A transmission problem for waves under time-varying delay and nonlinear weight
aa r X i v : . [ m a t h . A P ] F e b A TRANSMISSION PROBLEM FOR WAVES UNDER TIME-VARYING DELAY ANDNONLINEAR WEIGHT
Carlos A. S. Nonato, Carlos A. Raposo & Waldemar D. Bastos
Abstract
This manuscript focus on in the transmission problem for one dimensional waves with nonlinear weightson the frictional damping and time-varying delay. We prove global existence of solutions using Kato’svariable norm technique and we show the exponential stability by the energy method with the constructionof a suitable Lyapunov functional.
1. Introduction
In this paper we investigate global existence and decay properties of solutions for a transmission problem forwaves with nonlinear weights and time-varying delay. We consider the following system(1.1) u tt ( x, t ) − au xx ( x, t ) + µ ( t ) u t ( x, t ) + µ ( t ) u t ( x, t − τ ( t )) = 0 in Ω × ]0 , ∞ [ ,v tt ( x, t ) − bv xx ( x, t ) = 0 in ] L , L [ × ]0 , ∞ [ , where 0 < L < L < L , Ω =]0 , L [ ∪ ] L , L [ and a , b are positive constants. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Part with delay Elastic Part Part with delay ✛ ✲✛ ✲✛ ✲ L L L u ( x, t ) v ( x, t ) u ( x, t )The system (1.1) is subjected to the transmission conditions(1.2) u ( L i , t ) = v ( L i , t ) , i = 1 , au x ( L i , t ) = bv x ( L i , t ) , i = 1 , , the boundary conditions(1.3) u (0 , t ) = u ( L , t ) = 0and initial conditions(1.4) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) on Ω ,u t ( x, t − τ (0)) = f ( x, t − τ (0)) in Ω × ]0 , τ (0)[ ,v ( x,
0) = v ( x ) , v t ( x,
0) = v ( x ) on ] L , L [ , where the initial datum ( u , u , v , v , f ) belongs to a suitable Sobolev space.Here 0 < τ ( t ) is the time-varying delay and µ ( t ) and µ ( t ) are nonlinear weights acting on the frictionaldamping. As in [ ], we assume that(1.5) τ ( t ) ∈ W , ∞ ([0 , T ]) , ∀ T > Mathematics Subject Classification.
Primary 35L20; 35B40; Secondary 93D15.
Key words and phrases.
Transmission problem, Time-variable delay, Nonlinear weights, Exponential stability. and that there exist positive constants τ , τ and d satisfying(1.6) 0 < τ ≤ τ ( t ) ≤ τ , τ ′ ( t ) ≤ d < , ∀ t > . We are interested in proving the exponential stability for the problem (1.1)-(1.4). In order to obtain this, wewill assume that(1.7) max { , ab } < L + L − L L − L ) . As described in [ ], the assumption (1.7) gives the relationship between the boundary regions and the trans-mission permitted. It can be also seen as a restriction on the wave speeds of the two equations and the dampedpart of the domain. It is known that for Timoshenko systems [ ] and Bresse systems [ ] the wave speedsalways control the decay rate of the solution. It is an interesting open question to investigate the behavior ofthe solution when (1.7) is not satisfied.Time delay is the property of a physical system by which the response to an applied force is delayed in itseffect, and the central question is that delays source can destabilize a system that is asymptotically stable inthe absence of delays, see [ , , , ].Transmission problems are closely related to the design of material components, attracting considerableattention in recent years, e.g., in the analysis of damping mechanisms in the metallurgical industry or smartmaterials technology, see [ , ] and the references therein. Studies of fluid structure interaction and the addedmass effect, also known as virtual mass effect, hydrodynamic mass, and hydroelastic vibration of structures,started with H. Lamb [ ] who investigated the vibrations of a thin elastic circular plate in contact with water.Experimental study of impact on composite plates with fluid-structure interaction was investigated in [ ].From the mathematical point of view a transmission problem for wave propagation consists on a hyperbolicequation for which the corresponding elliptic operator has discontinuous coefficients.We consider the wave propagation over bodies consisting of two physically different materials, one purelyelastic and another subject to frictional damping. The type of wave propagation generated by mixed materialsoriginates a transmission (or diffraction) problem.To the best of our knowledge, the first contribution in literature for transmission problem with a time delaywas given by A. Benseghir in [ ]. More precisely, in [ ] the transmission problem(1.8) u tt − au xx + µ u t ( x, t ) + µ u t ( x, t − τ ) = 0 , in Ω × ]0 , ∞ [ ,v tt − bv xx = 0 , in ] L , L [ × ]0 , ∞ [ . with constant weights µ , µ and time delay τ > µ < µ ), it was proved the well-posedness of the system and, under condition (1.7), itwas established an exponential decay result.The result in [ ] were improved by S. Zitouni et al. [ ]. There, the authors considered the problem with atime-varying delay τ ( t ) of the form(1.9) u tt − au xx + µ u t ( x, t ) + µ u t ( x, t − τ ( t )) = 0 , in Ω × ]0 , ∞ [ ,v tt − bv xx = 0 , in ] L , L [ × ]0 , ∞ [and proved the global existence and exponential stability under suitable assumptions on the delay term andassumption (1.7). Without delay, systems (1.8), (1.9) has been investigated in [ ].The transmission problem with history and delay was considered by G. Li et al., [ ] where the equationswere expressed as(1.10) u tt − au xx + Z ∞ g ( s ) u xx ( x, t − s ) ds + µ u t ( x, t )+ µ u t ( x, t − τ ) = 0 , in Ω × ]0 , ∞ [ ,v tt − bv xx = 0 , in ] L , L [ × ]0 , ∞ [ . Under suitable assumptions on the delay term and on the function g , the authors proved an exponential stabilityresult for two cases. In the first case, they considered µ < µ and for second case, they assumed that µ = µ .S. Zitouni et al., [ ] extended the result in [ ] for varying delay. In [ ] was proved existence and theuniqueness of the solution by using the semigroup theory and the exponential stability of the solution by the RANSMISSION PROBLEM WITH DELAY AND WEIGHT 3 energy method for the following problem(1.11) u tt − au xx + Z ∞ g ( s ) u xx ( x, t − s ) ds + µ u t ( x, t )+ µ u t ( x, t − τ ( t )) = 0 , in Ω × ]0 , ∞ [ ,v tt − bv xx = 0 , in ] L , L [ × ]0 , ∞ [ . Stability to localized viscoelastic transmission problem was considered by Mu˜noz Rivera et al., [ ] wherethey considered(1.12) ρφ tt + σ x = 0 ,σ ( x, t ) = α ( x ) ϕ x − k ( x ) ϕ xt − β ( x ) ϕ xt = 0 . In [ ] the authors investigated the effect of the positions of the dissipative mechanisms on a bar with threecomponent ]0 , L [ , ] L , L [ , ] L , L [, and showed that the system is exponentially stable if and only if the viscouscomponent is not in the center of the bar. In other case, they showed the lack of exponential stability, and thatthe solutions still decay but just polynomially to zero.The case of time-varying delay has already been considered in other works, such as [ , , , ]. Waveequations with time-varying delay and nonlinear weights was considered in the recent work of Barros et al., [ ]where was studied the equation given by(1.13) u tt − u xx + µ ( t ) u t + µ ( t ) u t ( x, t − τ ( t )) = 0 , in ]0 , L [ × ]0 , + ∞ [ . Under proper conditions on nonlinear weights µ ( t ) , µ ( t ) and τ ( t ), authors proved global existence and anestimate for the decay rate of the energy.In the present work we improve the results in [ ] where, for constant weights µ ( t ) = µ , µ ( t ) = µ andunder adequate assumptions regarding the weight and time-varying delay, was proved the well posedness andsingularity of solutions by using the semigroup theory. Authors also showed exponential stability by introducingan appropriate Lyapunov functional.Here we consider a transmission problem with nonlinear weights and time-varying delay, which is the maincharacteristic of this work. Although there are some works on laminated beam and on Timoshenko system withdelay, all of them consider constant weights, i.e., µ and µ are constants. To the best of our knowledge, thereis no result for these systems with nonlinear weights. Moreover, since the weights are nonlinear, a difficultycomes in: the operator is nonautonomous. This makes hard the use semigroup theory to study well-posedness.To overcome it we use the Kato’s variable norm technique together with semigroup theory to show that thesystem is well-posed.The remainder of this paper is organized as follows. In section 2 we introduce some notations and prove thedissipative property for the energy of the system. In the section 3, by using Kato’s variable norm techniqueand under some restriction on the non-linear weights and the time-varying delay, the system is shown to bewell-posed. In section 4, we present the result of exponential stability by energy methods, and by using suitablesophisticated estimates for multipliers to construct an appropriated Lyapunov functional.
2. Notation and preliminaries
We start by setting the following hypothesis: (H1) µ : R + → ]0 , + ∞ [ is a non-increasing function of class C ( R + ) satisfying(2.1) (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( t ) µ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M , ∀ t ≥ , where M > (H2) µ : R + → R is a function of class C ( R + ), which is not necessarily positive or monotone, such that | µ ( t ) | ≤ βµ ( t ) , (2.2) | µ ′ ( t ) | ≤ M µ ( t ) , (2.3)for some 0 < β < √ − d and M > ] we introduce the new variable(2.4) z ( x, ρ, t ) = u t ( x, t − τ ( t ) ρ ) , ( x, ρ ) ∈ Ω × ]0 , , t > . CARLOS A. S. NONATO, CARLOS A. RAPOSO & WALDEMAR D. BASTOS
It is easily verified that the new variable satisfies τ ( t ) z t ( x, ρ, t ) + (1 − τ ′ ( t ) ρ ) z ρ ( x, ρ, t ) = 0and the problem (1.1) is equivalent to(2.5) u tt ( x, t ) − au xx ( x, t ) + µ ( t ) u t ( x, t ) + µ ( t ) z ( x, , t ) = 0 in Ω × ]0 , ∞ [ ,v tt ( x, t ) − bv xx ( x, t ) = 0 in ] L , L [ × ]0 , ∞ [ ,τ ( t ) z t ( x, ρ, t ) + (1 − τ ′ ( t ) ρ ) z ρ ( x, ρ, t ) = 0 in Ω × ]0 , × ]0 , ∞ [ . This system is subject to the transmission conditions(2.6) u ( L i , t ) = v ( L i , t ) , i = 1 , ,au x ( L i , t ) = bv x ( L i , t ) , i = 1 , , the boundary conditions(2.7) u (0 , t ) = u ( L , t ) = 0and the initial conditions(2.8) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) on Ω ,v ( x,
0) = v ( x ) , v t ( x,
0) = v ( x ) on Ω ,z ( x, ρ,
0) = u t ( x, − τ (0) ρ ) = f ( x, − τ (0) ρ ) , ( x, ρ ) in Ω × ]0 , . For any regular solution of (2.5), we define the energy as E ( t ) = 12 Z Ω (cid:0) | u t ( x, t ) | + a | u x ( x, t ) | (cid:1) dx,E ( t ) = 12 Z L L (cid:0) | v t ( x, t ) | + b | v x ( x, t ) | (cid:1) dx. The total energy is defined by(2.9) E ( t ) = E ( t ) + E ( t ) + ξ ( t ) τ ( t )2 Z Ω Z z ( x, ρ, t ) dρ dx, where(2.10) ξ ( t ) = ¯ ξµ ( t )is a non-increasing function of class C ( R + ) and ¯ ξ be a positive constant such that(2.11) β √ − d < ¯ ξ < − β √ − d . Our first result states that the energy is a non-increasing function.
