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On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback

  • Mohammad M. Al-Gharabli EMAIL logo
Published/Copyright: August 14, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we study a mathematical model for a one-dimensional suspension bridge problem with nonlinear damping. The model takes into consideration the vibration of the bridge deck in the vertical plane and main cable from which the bridge deck is suspended by the suspenders. We use the multiplier method to establish explicit and generalized decay results, without imposing restrictive growth assumption near the origin on the damping terms. Our results substantially improve, extend, and generalize some earlier related results in the literature.

Keywords: suspension bridge; coupled; nonlinear damping; general decay
MSC 2010: 35L51; 35B35; 35Q72; 35B41; 93D05

1 Introduction

Suspension bridges are known for their ability to span long distances, making them suitable for crossing deep valleys, wide rivers, or other challenging terrains. They are also famous for their graceful, iconic designs, which often include sweeping curves and tall towers. Some of the world most famous suspension bridges include the Golden Gate Bridge in San Francisco, the Brooklyn Bridge in New York City, and the Akashi Kaikyo Bridge in Japan.

Suspension bridges are generally well engineered and designed to be safe for public use, but accidents can occur due to a variety of factors, including natural disasters, human error, and structural issues. Some notable accidents involving suspension bridges such as Tacoma Narrows Bridge (1940): The original Tacoma Narrows Bridge in Washington, known as Galloping Gertie, famously collapsed on November 7, 1940, due to strong winds. The structural design of bridge and its susceptibility to aerodynamic forces caused it to twist and eventually fail. Fortunately, there were no casualties.

It is important to note that these accidents are relatively rare when compared to the vast number of suspension bridges worldwide. Engineering and construction practices have improved significantly over the years to enhance the safety and reliability of these structures. Comprehensive inspections, maintenance, and safety measures are typically in place to prevent accidents and ensure the structural integrity of suspension bridges.

When the displacement of the bridge deck (road bed) and the main suspension cable are taking into consideration, early results by McKenna and co-authors [1–5] modeled and investigated the structural behaviour of suspension bridges through the coupled system

(1.1) u t t + u x x x x + k [ u − v ] + + δ 1 u t = f , v t t − v x x − k [ u − v ] + + δ 2 v t = g ,

where u = u ( x , t ) and v = v ( x , t ) represents the displacement of the bridge and the main cable, respectively. The first equation models the vibration of the bridge (road bed) in the vertical plane, while the second equation describes the vibration of the main cable from which the bridge is suspended in the vertical plane. The constants δ 1 , δ 2 > 0 are frictional coefficients, and the term [ u − v ] + = max { u − v , 0 } stands for the nonlinear response of the suspension cables connecting the beams and the strings, with [ u − v ] + = 0 , indicating when the cable is slacking. The model in (1.1) was suggested by Lazer and Mckenna [1], which was motivated on the fact that it is unnatural to ignore the displacement of the main sustaining cable. They treat the main cable as a vibrating string, coupled with the vibrating beam (roadbed) by a piecewise linear springs with spring constant k , if expanded, but no restoring force if compressed. The stays connecting the beam and the string act so as to pull the cable down, therefore the minus sign in the second equation, and to hold the roadbed up, therefore causing a plus sign in the first equation. Recently, Mukiawa et al. [6] considered a mathematical model for a one-dimensional suspension bridge problem:

(1.2) m 1 u t t + E I u x x x x + k [ u − v ] + + δ 1 ∣ u t ∣ m − 2 u t = 0 , in ( 0 , ℓ ) × ( 0 , T ) , m 2 v t t − T f v x x − k [ u − v ] + + δ 2 ∣ v t ∣ r − 2 v t = 0 , in ( 0 , ℓ ) × ( 0 , T ) , u ( 0 , t ) = u x x ( 0 , t ) = u ( ℓ , t ) = u x x ( ℓ , t ) = 0 , for t ∈ ( 0 , T ) , v ( 0 , t ) = v ( ℓ , t ) = 0 , for t ∈ ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , for x ∈ ( 0 , ℓ ) , v ( x , 0 ) = v 0 ( x ) , v t ( x , 0 ) = v 1 ( x ) , for x ∈ ( 0 , ℓ ) ,

where the constant parameters of the problem in (1.3) are defined as follows: m 1 and m 2 represent, respectively, the mass per unit length of the bridge deck and the cable; EI and T f represent, respectively, the deck’s bending stiffness and the tensile strength of the cable; k represents the stiffness of the suspenders, which are defined as the vertical cables suspending the bridge deck to the main cable. m and r are constant exponents satisfying 2 ≤ m , r < ∞ . They proved the global existence of the solution using the Galerkin method, established, using the multiplier method, exponential and polynomial decay results, and provided some numerical tests to illustrate their theoretical decay results.

