Long-Term Damped Dynamics of the Extensible Suspension Bridge

Abstract

This work is focused on the doubly nonlinear equation 𝜕_𝑡𝑡𝑢+𝜕_{𝑥𝑥𝑥𝑥}𝑢+(𝑝−‖𝜕_𝑥𝑢‖²_𝐿2(0,1))𝜕_𝑥𝑥𝑢+𝜕_𝑡𝑢+𝑘²𝑢⁺=𝑓, whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness 𝑘². When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load 𝑝 and stiffness 𝑘². For a general external source 𝑓, we prove the existence of bounded absorbing sets. When 𝑓 is time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.

1. Introduction

1.1. The Model Equation

In this paper, we scrutinize the longtime behavior of a nonlinear evolution problem describing the damped oscillations of an extensible elastic bridge of unitary natural length suspended by means of flexible and elastic cables. The model equation ruling its dynamics can be derived from the standard modeling procedure, which relies on the basic assumptions of continuous distribution of the stays' stiffness along the girder and of the dominant truss behavior of the bridge (see, e.g., [1]).

In the pioneer papers by McKenna and coworkers (see [2–4]), the dynamics of a suspension bridge is given by the well-known damped equation

𝜕_𝑡𝑡𝑢+𝜕_{𝑥𝑥𝑥𝑥}𝑢+𝜕_𝑡𝑢+𝑘²𝑢⁺=𝑓, (1.1) where 𝑢=𝑢(𝑥,𝑡)∶[0,1]×ℝ→ℝ accounts for the downward deflection of the bridge in the vertical plane, and 𝑢⁺ stands for its positive part, namely,

𝑢⁺=𝑢if𝑢≥0,0if𝑢<0. (1.2) Our model is derived here by taking into account the midplane stretching of the road bed due to its elongation. As a consequence, a geometric nonlinearity appears into the bending equation. This is achieved by combining the pioneering ideas of Woinowsky-Krieger on the extensible elastic beam [5] with (1.1). Setting for simplicity all the positive structural constants of the bridge equal to 1, we have

𝜕_𝑡𝑡𝑢+𝜕_{𝑥𝑥𝑥𝑥}𝑢+(𝑝−‖𝜕_𝑥𝑢‖²_𝐿2(0,1))𝜕_𝑥𝑥𝑢+𝜕_𝑡𝑢+𝑘²𝑢⁺=𝑓, (1.3) where 𝑓=𝑓(𝑥,𝑡) is the (given) vertical dead load distribution. The term −𝑘²𝑢⁺ models a restoring force due to the cables, which is different from zero only when they are being stretched, and 𝜕_𝑡𝑢 accounts for an external resistant force linearly depending on the velocity. The real constant 𝑝 represents the axial force acting at the ends of the road bed of the bridge in the reference configuration. Namely, 𝑝 is negative when the bridge is stretched, positive when compressed.

As usual, 𝑢 and 𝜕_𝑡𝑢 are required to satisfy initial conditions as follows:

𝑢(𝑥,0)=𝑢₀𝜕(𝑥),𝑥∈[0,1],_𝑡𝑢(𝑥,0)=𝑢₁(𝑥),𝑥∈[0,1]. (1.4) Concerning the boundary conditions, we consider here the case when both ends of the bridge are pinned. Namely, for every 𝑡∈ℝ, we assume

𝑢(0,𝑡)=𝑢(1,𝑡)=𝜕_𝑥𝑥𝑢(0,𝑡)=𝜕_𝑥𝑥𝑢(1,𝑡)=0. (1.5) This is the simpler choice. However, other types of boundary conditions with fixed ends are consistent with the extensibility assumption as well; for instance, when both ends are clamped, or when one end is clamped and the other one is pinned. We address the reader to [6] for a more detailed discussion. Assuming (1.5), the domain of the differential operator 𝜕_{𝑥𝑥𝑥𝑥} acting on 𝐿²(0,1) is

𝒟(𝜕_{𝑥𝑥𝑥𝑥})={𝑤∈𝐻⁴(0,1)∶𝑤(0)=𝑤(1)=𝜕_𝑥𝑥𝑤(0)=𝜕_𝑥𝑥𝑤(1)=0}. (1.6) This operator is strictly positive selfadjoint with compact inverse, and its discrete spectrum is given by 𝜆_𝑛=𝑛⁴𝜋⁴, 𝑛∈ℕ. Thus, 𝜆₁=𝜋⁴ is the smallest eigenvalue. Besides, the peculiar relation

(𝜕_{𝑥𝑥𝑥𝑥})^1/2=−𝜕_𝑥𝑥 (1.7) holds true, with Dirichlet boundary conditions and

𝒟(−𝜕_𝑥𝑥)=𝐻²(0,1)∩𝐻¹₀(0,1). (1.8) Hence, if pinned ends are considered, the initial-boundary value problem (1.3)–(1.5) can be described by means of a single operator 𝐴=𝜕_{𝑥𝑥𝑥𝑥}, which enters the equation at the powers 1 and 1/2. Namely,

𝜕_𝑡𝑡𝑢+𝐴𝑢+𝜕_𝑡𝑢−(𝑝−‖𝑢‖²₁)𝐴^1/2𝑢+𝑘²𝑢⁺=𝑓, (1.9) where ‖⋅‖₁ is the norm of 𝐻¹₀(0,1). This fact is particularly relevant in the analysis of the critical buckling load 𝑝_𝑐, that is, the magnitude of the compressive axial force 𝑝>0 at which buckled stationary states appear.

As we shall show throughout the paper, this model leads to exact results which are rather simple to prove and, however, are capable of capturing the main behavioral dynamic characteristics of the bridge.

