International Journal of Analysis and Applications ISSN 2291-8639 Volume 11, Number 1 (2016), 43-53 http://www.etamaths.com EXPONENTIAL STABILITY OF THE HEAT EQUATION WITH BOUNDARY TIME-VARYING DELAYS MOUATAZ BILLAH MESMOULI1, ABDELOUAHEB ARDJOUNI1,2,∗ AND AHCENE DJOUDI1 Abstract. In this paper, we consider the heat equation with a time-varying delays term in the boundary condition in a bounded domain of Rn, the boundary Γ is a class C2 such that Γ = ΓD ∪ΓN , with ΓD ∩ ΓN = ∅, ΓD 6= ∅ and ΓN 6= ∅. Well-posedness of the problems is analyzed by using semigroup theory. The exponential stability of the problem is proved. This paper extends in n- dimensional the results of the heat equation obtained in [11]. 1. Introduction Time-delay often appears in many biological, electrical engineering systems and mechanical applica- tions, and in many cases, delay is a source of instability [3]. In the case of distributed parameter system- s, even arbitrarily small delays in the feedback may destabilize the system (see e.g. [1, 2, 8, 9, 10, 14]). The stability issue of systems with delay is, therefore, of theoretical and practical importance. In present paper, we are interested in the effect of a time-varying delays in boundary stabilization of the heat equation in domains of Rn. Let Ω ⊂ Rn be an open bounded set with boundary Γ of class C2. We assume that Γ is divided into two parts ΓN and ΓD; i.e., Γ = ΓD ∪ ΓN with ΓD ∩ ΓN = ∅, ΓD 6= ∅ and ΓN 6= ∅. In this domain Ω, we consider the initial boundary value problem ut (x,t) − ∆u (x,t) = 0 in Ω × (0,∞) ,(1.1) u (x,t) = 0 on ΓD × (0,∞) ,(1.2) ∂u ∂ν (x,t) = −µ1u (x,t) −µ2u (x,t− τ (t)) on ΓN × (0,∞) ,(1.3) u (x, 0) = u0 (x) in Ω,(1.4) u (x,t− τ (0)) = f0 (x,t− τ (0)) on ΓN × (0,τ (0)) ,(1.5) where ν (x) denotes the outer unit normal vector to the point x ∈ Γ and ∂u ∂ν is the normal derivative. Moreover, τ (t) > 0, µ1,µ2 ≥ 0 are fixed nonnegative real numbers, the initial datum (u0,f0) belongs to a suitable space. On the functions τ (·) we assume that there exists a positive constants τ, such that (1.6) 0 < τ0 ≤ τ (t) ≤ τ, ∀t > 0, Moreover, we assume (1.7) τ′ (t) < 1, ∀t > 0, and (1.8) τ ∈ W 2,∞ ([0,T]) , ∀t > 0. Note that , if t < τ (t), then u (x,t− τ (t)) is in the past and we need an initial value in the past. Moreover, by (1.7) and the mean value theorem, we have τ (t) − τ (0) < t, 2010 Mathematics Subject Classification. 35L05, 93D15. Key words and phrases. Heat equation; delay feedbacks; stabilization; Lyapunov method. