ENERGY DECAY OF SOLUTIONS FOR THE WAVE EQUATION WITH A TIME-VARYING DELAY TERM IN THE WEAKLY NONLINEAR INTERNAL FEEDBACKS

. In this paper, we investigate the nonlinear wave equation in a bounded domain with a time-varying delay term in the weakly nonlinear inter- nal feedback The asymptotic behavior of solutions is studied by using an appropriate Lya- punov functional. Moreover, we extend and improve the previous results in the literature.


1.
Introduction. This paper is concerned with the decay properties of solutions for the initial boundary value problem of a nonlinear wave equation of the form g(t − s)Lu(s)ds +µ 1 ψ(u t (x, t)) + µ 2 ψ(u t (x, t − τ (t))) = 0, in Ω×]0, +∞[, u(x, t) = 0, on Γ×]0, +∞[, u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) in Ω, u t (x, t − τ (0)) = f 0 (x, t − τ (0)), in Ω×]0, τ (0)[, (1) where Ω is a bounded domain in IR n , (n ∈ IN * ) with a smooth boundary ∂Ω = Γ. Moreover, τ (t) > 0 is a time-varying delay and µ 1 , µ 2 are positive real numbers. The initial datum (u 0 ; u 1 ; f 0 ) belongs to a suitable space, where Lu = −div(A∇u) = − N i,j=1 a i,j (x) ∂u ∂xi and A = (a i,j (x)) is a matrix that will be specified later. In the absence of the delay time (µ 2 = 0), the problem of existence and the decay of the energy has been extensively studied by several authors and many energy estimates have been derived for arbitrary growing feedbacks (see [3], [5], [6], [9], [12], [13], [17], [23]). The decay rate of the energy (when t goes to infinity) depends on the function H which represents the growth at the origin of ψ. Time delay is the property of a physical system by which the response to an applied force is delayed in its effect (see [25]). Time delays so often arise in many physical, chemical, biological and economical phenomena. In recent years, the control of PDEs with time delay effects has become an intensive area of research ( see [1], [26], [28]) and the references therein. In [7], the authors showed that a small delay in a boundary control could turn such well-behaved hyperbolic system into a wild one and therefore, delay becomes a source of instability. However, sometimes it also can improve the performance of the systems (see [26]). In order to stabilize a hyperbolic system involving input delay terms, additional control terms are necessary (see [18], [19], [27]). For instance in [18] the authors studied the wave equation with linear internal damping term with constant delay (ψ linear, τ (t) = constant in the problem (1)). They 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 (1) will be instable if µ 2 ≥ µ 1 . The main approach used in [18] is an observability inequality obtained with a Carleman estimate. The same results were obtained if both the damping and the delay are acting in the boundary. We also recall the result by C.Q. Xu, S.P. Yung and L.K. Li [27], where the authors proved a result similar to the one in [18] for the one-space dimension by adopting the spectral analysis approach. The case of time-varying delay in the wave equation has been investigated recently by S. Nicaise, J. Valein and E. Fridman [22] in one-space dimension and in the linear case (ψ linear in problem (1)) and proved an exponential stability result under the condition where the constant d satisfies In [21] S. Nicaise, C. Pignotti and J. Valein extended the above result to higherspace dimension and established an exponential decay results. Based on the previous works, our purpose in this paper is to give an energy decay estimate of the solution to the problem (1) for a weakly nonlinear damping and in the presence of a time-varying delay term by using a suitable energy and Lyapunov functionals and some properties of convex functions. These arguments of convexity were introduced and developed by I. Lasiecka et al. ([4], [13] ) and used by W.J. Liu, E. Zuazua [15] and M. Eller et al [8].
2. Preliminary results. In this section, we present some material for the proof of our main results. (A 1 ) : The matrix A = (a i,j (x)), where a i,j ∈ C 1 (Ω), is symmetric and there exists a constant a 01 > 0 such that for all x ∈ Ω and δ = (δ 1 , δ 2 , ....., δ N ) ∈ IR N , we have where and there exists a non-increasing differentiable function : ζ : IR + → IR + such that g (t) ≤ −ζ(t)g(t). where where τ 0 and τ 1 are two positive constants. (A 5 ) : The weight of dissipation and the delay satisfy We now state some Lemmas needed later.
Like in [18] we introduce the auxiliary unknown Then, we have Therefore, problem (1) is equivalent to Let ξ be a positive constant such that ξ satisfies We define the energy associated to the solution of the problem (13) by such that α(t) = ξζ(t).
In this way the proof of Lemma 2.2 is completed ENERGY DECAY OF SOLUTIONS 497 3. Asymptotic behavior. In this section, we prove the energy decay result by constructing a suitable Lyapunov functional. We denote by c various positive constants which may be different at variant occurrences. Now we define the following functional where and We need also the following Lemmas Lemma 3.1. Let (u, z) be a solution of problem (13). Then there exists two positive constants λ 1 , λ 2 such that for M sufficiently large.
Proof. By applying the Hölder inequality and Young's inequality and Lemma 2.2, we easily see that and hence it follows from (32) that ∀c > 0 we have Hence, combining (35)-(37) we deduce Finally, we get where c 5 = max(c 1 , c 2 , c 3 , c 4 ). From the definition of E(t) and selecting M sufficiently large, we obtain Proof. We take the derivative of φ(t). It follows from (30) that using the problem (13), then we have following the same idea in [31], yields by applying Hölder and Young's inequalities for the forth and fifth term in (43) for any δ > 0 we get and Using (44)-(46) then (43) becomes This completes the proof. Lemma 3.3. Let (u, z) be the solution of (13). Then ϕ(t) satisfies for any δ > 0 Proof. Now Taking the derivatives of ϕ(t) and using the problem (13), we obtain Using Young's inequality and the embedding H 1 0 (Ω) → L 2 (Ω), we infer Next, we will estimate the right hand side of (49). Applying the Hölder inequality and Young's inequality and the assumptions (A 1 ) − (A 2 ), we have for any t 0 > 0 Invoking (52) we get the following estimates and and A substitution of (50)-(55) into (49) yields This completes the proof.
Proof. Differentiating (32) with respect to t and using the second equation in (13), we have The proof is hence complete.
A simple integration over (0, t) yields exploiting the fact that H −1 1 is decreasing, we infer the equivalence of L(t), F 1 (t) and E(t) yields the estimate Which completes the proof.