This work is focused on the doubly nonlinear equation , whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness . When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load and stiffness . For a general external source , we prove the existence of bounded absorbing sets. When 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 [2โ€“4]), the dynamics of a suspension bridge is given by the well-known damped equation

where accounts for the downward deflection of the bridge in the vertical plane, and stands for its positive part, namely,

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

where is the (given) vertical dead load distribution. The term models a restoring force due to the cables, which is different from zero only when they are being stretched, and accounts for an external resistant force linearly depending on the velocity. The real constant represents the axial force acting at the ends of the road bed of the bridge in the reference configuration. Namely, is negative when the bridge is stretched, positive when compressed.

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

Concerning the boundary conditions, we consider here the case when both ends of the bridge are pinned. Namely, for every , we assume

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 acting on is

This operator is strictly positive selfadjoint with compact inverse, and its discrete spectrum is given by , . Thus, is the smallest eigenvalue. Besides, the peculiar relation

holds true, with Dirichlet boundary conditions and

Hence, if pinned ends are considered, the initial-boundary value problem (1.3)โ€“(1.5) can be described by means of a single operator , which enters the equation at the powers 1 and 1/2. Namely,

where is the norm of . This fact is particularly relevant in the analysis of the critical buckling load , that is, the magnitude of the compressive axial force 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., [11โ€“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,

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 to the original conservative model, it becomes

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 , the exponential decay of solutions to the trivial equilibrium state has been shown. The global dynamics of solutions for a general 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 , which turns out to be a strongly continuous semigroup in the autonomous case. The existence of an absorbing set for the solution operator 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 . 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)โ€“(1.5). First, we proceed with some preliminary estimates and prove the exponential stability of the system provided that the axial force is smaller than . 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)โ€“(1.5) into an abstract setting in order to establish more general results.

Let be a real Hilbert space, and let be a strictly positive selfadjoint operator with compact inverse. For , we introduce the scale of Hilbert spaces generated by the powers of

When , the index is omitted. The symbol will also be used to denote the duality product between and its dual space . In particular, we have the compact embeddings , along with the generalized Poincarรฉ inequalities

where is the first eigenvalue of . Finally, we define the product Hilbert spaces

For , we consider the following abstract Cauchy problem on in the unknown variable :

Problem (1.3)โ€“(1.5) is just a particular case of the abstract system (2.4), obtained by setting and with the boundary condition (1.5).

The following well-posedness result holds.

Proposition 2.1. Assume that Then, for all initial data , problem (2.4) admits a unique solution 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


Besides, for every , the map belongs to . Actually, it is a standard matter to verify the joint continuity

Remark 2.2. In the autonomous case, namely, when is time-independent, the semigroup property holds for all . Thus, is a strongly continuous semigroup of operators on which continuously depends on the initial data: for any initial data , is the unique weak solution to (2.4), with related norm given by

For any , we define the energy corresponding to as

and, abusing the notation, we denote by for each given initial data . Multiplying the first equation in (2.4) by , because of the relation

we obtain the energy identity

In particular, for every , there exists a positive increasing function such that

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 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 (referred to the initial time ) is a bounded set with the following property: for every , there is an entering time such that

whenever . In fact, we are able to establish a more general result.

Theorem 3.1. Let , and let be a translation bounded function in , that is, Then, there exists with the following property: in correspondence of every , there is such that whenever . Both and can be explicitly computed.

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

Lemma 3.2 (see [28, ]). Let be an absolutely continuous function satisfying, for some , , the differential inequality where is any locally summable function such that Then, there exist and such that, for every , it follows that whenever . Both and can be explicitly computed in terms of and .

Proof of Theorem 3.1. Here and in the sequel, we will tacitly use several times the Young and the Hรถlder inequalities, besides the usual Sobolev embeddings. The generic positive constant appearing in this proof may depend on and .
On account of (2.13), by means of the functional we introduce the function which satisfies the differential equality Because of the control we obtain the differential inequality Next, we consider the auxiliary functional and, regarding as , we have Noting that we are led to Precisely, we end up with Finally, we set where . We first observe that satisfies In order to estimate from below, a straightforward calculation leads to where we take advantage of The upper bound for can be easily achieved as follows: by virtue of Going back to differential equation and making use of (3.9) and (3.15), the function satisfies the identity and, as a consequence, we obtain the estimate Now, using (3.21) and (3.10), we have where . Thus, by virtue of (3.2) and (3.17), Lemma 3.2 yields

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 . 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 (see [14]).