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 43 44 MESMOULI, ARDJOUNI AND DJOUDI which implies t− τ (t) > −τ (0) , we thus obtain the initial condition (1.5). The last boundary-value problem describes the propagation of heat in a homogeneous n-dimensional rod. Here a stands for the heat conduction coefficient, u(x,t) is the value of the temperature field of the plant at time moment t and location x along the rod. In the sequel, the state dependence on time t and spatial variable x is suppressed whenever possible. The above problem, with both µ1,µ2 > 0 and a time-varying delay, has been studied in one space dimension by Nicaise, Valein and Fridman [12]. In [12] an exponential stability result is given, under the condition (1.9) µ2 < √ 1 −dµ1, where d is a constant such that (1.10) τ′ (t) ≤ d < 1, ∀t > 0, We are interested in giving an exponential stability result for such a problem. Let us denote by 〈v,w〉 the Euclidean inner product between two vectors (v,w) ∈ Rn. Under a suitable relation between the above coefficients we can give a well-posedness result and an exponential stability estimate for problem (1.1)–(1.5). 2. Well-posedness of the problem Using semigroup theory we can give the well-posedness of problem (1.1)-(1.5). Let us stand z (x,ρ,t) = u (x,t− τ (t) ρ) , x ∈ ΓN, ρ ∈ (0, 1) , t > 0. Then, the problem (1.1)-(1.5) is equivalent to ut (x,t) − ∆u (x,t) = 0 in Ω × (0,∞) ,(2.1) τ (t) zt (x,ρ,t) + (1 − τ′ (t) ρ) zρ (x,ρ,t) = 0 in ΓN × (0, 1) × (0,∞) ,(2.2) u (x,t) = 0 on ΓD × (0,∞) ,(2.3) ∂u ∂ν (x,t) = −µ1u (x,t) −µ2z (x, 1, t) on ΓN × (0,∞) ,(2.4) z (x, 0, t) = u (x,t) , x ∈ ΓN, t > 0,(2.5) u (x, 0) = u0 (x) in Ω,(2.6) z (x,ρ, 0) = f0 (x,−τ (0) ρ) , x ∈ ΓN, ρ ∈ (0, 1) .(2.7) If we denote by U := (u,z) T , then U′ = ( ut zt ) = ( ∆u (τ′(t)ρ−1) τ(t) zρ ) . Therefore, problem (2.1)–(2.7) can be rewritten as (2.8) { U′ = A(t) U, U (0) = (u0,f0 (·,−· τ (0))) T , in the Hilbert space H defined by (2.9) H := L2(Ω) ×L2(ΓN × (0, 1)), equipped with the standard inner product〈( u z ) , ( ũ z̃ )〉 H := ∫ Ω u(x)ũ(x)dx + ∫ ΓN ∫ 1 0 z(x,ρ)z̃(x,ρ)dρdΓ. EXPONENTIAL STABILITY OF THE HEAT EQUATION 45 The time varying operator A(t) is defined by A(t) ( u z ) := ( ∆u (τ′(t)ρ−1) τ(t) zρ ) , with domain D (A(t)) : = { (u,z) T ∈ ( E(∆,L2(Ω)) ∩V ) ×L2 ( ΓN,H 1 (0, 1) ) : ∂u ∂ν = −µ1u−µ2z (·, 1) on ΓN, u = z (·, 0) on ΓN } , where, V = H1ΓD = { u ∈ H1 (Ω) , u = 0 on ΓD } , and E(∆,L2(Ω)) = {u ∈ H1(Ω) : ∆u ∈ L2(Ω)}. Recall that for a function u ∈ E(∆,L2(Ω)), ∂u ∂ν belongs to H−1/2(ΓN ) and the next