New decay rates for a Cauchy thermelastic laminated Timoshenko problem with interfacial slip under Fourier or Cattaneo laws
aa r X i v : . [ m a t h . C A ] F e b NEW DECAY RATES FOR A CAUCHY THERMOELASTIC LAMINATEDTIMOSHENKO PROBLEM WITH INTERFACIAL SLIP UNDER FOURIER ORCATTANEO LAWS
AISSA GUESMIA
Institut Elie Cartan de Lorraine, UMR 7502, Universit´e de Lorraine3 Rue Augustin Fresnel, BP 45112, 57073 Metz Cedex 03, France
Abstract.
The objective of the present paper is to investigate the decay of solutions for a laminatedTimoshenko beam with interfacial slip in the whole space R subject to a thermal effect acting only on onecomponent modelled by either Fourier or Cattaneo law. When the thermal effect is acting via the secondor third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of theslip), we obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo systems, satisfy thesame polynomial stability estimates in the L -norm of the solution and its higher order derivatives withrespect of the space variable. The decay rate depends on the regularity of the initial data. In addition,the presence and absence of the regularity-loss type property are determined by some relations betweenthe parameters of systems. However, when the thermal effect is acting via the first comoponent ofthe system (transversal displacement), a new stability condition is introduced for both Timoshenko-Fourier and Timoshenko-Cattaneo systems. This stability condition is in the form of threshold betweenpolynomial stability and convergence to zero. To prove our results, we use the energy method in Fourierspace combined with judicious choices of weight functions to build appropriate Lyapunov functionals. Keywords:
Timoshenko beam, Interfacial slip, Heat conduction, Fourier law, Cattaneo law,Energy method, Fourier analysis.
MSC2010:
Introduction
In this paper, we investigate the decay properties of a thermoelastic laminated Timoshenko beam withinterfacial slip in the whole space R where the thermal effect is modelled by Fourier law or Cattaneo law.The first system we consider is the coupling of a laminated Timoshenko system with a heat conductiondescribed by Fourier law and given by ϕ tt − k ( ϕ x + ψ + w ) x + τ γη x = 0 ,ψ tt − k ψ xx + k ( ϕ x + ψ + w ) + τ γη x = 0 ,w tt − k w xx + k ( ϕ x + ψ + w ) + τ γη x = 0 ,η t − k η xx + γ ( τ ϕ xt + τ ψ xt + τ w xt ) = 0 , (1.1)and the second system of interest is the coupling between a laminated Timoshenko system with a heatconduction described by Cattaneo law and given by ϕ tt − k ( ϕ x + ψ + w ) x + τ γη x = 0 ,ψ tt − k ψ xx + k ( ϕ x + ψ + w ) + τ γη x = 0 ,w tt − k w xx + k ( ϕ x + ψ + w ) + τ γη x = 0 ,η t + k q x + γ ( τ ϕ xt + τ ψ xt + τ w xt ) = 0 ,q t + k q + k η x = 0 , (1.2)where k , k , k , k , k , γ > ϕ = ϕ ( x, t ), ψ = ψ ( x, t ), η = η ( x, t ) and q = q ( x, t ) denoting the transversaldisplacement and the rotation angle of the beam, the temperature and the heat flow, respectively, w = w ( x, t ) is proportional to the amount of slip along the interface, so the third equation in (1.1) and (1.2) E-mail addresse: [email protected]. describes the dynamics of the slip, x ∈ R and t >
0. The termal effect γη x is acting only on one equationof the laminated Timoshenko system, so(1.3) ( τ , τ , τ ) ∈ { (1 , , , (0 , , , (0 , , } . Systems (1.1) and (1.2) are, respectively, subject to the initial conditions ( ( ϕ, ψ, w, η )( x,
0) = ( ϕ , ψ , w , η )( x ) , ( ϕ t , ψ t , w t )( x,
0) = ( ϕ , ψ , w )( x )(1.4)and ( ( ϕ, ψ, w, η, q )( x,
0) = ( ϕ , ψ , w , η , q )( x ) , ( ϕ t , ψ t , w t )( x,
0) = ( ϕ , ψ , w )( x ) . (1.5)The main purpose of this paper is to investigate the capacity of the dissipation, generated by the heatconduction γη x via only one equation of the laminated Timoshenko system, to stabilize (1.1) and (1.2),and to determine its influence on the decay rate of solutions. We will show that the two cases( τ , τ , τ ) = (1 , ,
0) and ( τ , τ , τ ) ∈ { (0 , , , (0 , , } are completely different in the following sense: Case ( τ , τ , τ ) = (1 , , k = k , and when k = k , the following polynomial stability result holds true for (1.1) and (1.2): for any N, ℓ ∈ N ∗ with ℓ ≤ N , j ∈ { , . . . , N − ℓ } and U ∈ H N ( R ) ∩ L ( R ), there exists c > k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c (1 + t ) − ℓ/ k ∂ j + ℓx U k L ( R ) , ∀ t ∈ R + , where ∂ jx = ∂ j ∂x j , and U and U are defined in section 2. Case ( τ , τ , τ ) ∈ { (0 , , , (0 , , } : when the three speeds of wave propagations of the laminatedTimoshenko system are equal; that is(1.8) k = k = k , systems (1.1) and (1.2) are stable with the following decay rate: for any N, ℓ ∈ N ∗ with ℓ ≤ N , j ∈{ , . . . , N − ℓ } and U ∈ H N ( R ) ∩ L ( R ), there exists c , ˜ c > k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c e − ˜ c t k ∂ j + ℓx U k L ( R ) , ∀ t ∈ R + . If (1.8) is not satisfied, then the following estimate holds true for (1.1) and (1.2):(1.10) k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c (1 + t ) − ℓ/ k ∂ j + ℓx U k L ( R ) , ∀ t ∈ R + . It is well known in the literature that the behavior of the Fourier transform of U in the low frequencyregion determines the rate of decay of U , while its behavior in the high frequency rigion imposes aregularity restriction on the initial data known as the regularity-loss property; see [6, 14, 15, 24, 26, 27].It seems that the dissipation generated by the heat conduction is so weak in the high frequency regionthat it leads to the regularity-loss property in the estimates (1.7) and (1.10). On the other hand, therestriction (1.6) and the fact that the decay rate in (1.7) is smaller than the one in (1.9) and (1.10)indicate that the effect of the heat conduction is better propagated to the whole system from the secondor third equation of the laminated Timoshenko system than from the first one.A model describing laminated Timoshenko beams with interfacial slip based on the Timoshenko theory(see, for example, [12, 13, 18]) is given by ϕ tt − k ( ϕ x + ψ + w ) x = 0 ,ψ tt − k ψ xx + k ( ϕ x + ψ + w ) = 0 ,w tt − k w xx + k ( ϕ x + ψ + w ) = 0(1.11) AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 3 and can be derived from the following more general model of Bresse-type: ϕ tt − k ( ϕ x + ψ + lw ) x − ˜ lk ( w x − ˜ lϕ ) = 0 ,ψ tt − k ψ xx + k ( ϕ x + ψ + lw ) = 0 ,w tt − k ( w x − ˜ lϕ ) x + l k ( ϕ x + ψ + lw ) = 0 , (1.12)where l and ˜ l are positive constants. The system (1.12) coincides with (1.11) when l = 1 and ˜ l = 0.When w = l = ˜ l = 0, the system (1.12) is reduced to the following Timoshenko-type system: ( ϕ tt − k ( ϕ x + ψ ) x = 0 ,ψ tt − k ψ xx + k ( ϕ x + ψ ) = 0 . (1.13)The well-posedness as well as the stability questions for (1.11), (1.12) and (1.13) have been the subjectof various studies in the literature, where different controls (dampings, memories, heat conduction, ...)and/or boundary conditions (Dirichlet, Neaumann, mixed, ...) have been used to force the solution