Motivated by the aforementioned works, we consider a nonlinear one-dimensional coupled suspension bridge problem with nonlinear damping of general form, precisely, we consider the following problem:

(1.3) m 1 u t t + E I u x x x x + k [ u − v ] + + q 1 ( u t ) = 0 , in ( 0 , ℓ ) × ( 0 , T ) , m 2 v t t − T f v x x − k [ u − v ] + + q 2 ( v t ) = 0 , in ( 0 , ℓ ) × ( 0 , T ) , u ( 0 , t ) = u x x ( 0 , t ) = u ( ℓ , t ) = u x x ( ℓ , t ) = 0 , for t ∈ ( 0 , T ) , v ( 0 , t ) = v ( ℓ , t ) = 0 , for t ∈ ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , for x ∈ ( 0 , ℓ ) , v ( x , 0 ) = v 0 ( x ) , v t ( x , 0 ) = v 1 ( x ) , for x ∈ ( 0 , ℓ ) ,

where q 1 and q 2 are specific functions.

The positive coefficients m 1 , m 2 , E I , T f , k in problem (1.3) have no effect on the calculations performed in Section 4. Therefore, for simplicity, we study the following version of problem (1.3) given by:

(1.4) u t t + u x x x x + [ u − v ] + + q 1 ( u t ) = 0 , in ( 0 , ℓ ) × ( 0 , T ) , v t t − v x x − [ u − v ] + + q 2 ( v t ) = 0 , in ( 0 , ℓ ) × ( 0 , T ) , u ( 0 , t ) = u x x ( 0 , t ) = u ( ℓ , t ) = u x x ( ℓ , t ) = 0 , for t ∈ ( 0 , T ) , v ( 0 , t ) = v ( ℓ , t ) = 0 , for t ∈ ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , for x ∈ ( 0 , ℓ ) , v ( x , 0 ) = v 0 ( x ) , v t ( x , 0 ) = v 1 ( x ) , for x ∈ ( 0 , ℓ ) .

We consider the following hypothesis on the nonlinear functions q 1 and q 2 :

  1. q i : R → R (for i = 1 , 2 ) are nondecreasing C 1 functions such that

    Q i ( ∣ s ∣ ) ≤ ∣ q i ( s ) ∣ ≤ Q i − 1 ( ∣ s ∣ ) , for all ∣ s ∣ ≤ m , i = 1 , 2 , c 1 ∣ s ∣ ≤ ∣ q i ( s ) ∣ ≤ c 2 ∣ s ∣ p , for all ∣ s ∣ ≥ m , i = 1 , 2 ,

    where Q 1 and Q 2 are strictly increasing C 1 functions on [ 0 , ∞ ) , Q 1 ( 0 ) = Q 2 ( 0 ) = 0 , the constants m , c 1 , and c 2 are positive, and p ≥ 1 .

Remark 1.1

Problem (1.4) has been widely studied when

Q i ( s ) = c s p , i = 1 , 2 , with some p ≥ 1 .

Mukiawa et al. [6] proved that the energy decays exponentially if p = 1 and in a polynomial way if p > 1 .

The goal of this work is to provide an explicit decay estimate of the energy even if q i , i = 1 , 2 , have not a polynomial behaviour in zero, for instance, if q i ( s ) = e − 1 s , i = 1 , 2 , s ∈ ( 0 , 1 ] . Our aim is to establish explicit and generalized decay rate results for the energy of this system, without imposing any restrictive growth assumption near the origin on the damping terms. The results of this article allow a larger class of nonlinear functions of frictional damping q 1 and q 2 , from which the energy decay rates are not necessarily of exponential or polynomial types. The proofs of our results are done basically in two steps. In the first step, we use the multiplier method to choose the right multipliers. In the second step, we follow, with necessary modifications dictated by the nature of our systems, the method introduced and used by Martinez [7] to study the wave equations.

The rest of the article is organized as follows: In Section 2, we give some literature review about suspension bridges. In Section 3, we present some useful material and state the main stability result with some examples. In Section 4, we prove the main result Theorem 3.1, concerning the stability of problem (1.4).

