Existence and exponential stability of the damped wave equation with a dynamic boundary condition and a delay term.

Existence and exponential stability of the damped wave equation with a dynamic boundary condition and a delay term. Stéphane Gerbi LAMA, Université de Savoie, Chambéry, France Jeudi 24 avril 2014 Joint work with Belkacem Said-Houari, Alhosn University, Abu Dhabi, UAE S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 1 / 33

Outline of the talk 1 Introdution 2 Well-posedness of the problem : existence and uniqueness. Setup and notations Semigroup formulation : existence and uniqueness. 3 Asymptotic behavior 4 Some remarks S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 2 / 33

Outline 1 Introdution 2 Well-posedness of the problem : existence and uniqueness. Setup and notations Semigroup formulation : existence and uniqueness. 3 Asymptotic behavior 4 Some remarks S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 3 / 33

Position of the problem color code : Yellow : dynamic boundary conditions, red : time delay Consider the damped wave equation, with dynamic boundary conditions and time delay : u tt u α u t = 0, x Ω, t > 0, u(x, t) = 0, x Γ 0, t > 0, ( ) u α ut u tt(x, t) = (x, t) + (x, t) + µ1ut(x, t) + µ2ut(x, t τ ) ν ν x Γ 1, t > 0, u(x, 0) = u 0(x) x Ω, u t(x, 0) = u 1(x) x Ω, u t(x, t τ ) = f 0(x, t τ ) x Γ 1, t (0, τ ), (1) where u = u(x, t), t 0, x Ω which is a bounded regular domain of R N, (N 1), Ω = Γ 0 Γ 1, mes(γ 0) > 0, Γ 0 Γ 1 =, α, µ 1, µ 2 > 0 and u 0, u 1, f 0 are given functions. Moreover, τ > 0 represents the time delay Questions to be asked : 1 Existence, uniqueness and global existence? 2 Is the stationary solution u = 0 stable and what is the rate of the decay? S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 4 / 33

Dynamic boundary condition u tt u α u t = 0, x Ω, t > 0, u(x, t) = 0, x Γ 0, t > 0, ( ) u α ut u tt(x, t) = (x, t) + (x, t) + µ1ut(x, t)) ν ν x Γ 1, t > 0, u(x, 0) = u 0(x) x Ω, u t(x, 0) = u 1(x) x Ω, Longitudinal vibrations in a homogeneous bar in which there are viscous effects, and spring-mass system, Pellicer and Sola-Morales, 90 s Artificial boundary condition for unbounded domain : transparent and absorbing, and a lot of mix between these two types, Majda-Enquist 80 s, Ω is an exterior domain of R 3 in which homogeneous fluid is at rest except for sound waves. Each point of the boundary is subjected to small normal displacements into the obstacle. This type of dynamic boundary conditions are known as acoustic boundary conditions, Beale, 80 s Wentzell boundary conditions for PDE, Jérôme Goldstein, Gisèle Ruiz-Goldstein and co workers, 2000 s S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 5 / 33

Related works : damped waves with dynamic boundary conditions In the absence of delay, and with a nonlinear source terms, Gerbi and Said-Houari [GS2008, GS2011] showed the local existence, an exponential decay when the initial energy is small enough, an exponential growth when the initial energy is large enough and a blow-up phenomenon for linear boundary conditions (m = 2) u tt u α u t = u p 2 u, x Ω, t > 0 u(x, t) = 0, x Γ 0, t > 0 [ ] u α ut u tt(x, t) = (x, t) + ν ν (x, t) + r ut m 2 u t(x, t) x Γ 1, t > 0 u(x, 0) = u 0(x), u t(x, 0) = u 1(x) x Ω. [GS2008] S. Gerbi and B. Said-Houari, Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Advances in Differential Equations Vol. 13, No 11-12, pp. 1051-1074, 2008. [GS2011] S. Gerbi and B. Said-Houari, Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Analysis: Theory, Methods & Applications Vol. 74, pp. 7137-7150, 2011. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 6 / 33