toconverge to zero when time t goes to infinity, and get information on its speed of convergence.In case of bounded domains, we refer the reader to, for example, [1, 2, 3, 4, 8, 9, 10, 11, 17, 18, 19, 20,21, 22, 29] and the refereces therein.In case of unbounded domains, the stability of (1.12) and (1.13) has been also treated in the literaturefor the last few years. In this direction, we mention the papers [5, 7, 15, 16, 23, 25] (see also thereferences therein), where some polynomial stability estimates for L -norm of solutions have been provedusing frictional dampings, heat conduction effects or memory controls. In some particular cases, theoptimality of the decay rate was also proved.Our results in the present paper give extensions from the bounded to the unbounded domain case. Theproof is based on the energy method combined with the Fourier analysis (by using the transformation inthe Fourier space) and well chosen weight functions.The paper is organized as follows. In Section 2, we formulate (1.1) and (1.2) as first order Cauchysystems and give some preliminaries. In Sections 3 and 4, we prove our polynomial stability estimatesfor (1.1) and (1.2), respectively. We end our paper by some general comments and other related issuesin Section 5. 2. Formulation of the problems
To formulate (1.1) and (1.2) in abstract first order systems, we introduce the new variables(2.1) u = ϕ t , y = ψ t , θ = w t , v = ϕ x + ψ + w, z = ψ x and φ = w x . Then, the systems (1.1) and (1.2) can be rewritten, respectively, in the forms v t − u x − y − θ = 0 ,u t − k v x + τ γ η x = 0 ,z t − y x = 0 ,y t − k z x + k v + τ γ η x = 0 ,φ t − θ x = 0 ,θ t − k φ x + k v + τ γη x = 0 ,η t − k η xx + γ ( τ u x + τ y x + τ θ x ) = 0 . (2.2) A. GUESMIA and v t − u x − y − θ = 0 ,u t − k v x + τ γ η x = 0 ,z t − y x = 0 ,y t − k z x + k v + τ γ η x = 0 ,φ t − θ x = 0 ,θ t − k φ x + k v + τ γη x = 0 ,η t + k q x + γ ( τ u x + τ y x + τ θ x ) = 0 ,q t + k q + k η x = 0 . (2.3)Now, we define the variable U and its initial data U by U = ( ( v, u, z, y, φ, θ, η ) T for (2.2) , ( v, u, z, y, φ, θ, η, q ) T for (2.3) and U = ( ( v, u, z, y, φ, θ, η ) T ( · ,
0) for (2.2) , ( v, u, z, y, φ, θ, η, q ) T ( · ,
0) for (2.3) . The systems (2.2) and (2.3) with the initial conditions (1.4) and (1.5), respectively, are equivalent to ( U t ( x, t ) + A U xx ( x, t ) + A U x ( x, t ) + A U ( x, t ) = 0 ,U ( x,
0) = U ( x ) , (2.4)where, for (2.2),(2.5) A U xx = − k η xx , A U x = − u x − k v x + τ γη x − y x − k z x + τ γη x − θ x − k φ x + τ γη x γ ( τ u x + τ y x + τ θ x ) and A U = − y − θ k v k v , and for (2.3)(2.6) A = 0 , A U x = − u x − k v x + τ γη x − y x − k z x + τ γη x − θ x − k φ x + τ γη x k q x + γ ( τ u x + τ y x + τ θ x ) k η x and A U = − y − θ k v k v k q . For a function h : R → C , Re h , Im h , ¯ h and b h denote, respectively, the real part of h , the imaginarypart of h , the conjugate of h and the Fourier transformation of h . Using the Fourier transformation (with AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 5 respect to the space variable x ) to transform (2.4) in the Fourier space, we obtain the following first orderCauchy system:(2.7) ( b U t ( ξ, t ) − ξ A b U ( ξ, t ) + i ξ A b U ( ξ, t ) + A b U ( ξ, t ) = 0 , ξ ∈ R , t > , b U ( ξ,
0) = b U ( ξ ) , ξ ∈ R . The solution of (2.7) is given by(2.8) b U ( ξ, t ) = e − ( − ξ A + i ξ A + A ) t b U ( ξ ) . The energy b E associated with (2.7) is defined by(2.9) b E ( ξ, t ) = 12 h k | b v | + | b u | + k | b z | + | b y | + k | b φ | + | b θ | + | b η | i in case (2.2), and b E ( ξ, t ) = 12 h k | b v | + | b u | + k | b z | + | b y | + k | b φ | + | b θ | + | b η | + | b q | i (2.10)in case (2.3). System (2.7) is dissipative, since(2.11) ddt b E ( ξ, t ) = − k ξ | b η | in case (2.2), and(2.12) ddt b E ( ξ, t ) = − k | b q | in case (2.3). Indeed, first, we remember the following two trivial identities which will be frequently usedin this paper: for any two differentiable functions h, d : R → C , we have(2.13) ddt Re ( h ¯ d ) = Re ( h t ¯ d + d t ¯ h )and(2.14) ddt Re ( ih ¯ d ) = Re (cid:2) i ( h t ¯ d − d t ¯ h ) (cid:3) . In case (2.5), the first equation in (2.7) is equivalent to b v t − iξ b u − b y − b θ = 0 , b u t − ik ξ b v + iτ γ ξ b η = 0 , b z t − iξ b y = 0 , b y t − ik ξ b z + k b v + iτ γ ξ b η = 0 , b φ t − iξ b θ = 0 , b θ t − ik ξ b φ + k b v + iτ γ ξ b η = 0 , b η t + k ξ b η + iγξ ( τ b u + τ b y + τ b θ ) . (2.15)Multiplying the equations in (2.15) by k ¯ b v , ¯ b u , k ¯ b z , ¯ b y , k ¯ b φ , ¯ b θ and ¯ b η , respectively, adding the obtainedequations, taking the real part of the resulting expression and using (2.13), we obtain (2.11). Similarily,in case (2.6), the first equation of (2.7) is reduced to b v t − iξ b u − b y − b θ = 0 , b u t − ik ξ b v + iτ γ ξ b η = 0 , b z t − iξ b y = 0 , b y t − ik ξ b z + k b v + iτ γ ξ b η = 0 , b φ t − iξ b θ = 0 , b θ t − ik ξ b φ + k b v + iτ γ ξ b η = 0 , b η t + ik ξ b q + iγξ ( τ b u + τ b y + τ b θ ) = 0 , b q t + k b q + ik ξ b η = 0 . (2.16) A. GUESMIA
Multiplying the equations in (2.16) by k ¯ b v , ¯ b u , k ¯ b z , ¯ b y , k ¯ b φ , ¯ b θ , ¯ b η and ¯ b q , respectively, adding the obtainedequations, taking the real part of the resulting expression and using (2.13), we get (2.12).It is clear that the energy b E is equivalent to | b U | defined in case (2.15) by | b U ( ξ, t ) | = | b v | + | b u | + | b z | + | b y | + | b φ | + | b θ | + | b η | , and in case (2.16) by | b U ( ξ, t ) | = | b v | + | b u | + | b z | + | b y | + | b φ | + | b θ | + | b η | + | b q | . Since, for α = min { k , k , k , } and α = max { k , k , k , } , we have(2.17) α | b U ( ξ, t ) | ≤ b E ( ξ, t ) ≤ α | b U ( ξ, t ) | , ∀ ξ ∈ R , ∀ t ∈ R + . We finish this section by proving two lemmas, which will be also frequently used in the proof of ourstability results.
Lemma 2.1.
Let σ , p and r be real numbers such that σ > − and p, r > . Then there exists C σ,p,r > such that (2.18) Z ξ σ e − r t ξ p dξ ≤ C σ,p,r (1 + t ) − ( σ +1) /p , ∀ t ∈ R + . Proof.
For 0 ≤ t ≤
1, (2.18) is evident, for any C σ,p,r ≥ ( σ +1) /p σ +1 . For t >
1, we have Z ξ σ e − r t ξ p dξ = Z ξ σ +1 − p e − r t ξ p ξ p − dξ = Z ( ξ p ) ( σ +1 − p ) /p e − r t ξ p ξ p − dξ. Taking τ = r t ξ p . Then ξ p = τr t and ξ p − dξ = 1 p r t dτ. Substituting in the above integral, we find Z ( ξ p ) ( σ +1 − p ) /p e − r t ξ p ξ p − dξ = Z r t (cid:16) τr t (cid:17) ( σ +1 − p ) /p e − τ p r t dτ ≤ p ( r t ) ( σ +1) /p Z + ∞ τ ( σ +1 − p ) /p e − τ dτ ≤ ( σ +1) /p p r ( σ +1) /p C σ,p ( t + 1) − ( σ +1) /p , where C σ,p = Z + ∞ τ ( σ +1 − p ) /p e − τ dτ, which is a convergent integral, for any σ > − p >
0. This completes the proof of (2.18) with C σ,p,r = max (cid:26) ( σ +1) /p σ + 1 , ( σ +1) /p p r ( σ +1) /p C σ,p (cid:27) . (cid:3) Lemma 2.2.
For any positive real numbers σ , σ and σ , we have (2.19) sup | ξ |≥ | ξ | − σ e − σ t | ξ | − σ ≤ (1 + σ / ( σ σ )) σ /σ (1 + t ) − σ /σ , ∀ t ∈ R + . Proof.