4. The Global Attractor

In the remaining of the paper, we simplify the problem by assuming that the external force is time-independent. In which case, the operator is a strongly continuous semigroup on (see Remark 2.2). Having been proved in Section 3 the existence of the absorbing set , we could then establish here the existence of a global attractor by showing that the semigroup admits a bounded absorbing set in a more regular space and that it is uniformly compact for large values of (see, e.g., [27, Theoremโ€‰1.1]). In order to obtain asymptotic compactness, the -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 is the unique compact subset of which is at the same time

(i)attracting: for every bounded set , where denotes the usual Hausdorff semidistance in ,(ii)fully invariant:

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

Theorem 4.1. The semigroup acting on possesses a connected global attractor bounded in . Moreover, coincides with the unstable manifold of the set of the stationary points of , namely,

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

The set of the bridge equilibria under a vanishing lateral load consists of all the pairs such that the function is a weak solution to the equation

In particular, solves the following boundary value problem on the interval :

where we let , , and , . It is then apparent that is bounded in for every .

When , 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 is replaced by (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 , because of the spectral properties of the operator . On the contrary, for general values of , all critical values โ€œmove" when increases, as well as in [12]. Hence, it may happen that two different bifurcation values overlap for special values of , in which case they are referred as resonant values.

Assuming that , for instance, Figure 1 shows the bifurcation picture of solutions in dependence on the applied axial load . In particular, and

4.1. The Lyapunov Functional and Preliminary Estimates

We begin to prove the existence of a Lyapunov functional for , that is, a function satisfying the following conditions:

(i) if and only if ; (ii) is nonincreasing for any ;(iii) for all implies that .

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

Proof. Assertion (i) holds by the continuity of and by means of the estimates Using (3.9), we obtain quite directly that which proves the decreasing monotonicity of along the trajectories departing from . Finally, if is constant in time, we have that for all , which implies that is constant. Hence, for all , that is, .

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

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

Lemma 4.4. Given , for all and initial data with ,

Proof. Inequality (4.8) ensures that Moreover, taking into account that we obtain the estimate Finally, we have

Lemma 4.5. Let and . Then where

Proof. Because of the identity the thesis is trivial when . On the other hand, when we have

We are now in a position to prove the following.

Theorem 4.6. When , the solutions to (1.3)โ€“(1.5) decay exponentially, that is, with and being suitable positive constants, provided that .

Proof. Let be the functional where the constant is positive provided that . In view of applying Lemma 4.5, we remark that The first step is to prove the equivalence between and , that is, By virtue of (2.2), (4.20), and Lemma 4.5, the lower bound is provided by
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 : In particular, from (4.9) and (4.15), we find The last step is to prove the exponential decay of . To this aim, we obtain the identity where is given by (4.20). Exploiting the Young inequality and (4.22), we have from which it follows that Letting , by virtue of Lemma 3.2 (with ) and (4.22), we have The thesis follows by putting

The existence of a Lyapunov functional, along with the fact that 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 devised in [6] and following a general result (see [31, Lemmaโ€‰4.3]), tailored to our particular case.

4.2. The Semigroup Decomposition

By the interpolation inequality and (2.2), it is clear that

provided that is large enough and for some .

Again, is fixed and . Choosing such that (4.30) holds, according to the scheme first proposed in [6], we decompose the solution into the sum


solve the systems

The next three lemmas show the asymptotic smoothing property of , for initial data bounded by . We begin to prove the exponential decay of . Then, we prove the asymptotic smoothing property of .

Lemma 4.7. There is such that

Proof. After denoting we set , where is the first component of and In light of Lemma 4.4 and inequalities (4.30), we have the bounds Now, we compute the time-derivative of along the solutions to system (4.33) and we obtain The exponential decay of is entailed by exploiting the following Lemma 4.8 and then applying Lemmaโ€‰6.2 of [6]. From (4.37), the desired decay of follows.

Lemma 4.8. For any for every

Proof. After integrating (4.8) over and taking (4.9) into account, we obtain It follows that which, thanks to the Hรถlder inequality, yields for any .

The next result provides the boundedness of in a more regular space.

Proof. Lemma 4.9 (see [18, Lemmaโ€‰6.3]). The estimate holds for every .As well as in [6], we use here the interpolation inequality Jointly with (which follows by comparison from (4.9) and Lemma 4.7), this entails where Letting we have the bounds Taking the time-derivative of , we find Using (4.9) and (4.45), we control the rhs by and we obtain Since , the standard Gronwall lemma yields the boundedness of . Then, by virtue of (4.48), we obtain the desired estimate for .

By collecting previous results, Lemmaโ€‰4.3 in [31] can be applied to obtain the existence of the attractor 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 with finite fractal dimension in . 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 in is finite as well.


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