1.2. Earlier Contributions

In recent years, an increasing attention was paid to the analysis of buckling, vibrations, and postbuckling dynamics of nonlinear beam models, especially in connection with industrial applications [7, 8] and suspension bridges [9, 10]. As far as we know, most of the papers in the literature deal with approximations and numerical simulations, and only few works are able to derive exact solutions, at least under stationary conditions (see, e.g., [11–14]). In the sequel, we give a brief sketch of earlier contributions on this subject.

In the fifties, Woinowsky-Krieger [5] proposed to modify the theory of the dynamic Euler-Bernoulli beam, assuming a nonlinear dependence of the axial strain on the deformation gradient. The resulting motion equation,

𝜕_𝑡𝑡𝑢+𝜕_{𝑥𝑥𝑥𝑥}𝑢+(𝑝−‖𝜕_𝑥𝑢‖²_𝐿2(0,1))𝜕_𝑥𝑥𝑢=0, (1.10) has been considered for hinged ends in the papers [15, 16], with particular reference to well-posedness results and to the analysis of the complex structure of equilibria. Adding an external viscous damping term 𝜕_𝑡𝑢 to the original conservative model, it becomes

𝜕_𝑡𝑡𝑢+𝜕_{𝑥𝑥𝑥𝑥}𝑢+𝜕_𝑡𝑢+(𝑝−‖𝜕_𝑥𝑢‖²_𝐿2(0,1))𝜕_𝑥𝑥𝑢=0. (1.11) Stability properties of the unbuckled (trivial) and the buckled stationary states of (1.11) have been established in [17, 18] and, more formally, in [19]. In particular, if 𝑝<𝑝_𝑐, the exponential decay of solutions to the trivial equilibrium state has been shown. The global dynamics of solutions for a general 𝑝 has been first tackled in [20] and improved in [21], where the existence of a global attractor for (1.11) subject to hinged ends was proved relying on the construction of a suitable Lyapunov functional. In [22] previous results are extended to a more general form of the nonlinear term by virtue of a suitable decomposition of the semigroup first introduced in [6].

A different class of problems arises in the study of vibrations of a suspension bridge. The dynamic response of suspension bridges is usually analyzed by linearizing the equations of motion. When the effects of extensibility of the girder are neglected and the coupling with the main cable motion is disregarded, we obtain the well-known Lazer-McKenna equation (1.1). Free and forced vibrations in models of this type, both with constant and non constant loads, have been scrutinized in [10, 23]. The existence of strong solutions and global attractors for (1.1) has been recently obtained in [24].

In certain cases, Lazer-McKenna's model becomes inadequate and the effects of extensibility of the girder have to be taken into account. This can be done by introducing into the model equation (1.1) a geometric nonlinear term like that appearing in (1.10). Such a term is of some importance in the modeling of cable-stayed bridges (see, e.g., [1, 25]), where the elastic suspending cables are not vertical and produce a well-defined axial compression on the road bed.

Several studies have been devoted to the nonlinear vibrational analysis of mechanical models close to (1.3). Abdel-Ghaffar and Rubin [9, 26] presented a general theory and analysis of the nonlinear free coupled vertical-torsional vibrations of suspension bridges. They developed approximate solutions by using the method of multiple scales via a perturbation technique. If torsional vibrations are ignored, their model reduces to (1.3). Exact solutions to this problem, at least under stationary conditions, have been recently exhibited in [14].

1.3. Outline of the Paper

In the next Section 2, we formulate an abstract version of the problem. We observe that its solutions are generated by a solution operator 𝑆(𝑡), which turns out to be a strongly continuous semigroup in the autonomous case. The existence of an absorbing set for the solution operator 𝑆(𝑡) is proved in Section 3 by virtue of a Gronwall-type Lemma. Section 4 is focused on the autonomous case and contains our main result. Namely, we establish the existence of the regular global attractor for a general 𝑝. In particular, we prove this by appealing to the existence of a Lyapunov functional and without requiring any assumption on the strength of the dissipation term. A characterization of the global attractor is given in terms of the steady states of the system (1.3)–(1.5). First, we proceed with some preliminary estimates and prove the exponential stability of the system provided that the axial force 𝑝 is smaller than 𝑝_𝑐. Finally, the smoothing property of the semigroup generated by the abstract problem is stated via a suitable decomposition first devised in [6].

2. The Dynamical System

In the sequel, we recast problem (1.3)–(1.5) into an abstract setting in order to establish more general results.

Let (𝐻,⟨⋅,⋅⟩,‖⋅‖) be a real Hilbert space, and let 𝐴∶𝒟(𝐴)⋐𝐻→𝐻 be a strictly positive selfadjoint operator with compact inverse. For 𝑟∈ℝ, we introduce the scale of Hilbert spaces generated by the powers of 𝐴

𝐻^𝑟=𝒟(𝐴^𝑟/4),⟨𝑢,𝑣⟩_𝑟=⟨𝐴^𝑟/4𝑢,𝐴^𝑟/4𝑣⟩,‖𝑢‖_𝑟=‖𝐴^𝑟/4𝑢‖. (2.1) When 𝑟=0, the index 𝑟 is omitted. The symbol ⟨⋅,⋅⟩ will also be used to denote the duality product between 𝐻^𝑟 and its dual space 𝐻^−𝑟. In particular, we have the compact embeddings 𝐻^𝑟+1⋐𝐻^𝑟, along with the generalized Poincaré inequalities

𝜆₁‖𝑢‖⁴_𝑟≤‖𝑢‖⁴_𝑟+1,∀𝑢∈𝐻^𝑟+1, (2.2) where 𝜆₁>0 is the first eigenvalue of 𝐴. Finally, we define the product Hilbert spaces

ℋ^𝑟=𝐻^𝑟+2×𝐻^𝑟. (2.3) For 𝑝∈ℝ, we consider the following abstract Cauchy problem on ℋ in the unknown variable 𝑢=𝑢(𝑡):

𝜕_𝑡𝑡𝑢+𝐴𝑢+𝜕_𝑡𝑢−(𝑝−‖𝑢‖²₁)𝐴^1/2𝑢+𝑘²𝑢⁺=𝑓(𝑡),𝑡>0,𝑢(0)=𝑢₀,𝜕_𝑡𝑢(0)=𝑢₁. (2.4) Problem (1.3)–(1.5) is just a particular case of the abstract system (2.4), obtained by setting 𝐻=𝐿²(0,1) and 𝐴=𝜕_{𝑥𝑥𝑥𝑥} with the boundary condition (1.5).