It is clear that (2.19) is satisfied for t = 0. Let t > h ( x ) = x − σ e − σ t x − σ , for x ≥ h ′ ( x ) = ( σ σ tx − σ − σ ) x − σ − e − σ t x − σ . If t ≥ σ / ( σ σ ), then h ( x ) ≤ h ((( σ σ t ) /σ ) /σ ) = (( σ σ ) /σ ) − σ /σ e − σ /σ (1 + 1 /t ) σ /σ (1 + t ) − σ /σ ≤ (( σ σ ) /σ ) − σ /σ (1 + ( σ σ ) /σ ) σ /σ (1 + t ) − σ /σ = (1 + σ / ( σ σ )) σ /σ (1 + t ) − σ /σ , AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 7 which gives (2.19) by taking x = | ξ | . If 0 < t < σ / ( σ σ ), then h ( x ) ≤ h (1) = e − σ t (1 + t ) σ /σ (1 + t ) − σ /σ ≤ (1 + σ / ( σ σ )) σ /σ (1 + t ) − σ /σ , which implies (2.19), for x = | ξ | . (cid:3) Stability: Fourier law (1.1)This section is dedicated to the investigation of the asymptotic behavior, when time t goes to infinity,of the solution U of (2.4) in case of Fourier law (1.1). We will prove (1.7), (1.9) and (1.10) by showing,first, that | b U | converges exponentially to zero with respect to time t . Then, we prove that the solution(2.8) of (2.7) does not converge to zero when t goes to infinity if ( τ , τ , τ ) = (1 , ,
0) and k = k .In this section and in the next one, C denotes a generic positive constant, and C ε denotes a genericpositive constant depending on some positive constant ε . These generic constants can be different fromline to line. Before distinguishing between the three cases (1.3), we prove several identities, which willplay a crucial role in the proofs.Multiplying (2.15) and (2.15) by i ξ b z and − i ξ b y , respectively, adding the resulting equations, takingthe real part and using (2.14), we obtain(3.1) ddt Re (cid:16) i ξ b y b z (cid:17) = ξ (cid:0) | b y | − k | b z | (cid:1) − k Re (cid:16) i ξ b v b z (cid:17) + τ γξ Re (cid:16)b η b z (cid:17) . Multiplying (2.15) and (2.15) by i ξ b v and − i ξ b u , respectively, adding the resulting equations, takingthe real part and using (2.14), we find(3.2) ddt Re (cid:16) i ξ b u b v (cid:17) = ξ (cid:0) | b u | − k | b v | (cid:1) − Re (cid:16) i ξ b y b u (cid:17) − Re (cid:16) i ξ b θ b u (cid:17) + τ γξ Re (cid:16)b η b v (cid:17) . Multiplying (2.15) and (2.15) by i ξ b φ and − i ξ b θ , respectively, adding the resulting equations, takingthe real part and using (2.14), we get(3.3) ddt Re (cid:16) i ξ b θ b φ (cid:17) = ξ (cid:16) | b θ | − k | b φ | (cid:17) − k Re (cid:16) i ξ b v b φ (cid:17) + τ γξ Re (cid:16)b η b φ (cid:17) . Multiplying (2.15) and (2.15) by − ξ b v and − ξ b θ , respectively, adding the resulting equations, takingthe real part and using (2.13), we have ddt Re (cid:16) − ξ b θ b v (cid:17) = ξ (cid:16) k | b v | − | b θ | (cid:17) − ξ Re (cid:16) i ξ b u b θ (cid:17) − k ξ Re (cid:16) i ξ b φ b v (cid:17) − ξ Re (cid:16)b y b θ (cid:17) + τ γ ξ Re (cid:16) iξ b η b v (cid:17) . (3.4)Multiplying (2.15) and (2.15) by − ξ b v and − ξ b y , respectively, adding the resulting equations, takingthe real part and using (2.13), we infer that ddt Re (cid:16) − ξ b y b v (cid:17) = ξ (cid:0) k | b v | − | b y | (cid:1) − ξ Re (cid:16) i ξ b u b y (cid:17) − k ξ Re (cid:16) i ξ b z b v (cid:17) − ξ Re (cid:16)b θ b y (cid:17) + τ γ ξ Re (cid:16) iξ b η b v (cid:17) . (3.5)Multiplying (2.15) and (2.15) by i ξ b θ and − i ξ b z , respectively, adding the resulting equations, takingthe real part and using (2.14), we entail(3.6) ddt Re (cid:16) i ξ b z b θ (cid:17) = − ξ Re (cid:16)b y b θ (cid:17) + k ξ Re (cid:16) b φ b z (cid:17) + k Re (cid:16) i ξ b v b z (cid:17) − τ γξ Re (cid:16)b η b z (cid:17) . Mltiplying (2.15) and (2.15) by i ξ b y and − i ξ b φ , respectively, adding the resulting equations, taking thereal part and using (2.14), we arrive at(3.7) ddt Re (cid:16) i ξ b φ b y (cid:17) = − ξ Re (cid:16)b θ b y (cid:17) + k ξ Re (cid:16)b z b φ (cid:17) + k Re (cid:16) i ξ b v b φ (cid:17) − τ γξ Re (cid:16)b η b φ (cid:17) . A. GUESMIA
Multiplying (2.15) and (2.15) by − b z and − b u , respectively, adding the resulting equations, taking thereal part and using (2.13), it follows that(3.8) ddt Re (cid:16) − b u b z (cid:17) = − k Re (cid:16) i ξ b v b z (cid:17) − Re (cid:16) i ξ b y b u (cid:17) + τ γ Re (cid:16) iξ b η b z (cid:17) . Finally, multiplying (2.15) and (2.15) by − b φ and − b u , respectively, adding the resulting equations,taking the real part and using (2.13), it appears that(3.9) ddt Re (cid:16) − b u b φ (cid:17) = − k Re (cid:16) i ξ b v b φ (cid:17) − Re (cid:16) i ξ b θ b u (cid:17) + τ γ Re (cid:16) iξ b η b φ (cid:17) . Case 1: ( τ , τ , τ ) = (1 , , . We start by presenting the exponential stability result for (2.7) inthe next lemma.
Lemma 3.1.
Assume that k = k . Let b U be the solution (2.8) of (2.7) . Then there exist c, e c > suchthat (3.10) | b U ( ξ, t ) | ≤ e c e − c f ( ξ ) t | b U ( ξ ) | , ∀ ξ ∈ R , ∀ t ∈ R + , where (3.11) f ( ξ ) = ξ ξ + ξ + ξ + ξ . Proof.
Multiplying (2.15) and (2.15) by i ξ b η and − i ξ b u , respectively, adding the resulting equations,taking the real part and using (2.14), we get(3.12) ddt Re (cid:16) i ξ b u b η (cid:17) = γξ (cid:0) | b η | − | b u | (cid:1) + k ξ Re (cid:16) i ξ b η b u (cid:17) − k ξ Re (cid:16)b v b η (cid:17) . Similarily, multiplying (2.15) and (2.15) by b η and b θ , respectively, adding the resulting equations, takingthe real part and using (2.13), we find(3.13) ddt Re (cid:16)b η b θ (cid:17) = γRe (cid:16) i ξ b θ b u (cid:17) − k ξ Re (cid:16)b η b θ (cid:17) + k Re (cid:16) iξ b φ b η (cid:17) − k Re (cid:16)b v b η (cid:17) . Also, multiplying (2.15) and (2.15) by b η and b y , respectively, adding the resulting equations, taking thereal part and using (2.13), we obtain(3.14) ddt Re (cid:16)b η b y (cid:17) = − γRe (cid:16) i ξ b u b y (cid:17) − k ξ Re (cid:16)b η b y (cid:17) + k Re (cid:16) iξ b z b η (cid:17) − k Re (cid:16)b v b η (cid:17) . We define the functional F as follows: F ( ξ, t ) = Re (cid:20) i ξ (cid:18) λ b y b z + λ b θ b φ + iλ ξ b θ b v − ( λ + 1) k k − k b z b θ + ( λ + 1) k k − k b φ b y (cid:19)(cid:21) + Re (cid:16) − ξ b y b v + i λ ξ b u b v (cid:17) , (3.15)where λ , λ , λ and λ are positive constants to be defined later ( F is well defined since k = k ). Bymultiplying (3.1)-(3.4), (3.6) and (3.7) by λ , λ , λ , λ , − ( λ +1) k k − k and ( λ +1) k k − k , respectively, addingthe obtained equations and adding (3.5), we deduce that ddt F ( ξ, t ) = − ξ (cid:16) k λ | b z | + k λ | b φ | + (1 − λ ) | b y | + ( λ − λ ) | b θ | + ( k λ − k λ − k ) | b v | (cid:17) + I Re ( iξ b v b z ) + I Re ( iξ b v b φ ) + ξ h λ | b u | − Re (cid:16) iξ (cid:16) λ b u b θ + b u b y (cid:17)(cid:17)i − λ Re h iξ (cid:16)b y b u + b θ b u + iγξ b η b v (cid:17)i , (3.16)where I = k ξ − k λ − ( λ + 1) k k k − k and I = k λ ξ − k λ + ( λ + 1) k k k − k . We put F ( ξ, t ) = ξ (cid:20) F ( ξ, t ) − k Re (cid:16) I b u b z + I b u b φ (cid:17)(cid:21) . (3.17) AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 9