Green formula is valid (see section 1.5 of [4]) (2.10) ∫ Ω ∇u∇ϕdx = − ∫ Ω ∆uϕdx + 〈 ∂u ∂ν ,ϕ〉ΓN , ∀ϕ ∈ H 1 ΓD (Ω), where 〈·; ·〉ΓN means the duality pairing between H−1/2(ΓN ) and H1/2(ΓN ). Observe that the domain of A(t) is independent of the time t, i.e., (2.11) D(A(t)) = D(A(0)), t > 0. Note further that for (u,z) T ∈D(A(t)), ∂u/∂ν belongs to L2(ΓN ), since z (x, 1) is in L2(ΓN ). A general theory for equations of type (2.8) has been developed using semigroup theory [6, 7, 13]. The simplest way to prove existence and uniqueness results is to show that the triplet {A,H,D(A(0))}, with A = {A(t) : t ∈ [0,T]}, for some fixed T > 0, forms a CD-system (or constant domain system, see [6, 7]). More precisely, we can obtain a well-posedness result using semigroup arguments by Kato [5, 6, 13]. The following result is proved in [5, Theorem 1.9]. Theorem 1. Assume that (i) D(A(0)) is a dense subset of H, (ii) D(A(t)) = D(A(0)) for all t > 0, (iii) for all t ∈ [0,T], A(t) generates a strongly continuous semigroup on H and the family A = {A(t) : t ∈ [0,T]} is stable with stability constants C and m independent of t (i.e. the semigroup (St(s))s≥0 generated by A(t) satisfies ‖St(s)u‖H ≤ Cems‖u‖H, for all u ∈H and s ≥ 0), (iv) ∂tA belongs to L∞∗ ([0,T],B(D(A(0)),H)), the space of equivalent classes of essentially bound- ed, strongly measurable functions from [0,T] into the set B(D(A(0)),H) of bounded operators from D(A(0)) into H. Then, problem (2.8) has a unique solution U ∈ C([0,T],D(A(0))) ∩ C1([0,T],H) for any initial datum in D(A(0)). Our goal is then to check the above assumptions for problem (2.8). Lemma 1. D(A(0)) is dense in H. Proof. Let (f,h)T ∈H be orthogonal to all elements of D(A(0)), that is, 0 = 〈( u z ) , ( g h )〉 H = ∫ Ω u(x)g(x)dx + ∫ ΓN ∫ 1 0 z(x,ρ)h(x,ρ)dρdΓ, for all (u,z)T ∈D(A(0)). We first take u = 0 and z ∈D(ΓN ×(0, 1)). As (0,z)T ∈ D(A(0)), we obtain∫ ΓN ∫ 1 0 z(x,ρ)h(x,ρ)dρdΓ = 0. Since D(ΓN × (0, 1)) is dense in L2(ΓN × (0, 1), we deduce that h = 0. In the same way, by taking z = 0 and u ∈D(Ω) we see that g = 0. � 46 MESMOULI, ARDJOUNI AND DJOUDI Assuming (1.9) and (1.10) hold. Let ξ be a positive constant that satisfies (2.12) µ2√ 1 −d ≤ ξ ≤ 2µ1 − µ2√ 1 −d . Note that this choice of ξ is possible from assumption (1.9). We define on the Hilbert space H the time dependent inner product (2.13) 〈( u z ) , ( ũ z̃ )〉 t := ∫ Ω u (x) ũ (x) dx + ξτ (t) ∫ ΓN ∫ 1 0 z (x,ρ) z̃ (x,ρ) dpdΓ. Using this time dependent inner product and Theorem 1, we can deduce a well-posedness