Related works : delay term in 1D Datko [Dat91], showed that solutions of : w tt w xx aw xxt = 0, x (0, 1), t > 0, w (0, t) = 0, w x (1, t) = kw t (1, t τ ), t > 0, a, k, τ > 0 become unstable for an arbitrarily small values of τ and any values of a and k. Datko et al [DLP86] treated the following one dimensional problem: u tt(x, t) u xx(x, t) + 2au t(x, t) + a 2 u(x, t) = 0, 0 < x < 1, t > 0, u(0, t) = 0, t > 0, (2) u x(1, t) = ku t(1, t τ ), t > 0, If k e2a + 1 < 1 then the delayed feedback system is stable for all sufficiently small e 2a 1 delays. If k e2a + 1 > 1, then there exists a dense open set D in (0, ) such that for e 2a 1 each τ D, system (2) admits exponentially unstable solutions. [Dat91] R. Datko. Two questions concerning the boundary control of certain elastic systems. J. Differential Equations, 92(1):27 44, 1991. [DLP86] R. Datko, J. Lagnese, and M. P. Polis. An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim., 24(1):152 156, 1986. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 7 / 33

Related works : wave equations and boundary feedback delay Nicaise and Pignotti,[NP06], examined a system of wave equation with a linear boundary damping term with a delay: u tt u = 0, x Ω, t > 0, u(x, t) = 0, x Γ 0, t > 0, u (x, t) = µ1ut(x, t) + µ2ut(x, t τ ) x Γ1, t > 0, ν u(x, 0) = u 0(x), x Ω, u t(x, 0) = u 1(x) x Ω, u t(x, t τ ) = g 0(x, t τ ) x Ω, τ > 0, and proved under the assumption µ 2 < µ 1 that null stationary state is exponentially stable. They also proved instability if this condition fails. They also studied [NP08, NVF09], internal feedback, time-varying delay and distributed delay. [NP06] [NP08] S. Nicaise and 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):1561 1585, 2006. S. Nicaise and C. Pignotti. Stabilization of the wave equation with boundary or internal distributed delay. Diff. Int. Equs., 21(9-10):935 958, 2008. [NVF09] S. Nicaise, J. Valein, and E. Fridman. Stabilization of the heat and the wave equations with boundary time-varying delays. DCDS-S., S2(3):559 581, 2009. (3) S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 8 / 33

Main results Suppose that : Coefficient condition case 1: µ 1 > µ 2 or case 2: µ 1 µ 2 and α > ( µ 2 1 µ 2 2) 2µ 1 1 β β < 0 defined later then Problem (1) has a unique global solution, this solution decays exponentially to the null solution. Remark 1 If µ 1 > µ 2, as in the works of Nicaise and Pignotti, we can choose α = 0 so that no strong damping is necessary. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 9 / 33

Setup and notations First we reformulate the boundary delay problem, then by a semigroup approach and using the Lumer-Phillips theorem we will prove the global existence. Notations H 1 Γ 0 (Ω) = { u H 1 (Ω)/ u Γ0 = 0 } γ 1 the trace operator from H 1 Γ 0 (Ω) on L 2 (Γ 1 ) H 1/2 (Γ 1 ) = γ 1 ( H 1 Γ0 (Ω) ). E(, L 2 (Ω)) = { u H 1 (Ω) such that u L 2 (Ω) } For u E(, L 2 (Ω)), u ν H 1/2 (Γ 1 ) and we have Green s formula: u u(x) v(x)dx = u(x)v(x)dx + Ω Ω ν ; v v HΓ 1 0 (Ω), Γ 1 where.;. Γ1 means the duality pairing H 1/2 (Γ 1 ) and H 1/2 (Γ 1 ). S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 12 / 33

Reformulation of the delay term As in [NP06], we add the new variable: z (x, ρ, t) = u t (x, t τ ρ), x Γ 1, ρ (0, 1), t > 0. (4) Then, we have τ z t (x, ρ, t) + z ρ (x, ρ, t) = 0, in Γ 1 (0, 1) (0, + ). (5) Therefore, problem (1) is equivalent to: u tt u α u t = 0, x Ω, t > 0, τ z t(x, ρ, t) + z ρ(x, ρ, t) = 0, x Γ 1, ρ (0, 1), t > 0 u(x, t) = 0, x Γ 0, t > 0, ( ) u ut u tt(x, t) = (x, t) + α (x, t) + µ1ut(x, t) + µ2z(x, 1, t) x Γ 1, t > 0, ν ν z(x, 0, t) = u t(x, t) x Γ 1, t > 0, u(x, 0) = u 0(x) x Ω, u t(x, 0) = u 1(x) x Ω, z(x, ρ, 0) = f 0(x, τ ρ) x Γ 1, ρ (0, 1). (6) S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 13 / 33