Lemma 2.1.
Let ( u, v, z ) be a solution to the system (2.5) - (2.8) . Then the energy functional defined by (2.9) satisfies E ′ ( t ) ≤ − µ ( t ) (cid:18) − ¯ ξ − β √ − d (cid:19) Z Ω u t dx (2.12) − µ ( t ) (cid:18) ¯ ξ (1 − τ ′ ( t ))2 − β √ − d (cid:19) Z Ω z ( x, ρ, t ) dx ≤ . Proof.
Multiplying the first and second equations of (2.5) by u t ( x, t ) and v t ( x, t ), integrating on Ω and] L , L [ respectively and using integration by parts, we get12 ddt Z Ω (cid:0) u t + au x (cid:1) dx = − µ ( t ) Z Ω u t dx − µ ( t ) Z Ω z ( x, , t ) u t dx + a [ u x u t ] ∂ Ω , (2.13) RANSMISSION PROBLEM WITH DELAY AND WEIGHT 5 (2.14) 12 ddt Z L L (cid:0) v t + bv x (cid:1) dx = b [ v x v t ] L L . Now multiplying the third equation of (2.8) by ξ ( t ) z ( x, ρ, t ) and integrating on Ω × ]0 , τ ( t ) ξ ( t ) Z Ω Z z t ( x, ρ, t ) z ( x, ρ, t ) dρ dx = − ξ ( t )2 Z Ω Z (1 − τ ′ ( t ) ρ ) ∂∂ρ ( z ( x, ρ, t )) dρ dx. Consequently, ddt (cid:18) ξ ( t ) τ ( t )2 Z Ω Z z ( x, ρ, t ) dρ dx (cid:19) = ξ ( t )2 Z Ω ( z ( x, , t ) − z ( x, , t )) dx (2.15) + ξ ( t ) τ ′ ( t )2 Z Ω Z z ( x, , t ) dρ dx + ξ ′ ( t ) τ ( t )2 Z Ω Z z ( x, ρ, t ) dρ dx. From (2.9), (2.13), (2.14), (2.15) and using the conditions (2.6) and (2.7), we know that E ′ ( t ) = ξ ( t )2 Z Ω u t dx − ξ ( t )2 Z Ω z ( x, , t ) dx + ξ ( t ) τ ′ ( t )2 Z Ω z ( x, , t ) dx (2.16) + ξ ′ ( t ) τ ( t )2 Z Ω Z z ( x, ρ, t ) dρ dx − µ ( t ) Z Ω u t dx − µ ( t ) Z Ω z ( x, , t ) u t dx. Due to Young’s inequality, we have µ ( t ) Z Ω z ( x, , t ) u t dx ≤ | µ ( t ) | √ − d Z Ω u t dx + | µ ( t ) | √ − d Z Ω z ( x, , t ) dx. (2.17)Inserting (2.17) into (2.16), we obtain E ′ ( t ) ≤ − (cid:18) µ ( t ) − ξ ( t )2 − | µ ( t ) | √ − d (cid:19) Z Ω u t dx − (cid:18) ξ ( t )2 − ξ ( t ) τ ′ ( t )2 − | µ ( t ) | √ − d (cid:19) Z Ω z ( x, , t ) dx + ξ ′ ( t ) τ ( t )2 Z Ω Z z ( x, ρ, t ) dρ dx ≤ − µ ( t ) (cid:18) − ¯ ξ − β √ − d (cid:19) Z Ω u t dx − µ ( t ) (cid:18) ¯ ξ (1 − τ ′ ( t ))2 − β √ − d (cid:19) Z Ω z ( x, , t ) dx ≤ . Hence, the proof is complete. q.e.d.
