ENERGY DECAY FOR THE DAMPED WAVE EQUATION ON AN UNBOUNDED NETWORK

. We study the wave equation on an unbounded network of N,N ∈ N ∗ , ﬁnite strings and a semi-inﬁnite one with a single vertex identiﬁed to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the ﬁnite edges . The dissipation is given by a damping constant α > 0 via the condition (cid:80) Nj =0 ∂ x u j (0 ,t ) = α∂ t u 0 (0 ,t ) . We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for α (cid:54) = N + 1 we have an exponential decay of the energy and we give an explicit formula for the decay rate when the ﬁnite edges have the same length.


1.
Introduction. In the study of wave propagation in physical multi-structures such as networks, graphs or trees, a lot of questions are asked and considered from different points of view. The approaches used depend on the structures properties (open sets, networks, compact, unbounded, etc...) and also on the conditions imposed in the mathematical formulation of the physical problem. One of the most interesting question is about the energy of a wave that propagates along a network of strings (bounded or unbounded). The knowledge of the behaviour of the local or global energy gives a direct response to the question of stabilization or the existence of an equilibrium state of the considered system. For a network with a finite number of edges, say N ≥ 2, many developments on the previous question were obtained in different contexts. For compact (or bounded) networks, a deep investigation was done and a big amount of answers were given. In [1,3,4,5,22,11,15], the authors considered the wave equation ∂ 2 t u − c 2 ∆u = 0 on bounded networks with a variety of boundary (external) and vertex (internal) conditions. Namely the external conditions are Dirichlet, Neumann or Robin conditions. At the internal vertices, the continuity is imposed and a damping or Kirchhoff condition is considered. A formulation of the problem in a simple context is the following: Given N ≥ 2 positive numbers 1 , . . . , N and a real constant α, we let Γ be a network of N edges e 1 , . . . , e N with lengths 1 , . . . , N respectively. We consider the case where all of the edges are connected at a single vertex v. We identify each edge e j to the interval [0, j ] and the common vertex to 0. The free endpoints of the network are the points x = j , j = 1, . . . , N. The propagation of a wave in this structure with speed c = 1 is mathematically formulated by the system x u j = 0, ∀x ∈ [0, j ], j = 1, . . . , N u j (0, t) = u i (0, t), ∀1 ≤ i, j ≤ N, t ≥ 0 B.C., I.C. N j=1 ∂ x u j (0, t) = α∂ t u 1 (0, t), ∀t ≥ 0 (1) where B.C. are the conditions imposed at the points x = j while I.C. are the initial data given at t = 0. The internal condition N j=1 ∂ x u j (0, t) = α∂ t u 1 (0, t), ∀t ≥ 0 is a dissipation condition at x = 0 when α > 0, a conservation condition when α = 0 and an amplification one if α < 0. The solution to (1) is a vector valued function u(x, t) = (u 1 (x, t), . . . , u N (x, t)) lying in some functional space X. The most frequently asked question is about the comparison between the norm in X of the solution at time t and the norm of the given initial data. The energy is in some sense this considered norm. The stabilization of the system (1) and some more complicated configurations has been considered in a lot of papers [1,3,5,22,15],. . . . The decay of the energy and its rate have been established under some assumptions on edge lengths and on the damping amount term α. Negative answer for the finitetime stabilization was given in [1] if α takes some particular values related to the number of edges where the damping is acting.
The aim of this paper is to investigate the situation where the network is unbounded. More precisely we consider the configuration where an infinite edge e 0 , identified to the interval e 0 := [0, +∞[ is attached to a compact network with N finite edges. We put a modified Kirchhoff condition at the vertex 0. Precisely this is When α ≥ 0, the energy is a non increasing function of t. The case α = 0 is a conservative one and was considered in [6]. In that paper the authors have studied the local energy behaviour. By the way of two different approaches, they have proved that the local energy is exponentially decaying and the exact decay rate was given. It equals γ : where L is the total length of the compact part of the network, i.e L = N j=1 j . Let's stress that the given results were obtained for a network with equal edge lengths. In a recent work by Assel-Khenissi-Royer [7], the case of a network with different edge lengths 1 , . . . , N is studied and some results on the energy decay are obtained. Namely, if the ratios i j , i = j are not rational numbers, then exponential decay fails. Otherwise, the exponential decay takes place with a rate that can be computed.
In this paper, we deal with a damping acting at the vertex 0 via a constant α > 0 and given by the condition (2). To our best knowledge this is the first work studying the energy decay in this unbounded configuration. Thus, in order to obtain a complete and clear description and to go further with explicit formulae, we study the case of a network with N ≥ 2 finite edges with equal lengths and one infinite edge attached to them at the vertex 0. We transform the problem into a Cauchy initial value problem and we compute the spectrum of the associated linear operator. We write the wave propagator U(t) in terms of the resolvent and by a contour deformation technique we demonstrate an asymptotic expansion of U(t). We use this expansion to prove the main result of this paper, namely that for α = N + 1 the energy decays exponentially and the exact decay rate is obtained.