Semigroup formulation Let V := (u, u t, γ 1(u t), z) T ; then V satisfies the problem: { V (t) = (u t, u tt, γ 1(u tt), z t) T = AV (t), t > 0, V (0) = V 0, (7) where denotes the derivative with respect to time t, V 0 := (u 0, u 1, γ 1(u 1), f 0(.,.τ )) T and the operator A is defined by: v u u + α v v A = w u ν α v µ1v µ2z (., 1) ν z 1 τ zρ S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 15 / 33

Domain and energy space Energy space: H = H 1 Γ 0 (Ω) L 2 (Ω) L 2 (Γ 1 ) L 2 (Γ 1 ) L 2 (0, 1), H is a Hilbert space with respect to the inner product V, Ṽ = u. ũdx + vṽdx + w wdσ + ξ H Ω Ω Γ 1 Γ 1 for V = (u, v, w, z) T, Ṽ = (ũ, ṽ, w, z) T and ξ defined later. The domain of A is the set of V = (u, v, w, z) T such that: 1 0 z zdρdσ (u, v, w, z) T H 1 Γ 0 (Ω) H 1 Γ 0 (Ω) L 2 (Γ 1 ) L 2 ( Γ 1 ; H 1 (0, 1) ) (8) u + αv E(, L 2 (u + αv) (Ω)), L 2 (Γ 1 ) ν (9) w = γ 1 (v) = z(., 0) on Γ 1 (10) S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 16 / 33

The constant β For β R, define : C(β) = u 2 2 inf + β u 2 2,Γ 1 u HΓ 1 (Ω) 0 u 2 2 (11) C(β) is the first eigenvalue of the operator under the Dirichlet-Robin boundary conditions : { u(x) = 0, x Γ0 βu(x) + u ν (x) = 0 x Γ 1 From Kato s perturbation theory, C(β) is a continuous decreasing function and as C(0) > 0, it exists β < 0 such that C(β ) = 0. Definition of β it exists β < 0 such that β > β, C(β) > 0 S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 17 / 33

Existence result Suppose that : Coefficient condition case 1: µ 1 > µ 2 or case 2: µ 1 µ 2 and α > Theorem 1 ( µ 2 1 µ 2 2) 2µ 1 1 β β < 0 Let V 0 H, then there exists a unique solution V C (R + ; H) of problem (7). Moreover, if V 0 D (A), then V C (R + ; D (A)) C 1 (R + ; H). S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 18 / 33

Sketch of the proof. First step : A is dissipative 1 Let V = (u, v, w, z) T D (A), we have: AV, V H = u. vdx + v ( u + α v) dx Ω ( Ω + u ν α v ) µ1v µ2z (σ, 1) dσ ξ ν τ Γ 1 w Γ 1 1 0 zz ρdρdσ. Since u + αv E(, L 2 (u + αv) (Ω)) and L 2 (Γ 1), using Green s formula and the ν compatibility condition (10) gives: AV, V H = µ 1 v Γ 2 dσ µ 2 z (σ, 1) vdσ α 1 Γ 1 But from the compatibility condition (10), we get: ξ τ Γ 1 1 0 Ω v 2 dx ξ τ z ρz dρ dσ = ξ ( v 2 z 2 (σ, 1, t) ) dσ. 2τ Γ 1 Γ 1 1 0 z ρzdρdσ. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 19 / 33

Sketch of the proof. First step : A is dissipative 2 AV, V H ( = α v 2 dx µ 1 ξ ) v 2 dσ ξ Ω 2τ Γ 1 2τ Γ 1 µ 2 Γ 1 v(σ, t)z (σ, 1) dσ 1 0 z 2 (σ, 1, t)dσ Fix δ > 0, Young s inequality gives : v(σ, t)z (σ, 1) dσ δ z 2 (σ, 1) dσ + 1 v 2 (σ, t)dσ Γ 1 2 Γ 1 2δ Γ 1 Finally AV, V H +α ( Ω ξ 2τ δµ2 2 ( v 2 dx + ) µ 1 ξ 2τ µ2 2δ Γ 1 z 2 (σ, 1, t)dσ 0 ) Γ 1 v 2 dσ+ S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 20 / 33

Sketch of the proof. First step : A is dissipative 3 Fix δ and ξ Choose δ = µ1 and ξ = µ2 2τ µ 2 µ 1 AV, V H + α case 1: µ 1 > µ 2. For all α 0 Ω v 2 dx + µ2 1 µ 2 2 2µ 1 V H AV, V H 0. Γ 1 v 2 dσ 0 case 2: µ 1 µ 2, α > 0. Set β = µ2 1 µ 2 2 2αµ 1. Suppose : α > ( µ 2 1 µ 2 2) AV, V H + C(β) u 2 2 0 1. Thus C(β) > 0 and we get : 2µ 1 β V H AV, V H 0. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 21 / 33