3. Global solution
In this section, our goal is to prove existence and uniqueness of solutions to the system (2.5) - (2.8). This isthe content of Theorem 3.2.We begin by introducing the vector function U = ( u, v, ϕ, ψ, z ) T , where ϕ ( x, t ) = u t ( x, t ) and ψ ( x, t ) = v t ( x, t ). The system (2.5)-(2.8) can be written as(3.1) (cid:26) U t − A ( t ) U = 0 ,U (0) = U = ( u , v , u , v , f ( · , − , τ (0))) T , CARLOS A. S. NONATO, CARLOS A. RAPOSO & WALDEMAR D. BASTOS where the operator A ( t ) is defined by(3.2) A ( t ) U = ϕ ( x, t ) ψ ( x, t ) au xx ( x, t ) − µ ( t ) ϕ ( x, t ) − µ ( t ) z ( x, , t ) bv xx ( x, t ) − − τ ′ ( t ) ρτ ( t ) z ρ ( x, ρ, t ) . Now, taking into account the conditions (1.2)-(1.3), as well as previous results presented in [ , , , ],we introduce the set X ∗ = { ( u, v ) ∈ H (Ω) × H (] L , L [) /u (0) = u ( L ) = 0 , u ( L i ) = v ( L i ) , au x ( L i ) = bv x ( L i ) , i = 1 , } . We define the phase space as H = X ∗ × L (Ω) × L (] L , L [) × L (Ω × ]0 , h U, ˆ U i H = Z Ω ( ϕ ˆ ϕ + au x ˆ u x ) dx + Z L L (cid:16) ψ ˆ ψ + bv x ˆ v x (cid:17) dx + ξ ( t ) τ ( t ) Z Ω Z z ˆ z dρ dx, (3.3)for U = ( u, v, ϕ, ψ, z ) T and ˆ U = (ˆ u, ˆ v, ˆ ϕ, ˆ ψ, ˆ z ) T .The domain D ( A ( t )) of A ( t ) is defined by D ( A ( t )) = { ( u, v, ϕ, ψ, z ) T ∈ H / ( u, v ) ∈ (cid:0) H (Ω) × H (] L , L [) (cid:1) ∩ X ∗ , (3.4) ϕ ∈ H (Ω) , ψ ∈ H (] L , L [) , z ∈ L (cid:0) ]0 , L [; H (]0 , (cid:1) , ϕ = z ( · , } . Notice that the domain of the operator A ( t ) does not dependent on time t , i.e.,(3.5) D ( A ( t )) = D ( A (0)) , ∀ t > . A general theory for not autonomous operators given by equations of type (3.1) has been developed usingsemigroup theory, see [ ], [ ] and [ ]. The simplest way to prove existence and uniqueness results is toshow that the triplet { ( A , H , Y ) } , with A = {A ( t ) /t ∈ [0 , T ] } , for some fixed T > Y = A (0), forms aCD-systems (or constant domain system, see [ ] and [ ]). More precisely, the following theorem, which iddue to Tosio Kato (Theorem 1:9 of [ ]) gives the existence and uniqueness results and is proved in Theorem1 . ] (see also Theorem 2 .
13 of [ ] or [ ]). For convenience let states Kato’s result here. Theorem 3.1.
Assume that (i) Y = D ( A (0)) is dense subset of H , (ii) (3.5) holds, (iii) for all t ∈ [0 , T ] , A ( t ) generates a strongly continuous semigroup on H and the family A ( t ) = {A ( t ) /t ∈ [0 , T ] } is stable with stability constants C and m independent of t (i.e., the semigroup ( S t ( s )) s ≥ generated by A ( t ) satisfies k S t ( s ) u k H ≤ Ce ms k u k H , for all u ∈ H and s ≥ ), (iv) ∂ t A ( t ) belongs to L ∞∗ ([0 , T ] , B ( Y, H )) , which is the space of equivalent classes of essentially bounded,strongly measurable functions from [0 , T ] into the set B ( Y, H ) of bounded linear operators from Y into H .Then, problem (3.1) has a unique solution U ∈ C ([0 , T ] , Y ) ∩ C ([0 , T ] , H ) for any initial datum in Y . Using the time-dependent inner product (3.3) and the Theorem 3.1 we get the following result of existenceand uniqueness of global solutions to the problem (3.1).
Theorem 3.2. [Global solution] For any initial datum U ∈ H there exists a unique solution U satisfying U ∈ C ([0 , + ∞ [ , H ) for problem (3.1) .Moreover, if U ∈ D ( A (0)) , then U ∈ C ([0 , + ∞ [ , D ( A (0))) ∩ C ([0 , + ∞ [ , H ) . RANSMISSION PROBLEM WITH DELAY AND WEIGHT 7
Proof.