The following well-posedness result holds.

Proposition 2.1. Assume that 𝑓∈𝐿¹_loc(0,𝑇;𝐻). Then, for all initial data 𝑧=(𝑢₀,𝑢₁)∈ℋ, problem (2.4) admits a unique solution (𝑢(𝑡),𝜕_𝑡𝑢(𝑡))∈𝒞(0,𝑇;ℋ), (2.5) which continuously depends on the initial data.

We omit the proof of this result, which is based on a standard Galerkin approximation procedure (see, e.g., [15, 17]), together with a slight generalization of the usual Gronwall lemma. In particular, the uniform-in-time estimates needed to obtain the global existence are exactly the same as those we use in proving the existence of an absorbing set.

In light of Proposition 2.1, we define the solution operator

𝑆(𝑡)∈𝒞(ℋ,ℋ),∀𝑡≥0, (2.6)

as

𝑧=(𝑢₀,𝑢₁)↦𝑆(𝑡)𝑧=(𝑢(𝑡),𝜕_𝑡𝑢(𝑡)). (2.7) Besides, for every 𝑧∈ℋ, the map 𝑡↦𝑆(𝑡)𝑧 belongs to 𝒞(ℝ⁺,ℋ). Actually, it is a standard matter to verify the joint continuity

(𝑡,𝑧)↦𝑆(𝑡)𝑧∈𝒞(ℝ⁺×ℋ,ℋ). (2.8)

Remark 2.2. In the autonomous case, namely, when 𝑓 is time-independent, the semigroup property 𝑆(𝑡+𝜏)=𝑆(𝑡)𝑆(𝜏) (2.9) holds for all 𝑡,𝜏≥0. Thus, 𝑆(𝑡) is a strongly continuous semigroup of operators on ℋ which continuously depends on the initial data: for any initial data 𝑧∈ℋ, 𝑆(𝑡)𝑧 is the unique weak solution to (2.4), with related norm given by ℰ(𝑧)=‖𝑧‖²_ℋ=‖𝑢‖²₂+‖𝑣‖². (2.10)

For any 𝑧=(𝑢,𝑣)∈ℋ, we define the energy corresponding to 𝑧 as

1𝐸(𝑧)=ℰ(𝑧)+2(‖𝑢‖²₁−𝑝)²+𝑘²‖𝑢⁺‖², (2.11) and, abusing the notation, we denote 𝐸(𝑆(𝑡)𝑧) by 𝐸(𝑡) for each given initial data 𝑧∈ℋ. Multiplying the first equation in (2.4) by 𝜕_𝑡𝑢, because of the relation

𝑘²⟨𝑢⁺,𝜕_𝑡𝑘𝑢⟩=²2dd𝑡(‖𝑢⁺‖²), (2.12)

we obtain the energy identity

dd𝑡𝐸+2‖𝜕_𝑡𝑢‖²=2⟨𝜕_𝑡𝑢,𝑓⟩. (2.13) In particular, for every 𝑇>0, there exists a positive increasing function 𝒬_𝑇 such that

𝐸(𝑡)≤𝒬_𝑇(𝐸(0)),∀𝑡∈[0,𝑇]. (2.14)

3. The Absorbing Set

It is well known that the absorbing set gives a first rough estimate of the dissipativity of the system. In addition, it is the preliminary step to scrutinize its asymptotic dynamics (see, for instance, [27]). Here, due to the joint presence of geometric and cable-response nonlinear terms in (2.4), a direct proof of the existence of the absorbing set via explicit energy estimates is nontrivial. Indeed, the double nonlinearity cannot be handled by means of standard arguments, as either in [3] or in [24]. Dealing with a given time-dependent external force 𝑓 fulfilling suitable translation compactness properties, a direct proof of the existence of an absorbing set is achieved here by means of a generalized Gronwall-type lemma devised in [28].

An absorbing set for the solution operator 𝑆(𝑡) (referred to the initial time 𝑡=0) is a bounded set 𝔅_ℋ⊂ℋ with the following property: for every 𝑅≥0, there is an entering time 𝑡_𝑅≥0 such that

_{𝑡≥𝑡𝑅}𝑆(𝑡)𝑧⊂𝔅_ℋ, (3.1) whenever ‖𝑧‖_ℋ≤𝑅. In fact, we are able to establish a more general result.

Theorem 3.1. Let 𝑓∈𝐿^∞(ℝ⁺,𝐻), and let 𝜕_𝑡𝑓 be a translation bounded function in 𝐿²_loc(ℝ⁺,𝐻⁻²), that is, sup_𝑡≥0𝑡^𝑡+1‖𝜕_𝑡𝑓(𝜏)‖²₋₂d𝜏=𝑀<∞. (3.2) Then, there exists 𝑅₀>0 with the following property: in correspondence of every 𝑅≥0, there is 𝑡₀=𝑡₀(𝑅)≥0 such that 𝐸(𝑡)≤𝑅₀,∀𝑡≥𝑡₀, (3.3) whenever 𝐸(0)≤𝑅. Both 𝑅₀ and 𝑡₀ can be explicitly computed.

We are able to establish Theorem 3.1, leaning on the following lemma.