Multiplying (3.8) and (3.9) by I k and I k , respectively, adding the obtained equations, adding (3.16) andmultiplying the resulting formula by ξ , we arrive at ddt F ( ξ, t ) = − ξ (cid:16) k λ | b z | + k λ | b φ | + (1 − λ ) | b y | + ( λ − λ ) | b θ | + ( k λ − k λ − k ) | b v | (cid:17) + λ ξ | b u | + γξ Re (cid:20) iξ (cid:18) I k b η b z + I k b η b φ (cid:19) + λ ξ b η b v (cid:21) + ξ Re (cid:16) iI ξ b u b θ + iI ξ b u b y (cid:17) , (3.18)where I = λ + 1 k I − λ ξ and I = λ + 1 k I − ξ . Let λ > F ( ξ, t ) = F ( ξ, t ) + λ ξ Re (cid:16) iξ b u b η (cid:17) + 1 γ I ξ Re (cid:16)b η b θ (cid:17) + 1 γ I ξ Re (cid:16)b η b y (cid:17) . (3.19)Multiplying (3.12), (3.13) and (3.14) by λ ξ , γ I ξ and γ I ξ , respectively, adding the obtained equa-tions and adding (3.18), we see that ddt F ( ξ, t ) = − ξ (cid:16) k λ | b z | + k λ | b φ | + (1 − λ ) | b y | + ( λ − λ ) | b θ | + ( k λ − k λ − k ) | b v | (cid:17) − ( γλ − λ ) ξ | b u | + γλ ξ | b η | + λ ξ Re (cid:16) ik ξ b η b u − k b v b η (cid:17) + 1 γ I ξ Re (cid:16) ik ξ b φ b η − k ξ b η b θ − k b v b η (cid:17) + 1 γ I ξ Re (cid:16) ik ξ b z b η − k ξ b η b y − k b v b η (cid:17) + γξ Re (cid:20) iξ (cid:18) I k b η b z + I k b η b φ (cid:19) + λ ξ b η b v (cid:21) . (3.20)Applying Young’s inequality for the terms depending on b η in (3.20), it follows that, for any ε > ddt F ( ξ, t ) ≤ − ( k λ − ε ) ξ | b z | − ( k λ − ε ) ξ | b φ | − (1 − λ − ε ) ξ | b y | − ( λ − λ − ε ) ξ | b θ | − ( k λ − k λ − k − ε ) ξ | b v | − ( γλ − λ − ε ) ξ | b u | + C ε ,λ , ··· ,λ (1 + ξ + ξ + ξ + ξ ) ξ | b η | . (3.21)We choose 0 < λ < λ > λ > γ λ , 0 < λ < λ < λ − < ε < min { k λ , k λ , − λ , λ − λ , k λ − k λ − k , γλ − λ } . Hence, using the definition (2.9) of b E , (3.21) leads to, for some positive constant c , ddt F ( ξ, t ) ≤ − c ξ b E ( ξ, t ) + C (cid:0) ξ + ξ + ξ + ξ (cid:1) ξ | b η | . (3.22)Now, we introduce the Perturbed Energy L as follows:(3.23) L ( ξ, t ) = λ b E ( ξ, t ) + 11 + ξ + ξ + ξ + ξ F ( ξ, t ) , where λ is a positive constant to be fixed later. Then from (2.11), (3.22) and (3.23) we have ddt L ( ξ, t ) ≤ − c f ( ξ ) b E ( ξ, t ) − ( k λ − C ) ξ | b η | , (3.24)where f is defined in (3.11). Moreover, using the definitions (2.9), (3.19) and (3.23) of b E, F and L ,respectively, we get, for some c > λ ),(3.25) | L ( ξ, t ) − λ b E ( ξ, t ) | ≤ c ( ξ + | ξ | + ξ )1 + ξ + ξ + ξ + ξ b E ( ξ, t ) ≤ c b E ( ξ, t ) . Therefore, for λ large enough so that λ > max n Ck , c o , we deduce from (3.24) and (3.25) that(3.26) ddt L ( ξ, t ) + c f ( ξ ) b E ( ξ, t ) ≤ and(3.27) c b E ( ξ, t ) ≤ L ( ξ, t ) ≤ c b E ( ξ, t ) , where c = λ − c > c = λ + 3 c >
0. Consequently, a combination of (3.26) and the secondinequality in (3.27) leads to, for c = c c ,(3.28) ddt L ( ξ, t ) + c f ( ξ ) L ( ξ, t ) ≤ . Finally, by integration (3.28) with respect to time t and using (2.17) and (3.27), (3.10) follows with e c = c α c α . (cid:3) Theorem 3.2.
Assume that k = k . Let N, ℓ ∈ N ∗ such that ℓ ≤ N , U ∈ H N ( R ) ∩ L ( R ) and U be the solution of (2.4) . Then, for any j ∈ { , . . . , N − ℓ } , there exists c > such that (3.29) k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c (1 + t ) − ℓ/ k ∂ j + ℓx U k L ( R ) , ∀ t ∈ R + . Proof.
From (3.11) we have (low and high frequences)(3.30) f ( ξ ) ≥ ξ if | ξ | ≤ , ξ − if | ξ | > . Applying Plancherel’s theorem and (3.10), we entail(3.31) k ∂ jx U k L ( R ) = (cid:13)(cid:13)(cid:13)(cid:13) d ∂ jx U ( x, t ) (cid:13)(cid:13)(cid:13)(cid:13) L ( R ) = Z R ξ j | b U ( ξ, t ) | dξ ≤ e c Z R ξ j e − c f ( ξ ) t | b U ( ξ ) | dξ ≤ e c Z | ξ |≤ ξ j e − c f ( ξ ) t | b U ( ξ ) | dξ + e c Z | ξ | > ξ j e − c f ( ξ ) t | b U ( ξ ) | dξ := J + J . Using (2.18) (with σ = 2 j , r = c and p = 6) and (3.30), it follows, for the low frequency region,(3.32) J ≤ C k b U k L ∞ ( R ) Z | ξ |≤ ξ j e − c t ξ dξ ≤ C (1 + t ) − (1+2 j ) k U k L ( R ) . For the high frequency region, using (3.30), we observe that J ≤ C Z | ξ | > | ξ | j e − c t ξ − | b U ( ξ, | dξ ≤ C sup | ξ | > n | ξ | − ℓ e − c t | ξ | − o Z R | ξ | j + ℓ ) | b U ( ξ, | dξ, then, using (2.19) (with σ = 2 l , σ = c and σ = 2), J ≤ C (1 + t ) − ℓ k ∂ j + ℓx U k L ( R ) , (3.33)and so, by combining (3.31)-(3.33), we get (3.29). (cid:3) We finish this subsection by proving that the condition k = k is necessary for the stability of (2.7)in case (2.5) with ( τ , τ , τ ) = (1 , , Theorem 3.3.
Assume that k = k . Then | b U ( ξ, t ) | doesn’t converge to zero when time t goes to infinity. AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 11
Proof.
We show that, for any ξ ∈ R , the matrix(3.34) A := − ( − ξ A + iξA + A )has at least a pure imaginary eigenvalue; that is ∀ ξ ∈ R , ∃ λ ∈ C : Re ( λ ) = 0 , Im ( λ ) = 0 and det ( λI − A ) = 0 , where I denotes the identity matrix. From (2.5) with ( τ , τ , τ ) = (1 , ,
0) and k = k , we have(3.35) λI − A = λ − iξ − − − ik ξ λ iγξ λ − iξ k − ik ξ λ λ − iξ k − ik ξ λ iγξ k ξ + λ . A direct computaion shows that det ( λI − A ) = 2 k λ ( λ + k ξ ) (cid:2) λ ( λ + k ξ ) + γ ξ (cid:3) +( λ + k ξ ) (cid:2) λ (cid:0) λ ( λ + k ξ ) + γ ξ (cid:1) + k ξ ( λ + k ξ ) (cid:3) . (3.36)It is clear that, if ξ = 0, then λ = i √ k ξ is a pure imaginary eigenvalue of A . If ξ = 0, then λ = i √ k is a pure imaginary eigenvalue of A . Consequently, according to (2.8) (see [28]), the solution of (2.7)doesn’t converge to zero when time t goes to infinity. (cid:3) Case 2: ( τ , τ , τ ) = (0 , , . We present, first, our exponential stability result for (2.7).
Lemma 3.4.
Let b U be the solution (2.8) of (2.7) . Then there exist c, e c > such that (3.10) is satisfiedwith f ( ξ ) = ( ξ ξ + ξ if k = k = k , ξ ξ + ξ + ξ + ξ if not. (3.37) Proof.