Our goal is then to check the above assumptions for problem (3.1). First, we show that D ( A (0))is dense in H . The proof we will follow method used in [ ] with the necessary modification imposed by thenature of our problem. Let ˆ U = (ˆ u, ˆ v, ˆ ϕ, ˆ ψ, ˆ z ) T ∈ H be orthogonal to all elements of D ( A (0)), namely0 = h U, ˆ U i H = Z Ω ( ϕ ˆ ϕ + au x ˆ u x ) dx + Z L L (cid:16) ψ ˆ ψ + bv x ˆ v x (cid:17) dx + ξ ( t ) τ ( t ) Z Ω Z z ˆ z dρ dx, (3.6)for U = ( u, v, ϕ, ψ, z ) T ∈ D ( A (0)).We first take u = v = ϕ = ψ = 0 and z ∈ C ∞ (Ω × ]0 , U = (0 , , , , z ) T ∈ D ( A (0)) and therefore,from (3.6), we deduce that Z Ω Z z ˆ z dρ dx = 0 . Since C ∞ (Ω × ]0 , L (Ω × ]0 , z = 0. Similarly, let ϕ ∈ C ∞ (Ω), then U = (0 , , ϕ, , T ∈ D ( A (0)), which implies from (3.6) that Z Ω ϕ ˆ ϕ dx = 0 . So, as above, it follows that ˆ ϕ = 0. In the same way, by taking ψ ∈ C ∞ (] L , L [), we get from (3.6) Z L L ψ ˆ ψ dx = 0and by density of C ∞ (] L , L [) in L (] L , L [), we obtain ˆ ψ = 0.Finally, for ( u, v ) ∈ C ∞ (Ω × ] L , L [) (then ( u x , v x ) ∈ C ∞ (Ω × ] L , L [)) we obtain a Z Ω u x ˆ u x dx + b Z L L v x ˆ v x dx = 0 . Since C ∞ (Ω × ] L , L [) is dense in L (Ω × ] L , L [), we deduce that (ˆ u x , ˆ v x ) = (0 ,
0) because (ˆ u, ˆ v ) ∈ X ∗ .We consequently have(3.7) D ( A (0)) is dense in H . Now, we show that the operator A ( t ) generates a C − semigroup in H for a fixed t .We calculate hA ( t ) U, U i t for a fixed t . Take U = ( u, v, ϕ, ψ, z ) T ∈ D ( A ( t )). Then hA ( t ) U, U i t = − µ ( t ) Z Ω ϕ dx − µ ( t ) Z Ω z ( x, ϕ dx − ξ ( t )2 Z Ω Z (1 − τ ′ ( t ) ρ ) ∂∂ρ z ( x, ρ ) dρ dx. Since (1 − τ ′ ( t ) ρ ) ∂∂ρ z ( x, ρ ) = ∂∂ρ (cid:0) (1 − τ ′ ( t ) ρ ) z ( x, ρ ) (cid:1) + τ ′ ( t ) z ( x, ρ ) , we have Z (1 − τ ′ ( t ) ρ ) ∂∂ρ z ( x, ρ ) dρ = (1 − τ ′ ( t )) z ( x, − z ( x,
0) + τ ′ ( t ) Z z ( x, ρ ) dρ. Whereupon hA ( t ) U, U i t = − µ ( t ) Z Ω ϕ dx − µ ( t ) Z Ω z ( x, ϕ dx + ξ ( t )2 Z Ω ϕ dx − ξ ( t )(1 − τ ′ ( t ))2 Z Ω z ( x, dx − ξ ( t ) τ ′ ( t )2 Z Ω Z z ( x, ρ ) dρ dx. Therefore, from (2.17), we deduce hA ( t ) U, U i t ≤ − µ ( t ) (cid:18) − ¯ ξ − β √ − d (cid:19) Z Ω ϕ dx − µ ( t ) (cid:18) ¯ ξ (1 − τ ′ ( t ))2 − β √ − d (cid:19) Z Ω z ( x, dx CARLOS A. S. NONATO, CARLOS A. RAPOSO & WALDEMAR D. BASTOS + ξ ( t ) | τ ′ ( t ) | τ ( t ) τ ( t ) Z Ω Z z ( x, ρ ) dρ dx. Then, we have hA ( t ) U, U i t ≤ − µ ( t ) (cid:18) − ¯ ξ − β √ − d (cid:19) Z Ω ϕ dx − µ ( t ) (cid:18) ¯ ξ (1 − τ ′ ( t ))2 − β √ − d (cid:19) Z Ω z ( x, dx + κ ( t ) h U, U i t , where κ ( t ) = p τ ′ ( t ) τ ( t ) . From (2.12) we conclude that(3.8) hA ( t ) U, U i t − κ ( t ) h U, U i t ≤ , which means that operator ˜ A ( t ) = A ( t ) − κ ( t ) I is dissipative.Now, we prove the surjectivity of the operator λI − A ( t ) for fixed t > λ >
0. For this purpose, let F = ( f , f , f , f , f ) T ∈ H . We seek U = ( u, v, ϕ, ψ, z ) T ∈ D ( A ( t )) which is solution of( λI − A ( t )) U = F, that is, the entries of U satisfy the system of equations λu − ϕ = f , (3.9) λv − ψ = f , (3.10) λϕ − au xx + µ ( t ) ϕ + µ ( t ) z ( x,
1) = f , (3.11) λψ − bv xx = f , (3.12) λz + 1 − τ ′ ( t ) ρτ ( t ) z ρ = f . (3.13)Suppose that we have found u and v with the appropriated regularity. Therefore, from (3.9) and (3.10) wehave ϕ = λu − f , (3.14) ψ = λv − f . (3.15)It is clear that ϕ ∈ H (Ω) and ψ ∈ H (] L , L [). Furthermore, if τ ′ ( t ) = 0, following the same approach as in[ ], we obtain z ( x, ρ ) = ϕ ( x ) e σ ( ρ,t ) + τ ( t ) e σ ( ρ,t ) Z ρ f ( x, s )1 − sτ ′ ( s ) e − σ ( s,t ) ds, where σ ( ρ, t ) = λ τ ( t ) τ ′ ( t ) ln(1 − ρτ ′ ( t )) , is solution of (3.13) satisfying(3.16) z ( x,
0) = ϕ ( x ) . Otherwise, z ( x, ρ ) = ϕ ( x ) e − λτ ( t ) ρ + τ ( t ) e − λτ ( t ) ρ Z ρ f ( x, s ) e λτ ( t ) s ds is solution of (3.13) satisfying (3.16). From now on, for pratical purposes we will consider τ ′ ( t ) = 0 (the case τ ′ ( t ) = 0 is analogous). Taking into account (3.14) we have z ( x,
