Exponential Stability of a Transmission Problem with History and Delay

In this paper, we consider a transmission problem in the presence of history and delay terms.Under appropriate assumptions, we prove well-posedness by using the semigroup theory. Our stability estimate proves that the unique dissipation given by the history term is strong enough to stabilize exponentially the system in presence of delay by introducing a suitable Lyaponov functional.

Transmission problems arise in several applications of physics and biology. We note that problem (1)-(3) is related to the wave propagation over a body which consists of two different type of materials: the elastic part and the viscoelastic part that has the past history and time delay effect.
For wave equations with various dissipations, many results concerning stabilization of solutions have been proved. Recently, wave equations with viscoelastic damping have been investigated by many authors, see [2,4,3,9,8,10,16,18] and the references therein. It is showed that the dissipation produced by the viscoelastic part can produce the decay of the solution. For example, A. Guesmia [6] studied the equation and under the condition: ∃δ > 0, g ′ (s) ≤ −δg(s) ∀ s ∈ R + the authors showed the exponential decay.
Messaoudi [12] investigated the following viscoelastic equation: in a bounded domain, and established a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.
In [7] the authors examined a system of wave equations with a linear boundary damping term with a delay: and under the assumption they proved that the solution is exponentially stable. On the contrary, if (5) does not hold, they found a sequence of delays for which the corresponding solution of (4) will be unstable. In [11], authors considered the equation and under the assumption they proved the well-posedness and the exponential decay of energy. Recently, in [19] Yadav and Jiwari considered Burgers'-Fisher equation: the authors proved existence and uniqueness of solution. Furthermore, they also presented finite element analysis and approximation.
The paper is organized as follows. The well-posedness of the problem is analyzed in Section 2 using the semigroup theory. In Section 3, we prove the exponential decay of the energy when time goes to infinity. N. BAHRI AND A. BENIANI 733

Preliminaries and assumptions
We assume that the function g satisfies the following: A1: We assume that the function g : R + −→ R + is of class C 1 satisfying: A2: There exists a positive constantδ, As in [14], we introduce the variable Then Following the ideal in [5], we set the boundary and transmission conditions (2) become and the initial conditions (3) become It is clear that for all s > 0.
Let V := (u, v, φ, ψ, z, η t ) T , then V satisfies the problem 734 EXPONENTIAL STABILITY where V 0 := (u 0 (·, 0), v 0 , u 1 , v 1 , f 0 (·, −τ ), η 0 ) T The operator A and B are linear and defined by and L 2 g (R + , H 1 (Ω)) denotes the Hilbert space of H 1 -valued functions on R + , endowed with the inner product We define the inner product in the energy space H , and D(B) = H The well-posedness of problem (10)-(11) is ensured by the following theorem.

Proof
We use the semigroup approach. So, first, we prove that the operator A is dissipative. In fact, for For the last term of the right side of (15), we obtain where we have used that Using Young's inequality, we have Consequently, taking (A2) into account, we conclude that which is equivalent to Assume that with the suitable regularity we have found u and v, then Using the equation in (17), we obtain From (18), we obtain It is easy to see that the last equation in (17) with w(x, 0) = 0 has a unique solution By using (17), (18) and (20), the functions u and v satisfy ( We just need to prove that (21) has a solution (u, v) ∈ X * and replace in (18), (19) and (20) to get V = (u, v, φ, ψ, z, w) T ∈ D(A ) satisfying (16). Consequently, problem (21) is equivalent to the problem where the bilinear form Φ : (X * , X * ) → R and the linear form l : Using the properties of the space X * , it is easy to see that Φ is continuous and coercive, and l is continuous.
Applying the Lax-Milgram theorem, we infer that for all (ω 1 , ω 2 ) ∈ X * , problem (22) has a unique solution Thence, the operator λI − A is surjective for any λ > 0. That mean A is maximal monotone operator.Then,using Lummer-Phillips theorem [15],we dedece that A is an infinitesimal generator of a linear C 0 -semigroup on H . On the other hand, it is clear that the linear operator B is Lipschitz continuous.Finally,also A + B is an infinitesimal generator of a linear C 0 -semigroup on H . Consequently (14) is well-posed in the sense of Theorem 1(see [15]).

Exponential stability
In this section, we consider a decay result of problem (1)-(3). In fact using the energy method to produce a suitable Lyapunov functional Theorem 2 Let (u, v) be the solution of (1)- (3). Assume that (A1),(A2) hold, and that then there exist two constants γ 1 , γ 2 > 0 such that, For the proof of Theorem 2, we need some lemmas. For a solution of (1)-(3), we define the energy

Lemma 1
Let (u, v, η, z) be the solution of (10)- (11). Then we have the inequality where we have used that Young's inequality gives us

Now, we define the functional
then we have the following lemma.

Lemma 2
The functional D(t) satisfies

Proof
Taking the derivative of D(t) with respect to t and using (10), we have where we used that By the boundary conditions (2), we have where L = max{L 1 , L 3 − L 2 }. By making use of Young's inequality and (31), for any ε > 0, we obtain Young's inequality, Hölder's inequality and (A2) imply that ∫ Inserting the estimates (32) and (33) into (30), then (29) is fulfilled.
Next, enlightened by [13], we introduce the functional It is easy to see that q(x) is bounded: then we have the following results.

Proof
Taking the derivative of F 1 (t) with respect to t and using (10), we obtain We pay attention to The last term in (36) can be treated as follows where we used that − Inserting (37) and (38) in (36), we arrive at Using Minkowski and Young's inequalities, we have Young's inequality gives us that for any ε 1 > 0, Inserting (40)-(42) into (39), we obtain (34). By the same method, taking the derivative of F 1 (t) with respect to t, we obtain ) .
Thus, the proof of Lemma 3 is complete.
We define the functional then we have the following estimate.

Lemma 4
The functionals F 3 (t) satisfies We define the functional then we have the following estimate.

Lemma 5
The functional F 4 (t) satisfies

Proof
We define the Lyapunov functional where N 1 , N 2 , N 4 , N 5 and N 6 are positive constants that will be fixed later.
Taking the derivative of (49) with respect to t and taking advantage of the above lemmas, we have At this moment, we wish all coefficients except the last two in (50) will be negative. We want to choose N 2 and N 4 to ensure that . For this purpose, since 8l(L2−L1) L1+L3−L2 < min{a, b} we first choose N 4 satisfying 8l(L 2 − L 1 ) Once N 4 is fixed, we pick N 2 satisfying Then we take ε,ε 1 and ε 1 small enough, and δ 2 < 1 2N6 we have Once ε and ε 1 are fixed, we take N 5 satisfying Further, we take δ 2 < g0 2 we choose N 6 satisfying Then we have Then, we pick δ 2 satisfying Finally, choosing N 1 large enough such that the first and the last coefficients in (50) is positive. From the above, we deduce that there exist two positive constants α 1 and α 2 such that (50) becomes

Conclusion
In this paper we study the following transmission system with a past history and a delay term. Under assumptions on initial data and boundary conditions, past history and a delay term, we focused our study on the existence and asymptotic behavior of solutions where we obtained exponential decay of solutions for transmission problems.