Existence and exponential stability of solutions for transmission system with varying delay in R

In the present paper we are going to consider in a one dimension bounded domain a transmission system with a varying delay. Under suitable assumptions on the weights of the damping and the delay terms, we prove the well-possedness and the uniqueness of solution using the semigroup theory. Also we show the exponential stability by introducing an appropriate Lyaponov functional.


Introduction
It is well known that the PDEs with time delay have been much studied during the last years and their results is by now rather developed.See [1], [5,6,7,14,17,18,19].In the classical theory of delayed wave equations, several main parts are joined in a fruitful way, it is very remarkable that the damped wave equation with varying delays occupies a similar position and arise in many applied problems.
Moreover, we assume that where d is a positive constant.
To motivate our work, let us mention the work [16], when the authors studied well-posedness and exponential stability of a problem with structural damping and boundary delay in both cases µ > 0 and µ = 0 in a bounded and smooth domain, where k 2 = 0.The analogous problem with boundary feedback has been introduced and studied by Xu, Yung, Li [19] in one-space dimension using a fine spectral analysis and in higher space dimension by the authors [14].The case of time-varying delay has been already studied in [15] in one space dimension and in general dimension, with a possibly degenerate delay, in [16].Both these papers deal with boundary feedback.
This paper improves the results in [4]; for τ (t) = τ , under suitable assumptions on the weight of the damping and the weight of the delay, he prove the existence and the uniqueness of the solution using the semigroup theory.Also he show the exponential stability of the solution by introducing a suitable Lyaponov functional.. Without delay, system (1) has been investigated in [3]; for Ω = [0, L 1 ], the authors showed the well-posedness and exponential stability of the total energy.Muñoz Rivera and Oquendo [13] studied the wave propagations over materials consisting of elastic and viscoelastic components; that is, with the boundary and initial conditions: where ρ 1 and ρ 2 are densities of the materials and α 1 , α 2 are elastic coefficients and g is positive exponential decaying function.They showed that the dissipation produced by the viscoelastic part is strong enough to produce an exponential decay of the solution, no matter how small is its size.Ma and Oquendo [9] considered transmission problem involving two Euler-Bernoulli equations modeling the vibrations of a composite beam.By using just one boundary damping term in the boundary, they showed the global existence and decay property of the solution.Marzocchi et al [10] investigated a 1-D semi-linear transmission problem in classical thermoelasticity and showed that a combination of the first, second and third energies of the solution decays exponentially to zero, no matter how small the damping subdomain is.A similar result has sheen shown by Messaoudi and Said-Houari [12], where a transmission problem in thermoelasticity of type III has been investigated.See also Marzocchi et al [11] for a multidimensional linear thermoelastic transmission problem.The effect of the delay in the stability of hyperbolic systems has been investigated by many people.See for instance [6,7].The aim of this article is to study effect of the varying delay in the stability of our system.

Well-posedness
Using the semigroup theory, we prove the existence and uniqueness of solution of system (1).As in [14], let us introduce the following new variable (7) z(x, ρ, t) = u t (x, t − τ (t) ρ).
Then, we obtain Therefore, the first equation in problem ( 1) is become as which can be written as where the operator A(t) is given by ( 9) where the space X * is defined by Remark 2.1.Noting that the domain of D(A)(t) is independent of the time t; i.e., (13) D(A (t)) = D(A (0)), t > 0. Let We define the standard inner product in H as follows: Using semigroup arguments by the literature, we can obtain a well-posedness result (see [8]).
Theorem 2.1.Assume that Therefore, we will check the above assumptions for system (2).
In the same way, by taking In order to deduce a well-posedness result, we define on H the time dependent inner product where ξ is the positive constant satisfying ( 16) Note that, from (15), such a constant ξ exists.
where d is a positive constant.
Proof.For all s, t ∈ [0, T ], we have Indeed, , where a, b ∈ (s, t), and thus, , τ is bounded on [0, T ] and therefore, recalling also (2), This complete the proof.Lemma 2.3.Under condition (16) the operator is dissipative, and , where Then, for a fixed t, Integrating by parts, we obtain We get Existence and exponential stability of solutions Using Young's inequality, the third condition of (1) and the equality ϕ (L 2 ) = ψ (L 2 ) ,we obtain where ( 23) 2τ (t) .
Consequently, using ( 16), we deduce that Which means that the operator is bounded on [0, T ] for all T > 0 (by ( 2) and (3) and we have the space of equivalence classes of essentially bounded, strongly measurable functions from [0, T ] into B(D(A(0)), H).
The system (25)-( 26) can be reformulated as Integrating by parts, we obtain (32) The problem (32) and ( 33) is equivalent to the problem, where the bilinear form Φ : (X * × X * ) → R and the linear form l : X * → R are defined by Using the properties of the space X * , it is clear that Φ is continuous and coercive, and l is continuous.So applying the Lax-Milgram theorem, we deduce that for all (ω, ω) ∈ X * , problem (34) admits a unique solution (u, v) ∈ X * .It follows from ( 32) and ( 33 In conclusion, we have found U = (u, v, ϕ, ψ, z) T ∈ D(A(t)), which verifies (25), and thus (λI − A(t)) is surjective for some λ > 0 and t > 0. Again as κ (t) > 0, this proves that for any λ > 0 and t > 0.
Proof.Results ( 17), ( 18) and (35) imply that the family A 1 = {A 1 (t) : t ∈ [0, T ]} is a stable family of generators in H with stability constants independent of t.Therefore, all assumptions of Theorem 2.1 are satisfied by ( 13), Lemma2.1-Lemma2.4,and thus, the problem The requested solution of ( 2) is then given by This concludes the proof.

Stability result
In this section we study the asymptotic behavior of the system (1).For any regular solution of (1), we give the total energy as where ξ is the positive constant defined by (16).Our next main result reads as.
Then there exist two positive constants W and w, such that To prove Theorem 3.1, we use the following lemmas.First, we will need an explicit formula of energy derivative.The following energy functional law holds.
Lemma 3.1.Let (u, v, z) be the solution of (1).Assume that µ 1 ≥ µ 2 .Then we have the inequality Proof.From (36) we have where Using system (2), and integrating by parts, we obtain applying Young's inequality On the other hand, where Using the fact that , (41), (42), using boundary conditions and applying Young's inequality, we show that (39) holds.The proof is complete.
Following [2], we define the functional e s−t u 2 t (x, s)dsdx, and state the following lemma.
Lemma 3.2.Let (u, v) be the solution of (2).Then Now, we define the functional D(t) as follows (44) Then, we have the following estimate.
Lemma 3.3.The functional D(t) satisfies Proof.Taking the derivative of D(t) with respect to t and using (1), we find that Using the boundary conditions , we have On the other hand, we have by Poincaré's and Young's inequalities, where c 0 is the Poincaé's constant.Consequently, plugging the above estimates into (46), we find (45).Now, inspired by [10], we introduce the functional Next, in order to construct the Lyapounov function, we define the functionals Then, we have the following estimates.Lemma 3.4.For any 2 > 0, we have the estimates and Proof.Taking the derivative of L 1 (t) with respect to t and using (1), we obtain Integrating by parts, (52) On the other hand, Substituting (52) and ( 53) in (51), we find that Using Young's inequality and (48), equation (54) becomes Since q(L 1 ) > 0 and q(L 2 ) < 0, by using the boundary condition, we have Taking into account ( 56) and (57), then (55) gives (49).
By the same method, taking the derivative of L 2 (t) with respect to t, we obtain which is exactly (50).