1) = ϕe σ (1 ,t ) + τ ( t ) e σ (1 ,t ) Z f ( x, s )1 − sτ ′ ( s ) e − σ ( s,t ) ds (3.17) = ( λu − f ) e σ (1 ,t ) + τ ( t ) e σ (1 ,t ) Z f ( x, s )1 − sτ ′ ( s ) e − σ ( s,t ) ds RANSMISSION PROBLEM WITH DELAY AND WEIGHT 9 = λue σ (1 ,t ) − f e σ (1 ,t ) + τ ( t ) e σ (1 ,t ) Z f ( x, s )1 − sτ ′ ( s ) e − σ ( s,t ) ds. Substituting (3.14) and (3.17) in (3.11), and (3.15) in (3.12), we obtain(3.18) (cid:26) ηu − au xx = g ,λ v − bv xx = g , where η := λ + λµ ( t ) + λµ ( t ) e σ (1 ,t ) ,g := f + λf + µ ( t ) f + µ ( t ) f e σ (1 ,t ) − µ ( t ) τ ( t ) e σ (1 ,t ) Z f ( x, s )1 − sτ ′ ( s ) e − σ ( s,t ) ds,g := f + λf . In order to solve (3.18), we use a standard procedure, considering variational problem(3.19) Υ(( u, v ) , (˜ u, ˜ v )) = L (˜ u, ˜ v ) , where the bilinear form Υ : X ∗ × X ∗ → R and the linear form L : X ∗ → R are defined byΥ(( u, v ) , (˜ u, ˜ v )) = η Z Ω u ˜ u dx + a Z Ω u x ˜ u x dx + λ Z L L v ˜ v dx + b Z L L v x ˜ v x dx − a [ u x ˜ u ] ∂ Ω − b [ v x ˜ v ] L L and L (˜ u, ˜ v ) = Z Ω g ˜ u dx + Z L L g ˜ v dx. It is easy to verify that Υ is continuous and coercive, and L is continuous, so by applying the Lax-MilgramTheorem, we obtain a solution for ( u, v ) ∈ X ∗ for (3.18). In addition, it follows from (3.11) and (3.12) that( u, v ) ∈ H (Ω) × H (] L , L [) and so ( u, v, ϕ, ψ, z ) ∈ D ( A ( t )).Therefore, the operator λI − A ( t ) is surjective for any λ > t >
0. Again as κ ( t ) >
0, this prove that(3.20) λI − ˜ A ( t ) = ( λ + κ ( t )) I − A ( t ) is surjectivefor any λ > t > k Φ k t k Φ k s ≤ e c τ | t − s | , t, s ∈ [0 , T ] , where Φ = ( u, v, ϕ, ψ, z ) T , c is a positive constant and k · k is the norm associated the inner product (3.3). Forall t, s ∈ [0 , T ], we have k Φ k t − k Φ k s e cτ | t − s | = (cid:16) − e cτ | t − s | (cid:17) "Z Ω (cid:0) ϕ + au x (cid:1) dx + Z L L (cid:0) ψ + bv x (cid:1) dx + (cid:16) ξ ( t ) τ ( t ) − ξ ( s ) τ ( s ) e cτ | t − s | (cid:17) Z Ω Z z ( x, ρ ) dρ dx. It is clear that 1 − e cτ | t − s | ≤
0. Now we will prove ξ ( t ) τ ( t ) − ξ ( s ) τ ( s ) e cτ | t − s | ≤ c >
0. In order todo this , first observe that τ ( t ) = τ ( s ) + τ ′ ( r )( t − s ) , for some r ∈ ( s, t ). Since ξ is a non increasing function and ξ >
0, we get ξ ( t ) τ ( t ) ≤ ξ ( s ) τ ( s ) + ξ ( s ) τ ′ ( r )( t − s ) , which implies ξ ( t ) τ ( t ) ξ ( s ) τ ( s ) ≤ | τ ′ ( r ) | τ ( s ) | t − s | . Using (1.5) and that τ ′ is bounded, we deduce ξ ( t ) τ ( t ) ξ ( s ) τ ( s ) ≤ cτ | t − s | ≤ e cτ | t − s | , which proves (3.21) and therefore ( iii ) follows.Moreover, as κ ′ ( t ) = τ ′ ( t ) τ ′′ ( t )2 τ ( t ) √ τ ′ ( t ) − τ ′ ( t ) √ τ ′ ( t ) τ ( t ) is bounded on [0 , T ] for all T > ddt A ( t ) U = − µ ′ ( t ) ϕ − µ ′ ( t ) z ( · , τ ′′ ( t ) τ ( t ) ρ − τ ′ ( t )( τ ′ ( t ) ρ − τ ( t ) z ρ , with τ ′′ ( t ) τ ( t ) ρ − τ ′ ( t )( τ ′ ( t ) ρ − τ ( t ) bounded on [0 , T ] by (1.5) and (2.11). Thus(3.22) ddt ˜ A ( t ) ∈ L ∞∗ ([0 , T ] , B ( D ( A (0)) , H )) , where L ∞∗ ([0 , T ] , B ( D ( A (0)) , H )) is the space of equivalence classes of essentially bounded, strongly measurablefunctions from [0 , T ] into B ( D ( A (0)) , H ). Here B ( D ( A (0)) , H ) is the set of bounded linear operators from D ( A (0)) into H .Then, (3.8), (3.20) and (3.21) imply that the family ˜ A = n ˜ A ( t ) : t ∈ [0 , T ] o is a stable family of generators in H with stability constants independent of t , by Proposition 1 . ]. Therefore, the assumptions ( i ) − ( iv )of Theorem 3.1 are verified by (3.5), (3.7), (3.8), (3.20), (3.21) and (3.22). Thus, the problem(3.23) (cid:26) ˜ U t = ˜ A ( t ) ˜ U , ˜ U (0) = U has a unique solution ˜ U ∈ C ([0 , + ∞ [ , D ( A (0))) ∩ C ([0 , + ∞ [ , H ) for U ∈ D ( A (0)). The requested solution of(3.1) is then given by U ( t ) = e R t κ ( s ) ds ˜ U ( t )because U t ( t ) = κ ( t ) e R t κ ( s ) ds ˜ U ( t ) + e R t κ ( s ) ds ˜ U t ( t )= e R t κ ( s ) ds (cid:16) κ ( t ) + ˜ A ( t ) (cid:17) ˜ U ( t )= A ( t ) e R t κ ( s ) ds ˜ U ( t )= A ( t ) U ( t )which concludes the proof. q.e.d.
4. Exponential stability
This section is dedicated to study of the asymptotic behavior. The main goal of this section is to studythe stability of solutions to the system (2.5)-(2.8). This is the content of Theorem 4.4 where we show thatthe solution of problem (2.5)-(2.8) is exponentially stable. Our effort consists in building a suitable Lyapunovfunctional by the energy method. For this goal we present several technical lemmas.
Lemma 4.1.
