LONGTIME DYNAMICS FOR A TYPE OF SUSPENSION BRIDGE EQUATION WITH PAST HISTORY AND TIME DELAY

. In this paper, we investigate a suspension bridge equation with past history and time delay eﬀects, deﬁned in a bounded domain Ω of R N . Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has ﬁnite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.


(Communicated by Alain Miranville)
Abstract. In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain Ω of R N . Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.
1. Introduction. In this paper, we investigate the following suspension bridge equation with past history and time delay where Ω is a bounded domain in R N with sufficiently smooth boundary ∂Ω, α is a positive constant, µ is the memory kernel to be stated later, τ > 0 is a time delay, µ 1 > 0 and µ 2 is a real number, k is spring constant, u + = max{u, 0} denotes the positive part of u, the term −ku + models a restoring force due to the cables, and the initial data (u 0 , u 1 , j 0 ) belong to a suitable function space. From the physics point of view, the suspension bridge equations describe the transverse deflection of the roadbed in the vertical plane. The suspension bridge equations were introduced by Lazer and McKenna [13] as new problems in the field of nonlinear analysis. In the absence of memory and delay, that is when µ(s) = 0 and µ 2 = 0 in (1.1), the problem (1.1) has been extensively studied and many results concerning the well-posedness and the global attractors can be founded (see [1,16,21,22,26] and the references therein). An [1] established the well-posedness of the weak solution and the decay rate of the solution. Ma and Zhong [16] proved the existence of global attractors in energy space, later the authors [26] improved the results of [16] by showing the existence of strong solutions and the regularity of the global attractors. Park and Kang [21,22] proved the existence of pullback attractor for the non autonomous suspension bridge equations and the existence of global attractors for the suspension bridge equations with nonlinear damping, respectively.
In recent years, the evolution equations with memory effects have been studied by many authors, there are many results on this aspect, we do not list the references here. Related to the suspension bridge equations with memory term, we can mention the work of Kang [10](for N = 2) and [11], where the following equation was considered In these papers, Kang estimated the well-posedness and the long-time behavior of the suspension bridge equation when the damping mechanism is given by the memory.
On the other hand, time-delay effect often appears in many applications depending not only on the present state but also on some past occurrences. The time delay effect is sometimes unavoidable, and may be a source of instability. For example, it was shown in [20,24] that an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used. Nicaise and Pignotti [20] considered the wave equation with time delay They obtained that the energy of the problem decays exponentially under the condition 0 < µ 2 < µ 1 , later they extended the result to the time varying delay case in [19]. We also refer the reader to [24], where the authors obtained the same results as in [20] for the one dimension space by use of the spectral analysis approach.
Kirane and Said-Houari [12] discussed the following viscoelastic wave equation with time delay with suitable initial-boundary value conditions. They obtained the well-posedness and the energy decay of solutions for the problem (1.2) under the restriction 0 < µ 2 ≤ µ 1 . Later, Dai and Yang [7] improved the result of [12] under some weaker conditions. Moreover, Yang [25] studied the following Euler-Bernoulli viscoelastic equation with time delay He also obtained the well-posedness and the energy decay even in the case µ 1 = 0 and |µ 2 | is sufficiently small. Here, we also refer the interesting paper Liu and Zhang [14] , where they considered the equation (1.2) with past history and adding the source term f (u) and the well-posedness and the exponential stability were obtained. Recently, Park [23] considered the suspension bridge equation with time delay as following Under some conditions, the existence of global attractor which has finite fractal dimension for (1.3) was proved. But, there are not many researches on attractors especially the exponential attractors for other delayed system, it is worth mentioning the related researches [8,15]. But, to our best knowledge, there is no work on suspension equations with both past history and time delay. Motivated by the above-mentioned results, in the present paper, we investigate the existence of longtime dynamics for the problem. We obtain the quasi-stability of the system and the global attractor with finite fractal dimension which can be characterized by the unstable manifold of the set of stationary solution. The existence of exponential attractor is also established. We also prove the exponential decay of the system. We end this section by introducing two new variable. Firstly, as in Araújo et al. [2] which comes from the argument of Dafermos [6], we shall add a new variable η t to the system with past history, namely By differentiation we obtain We set for simplicity α− ∞ 0 µ(s)ds = 1. Secondly, following the framework proposed in [20], we introduce the new variable Hence problem (1.1) can be transformed into the equivalent system with boundary conditions 6) and the initial conditions ∈ Ω × (0, 1).