The paper is organized as follows. In section 2, we introduce the problem, the functional spaces and we prove the existence, uniqueness and regularity of the solution using the semigroup theory. In section 3, we investigate the spectral properties of the linear operator found in section 2. We compute explicitly the point spectrum and the resolvent. In section 4, the results of Section 3 are used for the study of the wave propagator U(t). Using the Laplace transform with a contour deformation and the residue theorem, we find an asymptotic expansion of U(t) when t tends to +∞ in terms of spectral projections. Finally, we exploit the information on U(t) to prove the exponential decay and to give its rate. The problem is formulated by the following system : ( The functions u 0 j , u 1 j , j = 0, . . . , N are the initial data. Figure 1 describes the setting of the problem (3).
In order to guarantee the wellposedness of the problem, we introduce the functional spaces. For k = 1, 2 we denote by H k (Γ) the space where L 2 is the space of square integrable functions. The space X has the structure of a Hilbert space when equipped with the inner product We associate to the system (3) the linear operator A defined by its domain To be more precise, the operator A has the operator valued matrix representation for some given initial data (f, g) ∈ V × X that we can write in terms of the initial conditions satisfied by u. This correspondence between (3) and (5) allows us to use the abstract semigroup theory and especially the Hille-Yosida Theorem (see [16]). Proof. We start by proving that A is dissipative. Let (u, v) ∈ D(A). To simplify the writing we put 0 = +∞. We have The real part of the preceding is −α|v 0 (0)| 2 which is negative due to the prescribed sign of the damping coefficient α. Now we proceed to prove that A is maximal dissipative. For this to be done, we let λ be a positive number and we claim that λI − A is surjective from D(A) to X. We take (f, g) ∈ X and we look for . This writes as a system For j = 0, . . . , N we substitute the first equation in the system into the second one. This yields λ 2 u j − u j = g j + λf j . We multiply this identity by w where w ∈ V is an arbitrary function then we integrate over the intervals (0, j ), j = 0, . . . , N and we take the sum over j. We get After one integration per parts, the left hand side of (8) takes the form Thus we reformulate the identity (8) as a variational problem of finding u such that and It is straightforward to see that b is a continuous and coercive bilinear form on V and that F is a continuous linear form on V. Therefore, by the Lax-Milgram theorem the variational problem has a unique solution u in V. Going back to the equations λ 2 u j − u j = g j + λf j , j = 0, . . . , N, we claim that u is in the space H 2 (0, +∞) × N j=1 H 2 (0, j ) and so the couple (u, v) is a solution of (7) in D(A). According to the Lumer-Philips Theorem [9], the operator A is maximal dissipative.
Hence we have the following existence and regularity result : The aim of this paper is study the decay of the energy of the solution u. Definition 2.3. Let u = (u 0 , u 1 , . . . , u N ) be the unique solution of (3). We define : where R is a positive number.
The global and the local energy of u are defined for all t ≥ 0 and one can see The first result on the behaviour of the energy for large t is the following.
Proof. Let u = (u 0 , u 1 , . . . , u N ) be the solution of (3) and let t ≥ 0. According to dE u ds ds one needs to calculate the derivative of the functional E u with respect to t. Denoting again 0 = +∞, we have : Using the relation ∂ 2 t u j = ∂ 2 x u j , ∀j = 0, . . . , N, the boundary conditions and the conditions satisfied by u at the vertex 0, yields This ends the proof.
The previous result shows that the decay of the energy is governed by the sign of α. The energy of the solution is non increasing for α ≥ 0. Without the dissipation condition at the vertex 0, i.e when α = 0, the problem is conservative and the global energy of the solution is constant for all time t ≥ 0 and equals to the one of the initial data. However, if α > 0 the energy is decaying and for fixed t > 0 its dissipated amount is given by α ∂ t u 0 (0, .) 2 L 2 (0,t) . Our aim in the rest of this paper is to study the decaying behaviour of the energy when t tends to infinity. We will prove under some assumption on α that the energy is exponentially decaying and the rate at which this happens will be given. The main tool will be the full spectral analysis of the resolvent associated to the operator A and an explicit computation of its spectrum.
3. Spectral analysis. In this section we give the spectral properties of the operator A. We describe the spectrum and investigate the properties of the resolvent.