Let ( u, v, z ) be a solution of (2.5) - (2.8) , then for any ε > and c is the Poincar´e’s constant,we have the estimate ddt I ( t ) ≤ − (cid:0) a − µ (0) c ε (cid:1) Z Ω u x dx − b Z L L v x dx (4.1) RANSMISSION PROBLEM WITH DELAY AND WEIGHT 11 + (cid:18) ε (cid:19) Z Ω u t dx + Z L L v t dx + β ε Z Ω z ( x, , t ) dx, where (4.2) I ( t ) = Z Ω uu t dx + Z L L vv t dx. Proof.
Differentiating I ( t ) and using (2.5), we obtain ddt I ( t ) = Z Ω u t dx − a Z Ω u x dx − µ ( t ) Z Ω uu t dx − µ ( t ) Z Ω uz ( x, , t ) dx + Z L L v t dx − b Z L L v x dx ≤ Z Ω u t dx − a Z Ω u x dx + (cid:12)(cid:12)(cid:12)(cid:12) µ ( t ) Z Ω uu t dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) µ ( t ) Z Ω uz ( x, , t ) dx (cid:12)(cid:12)(cid:12)(cid:12) + Z L L v t dx − b Z L L v x dx. From hypothesis (H1) and (H2), we have ddt I ( t ) ≤ Z Ω u t dx − a Z Ω u x dx + µ (0) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uu t dx (cid:12)(cid:12)(cid:12)(cid:12) (4.3) + βµ (0) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uz ( x, , t ) dx (cid:12)(cid:12)(cid:12)(cid:12) + Z L L v t dx − b Z L L v x dx. By using the conditions (2.6) and (2.7), we obtain u ( x, t ) = (cid:18)Z x u x ( s, t ) ds (cid:19) ≤ L Z L u x ( x, t ) dx, x ∈ [0 , L ] ,u ( x, t ) ≤ ( L − L ) Z L L u x ( x, t ) dx, x ∈ [ L , L ] , which imply the following Poincar´e’s inequality(4.4) Z Ω u ( x, t ) dx ≤ c Z Ω u x dx, x ∈ Ω , where c = max { L , L − L } is the Poincar´e’s constant. Using Young’s inequality and (4.4), we have(4.5) µ (0) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uu t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε µ (0) c Z Ω u x dx + 12 ε Z Ω u t dx and(4.6) βµ (0) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uz ( x, , t ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε µ (0) c Z Ω u x dx + β ε Z Ω z ( x, , t ) dx. Substituting (4.5) and (4.6) in (4.3) we conclude the lemma. q.e.d.Now, inspired by [ ], we introduce the functional(4.7) q ( x ) = x − L , x ∈ [0 , L ] ,L − L − L L − L ) ( x − L ) + L , x ∈ [ L , L ] ,x − L + L , x ∈ [ L , L ] . It is easy to see that q ( x ) is bounded, i.e., | q ( x ) | ≤ M , where M = max (cid:26) L , L − L (cid:27) . We have the following result.
Lemma 4.2.
Let ( u, v, z ) be a solution of (2.5) - (2.8) , then for any ε > , the following estimates holds true ddt I ( t ) ≤ (cid:18)
12 + 12 ε (cid:19) Z Ω u t dx + (cid:16) a M µ (0) ε (cid:17) Z Ω u x dx + β ε Z Ω z ( x, , t ) dx (4.8) − (cid:2) L u t ( L , t ) + ( L − L ) u t ( L , t ) (cid:3) − a (cid:2) L u x ( L , t ) + ( L − L ) u x ( L , t ) (cid:3) , and ddt I ( t ) = L − L − L L − L ) Z L L v t dx + b Z L L v x dx ! + 14 (cid:2) L v t ( L , t ) + ( L − L ) v t ( L , t ) (cid:3) (4.9) + b (cid:2) L v x ( L , t ) + ( L − L ) v x ( L , t ) (cid:3) , where (4.10) I ( t ) = − Z Ω q ( x ) u x u t dx and I ( t ) = − Z L L q ( x ) v x v t dx. Proof.
Differentiating I ( t ) and using (2.5), we obtain ddt I ( t ) = − Z Ω q ( x ) u xt u t dx − a Z Ω q ( x ) u xx u x dx + µ ( t ) Z Ω q ( x ) u x u t dx + µ ( t ) Z Ω q ( x ) u x z ( x, , t ) dx. Integrating by parts and considering the hypothesis (H1) and (H2), we have ddt I ( t ) ≤ Z Ω q ′ ( x ) u t dx − (cid:2) q ( x ) u t (cid:3) ∂ Ω + a Z Ω q ′ ( x ) u x dx − a (cid:2) q ( x ) u x (cid:3) ∂ Ω (4.11) + µ (0) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω q ( x ) u x u t dx (cid:12)(cid:12)(cid:12)(cid:12) + βµ (0) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω q ( x ) u x z ( x, , t ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω u t dx − (cid:2) q ( x ) u t (cid:3) ∂ Ω + a Z Ω u x dx − a (cid:2) q ( x ) u x (cid:3) ∂ Ω + µ (0) M (cid:12)(cid:12)(cid:12)(cid:12)Z Ω u x u t dx (cid:12)(cid:12)(cid:12)(cid:12) + βµ (0) M (cid:12)(cid:12)(cid:12)(cid:12)Z Ω u x z ( x, , t ) dx (cid:12)(cid:12)(cid:12)(cid:12) . On the other hand, by using the boundary conditions (2.7), we have12 (cid:2) q ( x ) u t (cid:3) ∂ Ω = 14 (cid:2) L u t ( L , t ) + ( L − L ) u t ( L , t ) (cid:3) , − a (cid:2) q ( x ) u x (cid:3) ∂ Ω ≤ − a (cid:2) L u x ( L , t ) + ( L − L ) u x ( L , t ) (cid:3) . Inserting the above two equalities into (4.11) and by Young’s inequality, we conclude that (4.11) gives (4.8).By the same argument, taking the derivative of I ( t ), we obtain ddt I ( t ) = 12 Z L L q ′ ( x ) v t dx − (cid:2) q ( x ) v t (cid:3) L L + b Z L L q ′ ( x ) v x dx − b (cid:2) q ( x ) v x (cid:3) L L = L − L − L L − L ) Z L L v t dx + b Z L L v x dx ! + 14 (cid:2) L v t ( L , t ) + ( L − L ) v t ( L , t ) (cid:3) + b (cid:2) L v x ( L , t ) + ( L − L ) v x ( L , t ) (cid:3) Hence, the proof is complete. q.e.d.As in [ ], taking into account the last lemma, we introduce the functional(4.12) J ( t ) = ¯ ξτ ( t ) Z Ω Z e − τ ( t ) ρ z ( x, ρ, t ) dρ dx. For this functional we have the following estimate.