Multiplying (2.15) and (2.15) by i ξ b η and − i ξ b y , respectively, adding the resulting equations,taking the real part and using (2.14), we get(3.38) ddt Re (cid:16) i ξ b y b η (cid:17) = γξ (cid:0) | b η | − | b y | (cid:1) + k ξ Re (cid:16) i ξ b η b y (cid:17) − k ξ Re (cid:16)b z b η (cid:17) − k Re (cid:16) iξ b v b η (cid:17) . Similarily, multiplying (2.15) and (2.15) by b η and b u , respectively, adding the resulting equations, takingthe real part and using (2.13), we find(3.39) ddt Re (cid:16)b u b η (cid:17) = − γRe (cid:16) i ξ b y b u (cid:17) − k ξ Re (cid:16)b u b η (cid:17) + k Re (cid:16) iξ b v b η (cid:17) . Also, multiplying (2.15) and (2.15) by iξ b θ and − iξ b η , respectively, adding the resulting equations, takingthe real part and using (2.14), we obtain(3.40) ddt Re (cid:16) iξ b η b θ (cid:17) = γξ Re (cid:16)b y b θ (cid:17) − k ξ Re (cid:16) iξ b η b θ (cid:17) + k ξ Re (cid:16)b φ b η (cid:17) + k Re (cid:16) iξ b v b η (cid:17) . Let us define the functionals(3.41) F ( ξ, t ) = Re h i ξ (cid:16) λ b y b z − λ b u b v + λ b θ b φ (cid:17) − λ ξ b θ b v + ξ b y b v i , (3.42) F ( ξ, t ) = (cid:18) k k ξ + λ (cid:19) Re (cid:16)b u b z (cid:17) , (3.43) F ( ξ, t ) = k k k (cid:0) k λ ξ − k λ (cid:1) Re (cid:18) i ξ b z b θ − b u b z − i k k ξ b φ b y (cid:19) and(3.44) F ( ξ, t ) = − k k (cid:0) λ ξ + λ (cid:1) Re (cid:18) i ξ b z b θ − b u b z − i k k ξ b φ b y + k k b u b φ (cid:19) , where λ , λ , λ and λ are positive constants to be fixed later. Multiplying (3.1)-(3.5) by λ , − λ , λ , λ and −
1, respectively, and adding the obtained equations, it follows that(3.45) ddt F ( ξ, t ) = ( λ + 1) ξ | b y | + Re (cid:16) (1 − λ ) ξ b θ b y + ( ξ − λ ) iξ b u b y − iγξ b η b v + γλ ξ b η b z (cid:17) − ξ (cid:16) ( k − k λ − k λ ) | b v | + k λ | b z | + ( λ − λ ) | b θ | + λ | b u | + k λ | b φ | (cid:17) + (cid:0) k ξ + k λ (cid:1) Re (cid:16) i ξ b z b v (cid:17) + (cid:0) k λ ξ − k λ (cid:1) Re (cid:16) i ξ b v b φ (cid:17) + (cid:0) λ ξ + λ (cid:1) Re (cid:16) i ξ b θ b u (cid:17) . Multiplying (3.8) by − (cid:16) k k ξ + λ (cid:17) , we entail ddt F ( ξ, t ) = − (cid:0) k ξ + k λ (cid:1) Re (cid:16) i ξ b z b v (cid:17) + (cid:18) k k ξ + λ (cid:19) Re (cid:16) i ξ b y b u (cid:17) . (3.46)Multiplying (3.7) by − k k , adding (3.6) and (3.8), and multiplying the obtained equation by k k k (cid:0) k λ ξ − k λ (cid:1) , we arrive at ddt F ( ξ, t ) = k k k (cid:0) k λ ξ − k λ (cid:1) Re (cid:20)(cid:18) k k − (cid:19) ξ b y b θ − iξ b y b u + γk k ξ b η b φ (cid:21) − (cid:0) k λ ξ − k λ (cid:1) Re (cid:16) i ξ b v b φ (cid:17) . (3.47)Similarily, adding (3.7) and (3.9), multiplying by − k k , adding (3.6) and (3.8), and multiplying theobtained formula by − k k (cid:0) λ ξ + λ (cid:1) , we deduce that ddt F ( ξ, t ) = k k (cid:0) λ ξ + λ (cid:1) Re (cid:20)(cid:18) − k k (cid:19) ξ b y b θ + iξ b y b u − γk k ξ b η b φ (cid:21) − (cid:0) λ ξ + λ (cid:1) Re (cid:16) i ξ b θ b u (cid:17) . (3.48)Now, let us introduce the functional F (3.49) F ( ξ, t ) = F ( ξ, t ) + F ( ξ, t ) + F ( ξ, t ) + F ( ξ, t ) . By combining (3.45)-(3.48), we deduce that ddt F ( ξ, t ) = − ξ (cid:16) ( k − k λ − k λ ) | b v | + k λ | b z | + ( λ − λ ) | b θ | + λ | b u | + k λ | b φ | (cid:17) + F ( ξ, t ) , (3.50)where(3.51) F ( ξ, t ) = Re (cid:16) I ξ b y b θ + iI ξ b y b u − I ξ b η b φ − iγξ b η b v + γλ ξ b η b z (cid:17) + ( λ + 1) ξ | b y | ,I = 1 − λ + k k k (cid:18) k k − (cid:19) ( k λ ξ − k λ ) + (cid:18) k k − (cid:19) λ ξ + (cid:18) k k − (cid:19) λ ,I = (cid:20)(cid:18) k k − k k (cid:19) λ + k k − (cid:21) ξ + λ + (cid:18) k k + 1 (cid:19) λ + k k λ and I = γ (cid:18) − k k (cid:19) λ ξ + γ ( λ + λ ) . Let λ and λ be positive constants, and F and L be the functionals(3.52) F ( ξ, t ) = F ( ξ, t ) + λ Re (cid:16) iξ b y b η (cid:17) − γ I Re (cid:16) iξ b η b θ (cid:17) + 1 γ I Re (cid:16)b u b η (cid:17) and(3.53) L ( ξ, t ) = λ b E ( ξ, t ) + ξ ˜ f ( ξ ) F ( ξ, t ) , AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 13 where ˜ f ( ξ ) = ( ξ + ξ if k = k = k , ξ + ξ + ξ + ξ if not.(3.54)Multiplying (3.38), (3.39) and (3.40) by λ , γ I and − γ I , respectively, adding the obtained equationsand adding (3.50), it appears that ddt F ( ξ, t ) = − ξ (cid:16) ( k − k λ − k λ ) | b v | + k λ | b z | + ( λ − λ ) | b θ | + λ | b u | + k λ | b φ | (cid:17) − ξ ( γλ − ( λ + 1)) | b y | + F ( ξ, t ) , (3.55)where F ( ξ, t ) = γλ ξ | b η | + ξ Re (cid:16) γλ b η b z − iγξ b η b v − I b η b φ + ik λ ξ b η b y − k λ b η b z (cid:17) − k λ Re (cid:16) iξ b v b η (cid:17) − γ I Re (cid:16) k ξ b φ b η + ik ξ b v b η − ik ξ b η b θ (cid:17) + 1 γ I Re (cid:16) ik ξ b v b η − k ξ b η b u (cid:17) . (3.56)Noticing that, if k = k = k , then I , I and I are constants. Otherwise, I , I and I are of the form const ξ + const . Then, by applying Young’s inequality, we see that, for any ε >
0, we have(3.57) F ( ξ, t ) ≤ ε ξ (cid:16) | b y | + | b θ | + | b u | + | b φ | + | b v | + | b z | (cid:17) + C ǫ ,λ , ··· ,λ ˜ f ( ξ ) | b η | . Therefore, we conclude from (3.55) and (3.57) that(3.58) ddt F ( ξ, t ) ≤ C ε ,λ , ··· ,λ ˜ f ( ξ ) | b η | − ξ ( γλ − λ − − ε ) | b y | − ξ (cid:16) ( k − k λ − k λ − ε ) | b v | + ( k λ − ε ) | b z | + ( λ − λ − ε ) | b θ | + ( λ − ε ) | b u | + ( k λ − ε ) | b φ | (cid:17) . We choose 0 < λ , 0 < λ <
1, 0 < λ < λ < − λ , λ > γ ( λ + 1) and0 < ε < min { k − k λ − k λ , k λ , λ − λ , λ , k λ , γλ − λ − } . Thus, using the definition (2.9) of b E , (3.58) leads to, for some positive constant c ,(3.59) ddt F ( ξ, t ) ≤ − c ξ b E ( ξ, t ) + C ˜ f ( ξ ) | b η | . Then, from (2.11), (3.53) and (3.59), we infer that ddt L ( ξ, t ) ≤ − c f ( ξ ) b E ( ξ, t ) − ( k λ − C ) ξ | b η | , (3.60)where f is defined in (3.37). On the other hand, the definitions (2.9), (3.52) and (3.53) of b E , F and L ,respectively, imply that there exists c > λ ) such that (cid:12)(cid:12)(cid:12) L ( ξ, t ) − λ b E ( ξ, t ) (cid:12)(cid:12)(cid:12) ≤ c ξ + | ξ | + ξ + | ξ | ˜ f ( ξ ) b E ( ξ, t ) ≤ c b E ( ξ, t ) . So, we choose λ > max n Ck , c o , we get (3.26) and (3.27) with c = λ − c > c = λ + 4 c > (cid:3) Theorem 3.5.
Let
N, ℓ ∈ N ∗ such that ℓ ≤ N , U ∈ H N ( R ) ∩ L ( R ) and U be the solution of (2.4) . Then, for any j ∈ { , . . . , N − ℓ } , there exist c , ˜ c > such that, forany t ∈ R + , (3.61) k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c e − ˜ c t k ∂ j + ℓx U k L ( R ) if k = k = k , and (3.62) k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c (1 + t ) − ℓ/ k ∂ j + ℓx U k L ( R ) if not . Proof.
From (3.37) we have (low and high frequences)(3.63) f ( ξ ) ≥ ξ if | ξ | ≤ , if | ξ | > k = k = k , and(3.64) f ( ξ ) ≥ ξ if | ξ | ≤ , ξ − if | ξ | > . The proof of (3.62) is identical to the one of Theorem 3.2 by using (3.64) and applying (2.18) (with σ = 2 j , r = c and p = 4) and (2.19) (with σ = 2 l , σ = c and σ = 4). To get (3.61), noticing that thelow frequencies can be treated as for (3.62). For the high frequencies, we have just to remark that (3.63)implies that Z | ξ | > | ξ | j e − cf ( ξ ) t | b U ( ξ, | dξ ≤ Z | ξ | > | ξ | j e − c t | b U ( ξ, | dξ ≤ sup | ξ | > (cid:8) | ξ | − ℓ e − c t (cid:9) Z R | ξ | j + ℓ ) | b U ( ξ, | dξ ≤ e − c t k ∂ j + ℓx U k L ( R ) , so (3.61) holds true with ˜ c = c . (cid:3) Case 3: ( τ , τ , τ ) = (0 , , . In this case, we prove the same stability results for (2.7) and (2.4)that given in Subsection 3.2, and moreover, the proofs are very similar.