λi A is surjective for all λ > 0. Step 1 : formulation λi A is surjective for all λ > 0. Let F = (f 1, f 2, f 3, f 4) T H. We seek V = (u, v, w, z) T D (A) solution of which writes: λw + (u + αv) ν (λi A) V = F, λu v = f 1 (12) λv (u + αv) = f 2 (13) + µ 1v + µ 2z(., 1) = f 3 (14) λz + 1 zρ = f4 (15) τ To find V = (u, v, w, z) T D (A) solution of the system (12), (13), (14) and (15), we proceed as in [NP06], with two major changes: 1 the dynamic condition on Γ 1 which adds an unknown and an equation, 2 the presence of v = u t in this dynamic boundary condition. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 22 / 33

λi A is surjective for all λ > 0. Step 2: : knowing u, determine v, z, w Suppose u is determined with the appropriate regularity. Then from (12), we get: v = λu f 1. (16) Therefore, from the compatibility condition on Γ 1, (10), we determine z(., 0) by: z(x, 0) = v(x) = λu(x) f 1(x), for x Γ 1 (17) Thus, from (15), z is solution of the linear Cauchy problem: { ( ) z ρ = τ f 4(x) λz(x, ρ) for x Γ 1, ρ (0, 1) z(x, 0) = λu(x) f 1(x) The solution of the Cauchy problem (18) is given by: ρ z(x, ρ) = λu(x)e λρτ f 1e λρτ + τ e λρτ f 4(x, σ)e λστ dσ for x Γ 1, ρ (0, 1). So we have at the point ρ = 1, for x Γ 1, 0 1 z(x, 1) = λu(x)e λτ + z 1(x), z 1(x) = f 1e λτ + τ e λτ f 4(x, σ)e λστ dσ (20) Since f 1 HΓ 1 0 (Ω) and f 4 L 2 (Γ 1) L 2 (0, 1), z 1 L 2 (Γ 1). Knowing u, we may deduce v by (16), z by (19) and using (20), we deduce w = γ 1(v) by (14). 0 (18) (19) S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 23 / 33

λi A is surjective for all λ > 0. Step 3. u = u + αv Set u = u + αv. From equations (13) and (14), u must satisfy: λ 2 λ u u = f2 + f1 in Ω 1 + λα 1 + λα u = 0 on Γ 0 ) u λ (µ 2e λτ + (λ + µ 1 ν = u + f (x) for x Γ 1 1 + λα (21) with f 1 L 2 (Ω), f 2 L 2 (Ω), f L 2 (Γ 1). The variational formulation of problem (21) is to find u H 1 Γ 0 (Ω) such that: Ω λ 2 uω + u ωdx + 1 + λα ) λ (µ 2e Γ1 λτ + (λ + µ 1 u(σ)ω(σ)dσ = (22) Ω ( f 2 + 1 + λα λ ) 1 + λα f1 ωdx + f (σ)ω(σ)dσ ω HΓ 1 0 (Ω) Γ 1 S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 24 / 33

λi A is surjective for all λ > 0. End of proof Since λ > 0, µ 1 > 0, µ 2 > 0, the left hand side of (22) defines a coercive bilinear form on H 1 Γ 0 (Ω). Thus by applying the Lax-Milgram lemma, there exists a unique u H 1 Γ 0 (Ω) solution of (22). Now, choosing ω C c, by Green s formula u E(, L 2 (Ω)). We recover u, v, z and finally setting w = γ 1 (v), we have found V = (u, v, w, z) T D (A) solution of (λid A) V = F. The well-posedness result, Theroem 1, follows from the Lummer-Phillips theorem. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 25 / 33

E is decreasing along trajectories Let ξ > 0, we define the functional energy of the solution of problem (6) as E(t) = E(t, z, u) = 1 2 + ξ 2 E is greater than the usual one : E 1(t) = 1 2 Set β = µ2 1 µ 2 2 2αµ 1. [ u(t) 22 + ut(t) 22 + ut(t) 22,Γ1 ] Γ 1 1 0 z 2 (σ, ρ, t) dρ dσ. (23) [ u(t) 22 + ut(t) 22 + ut(t) 22,Γ1 ]. Lemma 2 For u solution of (6), and for any t 0, we have: de (t) dt αc(β) u t 2 2 Corollary 1 Suppose the damping coefficient condition is fulfilled (that is β > β, C(β) > 0). Then the energy E is decreasing along the trajectories. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 27 / 33