2.
Assumptions and the main results. In this section, we present some assumptions and state the main results. We first give some notations about function spaces. We denote the norm in the space X by · X . As usual, we denote the scalar product in L 2 (Ω) by (·, ·) and L p (Ω) norm by · p respectively. In the case p = 2 we write · instead of · 2 . Let , with the inner product and norm respectively. With respect to the relative displacement η as a new variable, we introduce the following µ-weighted Hilbert space endowed with the inner product and norm respectively. Finally, we introduce the phase space for the trajectory solutions equipped with the norm for all U = (u, v, η, z) ∈ H. Let λ 1 be the best constant in the Poincaré's inequality In this paper, we will omit x and t in the functions of x and t if there is no ambiguity, c and c i are used to denote generic positive constants. Next, we will give some assumptions used in this paper.
(H4) With respect to the memory kernel, we assume that (2.6) and there exist µ 0 , δ > 0 such that and µ (s) ≤ −δµ(s), ∀s ∈ R + . In order to establish the main results of this paper, we first state the existence result.
The proof can be obtained by the combining the arguments of [10,11,15,23].
Remark 2. The well-posedness of the problem (1.5)-(1.7) defines the evolution operator is the weak solution corresponding to initial data (u 0 , u 1 , η 0 , z 0 ). S(t) satisfies the semigroup properties and define a nonlinear C 0 -semigroup, which is locally Lipshitz continuous on H. Then, the long-time dynamic of the problem (1.5)-(1.7) can be studied by the continuous dynamical system (H, S(t)).
In the end of this section, we shall give the main results of this paper.

GONGWEI LIU, BAOWEI FENG AND XINGUANG YANG
Moreover, if the nonlinear source term f satisfies the further assumption 9) and no external force is acting, we can obtain the uniform exponential decay of the associated energy.
Theorem 2.3. Let (H1)-(H4) hold with h = 0 and m f = 0, f also satisfy (2.9) and 0 < k < νλ 1 . Then, for any initial data U ∈ H, the following decay estimate holds for some positive constants K and w with K dependent of U and w independent of U .
3. Nonlinear dynamical systems. In this section, for sake of completeness and further references we collect several known results in the theory of nonlinear dynamical systems. They can be found in, for instance, Babin and Vishik [3]. Below we follow more closely the book by Chueshov and Lasiecka [5].
A compact set A ⊂ H is a global attractor for a dynamical system (H, S(t)), if it is fully invariant and uniformly attracting, that is S(t)A = A for all t ≥ 0, and for every bounded subset B ⊂ H, where dist H is the Hausdorff semidistance in H. Given a compact set M in a metric space X, the fractal dimension of M is defined by where N (M,ε) is the minimal number of closed balls with radius ε > 0 which covers M . Define the unstable manifold M + (Y ) emanating from the set Y ⊂ H such that there exists a full trajectory γ = {z(t) : t ∈ R} with the properties We recall the properties of the gradient systems. A dynamical system (H, S(t)) is called a gradient system if there exists a strict Lyapunov function for (H, S(t)) on the whole phase space H, that is, (a) a continuous functional Φ(z) such that the function Now, we give some well-known results on the existence and structure of global attractors, see for instance, Chueshov and Lasiecka [5]. We recall the concept of quasi-stability. Let X, Y, Z be three reflexive Banach spaces with X compactly embedded in Y and put H = X × Y × Z, considering the dynamical system (H, S(t)) given by an evolution operator where the functions u and ξ have the regularity Then (H, S(t)) is called quasi-stable on a set B ⊂ H, if there exists a compact seminorm n X on X (i.e. if x j → 0in X one has n X (x j ) → 0) and nonnegative scalar functions a, b, c, with a, c locally bounded in [0, ∞) and b ∈ L 1 (R + ) satisfying lim t→∞ b(t) = 0, such that and for any z 1 , z 2 ∈ B.