result. Theorem 2. For any initial datum U0 ∈D(A(0)) there exists a unique solution U ∈ C([0, +∞),D(A(0))) ∩C1([0, +∞),H), of system (2.8). Proof. We first observe that (2.14) ‖φ‖t ‖φ‖s ≤ e c 2τ0 |t−s| , ∀t,s ∈ [0,T], where φ = (u,z)T and c is a positive constant. Indeed, for all s,t ∈ [0,T], we have ‖φ‖2t −‖φ‖ 2 se c τ0 |t−s| = ( 1 −e c τ0 |t−s| )∫ Ω u2dx + ξ ( τ(t) − τ(s)e c τ0 |t−s| )∫ ΓN ∫ 1 0 z2(x,ρ)dρdΓ. We notice that 1 − e c τ0 |t−s| ≤ 0. Moreover τ(t) − τ(s)e c τ0 |t−s| ≤ 0 for some c > 0. Indeed, τ(t) = τ(s) + τ′(a)(t−s), where a ∈ (s,t), and thus, τ(t) τ(s) ≤ 1 + |τ′(a)| τ(s) |t−s|. By (1.8), τ′ is bounded on [0,T] and therefore, recalling also (1.7), τ(t) τ(s) ≤ 1 + c τ0 |t−s| ≤ e c τ0 |t−s| , which proves (2.14). Now we calculate 〈A(t)U,U〉t for a fixed t. Take U = (u,z)T ∈D(A(t)). Then, 〈A(t)U,U〉t = 〈( ∆u τ′(t)ρ−1 τ(t) zρ ) , ( u z )〉 t = ∫ Ω u(x)∆u(x)dx− ξ ∫ ΓN ∫ 1 0 (1 − τ′(t)ρ) zρ (x,ρ) z (x,ρ) dρdΓ. So, by Green’s formula, 〈A(t)U,U〉t = ∫ ΓN ∂u (x) ∂ν u(x)dΓ − ∫ Ω |∇u(x)|2dx − ξ ∫ ΓN ∫ 1 0 (1 − τ′(t)ρ)zρ(x,ρ)z(x,ρ)dρdΓ.(2.15) Integrating by parts in ρ, we obtain∫ ΓN ∫ 1 0 zρ(x,ρ)z(x,ρ)(1 − τ′(t)ρ) dρdΓ = ∫ ΓN ∫ 1 0 1 2 ∂ ∂ρ z2(x,ρ)(1 − τ′(t)ρ)dρdΓ = τ′(t) 2 ∫ ΓN ∫ 1 0 z2(x,ρ)dρdΓ + 1 2 ∫ ΓN {z2(x, 1) (1 − τ′ (t)) −z2(x, 0)}dΓ.(2.16) EXPONENTIAL STABILITY OF THE HEAT EQUATION 47 Therefore, from (2.15) and (2.16), 〈A(t)U,U〉t = ∫ ΓN ∂u (x) ∂ν u(x)dΓ − ∫ Ω |∇u(x)|2dx − ξ 2 ∫ ΓN {z2(x, 1) (1 − τ′ (t)) −z2(x, 0)}dΓ − ξτ′(t) 2 ∫ ΓN ∫ 1 0 z2(x,ρ)dρdΓ = − ∫ ΓN [µ1u (x) + µ2z (x, 1)] u(x)dΓ − ∫ Ω |∇u(x)|2dx + ξ 2 ∫ ΓN u2(x)dΓ − ξ 2 ∫ ΓN {z2(x, 1) (1 − τ′ (t)) dΓ − ξτ′(t) 2 ∫ Γ1 ∫ 1 0 z2(x,ρ)dρdΓ = − ( µ1 − ξ 2 )∫ ΓN u2(x)dΓ −µ2 ∫ ΓN z (x, 1) u(x)dΓ − ∫ Ω |∇u(x)|2dx − ξ 2 ∫ ΓN {z2(x, 1) (1 − τ′ (t)) dΓ − ξτ′(t) 2 ∫ ΓN ∫ 1 0 z2(x,ρ)dρdΓ, from which, using Cauchy-Schwarz’s, Poincaré’s inequality and (1.10), it follows that 〈A(t)U,U〉t ≤ ( −µ1 + ξ 2 + µ2 2 √ 1 −d − 1 Cp )∫ ΓN u2(x)dΓ + ( µ2 √ 1 −d 2 − ξ 2 (1 −d) )∫ ΓN z2(x, 1)dΓ + κ(t)〈U,U〉t,(2.17) where (2.18) κ(t) = (τ′2 (t) + 1) 1 2 2τ(t) . Now, observe that from (2.12), −µ1 + ξ 2 + µ2 2 √ 1 −d ≤ 0, µ2 √ 1 −d 2 − ξ 2 (1 −d) ≤ 0. Then (2.19) 〈A(t)U,U〉t −κ(t)〈U,U〉t ≤ 0, which means that the operator Ã(t) = A(t) −κ(t)I is dissipative. Moreover, κ′(t) = τ′′(t)τ′(t) 2τ(t)(τ′2(t) + 1) 1 2 − τ′(t)(τ′2(t) + 1) 1 2 2τ(t)2 , is bounded on [0,T] for all T > 0 (by (1.6) and (1.7)) and we have d dt A(t)U = ( 0 τ′′(t)τ(t)ρ−τ′(t)(τ′(t)ρ−1) τ(t)2 zρ ) , with τ′′(t)τ(t)ρ−τ′(t)(τ′(t)ρ−1) τ(t)2 bounded on [0,T]. Thus (2.20) d dt Ã(t) ∈ L∞∗ ([0,T],B(D(A(0)),H)), the space of equivalence classes of essentially bounded, strongly measurable functions from [0,T] into B(D(A(0)),H). Now, we show that λI −A(t) is surjective for fixed t > 0 and λ > 0. Given (g,h)T ∈ H, we seek U = (u,z)T ∈D(A(t)) solution of (λI −A(t)) ( u z ) = ( g h ) , 48 MESMOULI, ARDJOUNI AND DJOUDI that is verifying (2.21) { λu− ∆u = g, λz + 1−τ′(t)ρ τ(t) zρ = h. Suppose that we have found u with the appropriate regularity. We can then determine z, indeed z satisfies the differential equation, λz(x,ρ) + 1 − τ′ (t) ρ τ (t) zρ(x,ρ) = h(x,ρ), for x ∈ Γ,ρ ∈ (0, 1) , and the boundary condition (2.22) z(x, 0) = u(x), for x ∈ ΓN. Therefore z is explicitly given by z(x,ρ) = u(x)e−λρτ(t) + τ(t)e−λρτ(t) ∫ ρ 0 h(x,σ)eλστ(t)dσ, if τ′(t) = 0, and z(x,ρ) = u(x)e λ τ(t) τ′(t) ln(1−τ ′(t)ρ) + e λ τ(t) τ′(t) ln(1−τ ′(t)ρ) ∫ ρ 0 h(x,σ)τ(t) 1 − τ′(t)σ e −λ τ(t) τ′(t) ln(1−τ ′(t)σ) dσ, otherwise. This means that once u is found with the appropriate properties, we can find z. In particular, if τ′(t) = 0, (2.23) z(x, 1) = u(x)e−λτ(t) + z0(x), x ∈ ΓN, with z0 ∈ L2(ΓN ) defined by (2.24) z0(x) = τ(t)e −λτ(t) ∫ 1 0 h(x,σ)eλστ(t)dσ, x ∈ ΓN, and, if τ′(t) 6= 0, (2.25) z(x, 1) = u(x)e λ τ(t) τ′(t) ln(1−τ ′(t)) + z0(x), x ∈ ΓN, with z0 ∈ L2(ΓN ) defined by (2.26) z0(x) = e λ τ(t) τ′(t) ln(1−τ ′(t)) ∫ 1 0 h(x,σ)τ(t) 1 − τ′(t)σ e −λ τ(t) τ′(t) ln(1−τ ′(t)σ) dσ, for x ∈ ΓN . Then, we have to find u. In view of the equation (2.27) λu− ∆u = g. Multiplying this identity by a test function φ and integrating in space (2.28) ∫ Ω (λuφ− ∆uφ) dx = ∫ Ω gφdx, ∀φ ∈ H1ΓD, using Green’s formula, we obtain∫ Ω (λuφ− ∆uφ) dx = ∫ Ω (λuφ + ∇u∇φ) dx− ∫ ΓN ∂u ∂ν φdΓ = ∫ Ω (λuφ + ∇u∇φ) dx + ∫ ΓN (µ1u + µ2z (x, 1)) φdΓ. By (2.23), we obtain ∫ Ω (λuφ− ∆uφ) dx = ∫ Ω (λuφ + ∇u∇φ) dx + ∫ ΓN ( µ1u + µ2 ( ue−λτ(t) + z0 )) φdΓ, EXPONENTIAL STABILITY OF THE HEAT EQUATION 49 if τ′(t) = 0, and by (2.25)∫ Ω (λuφ− ∆uφ) dx = ∫ Ω (λuφ + ∇u∇φ) dx + ∫ ΓN ( µ1u + µ2 ( ue λ τ(t) τ′(t) ln(1−τ ′(t)) + z0 )) φdΓ, otherwise. Therefore, (2.28) can be rewritten as (2.29) ∫ Ω (λuφ + ∇u∇φ) dx + ∫ ΓN ( µ1u + µ2 ( ue−λτ(t) + z0 )) φdΓ = ∫ Ω gφdx, if τ′(t) = 0, and∫ Ω (λuφ + ∇u∇φ) dx + ∫ ΓN ( µ1u + µ2 ( ue λ τ(t) τ′(t) ln(1−τ ′(t)) + z0 )) φdΓ = ∫ Ω gφdx,(2.30) otherwise. As the left-hand side of (2.29) or (2.30) is coercive on H1ΓD (Ω), the Lax-Milgram lemma guarantees the existence and uniqueness of a solution u ∈ H1ΓD (Ω) of (2.29), (2.30). If we consider φ ∈ D(Ω) in (2.29), (2.30), we have that u solves (2.27) in