L) = −N1(L − L0) we conclude that d dt J4(t) ≤ −C5E2(t). � Now we are in position to show the main result of this paper. Theorem 3.6. Let us denote by (u, v) a strong solution of system (1.1)–(1.5), as in Theorem 2.2. Then there exist positive constants C and ω, such that E(t) ≤ C E(0)e−ω t. EJDE-2007/60 TRANSMISSION PROBLEM FOR WAVES 9 Proof. Let us define L(t) = N2E(t) + J3(t) + J4(t). From Lemma 3.1 we have d dt E(t) = −α ∫ L0 0 |ut|2 dx. From Lemma 3.4 we have d dt J3(t) ≤ −C4E1(t) +N0C0 ∫ L0 0 |ut|2 dx. From Lemma 3.5 we have d dt J4(t) ≤ −C5E2(t). In fact we have d dt L(t) ≤ −C4E1(t)− C5E2(t) + (N0C0 −N2α) ∫ L0 0 |ut|2 dx. Taking N2 large enough it follows d dt L(t) ≤ −C6E(t) Since L(t) is equivalent to E(t), we conclude that there exist positive constants C and ω, such that E(t) ≤ C E(0)e−ω t. � Theorem 3.6 can be extended easily to weak solutions by using density arguments and the lower semicontinuity of the energy functional E(t). This is the content of the following corollary whose proof is omitted. Corollary 3.7. Under the same hypotheses of Theorem 3.6, there exists positive constants C̄ and ω̄, such that E(t) ≤ C̄ E(0)e−ω̄t. Acknowledgements. The authors are grateful to the anonymous referee for giving us valuable suggestions that improved our paper. References [1] R. A. Adams; Sobolev Spaces. Academic Press, New York (1975). [2] J. L. Lions; Contrôlabilité Exacte Pertubations et Stabilisation de Systèmes Distribués. Col- lection RMA - Tomo 1, Masson , Paris (1998). [3] C. A. Raposo; The Transmition Problem for Timoshenko Systems of Memory Type. Doctoral Thesis. Federal University of Rio de Janeiro - IM-UFRJ, Rio de Janeiro-Brazil (2001). [4] J. E. M. Rivera and H. P. Oquendo; The transmission Problem of Viscoelastic Waves. Acta Applicandae Mathematicae, 62: 1, pp. 1-21 (2000). [5] E. Zuazua; Stability and Decay for a Class of Nonlinear Hyperbolic Problems. Asymptotic Analysis, 1: pp. 161-185 (1988). Waldemar D. Bastos Universidade Estadual Paulista - UNESP, Departamento de Matemática, 15054-000 - São José do Rio Preto - SP, Brazil E-mail address: waldemar@ibilce.unesp.br 10 W. D. BASTOS, C. A. RAPOSO EJDE-2007/60 Carlos A. Raposo Universidade Federal de São João del-Rei - UFSJ, Departamento de Matemática, 36300- 000 - São João del-Rei, MG, Brazil E-mail address: raposo@ufsj.edu.br 1. Introduction 2. Existence of solutions 3. Exponential stability Acknowledgements References