Lemma 3.2 (see [28, Lemma2.5]). Let Λ∶ℝ⁺→ℝ⁺ be an absolutely continuous function satisfying, for some 𝑀≥0, 𝜀>0, the differential inequality dd𝑡Λ(𝑡)+𝜀Λ(𝑡)≤𝜑(𝑡), (3.4) where 𝜑∶ℝ⁺→ℝ⁺ is any locally summable function such that sup_𝑡≥0𝑡^𝑡+1𝜑(𝜏)d𝜏≤𝑀. (3.5) Then, there exist 𝑅₁>0 and 𝛾>0 such that, for every 𝑅≥0, it follows that Λ(𝑡)≤𝑅₁,∀𝑡≥𝑅^1/𝛾(1+𝛾𝑀)⁻¹, (3.6) whenever Λ(0)≤𝑅. Both 𝑅₁ and 𝛾 can be explicitly computed in terms of 𝑀 and 𝜀.

Proof of Theorem 3.1. Here and in the sequel, we will tacitly use several times the Young and the Hölder inequalities, besides the usual Sobolev embeddings. The generic positive constant 𝐶 appearing in this proof may depend on 𝑝 and ‖𝑓‖_𝐿^∞_(ℝ⁺_,𝐻).
On account of (2.13), by means of the functional ℒ(𝑧)=𝐸(𝑧)−2⟨𝑢,𝑓⟩, (3.7) we introduce the function ℒ(𝑡)=𝐸(𝑡)−2⟨𝑢(𝑡),𝑓(𝑡)⟩, (3.8) which satisfies the differential equality dd𝑡ℒ+2‖𝜕_𝑡𝑢‖²=−2⟨𝑢,𝜕_𝑡𝑓⟩. (3.9) Because of the control 2|⟨𝑢,𝜕_𝑡1𝑓⟩|≤2‖𝑢‖²₂+2‖𝜕_𝑡𝑓‖²₋₂, (3.10) we obtain the differential inequality dd𝑡ℒ+2‖𝜕_𝑡𝑢‖²≤12𝐸+2‖𝜕_𝑡𝑓‖²₋₂. (3.11) Next, we consider the auxiliary functional Υ(𝑧)=⟨𝑢,𝑣⟩ and, regarding Υ(𝑆(𝑡)𝑧) as Υ(𝑡), we have dd𝑡Υ+Υ+‖𝑢‖²₂+(‖𝑢‖²₁−𝑝)²+𝑝(‖𝑢‖²₁−𝑝)+𝑘²‖𝑢⁺‖²−⟨𝑢,𝑓⟩=‖𝜕_𝑡𝑢‖². (3.12) Noting that 12(‖𝑢‖²₁−𝑝)²+𝑝(‖𝑢‖²₁1−𝑝)=2‖𝑢‖⁴₁−12𝑝², (3.13) we are led to d1d𝑡Υ+Υ+2‖𝑢‖⁴₁+‖𝑢‖²₂+𝑘²‖𝑢⁺‖²+12(‖𝑢‖²₁−𝑝)²−⟨𝑢,𝑓⟩=‖𝜕_𝑡𝑢‖²+12𝑝². (3.14) Precisely, we end up with d1d𝑡Υ+Υ+23𝐸≤2‖𝜕_𝑡𝑢‖²+12𝜆₁‖𝑓‖²+12𝑝². (3.15) Finally, we set Λ(𝑧)=ℒ(𝑧)+Υ(𝑧)+𝐶, (3.16) where 𝐶=(2/𝜆₁)‖𝑓‖²+1/2𝜆₁+|𝑝|/2𝜆₁. We first observe that Λ(𝑧) satisfies 121ℰ(𝑧)≤2𝐸(𝑧)≤Λ(𝑧)≤2𝐸(𝑧)+𝑐. (3.17) In order to estimate Λ from below, a straightforward calculation leads to 1Λ(𝑧)≥𝐸(𝑧)−2|⟨𝑢,𝑓⟩|−|Υ(𝑧)|+𝐶≥𝐸(𝑧)−2‖𝑢‖²₂−12‖𝑣‖²−14(‖𝑢‖²₁−𝑝)²−2‖𝑓‖²₋₂−12𝜆₁−|𝑝|2𝜆₁=1+𝐶2‖𝑢‖²₂+12‖𝑣‖²+14(‖𝑢‖²₁−𝑝)²+𝑘²‖𝑢⁺‖²−2‖𝑓‖²₋₂−12𝜆₁−|𝑝|2𝜆₁≥1+𝐶22𝐸(𝑧)−𝜆₁‖𝑓‖²−12𝜆₁−|𝑝|2𝜆₁1+𝐶≥2𝐸(𝑧), (3.18) where we take advantage of 1|Υ(𝑧)|≤‖𝑢‖‖𝑣‖≤₄𝜆₁‖𝑢‖₁1‖𝑣‖≤2‖𝑣‖²+12𝜆₁‖𝑢‖²₁≤12‖𝑣‖²+12𝜆₁(‖𝑢‖²₁−𝑝)+|𝑝|2𝜆₁≤12‖𝑣‖²+14(‖𝑢‖²₁−𝑝)²+|𝑝|2𝜆₁+12𝜆₁. (3.19) The upper bound for Λ can be easily achieved as follows: Λ(𝑧)≤𝐸(𝑧)+‖𝑢‖²₂+‖𝑣‖²+12‖𝑢‖⁴₁+‖𝑓‖²₋₂+132𝜆₁1+𝐶≤2𝐸(𝑧)+𝜆₁‖𝑓‖²+132𝜆₁+𝐶≤2𝐸(𝑧)+𝑐, (3.20) by virtue of 1|Υ(𝑧)|≤‖𝑢‖‖𝑣‖≤₄𝜆₁‖𝑢‖₁‖𝑣‖≤‖𝑣‖²+14𝜆₁‖𝑢‖²₁≤‖𝑣‖²+12‖𝑢‖⁴₁+132𝜆₁. (3.21) Going back to differential equation and making use of (3.9) and (3.15), the function Λ(𝑡)=Λ(𝑆(𝑡)𝑧) satisfies the identity dΛd𝑡Λ+2+Υ2+12‖𝑢‖²₂+12‖𝜕_𝑡𝑢‖²+14‖𝑢‖⁴₁+12𝑘²‖𝑢⁺‖²=−2⟨𝑢,𝜕_𝑡𝑝𝑓⟩+²2, (3.22) and, as a consequence, we obtain the estimate dΛd𝑡Λ+2+12(Υ+‖𝑢‖²₂+‖𝜕_𝑡𝑢‖²+12‖𝑢‖⁴₁+4⟨𝑢,𝜕_𝑡𝑝𝑓⟩)≤²2. (3.23) Now, using (3.21) and (3.10), we have dΛd𝑡Λ+2≤2‖𝜕_𝑡𝑓‖²₋₂+𝑐, (3.24) where 𝑐=1/16𝜆₁+𝑝²/2. Thus, by virtue of (3.2) and (3.17), Lemma 3.2 yields 𝐸(𝑡)≤2Λ(𝑡)≤2𝑅₁(𝑀,𝑐). (3.25)