RANSMISSION PROBLEM WITH DELAY AND WEIGHT 13
Lemma 4.3 ([ , Lemma 3.7]) . Let ( u, v, z ) be a solution of (2.5) - (2.8) . Then the functional J ( t ) satisfies (4.13) ddt J ( t ) ≤ − J ( t ) + ¯ ξ Z Ω u t dx. Now we are in position to prove our result of stability.
Theorem 4.4.
Let U ( t ) = ( u ( t ) , v ( t ) , ϕ ( t ) , ψ ( t ) , z ( t )) be the solution of (2.5) - (2.8) with initial data U ∈ D ( A (0)) and E ( t ) the energy of U . Assume that the hypothesis (1.5) , (1.6) , (H1), (H2) and (4.14) max { , ab } < L + L − L L − L ) hold. Then there exist positive constants c and α such that (4.15) E ( t ) ≤ cE (0) e − αt , ∀ t ≥ . Proof.
Let us define the Lyapunov functional(4.16) L ( t ) = N E ( t )( t ) + X i =1 N i I i ( t ) + J ( t ) , where N , N i , i = 1 , , K such that(4.17) ddt E ( t ) ≤ − K (cid:20)Z Ω u t dx + Z Ω z ( x, , t ) dx (cid:21) . It follows from the transmission conditions (2.6) that(4.18) a u x ( L i , t ) = b v x ( L i , t ) , i = 1 , . Using the estimates (4.1), (4.8), (4.9), (4.13), (4.17) and the equation (4.18), we obtain ddt L ( t ) ≤ − (cid:20) KN − (cid:18) ε (cid:19) N − (cid:18)
12 + 12 ε (cid:19) N − ¯ ξ (cid:21) Z Ω u t dx (4.19) − (cid:18) KN − β ε N − β ε N (cid:19) Z Ω z ( x, , t ) dx − h(cid:0) a − µ (0) c ε (cid:1) N − (cid:16) a M µ (0) ε (cid:17) N i Z Ω u x dx + (cid:20) N + L − L − L L − L ) N (cid:21) Z L L v t dx − (cid:20) N − L − L − L L − L ) N (cid:21) b Z L L v x dx − ( N − N ) (cid:20) L u t ( L , t ) + L − L u t ( L , t ) (cid:21) − (cid:16) N − ab N (cid:17) a (cid:20) L u t ( L , t ) + L − L u t ( L , t ) (cid:21) − J ( t ) . Now we observe that under assumption (4.14), we can always find real constants N , N and N in such waythat N + L − L − L L − L ) N < , N > max n , ab o N , N > N . After that, we pick positive constants ε and ε small enough that µ (0) c ε N + M µ (0) ε N < a (cid:18) N − N (cid:19) . Finally, since ξ ( t ) τ ( t ) non-negative and limited, we choose N large enough that (4.19) is taken into the followingestimate ddt L ( t ) ≤ − η Z Ω (cid:0) u t + u x (cid:1) dx − η Z L L (cid:0) v t + v x (cid:1) dx − η Z Ω z ( x, ρ, t ) dx − η Z Ω z ( x, , t ) dx ≤ − η Z Ω (cid:0) u t + u x (cid:1) dx − η Z L L (cid:0) v t + v x (cid:1) dx − η Z Ω z ( x, ρ, t ) dx, for a certain positive constant η .This implies by (2.9) that there exists η > ddt L ( t ) ≤ − η E ( t ) , ∀ t ≥ . On the hand, it is not hard to see for N large enough that the L ( t ) ∼ E ( t ), i.e. there exists two positiveconstants γ and γ such that(4.21) γ E ( t ) ≤ L ( t ) ≤ γ E ( t ) , ∀ t ≥ . Combining (4.20) and (4.21), we obtain ddt L ( t ) ≤ − α L ( t ) , ∀ t ≥ L ( t ) ≤ L (0) e − αt , ∀ t ≥ . The desired result (4.15) follows by using estimates (4.21) and (4.22). Then, the proof of Theorem 4.4 iscomplete. q.e.d.
Acknowledgment.
The authors thanks CAPES(Brazil).
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E-mail address : [email protected]
Federal University of S˜ao Jo˜ao del-Rei, Mathematics Departament, S˜ao Jo˜ao del-Rei, 36307-352, Minas Gerais, Brazil
E-mail address : [email protected]
S˜ao Paulo State University, Mathematics Departament, S˜ao Jos´e do Rio Preto, 15054-352, S˜ao Paulo, Brazil