3.1. Computation of the point spectrum. According to the m-dissipativity of the operator A the spectrum is located in the set C − := {λ ∈ C; (λ) ≤ 0}. We look here for the eigenvalues of finite multiplicities with negative real parts. Thus let λ ∈ C − and let's find a non trivial ( This writes as a system v = λu u = λv . The problem is therefore reduced to a list of second order differential equations involving the functions u j , j = 0, . . . , N satisfying the boundary conditions at j , j = 1, . . . , N and coupled through the continuity and dissipation conditions at the vertex 0. More precisely, we have to find N +1 functions u 0 , . . . , u N such that (u 0 , . . . , u N ) ≡ 0 R N +1 , and (u, λu) ∈ D(A). It's clear that λ = 0 is not an eigenvalue since the only corresponding solution in D(A) is (0, 0). Let λ = 0. For every j = 0, . . . , N a fundamental system of solutions to equation (15) is {e λx , e −λx }, so there exist two constants α j , β j such that u j (x) = α j e λx + β j e −λx , for j = 1, . . . , N, x ∈ [0, j ] and a constant α 0 such that u 0 (x) = α 0 e λx , ∀x ∈ [0, +∞[. For 1 ≤ j ≤ N, the Dirichlet conditions at x = j gives β j = −e 2λ j α j and the continuity condition at the vertex 0 yields α 0 = (1−e 2λ j )α j .
Writing the dissipation condition at the vertex 0 gives The computation of the eigenvalues is equivalent to finding the values of λ ∈ C − for which the linear system has non trivial solution (α 1 , . . . , α N , α 0 ) ∈ C N +1 .
Lemma 3.1. Let c = 1 − α and a j = 1 − e 2λ j , b j = 1 + e 2λ j for j = 1, . . . , N. The determinant of the system (16) is given by Proof. For two given sets We have recursively We want to find the eigenvalues of A so we need to factorize δ N then we look for its roots. The dependence of δ N on λ is made through the coefficients a j , b j , j = 1, . . . , N. The formula (17) becomes j is the total length of the bounded component of the network.
To go further on in the factorization of δ N we need to make some assumptions on the lengths j , j = 1, . . . , N. We assume for instance that there exists a constant > 0 such that j = , ∀j = 1, . . . , N. The cases where the edge lengths satisfy the more general condition j k ∈ Q or the irrationality condition j k ∈ Q will be considered in a further paper.
In the previous relations, Arcoth and Argth are the reciprocal functions of the hyperbolic cotangent function and of the hyperbolic tangent function respectively.
If sinh(λ ) = 0, then the other roots of δ N are the solutions of the equation The denominator in the left hand side is positive since α > 0 and N ≥ 2. A direct computation of the solutions of (19) with a discussion on the sign of α − (N + 1) give the result.
The eigenvectors associated with the eigenvalues given in the last proposition are explicitly computed and we have the following result : We assume that j = , ∀j = 1, . . . , N with > 0. Then (1) Let n ∈ Z * and λ n = inπ . Then λ n is a simple eigenvalue of A and an associated eigenvector is (φ n , λ n φ n ) t where φ n (x) = sin (λ n x) (2) If α > N + 1, then ∀n ∈ Z, µ n := Arcoth(r) + inπ is a simple eigenvalue of A and an associated eigenvector is (ψ n , µ n ψ n ) where (3) If α < N +1, then ∀n ∈ Z, µ n := Argth(r) + i(2n + 1)π 2 is a simple eigenvalue of A and an associated eigenvector is (ψ n , µ n ψ n ) where Proof. For λ n = inπ , n ∈ Z * to be an eigenvalue of A with domain D(A) the solution of A(u, v) = λ(u, v) must satisfy u 0 = 0. The other components of u are easily calculated using the differential equations, the Dirichlet condition at x = and the two conditions at x = 0. For µ n , n ∈ Z, the computation of the coefficients α j , β j , j = 1, . . . , N and α 0 using the conditions at x = and x = 0 gives, using r = 1 − α N Remark 1.
• if α > 2, then A has its eigenvalues in the set • if α < 2, then A has its eigenvalues in the set 2. In the particular case when α = 1 and N ≥ 2, the eigenvalues are purely imaginary and are given by the sequence inπ 2 n∈Z * .

3.2.
Computation of the resolvent. Let λ ∈ C such that λ is in the resolvent set of A. We denote by R(λ) = (λI − A) −1 the resolvent of A defined as an operator from (H 1 × L 2 )(Γ) to D(A). We will calculate R(λ) and use it to study the wave semigroup associated to A by Theorem (2.1).