Remark 3.3. If the set of stationary solutions to (2.4) shrinks to a single element, the subsequent asymptotic behavior of the system becomes quite simple. Indeed, this occurs when 𝑝<𝑝_𝑐=𝜆₁. If this is the case, the only trivial solution exists and is exponentially stable, as it will be shown in Section 4. The more complex and then attractive situation occurs when the set of steady solutions contains a large (possibly infinite) amount of elements. To this end, we recall here that the set of the bridge stationary-solutions (equilibria) has a very rich structure, even when 𝑓=0 (see [14]).

4. The Global Attractor

In the remaining of the paper, we simplify the problem by assuming that the external force 𝑓 is time-independent. In which case, the operator 𝑆(𝑡) is a strongly continuous semigroup on ℋ (see Remark 2.2). Having been proved in Section 3 the existence of the absorbing set 𝔅, we could then establish here the existence of a global attractor by showing that the semigroup 𝑆(𝑡) admits a bounded absorbing set in a more regular space and that it is uniformly compact for large values of 𝑡 (see, e.g., [27, Theorem 1.1]). In order to obtain asymptotic compactness, the 𝛼-contraction method should be employed (see [20] for more details). If applied to (2.4), however, such a strategy would need a lot of calculations and, what is more, would provide some regularity of the attractor only if the dissipation is large enough (see [21]).

Noting that in the autonomous case, problem (2.4) becomes a gradient system, there is a way to overcome these difficulties by using an alternative approach which appeals to the existence of a Lyapunov functional in order to prove the existence of a global attractor. This technique has been successfully adopted in some recent papers concerning some related problems, just as the longterm analysis of the transversal motion of extensible viscoelastic [6] and thermoelastic [29] beams.

We recall that the global attractor 𝒜 is the unique compact subset of ℋ which is at the same time

(i)attracting: lim_𝑡→∞𝛿(𝑆(𝑡)𝔅,𝒜)→0, (4.1) for every bounded set 𝔅⊂ℋ, where 𝛿 denotes the usual Hausdorff semidistance in ℋ,(ii)fully invariant: 𝑆(𝑡)𝒜=𝒜,∀𝑡≥0. (4.2)

We address the reader to the books [20, 27, 30] for a detailed presentation of the theory of attractors.

Theorem 4.1. The semigroup 𝑆(𝑡) acting on ℋ possesses a connected global attractor 𝒜 bounded in ℋ². Moreover, 𝒜 coincides with the unstable manifold of the set 𝒮 of the stationary points of 𝑆(𝑡), namely, 𝒜={𝑧(0)∶𝑧isacomplete(bounded)trajectoryof𝑆(𝑡)∶lim_𝑡→∞‖𝑧(−𝑡)−𝒮‖_ℋ=0}. (4.3)

Remark 4.2. Due to the regularity and the invariance of 𝒜, we observe that 𝑆(𝑡)𝑧 is a strong solution to (2.4) whenever 𝑧∈𝒜.

The set 𝒮 of the bridge equilibria under a vanishing lateral load consists of all the pairs (𝑢,0)∈ℋ such that the function 𝑢 is a weak solution to the equation

𝐴𝑢−(𝑝−‖𝑢‖²₁)𝐴^1/2𝑢+𝑘²𝑢⁺=0. (4.4) In particular, 𝑢 solves the following boundary value problem on the interval [0,1]:

𝜕_{𝑥𝑥𝑥𝑥}𝑢+(𝑏𝜋²−‖𝜕_𝑥𝑢‖²_𝐿2(0,1))𝜕_𝑥𝑥𝑢+𝜅²𝜋⁴𝑢⁺=0,𝑢(0)=𝑢(1)=𝜕_𝑥𝑥𝑢(0)=𝜕_𝑥𝑥𝑢(1)=0, (4.5) where we let 𝑘=𝜅𝜋², 𝜅∈ℝ, and 𝑝=𝑏𝜋², 𝑏∈ℝ. It is then apparent that 𝒮 is bounded in 𝐻²(0,1)∩𝐻¹₀(0,1) for every 𝑏,𝜅∈ℝ.

When 𝜅=0, a general result has been established in [13] for a class of nonvanishing sources. In [11, 12], the same strategy with minor modifications has been applied to problems close to (4.5), where the term 𝑢⁺ is replaced by 𝑢 (unyielding ties).

The set of buckled solutions to problem (4.5) is built up and scrutinized in [14]. In order to have a finite number of solutions, we need all the bifurcation values to be distinct. This occurrence trivially holds when 𝜅=0, because of the spectral properties of the operator 𝜕_{𝑥𝑥𝑥𝑥}. On the contrary, for general values of 𝜅, all critical values “move" when 𝜅 increases, as well as in [12]. Hence, it may happen that two different bifurcation values overlap for special values of 𝜅, in which case they are referred as resonant values.

Assuming that 𝜅=1, for instance, Figure 1 shows the bifurcation picture of solutions in dependence on the applied axial load 𝑝=𝑏𝜋². In particular, 𝑢₀=0 and

Figure 1 

The bifurcation picture for 𝜅=1.

𝑢^±₁(𝑥)=𝐴^±₁sin(𝜋𝑥),𝐴⁻₁=−2(𝑏−1),𝐴⁺₁=2(𝑏−2). (4.6)

4.1. The Lyapunov Functional and Preliminary Estimates

We begin to prove the existence of a Lyapunov functional for 𝑆(𝑡), that is, a function ℒ∈𝐶(ℋ,ℝ) satisfying the following conditions:

(i)ℒ(𝑧)→∞ if and only if ‖𝑧‖_ℋ→∞; (ii)ℒ(𝑆(𝑡)𝑧) is nonincreasing for any 𝑧∈ℋ;(iii)ℒ(𝑆(𝑡)𝑧)=ℒ(𝑧) for all 𝑡>0 implies that 𝑧∈𝒮.

Proposition 4.3. If 𝑓 is time-independent, the functional ℒ defined in (3.7) is a Lyapunov functional for 𝑆(𝑡).

Proof. Assertion (i) holds by the continuity of ℒ and by means of the estimates 123𝐸(𝑧)−𝑐≤ℒ(𝑧)≤2𝐸(𝑧)+𝑐. (4.7) Using (3.9), we obtain quite directly that dd𝑡ℒ(𝑆(𝑡)𝑧)=−2‖𝜕_𝑡𝑢(𝑡)‖²≤0, (4.8) which proves the decreasing monotonicity of ℒ along the trajectories departing from 𝑧. Finally, if ℒ(𝑆(𝑡)𝑧) is constant in time, we have that 𝜕_𝑡𝑢=0 for all 𝑡, which implies that 𝑢(𝑡) is constant. Hence, 𝑧=𝑆(𝑡)𝑧=(𝑢₀,0) for all 𝑡, that is, 𝑧∈𝒮.

The existence of a Lyapunov functional ensures that 𝐸(𝑡) is bounded. In particular, bounded sets have bounded orbits.

Notation 1. Till the end of the paper, 𝑄∶ℝ⁺₀→ℝ⁺ will denote a generic increasing monotone function depending explicity only on 𝑅 and implicity on the structural constants of the problem. The actual expression of 𝑄 may change, even within the same line of a given equation.

Lemma 4.4. Given 𝑓∈𝐻, for all 𝑡>0 and initial data 𝑧∈ℋ with ‖𝑧‖_ℋ≤𝑅, ℰ(𝑡)≤𝑄(𝑅). (4.9)

Proof. Inequality (4.8) ensures that ℒ(𝑡)=ℒ(𝑆(𝑡)𝑧)≤ℒ(𝑧)≤𝑄(𝑅)∀𝑡≥0. (4.10) Moreover, taking into account that ‖𝑢(𝑡)‖²≤1𝜆₁‖𝑢(𝑡)‖²₂≤1𝜆₁ℰ(𝑡), (4.11) we obtain the estimate ℒ(𝑡)≥ℰ(𝑡)−2⟨𝑓,𝑢(𝑡)⟩≥ℰ(𝑡)−2‖𝑓‖²₋₂−12‖𝑢(𝑡)‖²₂≥122ℰ(𝑡)−𝜆₁‖𝑓‖². (4.12) Finally, we have 4ℰ(𝑡)≤2ℒ(𝑡)+𝜆₁‖𝑓‖²4≤2𝑄(𝑅)+𝜆₁‖𝑓‖²=𝑄(𝑅). (4.13)

Lemma 4.5. Let 𝑝<𝜆₁ and ℱ_𝑝(𝑢)=𝐴𝑢−𝑝𝐴^1/2𝑢. Then ⟨ℱ_𝑝(𝑢),𝑢⟩≥𝐶(𝑝)‖𝑢‖²₂, (4.14) where 𝐶(𝑝)=𝑝1,𝑝≤0,(1−𝜆₁),0<𝑝<𝜆₁. (4.15)

Proof. Because of the identity ⟨ℱ_𝑝(𝑢),𝑢⟩=‖𝑢‖²₂−𝑝‖𝑢‖²₁, (4.16) the thesis is trivial when 𝑝≤0. On the other hand, when 0<𝑝<𝜆₁ we have ⟨ℱ_𝑝(𝑢),𝑢⟩=‖𝑢‖²₂−𝑝‖𝑢‖²₁𝑝≥(1−𝜆₁)‖𝑢‖²₂. (4.17)

We are now in a position to prove the following.

Theorem 4.6. When 𝑓=0, the solutions to (1.3)–(1.5) decay exponentially, that is, ℰ(𝑡)≤𝑐₀ℰ(0)𝑒^−𝑐𝑡 (4.18) with 𝑐₀ and 𝑐 being suitable positive constants, provided that 𝑝<𝜆₁.