Lemma 3.6.
The result of Lemma 3.4 holds true also when ( τ , τ , τ ) = (0 , , .Proof. Multiplying (2.15) and (2.15) by i ξ b η and − i ξ b θ , respectively, adding the resulting equations,taking the real part and using (2.14), we get(3.65) ddt Re (cid:16) i ξ b θ b η (cid:17) = γξ (cid:16) | b η | − | b θ | (cid:17) + k ξ Re (cid:16) i ξ b η b θ (cid:17) − k ξ Re (cid:16) b φ b η (cid:17) − k Re (cid:16) iξ b v b η (cid:17) . Similarily, multiplying (2.15) and (2.15) by b η and b u , respectively, adding the resulting equations, takingthe real part and using (2.13), we find(3.66) ddt Re (cid:16)b u b η (cid:17) = − γRe (cid:16) i ξ b θ b u (cid:17) − k ξ Re (cid:16)b u b η (cid:17) + k Re (cid:16) iξ b v b η (cid:17) . Also, multiplying (2.15) and (2.15) by iξ b y and − iξ b η , respectively, adding the resulting equations, takingthe real part and using (2.14), we obtain(3.67) ddt Re (cid:16) iξ b η b y (cid:17) = γξ Re (cid:16)b θ b y (cid:17) − k ξ Re (cid:16) iξ b η b y (cid:17) + k ξ Re (cid:16)b z b η (cid:17) + k Re (cid:16) iξ b v b η (cid:17) . After, we define the functionals(3.68) F ( ξ, t ) = Re h i ξ (cid:16) λ b y b z − λ b u b v + λ b θ b φ (cid:17) + λ ξ b θ b v − ξ b y b v i , (3.69) F ( ξ, t ) = (cid:18) k k λ ξ + λ (cid:19) Re (cid:16)b u b φ (cid:17) , (3.70) F ( ξ, t ) = − k (cid:0) k ξ − k λ (cid:1) Re (cid:18) i ξ b z b θ − i k k ξ b φ b y + k k b u b φ (cid:19) and(3.71) F ( ξ, t ) = (cid:0) ξ + λ (cid:1) Re (cid:18) i ξ b z b θ − i k k ξ b φ b y + k k b u b φ − b u b z (cid:19) , AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 15 where λ , λ , λ and λ are positive constants to be fixed later. Multiplying (3.1)-(3.4) by λ , − λ , λ and − λ , respectively, adding the obtained equations and adding (3.5), we infer that(3.72) ddt F ( ξ, t ) = Re h iξ (cid:16)(cid:0) λ − λ ξ (cid:1) b θ b u − iξγλ b η b φ (cid:17) + ( λ − ξ b θ b y − iγλ ξ b η b v i +( λ + λ ) ξ | b θ | − ξ (cid:16) ( k λ − k λ − k ) | b v | + k λ | b z | + (1 − λ ) | b y | + λ | b u | + k λ | b φ | (cid:17) +( k ξ − k λ ) Re (cid:16) i ξ b v b z (cid:17) + ( k λ ξ + k λ ) Re (cid:16) i ξ b φ b v (cid:17) + ( ξ + λ ) Re (cid:16) i ξ b y b u (cid:17) . Multiplying (3.9) by − (cid:16) k k λ ξ + λ (cid:17) , we arrive at ddt F ( ξ, t ) = (cid:18) k k λ ξ + λ (cid:19) Re (cid:16) i ξ b θ b u (cid:17) − (cid:0) k λ ξ + k λ (cid:1) Re (cid:16) i ξ b φ b v (cid:17) . (3.73)Adding (3.7) and (3.9), multiplying the obtained equation by − k k , adding (3.6) and multiplying thereuslting equation by − (cid:16) k k ξ − λ (cid:17) , it follows that ddt F ( ξ, t ) = (cid:18) k k ξ − λ (cid:19) Re (cid:20) iξ (cid:18) − k k b θ b u − iγξ b η b z (cid:19) + (cid:18) − k k (cid:19) ξ b θ b y (cid:21) − (cid:0) k ξ − k λ (cid:1) Re (cid:16) i ξ b v b z (cid:17) . (3.74)Similarily, adding (3.7) and (3.9), multiplying the obtained equation by − k k , adding (3.6) and (3.8), andmultiplying the reuslting equation by ξ + λ , we entail ddt F ( ξ, t ) = (cid:0) ξ + λ (cid:1) Re (cid:20) iξ (cid:18) k k b θ b u + iγξ b η b z (cid:19) + (cid:18) k k − (cid:19) ξ b θ b y (cid:21) − (cid:0) ξ + λ (cid:1) Re (cid:16) i ξ b y b u (cid:17) . (3.75)Let F the functional defined by (3.49). A combination of (3.72)-(3.75) implies that ddt F ( ξ, t ) = − ξ (cid:16) ( k λ − k λ − k ) | b v | + k λ | b z | + (1 − λ ) | b y | + λ | b u | + k λ | b φ | (cid:17) + F ( ξ, t ) , (3.76)where(3.77) F ( ξ, t ) = Re (cid:16) iI ξ b θ b u − I ξ b η b z + I ξ b θ b y − iγλ ξ b η b v + γλ ξ b η b φ (cid:17) + ( λ + λ ) ξ | b θ | ,I = (cid:20)(cid:18) k k − (cid:19) λ + k k − k k (cid:21) ξ + k k λ + (cid:18) k k + 1 (cid:19) λ + λ ,I = γ (cid:20)(cid:18) − k k (cid:19) ξ + λ + λ (cid:21) and I = (cid:18) k k − (cid:19) ( ξ + λ ) + (cid:18) − k k (cid:19) (cid:18) k k ξ − λ (cid:19) + λ − . Let λ and λ be positive constants and L be the functional defined by (3.53), where ˜ f is defined by (3.54)and F is given by(3.78) F ( ξ, t ) = F ( ξ, t ) + λ Re (cid:16) iξ b θ b η (cid:17) + 1 γ I Re (cid:16)b u b η (cid:17) − γ I Re (cid:16) iξ b η b y (cid:17) . Multiplying (3.65)-(3.67) by λ , γ I and − γ I , respectively, adding the obtained equations and adding(3.76), we find ddt F ( ξ, t ) = − ξ (cid:16) ( k λ − k λ − k ) | b v | + k λ | b z | + (1 − λ ) | b y | + λ | b u | + k λ | b φ | (cid:17) − ( γλ − λ − λ ) ξ | b θ | + F ( ξ, t ) , (3.79) where F ( ξ, t ) = γλ ξ | b η | − ξ Re (cid:16) I b η b z + iγλ ξ b η b v − γλ b η b φ − ik λ ξ b η b θ + k λ b φ b η (cid:17) − k λ Re (cid:16) iξ b v b η (cid:17) − γ I Re (cid:16) k ξ b η b u − ik ξ b v b η (cid:17) + 1 γ I Re (cid:16) ik ξ b η b y − k ξ b η b z − ik ξ b v b η (cid:17) . (3.80)We remark that, if k = k = k , then I , I and I are constants. Otherwise, I , I and I are of theform const ξ + const . Then, by applying Young’s inequality, we see that, for any ε >
0, we have(3.81) F ( ξ, t ) ≤ ε ξ (cid:16) | b y | + | b θ | + | b u | + | b φ | + | b v | + | b z | (cid:17) + C ǫ ,λ , ··· ,λ ˜ f ( ξ ) | b η | , where ˜ f is defined in (3.54). Therefore, we conclude from (3.79) and (3.81) that(3.82) ddt F ( ξ, t ) ≤ C ε ,λ , ··· ,λ ˜ f ( ξ ) | b η | − ( γλ − λ − λ − ε ) ξ | b θ | − ξ (cid:16) ( k λ − k λ − k − ε ) | b v | + ( k λ − ε ) | b z | + (1 − λ − ε ) | b y | + ( λ − ε ) | b u | + ( k λ − ε ) | b φ | (cid:17) . We choose 0 < λ , 0 < λ < λ >
1, 0 < λ < λ − λ > γ ( λ + λ ) and0 < ε < min { k λ , λ , − λ , k λ , k λ − k λ − k , γλ − λ − λ } . Then, using the definition (2.9) of b E , (3.82) imlies (3.59), and then (3.60) holds true. Consequentely, theproof can be ended as for Lemma 3.4. (cid:3) Theorem 3.7.