2 Literature review

Mathematicians have not shown any interest in suspension bridges until recently. McKenna, in 1987, introduced first nonlinear models to study them from a theoretical point of view, then he was followed by several other mathematicians [1,8]. McKenna’s main idea was to consider the slackening of the hangers as a nonlinear phenomenon, a statement which is by now well known also among engineers [9,10]. The slackening phenomenon was analyzed in various complex beam models by several authors [11–13]. After that, Bochicchio et al. [14] managed to establish explicit stationary solutions of the following system modelling a beam connected to a cable through hangers:

(2.1) m c v t t − H v x x + δ c v t − k [ u − v ] + = q c + f c ( x , t ) , x ∈ ( 0 , π ) , t > 0 , m b u t t + E I u x x x x + δ b u t + [ γ − M ‖ u x ‖ L 2 ( 0 , π ) 2 ] u x x + k [ u − v ] + = q b + f b ( x , t ) ,

where v and u represent the vertical displacements of the cable and the beam, m b and m c are the masses, q c and q b are the dead loads and f b and f c are the external forces, and the other constants depend on the elasticity of the physical materials. We notice here that the coupling between the two unknowns is due to the nonlinear term [ u − v ] + , which represents the possible slackening of the hangers and the second (beam) equation is nonlocal and contains a term representing the pre-stressing of the beam. The nonlinear term [ u − v ] + is due to McKenna, while the nonlocal term was first suggested by Woinowsky-Krieger [15]. Since the Federal report [16] blamed the Tacoma collapse to the appearance of torsional oscillations, it was clear that a beam model is no longer appropriate and does not have enough degrees of freedom to give a full explanation of the collapse. For this reason, McKenna considered some more general models, which are able to describe torsional oscillations [3,17]. They were able to show numerically a sudden transition from vertical to torsional oscillations. Other related results have been obtained by Moore [18] for a slightly different model. The well posedness of a further interesting model was proved in [19], where the unknowns are the vertical oscillations of the deck, its torsional angle and the coupling with the two sustaining cables. Again, the hangers were seen as linear springs which may slacken. If the vibration of the cable is ignored, the authors in [1–5] reduced system (2.1) to the following fourth-order equation:

(2.2) u t t + u x x x x + k 2 [ u ] + + δ u t = f ,

and results of existence and stability of periodic solutions have been established by assuming the suspension bridge as a bending beam. Also, Bochicchio et al. [20] considered

(2.3) u t t + u x x x x + ( p − ‖ u x ‖ L 2 ( 0 , π ) 2 ) u x x + k 2 [ u ] + + δ u t = f

and showed the existence of a global attractor with optimal regularity. For recent studies on the existence and asymptotic behaviour of solutions for the mathematical model of suspension bridge, Feola et al. [21] considered the following a quasilinear beam-wave system of equations which provides a refined model for the time-evolution of suspension bridges

(2.4) y t t = ℬ y + F 1 ( x , y , y x , y x x , θ , θ x , θ x x ) + α y t + γ f b ( t ) , θ t t = W θ + F 2 ( x , y , y x , y x x , θ , θ x , θ x x ) + β θ t + δ f w ( t ) , y = y ( t , x ) , θ = θ ( t , x ) , t ∈ ℛ , x ∈ [ 0 , L ] ,

where ℬ and W are time-independent linear operators with variable coefficients. They proved a result of local well posedness in Sobolev regularity. Hajjej [22] considered the following Balakrishnan-Taylor problem which describes the deformation of the deck of either a footbridge or a suspension bridge

(2.5) ∣ z t ∣ k z t t + α Δ 2 z t t + Δ 2 − ( − ξ 1 + ξ 2 ‖ z x ‖ 2 2 + σ ( z x , z x t ) ) z x x + Δ 2 z t + γ ( x ) f ( z t ) + h ( z ) = 0 ,

with hinged boundary conditions along its short edges, as well as free boundary conditions along its remaining free edges. The author established the existence of solutions and the exponential decay of energy.