For quasi-stable systems, we state the following theorem, cf. [5], Prop.7.9.4 and Thm. 7.9.6. Theorem 3.2. Let (H, S(t)) be a dynamical system given by (3.1) and satisfying (3.2) and quasi-stable on positively invariant bounded subsets of H. Then (H, S(t)) is asymptotically smooth and its compact global attractor, if any, has finite fractal dimension.

4.
Global attractor and its fractal dimension. In this section, we will use the abstract results given in Section 3 to prove Theorem 2.2. We prove that the dynamical system (H, S(t)) is gradient and quasi-stable on bounded positively invariant sets.
First, we define the energy functional of solutions for problem (1.5)-(1.7) as where Using integration by substitution s = t − ρτ, we obtain It follows from (2.1) and (2.5) that Then, we deduce that Hence, (4.4) yields that We will prove Theorem 2.2 in two parts.
Step 1. Proceeding as in the proof of Lemma 4.1, we obtain Noticing that |u + −ũ + | ≤ |u −ũ| and ||w|| 2 ≤λ||w|| 2 2(p+1) , for any δ 0 > 0, we have Using the generalized Hölder's inequality with p 2(p+1) + 1 2(p+1) + 1 2 = 1, assumption (2.2) and Young's inequality, we have here and after C(.) denotes a generic constant depends on the variable, maybe different from line to line. Substituting these estimates into (4.14), we have On the other hand, by the similar argument as [23], we can show that there exist positive constants α 1 and α 2 such that (4.16) Step 2. Let us define the functional where M > 0, ε > 0 are constants to be determined later and Then, we can show that there exist positive constants α 3 and α 4 such that Indeed, it follows from Young's inequality and Poincaré's that Hence there exists a constant m 0 > 0 such that Then we can obtain (4.17) with α 3 = M − m 0 and α 4 = M + m 0 by choosing M > 0 such that M − m 0 > 0.
Step 3. We show the estimate of φ w (t). Taking the derivative of φ w (t), using Eq. (4.10), subtracting and adding E w (t), we can get As in the step 1, we can get the following estimates

GONGWEI LIU, BAOWEI FENG AND XINGUANG YANG
for any δ 1 > 0. Inserting these estimates into (4.18), we get Step 4. We show the estimate of ψ w (t). Taking the derivative of ψ w (t), using Eq. (4.10), we get Noticing the procedure used in Step 3, we can deduce the following estimates for any For the last term of (4.20), we have Inserting the above estimates into (4.20), we get Step 5. First, from (2.8), we have It follows from (4.15), (4.19) and (4.21) that Now, we fix δ 2 < 2µ 0 , and then take δ 1 > 0 sufficiently small such that For such δ 1 , we choose ε > 0 sufficiently small such that For fixed δ 1 and ε, we choose δ 2 sufficiently small such that Then we take M sufficiently large such that the coefficients of ∞ 0 µ (s) ζ t (s) 2 V ds, w t 2 and q(1, t) 2 are all positive. Hence, we arrive at G (t) ≤ −εE w (t) + C(B) w 2 2(p+1) . Combining the above inequality and the right hand side of (4.17), we obtain and so Using (4.17) again, we have Since w, w t , ζ t , q Hence, using (4.16), by renaming the constants, we can obtain (4.9). This completes the proof. Let B be a bounded positively invariant set of (H, S(t)). Let S(t)U 0 = (u, u t , η t , z) and S(t)Ũ 0 = (ũ,ũ t ,η t ,z) be the weak solution of problem (1.5)-(1.7) corresponding to the initial data U 0 = (u 0 , u 1 , η 0 , z 0 ) ∈ B andŨ 0 = (ũ 0 ,ũ 1 ,η 0 ,z 0 ) ∈ B, respectively. We consider the semi-norm given by By using the compact embedding V → → L 2(p+1) , we have that n V (·) is compact in V . Then, from (4.9), we can see that Finally, it is easy to see that b ∈ L 1 R + with lim t→∞ b(t) = 0 and c(t) is locally bounded on [0, ∞). So, stabilization inequality (3.4) holds. Therefore, the dynamical system (H, S(t) is quasi-stable on bounded positively invariant set B ⊂ H.