Sketch of the proof and asymptotic behavior We multiply the first equation in (6) by u t and perform integration by parts to get: 1 d ] [ u(t) 22 2 dt + ut(t) 22 + ut(t) 22,Γ1 + α u t(t) 2 2 (24) +µ 1 u t(t) 2 2,Γ 1 + µ 2 u t(σ, t)u t(σ, t τ )dσ = 0. Γ 1 By defintion of z, we have: u t(σ, t)u t(σ, t τ )dσ = u t(σ, t)z(σ, 1, t)dσ Γ 1 Γ 1 Fix δ > 0, Young s inequality gives : u t(σ, t)z(σ, 1, t)dσ δ z 2 (σ, 1) dσ + 1 ut 2 (σ, t)dσ Γ 1 2 Γ 1 2δ Γ 1 1 Differentiating the term z 2 (σ, ρ, t) dρ dσ with respect to t and using the fact that z t = zρ, we get τ Finally Γ 1 0 d E dt ( α u t 2 µ 1 ξ ) ( ) 2τ µ2 u t 2 ξ 2,Γ 2δ 1 2τ δµ2 z 2 (σ, 1, t)dσ 2 Γ 1 S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 28 / 33

Asymptotic behavior Fix δ and ξ Choose δ = µ 1 µ 2 and ξ = µ2 2 τ µ 1, set β = µ2 1 µ2 2 2αµ 1. de (t) dt αc(β) u t 2 2 The asymptotic stability result reads as follows: Theorem 3 Assume the damping coefficient relation is fulfiled. Then there exist two positive constants C and γ independent of t such that for u solution of problem (6), we have: E(t) Ce γt, t 0. (25) S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 29 / 33

Lyapunov function For ε > 0, to be chosen later, we define the Lyapunov function: L(t) = E(t) + ε u(x, t)u t(x, t) dx + ε u(σ, t)u t(σ, t) dσ Ω Γ 1 + εα u(x, t) 2 dx (26) 2 Ω + εξ 1 e 2τ ρ z 2 (σ, ρ, t) dρ dσ. 2 Γ 1 0 There exist two positive constants β 1 and β 2 > 0 depending on ε such that for all t 0 β 1E(t) L(t) β 2E(t). (27) By taking the time derivative of the function L defined by (26), using problem (6), performing several integration by parts, and using the previous inequality on the derivative of E and the same Young s inequality with δ = µ1 and ξ = µ2 2τ, we choose µ 2 µ 1 ε > 0 such that there exist two positive constants C and γ independent of t: L(t) C e γt, t 0. Consequently, by using (27) once again, we conclude that it exists C > 0 such that: E(t) Ce γt, t 0. S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 30 / 33

Some final remarks 1 Since the energy associated to (1) is less than the one associated to (6), it is obvious that the exponential stability of the solution associate to problem (6) implies the exponential stability of the one associated to (1). 2 The presence of the strong damping term α u t in equation (1) plays an essential role in the behavior of the system. The condition µ 1 < µ 2 is a necessary condition in the case α = 0, since Nicaise and Pignotti [NP06] showed an instability result if this condition fails. 3 Adapting the same method to the system with internal feedback: ( ( ) ) u tt u α u t + b (x) µ 1u t(x, t) + µ 2u t x, t τ = 0, x Ω, t > 0 u(x, t) = 0, x Γ 0, t > 0 [ ] u α ut u tt(x, t) = (x, t) + (x, t) ν ν x Γ 1, t > 0 u(x, 0) = u 0(x), u t(x, 0) = u 1(x) x Ω, u(x, t τ ) = f 0(x, t τ ) x Ω (0, τ) with b L (Ω) is a function which satisfies b (x) 0, a.e. in Ω and b (x) > b 0 > 0 a.e. in ω where ω Ω is an open neighborhood of Γ 1, the results are still valid. (28) S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 32 / 33

Instability result? Can we show that if µ 1 < µ 2 and α ( µ 2 1 µ 2 2) 2µ 1 1 β we can find solution with constant energy or energy that goes to infinity? Hint: Try to find a solution of the form: u(t, x) = e λt φ(x) with λ C, R(λ) > 0. [GS2012] S. Gerbi and B. Said-Houari. Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term Applied Mathematics and Computations, 218(1):11900 11910, 2012 Thank you for your attention S. Gerbi (LAMA, UdS, Chambéry) Dynamic boundary conditions and delay term Chambéry 2014 33 / 33