Motivated by the wonderful book of Rocard [23], where it was pointed out that the correct way to model a suspension bridge is through a thin plate, Ferrero and Gazzola [24] introduced the following hyperbolic problem:

(2.6) u t t ( x , y , t ) + η u t + Δ 2 u ( x , y , t ) + h ( x , y , u ) = f , in Ω × R + , u ( 0 , y , t ) = u x x ( 0 , y , t ) = u ( π , y , t ) = u x x ( π , y , t ) = 0 , ( y , t ) ∈ ( − l , l ) × R + , u y y ( x , ± l , t ) + σ u x x ( x , ± l , t ) = 0 , ( x , t ) ∈ ( 0 , π ) × R + , u y y y ( x , ± l , t ) + ( 2 − σ ) u x x y ( x , ± l , t ) = 0 , ( x , t ) ∈ ( 0 , π ) × R + , u ( x , y , 0 ) = u 0 ( x , y ) , u t ( x , y , 0 ) = u 1 ( x , y ) , in Ω × R + ,

where Ω = ( 0 , π ) × ( − l , l ) is a planar rectangular plate, σ is the well known Poisson ratio, η is the damping coefficient, h is the nonlinear restoring force of the hangers, and f is an external force. After the appearance of the aforementioned model, many mathematician showed interest in investigating variants of it, using different kinds of damping in the aim to obtain stability of the bridge modelled though the aforementioned problem. See, for instance, [25–41].

3 Preliminaries and the main result

In this section, we define a Sobolev space and state the main stability result with some examples to illustrate our decay results.

We introduce the following space:

(3.1) H ∗ 2 ( 0 , ℓ ) = H 2 ( 0 , ℓ ) ∩ H 0 1 ( 0 , ℓ )

together with the inner product

(3.2) ⟨ w , z ⟩ H ∗ 2 ( 0 , ℓ ) = ∫ 0 ℓ w x x z x x d x .

It is easy to check that the norm ‖ · ‖ H ∗ 2 ( 0 , ℓ ) given by

‖ w ‖ H ∗ 2 ( 0 , ℓ ) 2 = ⟨ w , w ⟩ H ∗ 2 ( 0 , ℓ )

is equivalent to the usual H 2 ( 0 , ℓ ) norm and the pair ( H ∗ 2 ( 0 , ℓ ) , ‖ · ‖ H ∗ 2 ( 0 , ℓ ) ) forms a Banach space. Throughout this article, c is used to denote a generic positive constant.

For completeness, we state the following existence result whose proof can be established similarly to that in [6]. In fact, this result can be proved using the Garlekin approximation method [42,43].

Proposition 3.1

Assume that ( A ) holds and ( u 0 , u 1 , v 0 , v 1 ) ∈ H * 2 ( 0 , ℓ ) × L 2 ( 0 , ℓ ) × H 0 1 ( 0 , ℓ ) × L 2 ( 0 , ℓ ) be given. Then problem (1.4) has a global unique weak solution

u ∈ L ∞ ( [ 0 , T ] ; H ∗ 2 ( 0 , ℓ ) ) , u t ∈ L ∞ ( [ 0 , T ] ; L 2 ( 0 , ℓ ) ) ,

v ∈ L ∞ ( [ 0 , T ] ; H 0 1 ( 0 , ℓ ) ) , v t ∈ L ∞ ( [ 0 , T ] ; L 2 ( 0 , ℓ ) ) ,

for any T > 0 .

The energy functional associated with problem (1.4) is defined by

(3.3) E ( t ) = 1 2 [ ‖ u t ‖ L 2 ( 0 , ℓ ) 2 + ‖ u x x ‖ L 2 ( 0 , ℓ ) 2 + ‖ v t ‖ L 2 ( 0 , ℓ ) 2 + ‖ v x ‖ L 2 ( 0 , ℓ ) 2 + ‖ [ u − v ] + ‖ L 2 ( 0 , ℓ ) 2 ] .

Our main stability result is the following.

Theorem 3.1

Suppose that condition ( A ) holds, ( u 0 , u 1 , v 0 , v 1 ) ∈ H * 2 ( 0 , ℓ ) × L 2 ( 0 , ℓ ) × H 0 1 ( 0 , ℓ ) × L 2 ( 0 , ℓ ) be given. Then, there exist positive constants c , c ˜ such that the energy functional (3.3) satisfies, for t large,

(3.4) E ( t ) ≤ c ¯ Q ˜ − 1 1 t 2 ,

where

Q ˜ ( s ) = s ( Q 1 − 1 + Q 2 − 1 ) − 1 ( s ) .

Moreover, if Q 1 and Q 2 are strictly convex on ( 0 , r ) , for some r > 0 , and Q 1 ′ ( 0 ) = Q 2 ′ ( 0 ) = 0 , then we have the improved estimate

(3.5) E ( t ) ≤ c ( Q 1 − 1 + Q 2 − 1 ) 1 t 2 .