Proof of Theorem 2.2. From Theorem 3.2 and Lemma 4.5, we see that the dynamical system (H, S(t)) is asymptotically smooth. Then by using the properties of gradient system (Lemma 4.1, 4.2 4.3), we can get that all the assumptions of Theorem 3.1 are satisfied. Hence (H, S(t)) has a compact global attractor given by A = M + (N ). Moreover, the dynamical system (H, S(t)) is quasi-stable on the attractor A (Lemma 4.5), then by applying again Theorem 3.2, we know that the attractor A has finite fractal dimension. The proof is complete. 5. Exponential attractor. In this section, we shall study the existence of exponential attractor to problem (1.5)-(1.7), and the main result is given in the following theorem.
Theorem 5.1. Assume the assumptions of Theorem 2.2 hold, the corresponding dynamical system (H, S(t) possesses a generalized exponential attractor. More precisely, for any given θ ∈ (0, 1], there exists a generalized exponential attractor A exp, θ ⊂ H, with finite fractal dimension in the extended spaceH −θ which is defined as the interpolation of where V denote the topological dual of V and Before proceeding, we first briefly introduce some basic theorems to exponential attractor. An exponential attractor of a dynamical system (H, S(t)) is a compact set A exp ⊂ H, that enjoys three characteristic properties: (i) it has finite fractal dimension, (ii) it is positively invariant, (iii) it attracts exponentially fast the trajectories from any bounded set of initial data. That is, for any bounded set D ⊂ H, there exist positive constants t D , C D , and γ D such that In this paper, we consider the concept of generalized exponential attractor as presented in Chueshov and Lasiecka [5], where the set A exp has finite fractal dimension in an extended phase spaceH ⊃ H. Thus we can consider the exponential attractors in weak phase spaces. In addition, we can show the existence of generalized exponential attractor if the dynamical system is quasi-stable.
To prove Theorem 5.1, we use the following abstract result given in Chueshov and Lasiecka [5] Theorem 7.9.9. Proposition 1. Let (H, S(t)) be a dynamical system satisfying (3.1) and (3.2) and quasi-stable on some bounded absorbing set B . In addition assume there exists an extended spaceH ⊇ H such that, for any T > 0.
where C BT > 0 and γ ∈ (0, 1] are constants. Then the dynamical (H, S(t)) has a generalized exponential attractor A exp ⊂ H with finite fractal dimension in H.
Proof of Theorem 5.1. Now let us take B = {U |Φ(U ) ≤ R}, where Φ is the strict Lyapunov functional considered in Lemma 4.1. Then we can have that for R large B is a positively invariant bounded absorbing set, which implies that the system is quasi-stable on set B. On the other hand, for the solution U (t) = (u(t), u t (t), η t , z(ρ, t)) ∈ H, we infer from the system (1.5)-(1.7) that u t (t), u tt (t), η t t , z t ∈ L 2 loc (R + ,H −1 ), Then for solution U (t) with initial data y = U (0) ∈ B, we conclude from the positive invariance of B that, for any T > 0, where C BT is a positive constant. This shows that for any y ∈ B the map t → S(t)y is 1 2 -Hölder continuous in the extended phase spaceH −1 , then Proposition 5.1 guarantees the existence of a generalized exponential attractor with finite fractal dimension inH −1 .
By the same argument in [4,9] using interpolation theorem, we can obtain the existence of exponential attractors inH −θ with θ ∈ (0, 1). The proof is complete.
6. Exponential decay. In this section, we study the exponential decay property of solution to problem (1.5)-(1.7) with h = 0. The energy functional corresponding to this system is defined as (4.1) with h = 0. Now, let us define a Lyapunov functional L(t) = E(t) + ε 1 ϕ(t) + ε 2 ψ(t) where ε 1 and ε 2 are two positive numbers to be determined later.
Proof of Theorem 2.3. To achieve our goal, we need the estimates about E (t), φ (t) and ψ (t). Some calculations are similar as the proof Lemma 4.4, Here, we only give the sketch and the different calculations.
Step 1. Taking the time derivative of φ(t), using Eq. (1.5), and integrating by part and subtracting and adding E(t), we can get where we also use −k