The stability result given in Theorem 3.5 is satisfied when ( τ , τ , τ ) = (0 , , .Proof. The proof is identical to the one of Theorem 3.5. (cid:3) Stability: Cattaneo law (1.2)This section concerns the stability of (2.4) in case of Cattaneo law (1.2). We will prove (1.7), (1.9)and (1.10). Moreover, we prove that (2.7) is not stable when ( τ , τ , τ ) = (1 , ,
0) and k = k .First, observe that (2.15) -(2.15) are identical to (2.16) -(2.16) , and (2.15) with k ξ b η replacedby ik ξ b q is equal to (2.16) . So (3.1)-(3.9) are still valide. Moreover, (3.12)-(3.14), (3.38)-(3.40) and(3.65)-(3.67) are satisfied with ik ξ b q instead of k ξ b η . On the other hand, we prove the next expressions,which take in consideration the last equation in (2.16).Multiplying (2.16) and (2.16) by iξ b q and − iξ b η , respectively, adding the resulting equations, takingthe real part and using (2.14), we find(4.1) ddt Re (cid:16) iξ b η b q (cid:17) = k ξ (cid:0) | b q | − | b η | (cid:1) + k Re (cid:16) iξ b q b η (cid:17) + γξ Re (cid:16) ( τ b u + τ b y + τ b θ ) b q (cid:17) . Multiplying (2.16) and (2.16) by i ξ b q and − i ξ b v , respectively, adding the resulting equations, takingthe real part and using (2.14), we get(4.2) ddt Re (cid:16) i ξ b v b q (cid:17) = − ξ Re (cid:16)b u b q (cid:17) + Re (cid:16) iξ b y b q + iξ b θ b q + ik ξ b q b v (cid:17) − k ξ Re (cid:16)b v b η (cid:17) . Similarily, using the multipliers b q and b v instead of i ξ b q and − i ξ b v , respectively, we obtain(4.3) ddt Re (cid:16)b v b q (cid:17) = Re (cid:16) iξ b u b q (cid:17) + Re (cid:16)b y b q + b θ b q + k b q b v (cid:17) + k Re (cid:16) iξ b v b η (cid:17) . Multiplying (2.16) and (2.16) by i ξ b q and − i ξ b φ , respectively, adding the resulting equations, takingthe real part and using (2.14), we arrive at(4.4) ddt Re (cid:16) i ξ b φ b q (cid:17) = − ξ Re (cid:16)b θ b q (cid:17) + k Re (cid:16) iξ b q b φ (cid:17) − k ξ Re (cid:16)b φ b η (cid:17) . Similarily, using the multipliers b q and b φ instead of i ξ b q and − i ξ b φ , respectively, we entail(4.5) ddt Re (cid:16)b φ b q (cid:17) = Re (cid:16) iξ b θ b q (cid:17) + k Re (cid:16)b q b φ (cid:17) + k Re (cid:16) iξ b φ b η (cid:17) . AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 17
Also, multiplying (2.16) and (2.16) by i ξ b q and − i ξ b z , respectively, adding the resulting equations,taking the real part and using (2.14), we infer that(4.6) ddt Re (cid:16) i ξ b z b q (cid:17) = − ξ Re (cid:16)b y b q (cid:17) + k Re (cid:16) iξ b q b z (cid:17) − k ξ Re (cid:16)b z b η (cid:17) . Similarily, using the multipliers b q and b z instead of i ξ b q and − i ξ b z , respectively, it appears that(4.7) ddt Re (cid:16)b z b q (cid:17) = Re (cid:16) iξ b y b q (cid:17) + k Re (cid:16)b q b z (cid:17) + k Re (cid:16) iξ b z b η (cid:17) . Case 1: ( τ , τ , τ ) = (1 , , . As in Subsection 3.1, we start by presenting the exponential stabilityresult of (2.7) in the next lemma.
Lemma 4.1.
The result of Lemma 3.1 is satisfied in case (2.16) when k = k and ( τ , τ , τ ) = (1 , , .Proof. We use the arguments used in Subsection 3.1. We define the functional F by (3.15) and we get(3.16) (because we used only the first six equations in (2.15) which are the same in (2.16)). We consider F and F defined by (3.17) and (3.19), and we find (3.20) with k ξ b η replaced by ik b q . We put, for λ > F ( ξ, t ) = F ( ξ, t ) + λ ξ Re (cid:16) iξ b η b q (cid:17) + 1 k I ξ Re (cid:16) iξ b v b q (cid:17) + 1 k I ξ Re (cid:16)b φ b q (cid:17) + 1 k I ξ Re (cid:16)b z b q (cid:17) , where I = ( γλ − k λ ) ξ − k γ ( I + I ) , I = γk I − k γ I and I = γk I − k γ I . Multiplying (4.1), (4.2), (4.5) and (4.7) by λ ξ , k I ξ , k I ξ and k I ξ , respectively, adding theobtained equations, adding (3.20) and applying Young’s inequality for the terms depending on b q , we find,for any ε > ddt ˜ F ( ξ, t ) ≤ − ( k λ − ε ) ξ | b z | − ( k λ − ε ) ξ | b φ | − (1 − λ − ε ) ξ | b y | − ( λ − λ − ε ) ξ | b θ | − ( k λ − k λ − k − ε ) ξ | b v | − ( γλ − λ − ε ) ξ | b u | − ( k λ − γλ − ε ) ξ | b η | + C ε ,λ , ··· ,λ (1 + ξ + ξ + ξ + ξ ) | b q | . (4.8)We choose 0 < λ < λ > λ > γ λ , λ > γk λ , 0 < λ < λ < λ − < ε < min { k λ , k λ , − λ , λ − λ , k λ − k λ − k , γλ − λ , k λ − γλ } . Hence, using the definition (2.10) of b E , (4.8) leads to, for some positive constant c , ddt ˜ F ( ξ, t ) ≤ − c ξ b E ( ξ, t ) + C (cid:0) ξ + ξ + ξ + ξ (cid:1) | b q | . (4.9)So we consider L given by (3.23), with ˜ F instead of F , and use (2.12) to find ddt L ( ξ, t ) ≤ − c f ( ξ ) b E ( ξ, t ) − ( k λ − C ) | b q | , (4.10)where f is defined by (3.11). Finally, the proof can be finished exactely as in the proof of (3.10). (cid:3) Theorem 4.2.
The result of Theorem 3.2 is satisfied in case (2.16) when k = k and ( τ , τ , τ ) =(1 , , .Proof. The proof is identical to the one of Theorem 3.2. (cid:3)
The third result of this subsection says that (2.7) is not stable if k = k . Theorem 4.3.
Assume that k = k . Then | b U ( ξ, t ) | doesn’t converge to zero when time t goes to infinity. Proof.
As in Subsection 3.1, we show that, for any ξ ∈ R , the matrix (3.34) has at least a pure imaginaryeigenvalue. From (2.6) with ( τ , τ , τ ) = (1 , ,
0) and k = k , we have(4.11) λI − A = λ − iξ − − − ik ξ λ iγξ
00 0 λ − iξ k − ik ξ λ λ − iξ k − ik ξ λ iγξ λ ik ξ ik ξ k + λ . A direct computaion shows that det ( λI − A ) = 2 k λ ( λ + k )( λ + k ξ ) (cid:0) λ + γ ξ (cid:1) + λ ( λ + k )( λ + k ξ ) (cid:0) λ + ( k + γ ) ξ (cid:1) − ik ξ ( λ + k ξ ) (cid:2) ik k ξ ( λ + k ξ ) + ik λ ξ ( λ + k ) + ik k λ ξ + ik k λξ (cid:3) . (4.12)We see that, if ξ = 0, then λ = i √ k ξ is a pure imaginary eigenvalue of A . If ξ = 0, then λ = i √ k is apure imaginary eigenvalue of A . Consequently (see [28]), the solution (2.8) of (2.7) doesn’t converge tozero when times t goes to infinity. (cid:3) Case 2: ( τ , τ , τ ) = (0 , , . We present, first, our exponential stability result for (2.7).
Lemma 4.4.
The result of Lemma 3.4 is satisfied in case (2.16) with ( τ , τ , τ ) = (0 , , .Proof. We addapt the arguments used in Subsection 3.2. We define F - F and F as in Subsection 3.2,and we get (3.55), where F is defined by (3.56) with k ξη replaced by ik q . Let λ > F ( ξ, t ) = ξ F ( ξ, t ) + λ ξ Re (cid:16) iξ b η b q (cid:17) + 1 k (cid:18) k γ ( I − I ) + ( k λ − γλ ) ξ (cid:19) ξ Re (cid:16)b v b q (cid:17) − k (cid:18) I + k γ I (cid:19) ξ Re (cid:16) iξ b φ b q (cid:17) + 1 k ( k λ − γλ ) ξ Re (cid:16) iξ b z b q (cid:17) . Multiplying (4.1), (4.3), (4.4), (4.6) and (3.55) by λ ξ , k (cid:18) k γ ( I − I ) + k λ − γ (cid:19) ξ , − k (cid:18) I + k γ I (cid:19) ξ , k ( γλ − k λ ) ξ and ξ , respectively, adding the obtained equations and applying Young’s inequality for the terms depending on b q , we find, for any ε > ddt ˜ F ( ξ, t ) ≤ C ε ,λ , ··· ,λ ˜ f ( ξ ) | b q | − ( γλ − λ − − ε ) ξ | b y | − ( k λ − γλ − ε ) ξ | b η | − ξ (cid:16) ( k − k λ − k λ − ε ) | b v | + ( k λ − ε ) | b z | + ( λ − λ − ε ) | b θ | + ( λ − ε ) | b u | + ( k λ − ε ) | b φ | (cid:17) , where ˜ f is defined in (3.54). We choose 0 < λ , 0 < λ <
1, 0 < λ < λ < − λ , λ > γ ( λ + 1), λ > γk λ and0 < ε < min { k − k λ − k λ , k λ , λ − λ , λ , k λ , γλ − λ − , k λ − γλ } . Thus, using the definition of b E , (4.13) implies that, for some positive constant c ,(4.14) ddt ˜ F ( ξ, t ) ≤ − c ξ b E ( ξ, t ) + C ˜ f ( ξ ) | b q | . Therefore, we introduce the functional(4.15) L ( ξ, t ) = λ b E ( ξ, t ) + 1˜ f ( ξ ) ˜ F ( ξ, t )and we deduce that, using (2.12) and (4.14) ddt L ( ξ, t ) ≤ − c f ( ξ ) b E ( ξ, t ) − ( k λ − C ) | b q | , (4.16)where f is defined in (3.37). The proof can be ended as for Lemma 3.1. (cid:3) AUCHY THERMOELASTIC LAMINATED TIMOSHENKO PROBLEM WITH INTERFACIAL SLIP 19
Theorem 4.5.