Proof. Let Φ be the functional 1Φ(𝑧)=𝐸(𝑧)+𝜀Υ(𝑧)−2𝑝², (4.19) where the constant 𝜀=min{𝜆₁𝐶(𝑝),1} (4.20) is positive provided that 𝑝<𝜆₁. In view of applying Lemma 4.5, we remark that Φ=⟨ℱ_𝑝(𝑢),𝑢⟩+‖𝜕_𝑡𝑢‖²+12‖𝑢‖⁴₁+𝜀⟨𝑢,𝜕_𝑡𝑢⟩+𝑘²‖𝑢⁺‖². (4.21) The first step is to prove the equivalence between ℰ and Φ, that is, 𝜀2𝜆₁ℰ≤Φ≤𝑄(‖𝑧‖_ℋ)ℰ. (4.22) By virtue of (2.2), (4.20), and Lemma 4.5, the lower bound is provided by 𝜀Φ≥(𝐶(𝑝)−2𝜆₁)‖𝑢‖²₂𝜀+(1−2)‖𝜕_𝑡𝑢‖²≥𝜀2𝜆₁ℰ. (4.23)
On the other hand, by applying Young inequality and using (2.2), we can write the following chain of inequalities which gives the upper bound of Φ: 𝑘Φ≤(𝐶(𝑝)+²𝜆₁+12𝜆₁)‖𝑢‖²₂𝜀+(1+²2)‖𝜕_𝑡𝑢‖²−𝑝‖𝑢‖²₁+12‖𝑢‖⁴₁𝑘≤(1+𝐶(𝑝)+²𝜆₁+12𝜆₁+𝜀²2)ℰ+‖𝑢‖²₁(12‖𝑢‖²₁−𝑝). (4.24) In particular, from (4.9) and (4.15), we find 𝑘Φ≤(2+²𝜆₁+12𝜆₁+𝜀²2+𝑄(‖𝑧‖_ℋ)𝜆₁)ℰ=𝑄(‖𝑧‖_ℋ)ℰ. (4.25) The last step is to prove the exponential decay of Φ. To this aim, we obtain the identity dd𝑡Φ+𝜀Φ+2(1−𝜀)‖𝜕_𝑡𝑢‖²+𝜀2‖𝑢‖⁴₁+𝜀(1−𝜀)⟨𝜕_𝑡𝑢,𝑢⟩=0, (4.26) where 𝜀 is given by (4.20). Exploiting the Young inequality and (4.22), we have dd𝑡Φ+𝜀Φ+(1−𝜀)‖𝜕_𝑡𝑢‖²≤𝜀²(1−𝜀)4𝜆₁‖𝑢‖²₂≤𝜀(1−𝜀)2Φ, (4.27) from which it follows that dd𝑡Φ+𝜀(1+𝜀)2Φ≤0. (4.28) Letting 𝑐=𝜀(1+𝜀)/2, by virtue of Lemma 3.2 (with 𝑀=0) and (4.22), we have 𝜀2𝜆₁ℰ(𝑡)≤Φ(𝑡)≤Φ(0)𝑒^−𝑐𝑡≤𝑄(‖𝑧‖_ℋ)ℰ(0)𝑒^−𝑐𝑡. (4.29) The thesis follows by putting 𝑐₀=2𝜆₁𝑄(‖𝑧‖_ℋ)/𝜀.

The existence of a Lyapunov functional, along with the fact that 𝒮 is a bounded set, allows us prove the existence of the attractor by showing a suitable (exponential) asymptotic compactness property of the semigroup, which will be obtained by exploiting a particular decomposition of 𝑆(𝑡) devised in [6] and following a general result (see [31, Lemma 4.3]), tailored to our particular case.

4.2. The Semigroup Decomposition

By the interpolation inequality ‖𝑢‖²₁≤‖𝑢‖‖𝑢‖₂ and (2.2), it is clear that

12‖𝑢‖²₂≤‖𝑢‖²₂−𝑝‖𝑢‖²₁+𝛼‖𝑢‖²≤𝑚‖𝑢‖²₂, (4.30) provided that 𝛼>0 is large enough and for some 𝑚=𝑚(𝑝,𝛼)≥1.

Again, 𝑅>0 is fixed and ‖𝑧‖_ℋ≤𝑅. Choosing 𝛼>0 such that (4.30) holds, according to the scheme first proposed in [6], we decompose the solution 𝑆(𝑡)𝑧 into the sum

𝑆(𝑡)𝑧=𝐿(𝑡)𝑧+𝐾(𝑡)𝑧, (4.31) where

𝐿(𝑡)𝑧=(𝑣(𝑡),𝜕_𝑡𝑣(𝑡)),𝐾(𝑡)𝑧=(𝑤(𝑡),𝜕_𝑡𝑤(𝑡)) (4.32) solve the systems

𝜕_𝑡𝑡𝑣+𝐴𝑣+𝜕_𝑡𝑣−(𝑝−‖𝑢‖²₁)𝐴^1/2𝑣+𝛼𝑣=0,(𝑣(0),𝜕_𝑡𝜕𝑣(0))=𝑧,_𝑡𝑡𝑤+𝐴𝑤+𝜕_𝑡𝑤−(𝑝−‖𝑢‖²₁)𝐴^1/2𝑤−𝛼𝑣+𝑘²𝑢⁺=𝑓,(𝑤(0),𝜕_𝑡𝑤(0))=0. (4.33) The next three lemmas show the asymptotic smoothing property of 𝑆(𝑡), for initial data bounded by 𝑅. We begin to prove the exponential decay of 𝐿(𝑡)𝑧. Then, we prove the asymptotic smoothing property of 𝐾(𝑡).

