Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback

In this paper, we consider the wave equation with a weak internal constant delay term: u′′(x, t)− ∆xu(x, t) + μ1(t) u′(x, t) + μ2(t) u′(x, t− τ) = 0 in a bounded domain. Under appropriate conditions on μ1 and μ2, we prove global existence of solutions by the Faedo–Galerkin method and establish a decay rate estimate for the energy using the multiplier method.


Introduction
In this paper we investigate the decay properties of solutions for the initial boundary value problem for the linear wave equation of the form on Ω, where Ω is a bounded domain in IR n , n ∈ IN * , with a smooth boundary ∂Ω = Γ, τ > 0 is a time delay and the initial data (u 0 , u 1 , f 0 ) belong to a suitable function space.
In absence of delay (µ 2 = 0), the energy of problem (P) is exponentially decaying to zero provided that µ 1 is constant, see, for instance, [3,4,7,8] and [12].On the contrary, if µ 1 = 0 and µ 2 > 0 (a constant weight), that is, there exists only the internal delay, the system (P) 2 A. Benaissa, A. Benguessoum and S. A. Messaoudi becomes unstable (see, for instance, [5]).In recent years, the PDEs with time delay effects have become an active area of research since they arise in many practical problems (see, for example, [1,19]).In [5], it was shown that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one and, therefore, delay becomes a source of instability.To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary (see [13,15,20]).For instance, the authors of [13] studied the wave equation with a linear internal damping term with constant delay (τ = const in the problem (P) and determined suitable relations between µ 1 and µ 2 , for which the stability or alternatively instability takes place.More precisely, they showed that the energy is exponentially stable if µ 2 < µ 1 and they also found a sequence of delays for which the corresponding solution of (P) will be unstable if µ 2 ≥ µ 1 .The main approach used in [13] is an observability inequality obtained with a Carleman estimate.The same results were obtained if both the damping and the delay are acting on the boundary.We also recall the result by Xu, Yung and Li [20], where the authors proved a result similar to the one in [13] for the one-space dimension by adopting the spectral analysis approach.
In [17], Nicaise, Pignotti and Valein extended the above result to higher space dimensions and established an exponential decay.
Our purpose in this paper is to give an energy decay estimate of the solution of problem (P) in the presence of a delay term with a weight depending on time.We use the Galerkin approximation scheme and the multiplier technique to prove our results.

Preliminaries and main results
First assume the following hypotheses: (H2) µ 2 : IR + → IR is a function of class C 1 (IR + ), which is not necessarily positive or monotone, such that ) for some 0 < β < 1 and M > 0.
We now state a Lemma needed later.
Assume that there exist σ > −1 and ω > 0 such that Wave equation with a constant weak delay and a weak internal feedback We introduce, as in [13], the new variable Then, we have (2.9) Therefore, problem (P) takes the form: Let ξ be a positive constant such that We define the energy of the solution by: where ξ(t) = ξµ 1 (t).
We have the following theorem.
Assume that (H1) and (H2) hold.Then problem (P) admits a unique global weak solution Moreover, for some positive constants c, ω, we obtain the following decay property: Lemma 2.3.Let (u, z) be a solution to the problem (2.10).Then, the energy functional defined by (2.12) satisfies Proof.Multiplying the first equation in (2.10) by u t (x, t), integrating over Ω and using Green's identity, we obtain: (2.15) We multiply the second equation in (2.10) by ξ(t)z and integrate over Ω × (0, 1) to obtain: This yields (2.17) Combination of (2.15) and (2.17) leads to Recalling the definition of E(t) in (2.12), we arrive at Due to Young's inequality, we have This completes the proof of the lemma.
We construct approximate solutions (u k , z k ), k = 1, 2, 3, . . ., in the form where g jk and h jk (j = 1, 2, . . ., k) are determined by the following system of ordinary differential equations: associated with the initial conditions and ) By virtue of the theory of ordinary differential equations, the system (3.1)-(3.5)has a unique local solution which is extended to a maximal interval [0, In the next step, we obtain a priori estimates for the solution of the system (3.1)-(3.5),so that it can be extended beyond [0, T k [ to obtain a solution defined for all t > 0.Then, we utilize a standard compactness argument for the limiting procedure.The first estimate.Since the sequences u 0k , u 1k and z 0k converge, then from (2.14) we can find a positive constant C independent of k such that where These estimates imply that the solution (u k , z k ) exists globally in [0, +∞[. ) ) )

.11)
The second estimate.We first estimate u k (0).Replacing w j by u k (t) in (3.1) and taking t = 0, we obtain: Wave equation with a constant weak delay and a weak internal feedback 7 Differentiating (3.1) with respect to t, we get Multiplying by g jk (t), summing over j from 1 to k, it follows that 1 2 (3.12) Differentiating (3.4) with respect to t, we get Multiplying by h jk (t), summing over j from 1 to k, it follows that Taking the sum of (3.12) and (3.13), we obtain that 1 2 Using (H1), (H2), Cauchy-Schwarz and Young's inequalities, we obtain 1 2 Integrating the last inequality over (0, t) and using (3.6), we get Using Gronwall's lemma, we deduce that for all t ∈ IR + , therefore, we conclude that ) for all T ≥ 0. We have to show that u is a solution of (P).From (3.15) we have that Therefore u ς → u strongly and a.e. in Q.
Using Young's inequality, we obtain 1 2 where c is a positive constant.Then integrating over (0, t), using Gronwall's lemma, we conclude that Hence, uniqueness follows.

Asymptotic behavior
From now on, we denote by c various positive constants which may be different at different occurrences.We multiply the first equation of (2.10) by φ E q u, where φ is a bounded function satisfying all the hypotheses of Lemma 2.1.We obtain Similarly, we multiply the second equation of (2.10) by E q φ ξ(t)e −2τρ z(x, ρ, t) and get 0 Taking their sum, we obtain Wave equation with a constant weak delay and a weak internal feedback 11 where A = 2 min{1, e −2τ 1 }.Recalling that ξ ≤ 0 and the definition of E, we have where we have also used the Cauchy-Schwarz inequality.Combining these estimates and choosing ε sufficiently small, we conclude from (4.1) that T S E q+1 φ dt ≤ CE q+1 (S).
This ends the proof of Theorem 2.2.