Remark 3.1

(The case of the polynomial growth) As a special case of ( A ) , when q 1 and q 2 have a polynomial growth, which means that there exist constants c 1 , c 2 > 0 and ζ 1 , ζ 2 ≥ 1 such that

(3.6) c 1 min { ∣ s ∣ , ∣ s ∣ ζ i } ≤ ∣ q i ( s ) ∣ ≤ c 2 max { ∣ s ∣ , ∣ s ∣ 1 ζ i } , i = 1 , 2 .

Then, according to Theorem 3.1, we have the following estimate:

E ( t ) ≤ c t 2 ζ ,

where ζ = max { ζ 1 , ζ 2 } . However, we can obtain a better decay rate as follows:

(3.7) E ( t ) ≤ c e − c t , if ζ 1 = ζ 2 = 1 , ( exponential decay ) E ( t ) ≤ c t 2 ζ − 1 , if ζ 1 , ζ 2 > 1 , ( polynomial decay ) .

The proof of (3.7) can be done using similar arguments as in the proof of Theorem 3.1.

Examples 3.2

We consider some examples to illustrate our decay results:

  1. Between polynomial and exponential growth

  2. Let Q 1 ( s ) = Q 2 ( s ) = e − ( ln s ) 2 near zero. Then, (3.4) gives

    E ( t ) ≤ c e − 2 ( ln t ) 1 2 .

  3. Exponential growth

  4. Let Q 1 ( s ) = Q 2 ( s ) = e − 1 ∕ s near zero. Then, (3.4) implies

    E ( t ) ≤ c ( ln ( t ) ) 2 .

  5. Faster than exponential growth

  6. If Q 1 ( s ) = Q 2 ( s ) = e − e 1 ∕ s near zero. Then, by using (3.4), we have

    E ( t ) ≤ c ( ln ( ln ( t ) ) ) 2 .

4 Proof of the main result

In this section, we prove Theorem (3.1). For this purpose, we establish several lemmas.

Lemma 4.1

[7] Let E : R + ⟶ R + be nonincreasing function and σ : R + ⟶ R + be an increasing C 1 -function, with σ ( t ) → + ∞ as t → + ∞ . Assume that there exist α , β ≥ 0 and c > 0 such that

∫ s ∞ σ ′ ( t ) E 1 + α ( t ) d t ≤ c E 1 + α ( s ) + c E ( s ) σ β , 1 ≤ s < ∞ .

Then, there exist positive constants k , ω > 0 such that for all t ≥ 1 , we have

E ( t ) ≤ k e − ω σ ( t ) , i f α = β = 0 , E ( t ) ≤ C ( σ ( t ) ) 1 + β β , i f α > 0 .

Lemma 4.2

The energy functional, defined by (3.3), satisfies

(4.1) d E ( t ) d t = − ∫ 0 ℓ u t q 1 ( u t ) d x − ∫ 0 ℓ v t q 2 ( v t ) d x ≤ 0 , ∀ t ≥ 0 .

Proof

Multiplying (1.4)1 and (1.4)2 by u t and v t , respectively, and integrating the results over ( 0 , ℓ ) , we obtain

(4.2) d d t 1 2 ‖ u t ‖ L 2 ( 0 , ℓ ) 2 + 1 2 ‖ u x x ‖ L 2 ( 0 , ℓ ) 2 + ∫ 0 ℓ [ u − v ] + u t d x + ∫ 0 ℓ u t q 1 ( u t ) d x = 0 , d d t 1 2 ‖ v t ‖ L 2 ( 0 , ℓ ) 2 + 1 2 ‖ v x ‖ L 2 ( 0 , ℓ ) 2 − ∫ 0 ℓ [ u − v ] + v t d x + ∫ 0 ℓ v t q 2 ( v t ) d x = 0 .

Adding the equations in (4.2), we obtain

(4.3) d d t 1 2 [ ‖ u t ‖ L 2 ( 0 , ℓ ) 2 + ‖ u x x ‖ L 2 ( 0 , ℓ ) 2 + ‖ v t ‖ L 2 ( 0 , ℓ ) 2 + ‖ v x ‖ L 2 ( 0 , ℓ ) 2 + ‖ [ u − v ] + ‖ L 2 ( 0 , ℓ ) 2 ] + ∫ 0 ℓ u t q 1 ( u t ) d x + ∫ 0 ℓ v t q 2 ( v t ) d x = 0 .