Lemma 4.7. There is 𝜔=𝜔(𝑅)>0 such that ‖𝐿(𝑡)𝑧‖_ℋ≤𝐶𝑒^−𝜔𝑡. (4.34)

Proof. After denoting ℰ₀(𝑡)=ℰ₀(𝐿(𝑡)𝑧)=‖𝐿(𝑡)𝑧‖²_ℋ=‖𝑣(𝑡)‖²₂+‖𝜕_𝑡𝑣(𝑡)‖², (4.35) we set Φ₀(𝑡)=Φ₀(𝐿(𝑡)𝑧,𝑢(𝑡)), where 𝑢(𝑡) is the first component of 𝑆(𝑡)𝑧 and Φ₀(𝐿(𝑡)𝑧,𝑢(𝑡))=ℰ₀(𝐿(𝑡)𝑧)−𝑝‖𝑣(𝑡)‖²₁1+(𝛼+2)‖𝑣(𝑡)‖²+‖𝑢(𝑡)‖²₁‖𝑣(𝑡)‖²₁+⟨𝜕_𝑡𝑣(𝑡),𝑣(𝑡)⟩. (4.36) In light of Lemma 4.4 and inequalities (4.30), we have the bounds 12ℰ₀≤Φ₀≤𝑄(𝑅)ℰ₀. (4.37) Now, we compute the time-derivative of Φ₀ along the solutions to system (4.33) and we obtain dΦd𝑡⁰+Φ₀=2⟨𝜕_𝑡𝑢,𝐴^1/2𝑢⟩‖𝑣‖²₁≤𝑄(𝑅)‖𝜕_𝑡𝑢‖Φ₀. (4.38) The exponential decay of Φ₀ is entailed by exploiting the following Lemma 4.8 and then applying Lemma 6.2 of [6]. From (4.37), the desired decay of ℰ₀ follows.

Lemma 4.8. For any 𝜀>0,^𝑡_𝜏‖𝜕_𝑡𝜀𝑢(𝑠)‖ds≤𝜀(𝑡−𝜏)+4+𝑄(𝑅)𝜀, (4.39) for every 𝑡≥𝜏≥0.

Proof. After integrating (4.8) over (𝜏,𝑡) and taking (4.9) into account, we obtain 12ℰ(𝑆(𝑡)𝑧)−2‖𝑓‖²𝜆₁≤ℒ(𝑆(𝑡)𝑧)+2^𝑡_𝜏‖𝜕_𝑡𝑢(𝑠)‖²ds=ℒ(𝑆(𝜏)𝑧)≤ℒ(𝑧). (4.40) It follows that ^𝑡_𝜏‖𝜕_𝑡𝑢(𝑠)‖²ds≤𝑄(𝑅), (4.41) which, thanks to the Hölder inequality, yields ^𝑡_𝜏‖𝜕_𝑡𝑢(𝑠)‖ds≤𝜀√𝑡−𝜏+𝑄(𝑅)𝜀𝜀≤𝜀(𝑡−𝜏)+4+𝑄(𝑅)𝜀, (4.42) for any 𝜀>0.

The next result provides the boundedness of 𝐾(𝑡)𝑧 in a more regular space.

Proof. Lemma 4.9 (see [18, Lemma 6.3]). The estimate ‖𝐾(𝑡)𝑧‖_ℋ²≤𝑄(𝑅) (4.43) holds for every 𝑡≥0.As well as in [6], we use here the interpolation inequality ‖𝑤‖²₃≤‖𝑤‖₂‖𝑤‖₄. (4.44) Jointly with ‖𝑤‖₂≤𝑄(𝑅) (which follows by comparison from (4.9) and Lemma 4.7), this entails 𝑝‖𝑤‖²₃≤12ℰ₁+𝑄(𝑅), (4.45) where ℰ₁(𝑡)=ℰ₁(𝐾(𝑡)𝑧)=‖𝐾(𝑡)𝑧‖²_ℋ2=‖𝑤(𝑡)‖²₄+‖𝜕_𝑡𝑤(𝑡)‖²₂. (4.46) Letting Φ₁=ℰ₁+(‖𝑢‖²₁−𝑝)‖𝑤‖²₃+⟨𝜕_𝑡𝑤,𝐴𝑤⟩−2⟨𝑓,𝐴𝑤⟩+2𝑘²⟨𝑢⁺,𝐴𝑤⟩, (4.47) we have the bounds 13ℰ₁−𝑄(𝑅)≤Φ₁≤𝑄(𝑅)ℰ₁+𝑄(𝑅). (4.48) Taking the time-derivative of Φ₁, we find dΦd𝑡¹+Φ₁=2⟨𝜕_𝑡𝑢,𝐴^1/2𝑢⟩‖𝑤‖²₃+2𝛼⟨𝐴^1/2𝑣,𝐴^1/2𝜕_𝑡𝑤⟩+[𝛼⟨𝐴^1/2𝑣,𝐴^1/2𝑤⟩−⟨𝑓,𝐴𝑤⟩]−𝑘²⟨𝐴𝑤,𝑢⁺⟩+2𝑘²⟨𝐴𝑤,𝜕_𝑡𝑢⁺⟩. (4.49) Using (4.9) and (4.45), we control the rhs by 18ℰ₁+𝑄(𝑅)ℰ₁1+𝑄(𝑅)≤4ℰ₁3+𝑄(𝑅)≤4Φ₁+𝑄(𝑅), (4.50) and we obtain dΦd𝑡¹+14Φ₁≤𝑄(𝑅). (4.51) Since Φ₁(0)=0, the standard Gronwall lemma yields the boundedness of Φ₁. Then, by virtue of (4.48), we obtain the desired estimate for ℰ₁.

By collecting previous results, Lemma 4.3 in [31] can be applied to obtain the existence of the attractor 𝒜 and its regularity. Within our hypotheses and by virtue of the decomposition (4.33), it is also possible to prove the existence of regular exponential attractors for 𝑆(𝑡) with finite fractal dimension in ℋ. This can be done by a procedure very close to that followed in [6]. Since the global attractor is the minimal closed attracting set, we can conclude that the fractal dimension of 𝒜 in ℋ is finite as well.

Acknowledgments

The authors are indebted to the anonymous referees for their valuable remarks and comments.