D′(Ω) and thus u ∈ E(∆,L2(Ω)). Using Green’s formula (2.10) in (2.29) and using (2.27), we obtain, if τ′(t) = 0∫ ΓN ( µ1 + µ2e −λτ(t) ) uφdΓ + 〈 ∂u ∂ν ,φ〉ΓN = −µ2 ∫ ΓN z0φdΓ, from which follows ∂u ∂ν + ( µ1 + µ2e −λτ(t) ) u = −µ2z0 on ΓN, which imply that ∂u ∂ν = −µ1u−µ2z (·, 1) on ΓN, where we have used (2.23) and (2.27). We find the same result if τ′(t) 6= 0. In conclusion, we have found (u,z)T ∈D(A), which verifies (2.21), and thus λI −A(t) is surjective for some λ > 0 and t > 0. Again as κ(t) > 0, this proves that (2.31) λI −Ã(t) = (λ + κ(t))I −A(t) is surjective, for any λ > 0 and t > 0. Then, (2.14), (2.19) and (2.31) imply that the family à = {Ã(t) : t ∈ [0,T]} is a stable family of generators in H with stability constants independent of t, by [6, Proposition 1.1]. Therefore, the assumptions (i)-(iv) of Theorem 1 are satisfied by (2.11), (2.14), (2.19), (2.31), (2.20) and Lemma 1, and thus, the problem { Ũ′ = Ã(t)Ũ, Ũ(0) = U0, has a unique solution Ũ ∈ C([0, +∞),D(A(0))) ∩ C1([0, +∞),H) for U0 ∈ D(A(0)). The requested solution of (2.8) is then given by U(t) = eβ(t)Ũ(t), with β(t) = ∫ t 0 κ(s)ds, because U′eβ(t)Ũ(t) + eβ(t)Ũ′(t) = κ(t)eβ(t)Ũ(t) + eβ(t)Ã(t)Ũ(t) = eβ(t)(κ(t)Ũ(t) + Ã(t)Ũ(t)) = eβ(t)A(t)Ũ(t) = A(t)eβ(t)Ũ(t) = A(t)U(t). 50 MESMOULI, ARDJOUNI AND DJOUDI This concludes the proof. � 3. The decay of the energy Let us choose the following energy (3.1) E (t) = 1 2 ∫ Ω u2 (x,t) dx + ξτ (t) 2 ∫ ΓN ∫ 1 0 u2 (x,t− τ (t) ρ) dpdΓ, where ξ is a suitable positive constant. Proposition 1. Let (1.9) and (1.10) be satisfied. Then for all regular solution of problem (2.8), the energy is decreasing and satisfies (3.2) E′ (t) ≤−C (∫ ΓN u2 (x,t) dΓ + ∫ ΓN u2 (x,t− τ (t)) dΓ ) . Proof. Differentiating (3.1), we get E′ (t) = ∫ Ω uutdx + ξτ′ (t) 2 ∫ ΓN ∫ 1 0 u2 (x,t− τ (t) ρ) dpdΓ + ξτ (t) ∫ ΓN ∫ 1 0 (1 − τ′ (t) ρ) u (x,t− τ (t) ρ) ut (x,t− τ (t) ρ) dpdΓ, then E′ (t) = ∫ Ω u∆udx + ξτ′ (t) 2 ∫ ΓN ∫ 1 0 u2 (x,t− τ (t) ρ) dpdΓ + ξτ (t) ∫ ΓN ∫ 1 0 (1 − τ′ (t) ρ) u (x,t− τ (t) ρ) ut (x,t− τ (t) ρ) dpdΓ. By Green’s formula and integrating by parts in ρ, we obtain E′ (t) = − ∫ Ω |∇u|2 dx + ∫ ΓN u ∂u ∂ν dΓ − ξ 2 ∫ ΓN u2 (x,t− τ (t)) (1 − τ′ (t)) dΓ + ξ 2 ∫ ΓN u2 (x,t) dΓ, and by (1.3), we obtain E′ (t) = − ∫ Ω |∇u|2 dx− ∫ ΓN [ µ1u 2 (x,t) + µ2u (x,t) u (x,t− τ (t)) ] dΓ − ξ 2 ∫ ΓN u2 (x,t− τ (t)) (1 − τ′ (t)) dΓ + ξ 2 ∫ ΓN u2 (x,t) dΓ. By Cauchy-Schwarz’s and Poincaré’s inequality, we get, E′ (t) ≤ ( − 1 Cp −µ1 + ξ 2 + µ2 2 √ 1 −d )∫ ΓN u2 (x,t) dΓ − ( ξ (1 −d) 2 + µ2 √ 1 −d 2 )∫ ΓN u2 (x,t− τ (t)) dΓ. Since the condition (2.12), we deduce that − 1 Cp −µ1 + ξ 2 + µ2 2 √ 1 −d ≤ 0. which concludes the proof. � EXPONENTIAL STABILITY OF THE HEAT EQUATION 51 4. Exponential stability In this section, we will give an exponential stability result for the problem (1.1)–(1.5) by using the following Lyapunov functional (4.1) E (t) = E (t) + γÊ (t) , where γ > 0 is a parameter that will be fixed small enough later on, E is the standard energy defined by (3.1) and Ê is defined by (4.2) Ê (t) = ξτ (t) ∫ ΓN ∫ 1 0 e−2τ(t)ρu2 (x,t− τ (t) ρ) dpdΓ. Note that, the functional Ê is equivalent to the energy E, that is there exist two positive constant d1, d2 such that (4.3) d1E (t) ≤E (t) ≤ d2E (t) . Theorem 3. Assume (1.6) and (1.7). Then, there exist positive constants C1, C2 such that for any solution of problem (1.1)–(1.5), E(t) ≤ C1E(0)e−C2t, ∀t ≥ 0. Proof. First, we differentiate Ê (t) to have d dt Ê (t) = τ′ (t) τ (t) Ê (t) + ξτ (t) ∫ ΓN ∫ 1 0 (−2τ′ (t) ρ) e−2τ(t)ρu2 (x,t− τ (t) ρ) dpdΓ + J, where J = 2ξτ (t) ∫ ΓN ∫ 1 0 e−2τ(t)ρ (1 − τ′ (t) ρ) ut (x,t− τ (t) ρ) u (x,t− τ (t) ρ) dpdΓ. Moreover, by noticing one more time that z (x,ρ,t) = u (x,t− τ (t) ρ) , x ∈ ΓN, ρ ∈ (0, 1) , t > 0, and by integrating by parts in ρ, we have J = −ξ ∫ ΓN ∫ 1 0 e−2τ(t)ρ (1 − τ′ (t) ρ) ∂ ∂ρ (z (x,ρ,t)) 2 dpdΓ = ξ ∫ ΓN ∫ 1 0 e−2τ(t)ρ [−2τ (t) (1 − τ′ (t) ρ) − τ′ (t)] z2 (x,ρ,t) dpdΓ −ξ ∫ ΓN e−2τ(t) (1 − τ′ (t)) z2 (x, 1, t) dΓ + ξ ∫ ΓN z2 (x, 0, t) dΓ = ξ ∫ ΓN ∫ 1 0 e−2τ(t)ρ [−2τ (t) (1 − τ′ (t) ρ) − τ′ (t)] u2 (x,t− τ (t) ρ) dpdΓ −ξ ∫ ΓN e−2τ(t) (1 − τ′ (t)) u2 (x,t− τ (t)) dΓ + ξ ∫ ΓN u2 (x,t) dΓ. Therefore, we have d dt Ê (t) = τ′ (t) τ (t) Ê (t) + ξ ∫ ΓN ∫ 1 0 e−2τ(t)ρ [−2τ (t) − τ′ (t)] u2 (x,t− τ (t) ρ) dpdΓ − ξ ∫ ΓN e−2τ(t) (1 − τ′ (t)) u2 (x,t− τ (t)) dΓ + ξ ∫ ΓN u2 (x,t) dΓ = −2Ê (t) − ξ ∫ ΓN e−2τ(t) (1 − τ′ (t)) u2 (x,t− τ (t)) dΓ + ξ ∫ ΓN u2 (x,t) dΓ. 52 MESMOULI, ARDJOUNI AND DJOUDI As τ′ (t) < 1, we obtain (4.4) d dt Ê (t) ≤−2Ê (t) + ξ ∫ ΓN u2 (x,t) dΓ. Consequently, gathering (3.2), (4.1) and (4.4), we obtain d dt E (t) = d dt E (t) + γ d dt Ê (t) ≤−2γÊ (t) + γξ ∫ ΓN u2 (x,t) dΓ −C ∫ ΓN ( u2 (x,t) + u2 (x,t− τ (t)) ) dΓ. Then, for γ sufficiently small, we can estimate (4.5) d dt E (t) ≤−2γÊ (t) −C ∫ ΓN ( u2 (x,t) + u2 (x,t− τ (t)) ) dΓ. Now, observe that by assumption (1.6) on τ (t), we can deduce Ê (t) ≥ ξτ (t) ∫ ΓN ∫ 1 0 e−2τρu2 (x,t− τ (t) ρ) dpdΓ ≥ kξτ (t) 2 ∫ ΓN ∫ 1 0 u2 (x,t− τ (t) ρ) dpdΓ,(4.6) for some positive constant k. 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