The result of Theorem 3.5 is staisfied in case (1.2) with ( τ , τ , τ ) = (0 , , .Proof. The proof is identical to the one of Theorem 3.5. (cid:3)
Case 3: ( τ , τ , τ ) = (0 , , . In this case, we prove the same stability results for (2.7) and (2.4)that given in Subsection 4.2, and moreover, the proofs are very similar.
Lemma 4.6.
The result of Lemma 3.4 holds true also in case (2.16) with ( τ , τ , τ ) = (0 , , .Proof. We define F - F and F as in Subsection 3.3 and we obtain (3.79), where F is defined in (3.80)with k ξη replaced by ik q . Let λ > F ( ξ, t ) = ξ F ( ξ, t ) + λ ξ Re (cid:16) iξ b η b q (cid:17) + 1 k (cid:18) γ ( k I − I ) + ( k λ − γλ ) ξ (cid:19) ξ Re (cid:16)b v b q (cid:17) + 1 k ( γλ − k λ ) ξ Re (cid:16) iξ b φ b q (cid:17) − k (cid:18) I + k γ I (cid:19) ξ Re (cid:16) iξ b z b q (cid:17) . Multiplying (4.1), (4.3), (4.4), (4.6) and (3.79) by λ ξ , k (cid:18) k γ ( I − I ) − ( k λ + γλ ) ξ (cid:19) ξ , k ( γλ − k λ ) ξ , − k (cid:18) I + k γ I (cid:19) ξ and ξ , respectively, adding the obtained equations and applying Young’s inequality for the terms depending on b q , we find, for any ε > ddt ˜ F ( ξ, t ) ≤ C ε ,λ , ··· ,λ ˜ f ( ξ ) | b q | − ξ ( γλ − λ − λ − ε ) | b θ | − ξ ( k λ − γλ − ε ) | b η | − ξ (cid:16) ( k λ − k λ − k − ε ) | b v | + ( k λ − ε ) | b z | + (1 − λ − ε ) | b y | + ( λ − ε ) | b u | + ( k λ − ε ) | b φ | (cid:17) , ˜ f is defined in (3.54). We choose 0 < λ , 0 < λ < λ >
1, 0 < λ < λ − λ > γ ( λ + λ ), λ > γk λ and0 < ε < min { k λ , λ , − λ , k λ , k λ − k λ − k , γλ − λ − λ , k λ − γλ } . Then, using the definition of b E , (4.17) imlies (4.14), and then (4.16) holds true. Consequentely, the proofcan be ended as for Lemma 4.4. (cid:3) Theorem 4.7.
The stability result given in Theorem 4.5 is satisfied when ( τ , τ , τ ) = (0 , , .Proof. The proof is identical to the one of Theorem 4.5. (cid:3) Comments and issues
1. The optimality of the obtained decay rates on k ∂ jx U k L ( R ) is an interesting open question. Thisquestion will be the focus of our attention in a future work.2. When ( τ , τ , τ ) ∈ { (0 , , , (0 , , } and k = k = k , the function f tends to 1 when ξ goes toinfinity, which avoid the regularity loss property; that is, (1.9) with j = ℓ = 0 gives the stability of (1.1)and (1.2) with a decay rate of k U k L ( R ) depending only on k U k L ( R ) and k U k L ( R ) . However, in theother cases, f tends to 0 when ξ goes to infinity, this means that the dissipation is very weak in the highfrequency region, which imposes the regularity loss property in the estimates because (1.7) and (1.10)with j = ℓ = 0 imply only the boundedness of k U k L ( R ) .3. The estimate (1.9) leads to a faster speed of convergence to zero of k ∂ jx U k L ( R ) than the oneguareented by (1.7) and (1.10). This can be explained by the fact that the Cattaneo law generates adissipation stronger than the one generated by the Fourier law. On the other hand, for both laws with( τ , τ , τ ) ∈ { (0 , , , (0 , , } , the situation is more favorable when k = k = k than in the oppositecase.4. From the mathematical point of view, one can take γ ∈ R ∗ in case (1.1), and γ, k ∈ R ∗ in case(1.2) (instead of γ, k > γ, k < (3.12), (3.38), (3.65) and (4.1) by −
1, and using the obtained identities instead of (3.12), (3.38), (3.65)and (4.1).5. The coupling terms(5.1) τ j γη x and γ ( τ ϕ xt + τ ψ xt + τ w xt )in (1.1) and (1.2) are of order one with respect to x . Mathematicaly, these coupling terms can be replacedby (order zero with respect to x )(5.2) τ j γη and − γ ( τ ϕ t + τ ψ t + τ w t ) , respectively, with γ ∈ R ∗ . In this case, the terms iτ j γξ b η and iγξ ( τ b u + τ b y + τ b θ ) in (2.15) and (2.16) arereplaced by τ j γ b η and − γ ( τ b u + τ b y + τ b θ ), respectively. On the other hand, (3.10) holds true with f ( ξ ) = ξ ξ + ξ + ξ + ξ + ξ instead of (3.11), and f ( ξ ) = ( ξ ξ + ξ + ξ if k = k = k , ξ ξ + ξ + ξ + ξ + ξ if notinstead of (3.37), and so we get the stability estimates k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c (1 + t ) − ℓ/ k ∂ j + ℓx U k L ( R ) , ∀ t ∈ R + instead of (3.29), k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c e − ˜ c t k ∂ j + ℓx U k L ( R ) if k = k = k instead of (3.61), and k ∂ jx U k L ( R ) ≤ c (1 + t ) − / − j/ k U k L ( R ) + c (1 + t ) − ℓ/ k ∂ j + ℓx U k L ( R ) if notinstead of (3.62). These stability estimates show that the decay rates in case (5.2) is smaller than theones obtained in case (5.1). Moreover, the non stability result when k = k and ( τ , τ , τ ) = (1 , ,
0) isstill valid using the same arguments of proof, since we get (3.36) and (4.12) with γ instead of γ ξ . References [1] M. S. Alves, P. Gamboa, G. C. Gorain, A. Rambaud and O. Vera, Asymptotic behavior of a flexible structure withCattaneo type of thermal effect, Indagationes Mathematicae, 27 (2016), 821-834.[2] C. F. Beards and I. M. A. Imam, The damping of plate vibration by interfacial slip between layers, Int. J. Mach. Tool.Des. Res., 18 (1978), 131-137.[3] X. G. Cao, D. Y. Liu and G. Q. Xu, Easy test for stability of laminated beams with structural damping and boundaryfeedback controls, J. Dynamical Control Syst., 13 (2007), 313-336.[4] M. M. Cavalcanti, V. N. Domingos Cavalcanti, F. A. Falcao Nascimento, I. Lasiecka and J. H. Rodrigues, Uniform decayrates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping,Z. Angew. Math. Phys., 65 (2014), 1189-1206.[5] L. Djouamai and B. Said-Houari, A new stability number of the Bresse-Cattaneo system, Math. Meth. Appl. Sci., 41(2018), 2827-2847.[6] L. H. Fatori, R. N. Monteiro and H. D. Fern´andez Sare, The Timoshenko system with history and Cattaneo law, AppliedMathematics and Computation, 228 (2014), 128-140.[7] T. E. Ghoul, M. Khenissi and B. Said-Houari, On the stability of the Bresse system with frictional damping, J. Math.Anal. Appl., 455 (2017), 1870-1898.[8] A. Guesmia, Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Medi.J. Math., 14 (2017), 19 pages.[9] A. Guesmia, Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the verticaldisplacement, Nonauton. Dyn. Syst., 4 (2017), 78-97.[10] A. Guesmia, Well-posedness and stability results for laminated Timoshenko beams with interfacial slip and infinitememory, IMA J. Math. Cont. Info., 37 (2020), 300-350.[11] A. Guesmia, S. Messaoudi and A. Soufyane, On the stabilization for a linear Timoshenko system with infinite historyand applications to the coupled Timoshenko-heat systems, Elec. J. Diff. Equa., 2012 (2012), 1-45.[12] S. W. Hansen, In control and estimation of distributed parameter systems: Non-linear phenomena, International Seriesof Numerical Analysis, 118 (1994), 143-170.[13] S. W. Hansen and R. Spies, Structural damping in a laminated beams due to interfacial slip, J. Sound Vibration, 204(1997), 183-202.