We look for (φ, ψ) ∈ D(A) such that (λI − A)(φ, ψ) = (f, g) is true. This relation can be written as system 20) in which the first equation has to be understood as a collection of the N + 1 differential equations −φ j + λ 2 φ j = g j + λf j while the second one is ψ j = λφ j − f j , for j = 0, . . . , N. Let j ∈ {0, 1, . . . , N } and h j = λf j + g j . In the following we give the solutions of the inhomogeneous second order differential equation (λf 0 (s) + g 0 (s)) sinh(λ(s − x))ds , and Here we have made the notationsF (λ) = Proof. Let (f, g) ∈ X. From (20) we see that the equation R(λ)(φ, ψ) = (f, g) is equivalent to the system We denote by h = λf + g so the equation for φ writes as −∆φ + λ 2 φ = h and we look for and for j = 0 The continuity condition at the single vertex gives Thus we need to calculate α j , j = 0, . . . , N and the solutions φ j will be in the form When 1 ≤ j ≤ n, the Dirichlet condition at the edges endpoints x = gives The dissipation condition Taking the sum over j in (26) and using (27) one obtains It is not difficult to see the C α (λ) does not vanish when λ ∈ C + . Now going back to (26) we get , for j = 1, . . . , N, 4. Energy decay. From the results given in Theorem (3.2) the resolvent R is a holomorphic function on the open set C + = {λ ∈ C; λ > 0} . We already know from ( [9]) that R is related to the dissipative semigroup U(t) by the Laplace transform Let χ ∈ C ∞ 0 (0, +∞). We denote by R χ (λ) = χR(λ)χ the cut-off resolvent of A. The function χ is a cut-off on the unbounded edge e 0 of Γ. Moreover, using the semigroup theory, the solution Ψ = (u, ∂ t u) of the Cauchy problem (5) is given by Ψ(t) = U(t)(f, g), ∀t ≥ 0 where (f, g) are the given initial data. Our aim is to find the decay properties of the wave group for large t and then to use them for the energy decay of the solution of (3). Hence we may use an inversion formula for the Laplace transform. Using the fact that A is a dissipative operator that generates a contraction semi-group, we may use the Hille-Yosida theorem (see [16]) and can write U(t) as a contour integral of R(λ). More precisely If we let γ be a strictly positive number then we have For (f, g) ∈ D(A), the solution u = (u 0 , . . . , u N ) of (3) and its derivative ∂ t u, with initial data (u 0 , u 1 ) = (f, g) are given ∀x ∈ Γ, t ≥ 0 by The function φ in the formulae for u and ∂ t u is the first component of the solution of the resolvent equation given by (3.2). We have added the variable λ to φ in order to mark its dependence on it. The approach we will use to study the decay of the energy of u is to make a contour deformation in the formulae (32) and then we apply the residue theorem. To achieve this, we need more precise informations on the resolvent R namely its singularities and useful estimates on the norm of R(λ) as an operator from X to D(A). (ii) if α > N + 1, then P = inπ , n ∈ Z * ∪ Arcoth(r) + inπ , n ∈ Z .
Proof. We use the results given by Theorem (3.2). It can be easily seen that φ and ψ depend holomorphically on λ except when C α (λ) or sinh(λ ) vanish. It is straightforward that the roots of C α (λ) and sinh(λ ) are given by the set P depending on the comparison of α and N + 1. All of these roots are simple.
Definition 4.2. Let α > 0 such that α = N + 1. The spectral abscissa of the operator A is defined as Remark 2. It is clear that ω(α) is a non positive number whenever it exists. In the case considered in this paper we have In the two cases, the spectral abscissa is a negative number. In the case α = N + 1 or α = 1, ω(α) does not exist. One might consider in this case that the spectrum is purely imaginary so the exponential decay of the energy does not take place. This case will be studied in a further paper.
If we pass to the limit m → +∞ in (34) we will have to study the behaviour of the contour integrals ((1), . . . , (4)) therein. According to (35), when m tends to +∞ we find from Let (f, g) ∈ D(A) and 0 ≤ j ≤ N.
For the first claim, we use integration per parts in (21) and (22) and hence we get λφ 0 (x) = λK 1 (x, λ)f 0 (0) + K f,g 2 (x, λ) where K 1 (x, λ) = α sinh(λ ) − N cosh(λ ) C α (λ) e λx − cosh(λx), and Q c,s u,v , Q c,s u,v are given by −v(s) sinh(λ(s − x)))ds Q s,c u,v (λ, x) = −v(s) cosh(λ(s − x)))ds, It was proven in [11] (Remark 4) that the operators Q c,s u,v , Q s,c u,v are uniformly bounded with respect to (λ) on the space H 1 × L 2 and more functional properties were given there. Thus, on one hand, the operator K 2 is uniformly bounded. On the other hand, to prove that λK 1 is bounded, we remark that the uniform gap between the eigenvalues λ n and µ n allows us to use the Proposition 3 in [11]. Hence, there exists C > 0 such that γ −γ e ±iamt e tx R(x ± ia m )dx ≤ C m , ∀m ∈ N * .
The limit energy E ∞ is the norm of the projection of the initial conditions on the eigenspaces associated to the purely imaginary eigenvalues of the operator A.