This leads to

(4.4) d E ( t ) d t = − ∫ 0 ℓ u t q 1 ( u t ) d x − ∫ 0 ℓ v t q 2 ( v t ) d x ≤ 0 , ∀ t ≥ 0 .

Therefore, the energy functional E is nonincreasing and bounded above by E ( 0 ) . □

Remark 4.1

[7,44]

  1. Let Q 0 ≔ ( Q 1 − 1 + Q 2 − 1 ) − 1 and

    (4.5) q ˜ ( t ) ≔ 1 + ∫ 1 t 1 Q 0 1 s d s , t ≥ t 0 ,

    for some t 0 > max { 1 , 1 m } . Then

    q ˜ ′ ( t ) = 1 Q 0 1 t > 0 , t ≥ t 0 , q ˜ ′ ( t ) → + ∞ as t → + ∞ ,

    and q ˜ ′ ( t ) is strictly increasing. Thus, q ˜ is a convex and strictly increasing C 2 -function, with q ˜ ( t ) → + ∞ as t → ∞ .

  2. If we set

    (4.6) γ 0 ≔ q ˜ − 1 , t ≥ t 0 ,

    then it is easy to check that γ 0 is strictly increasing and γ 0 ′ ( t ) = Q 0 1 γ 0 ( t ) is strictly decreasing. So γ 0 is a concave C 2 -function, with γ 0 ( t ) → + ∞ as t → + ∞ .

Proof of the main result

Multiplying (1.4)1 by γ 0 ′ u E ( t ) and (1.4)2 by γ 0 ′ v E ( t ) , respectively, then integrating the results over ( 0 , ℓ ) × ( s , T ) , for T > s and γ 0 ′ = Q 0 1 γ 0 which is defined in Remark 4.1, we obtain

∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u t t u + u x x x x u + [ u − v ] + u + u q 1 ( u t ) ) d x d t = 0 , ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( v t t v − v x x v − [ u − v ] + v + v q 2 ( v t ) ) d x d t = 0 .

By using integration by parts and routine calculations, we obtain

(4.7) ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( ( ( u u t ) t − u t 2 ) + u x x 2 + [ u − v ] + u + u q 1 ( u t ) ) d x d t = 0 , ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( ( ( v v t ) t − v t 2 ) + v x 2 − [ u − v ] + v + v q 2 ( v t ) ) d x d t = 0 ,

which gives

(4.8) ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u u t ) t d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u t 2 + u x x 2 + [ u − v ] + u ) d x d t − 3 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ u t 2 d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ u x x 2 d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ [ u − v ] + u d x d t + ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ u q 1 ( u t ) d x d t = 0 , ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( v v t ) t d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( v t 2 + v x 2 − [ u − v ] + v ) d x d t − 3 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ v t 2 d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ v x 2 d x d t − 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ [ u − v ] + v d x d t + ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ v q 2 ( v t ) d x d t = 0 .

Add the equations in (4.8), we obtain

(4.9) ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( ( u u t ) t + ( v v t ) t ) d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u t 2 + u x x 2 + v t 2 + v x 2 + [ u − v ] + 2 ) d x d t − 3 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u t 2 + v t 2 ) d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u x x 2 + v x 2 ) d x d t + 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ [ u − v ] + 2 d x d t + ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ u q 1 ( u t ) d x d t + ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ v q 2 ( v t ) d x d t ≤ 0 .

Recalling (3.3), we obtain

∫ s T γ 0 ′ E 2 ( t ) d t ≤ − ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( ( u u t ) t + ( v v t ) t ) d x d t + 3 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u t 2 + v t 2 ) d x d t − 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u x x 2 + v x 2 ) d x d t − 1 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ [ u − v ] + 2 d x d t − ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ u q 1 ( u t ) d x d t − ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ v q 2 ( v t ) d x d t ,

which implies, by discarding the negative terms,

(4.10) ∫ s T γ 0 ′ E 2 ( t ) d t ≤ − ∫ s T γ 0 ′ d d t ∫ 0 ℓ E ( t ) ( u u t + v v t ) d x d t ︸ I 1 + ∫ s T γ 0 ′ E ′ ( t ) ∫ 0 ℓ ( u u t + v v t ) d x d t ︸ I 2 + 3 2 ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ ( u t 2 + v t 2 ) d x d t ︸ I 3 − ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ u q 1 ( u t ) ︸ I 4 − ∫ s T γ 0 ′ E ( t ) ∫ 0 ℓ v q 2 ( v t ) d x d t ︸ I 5 .

Now, we estimate the terms I 1 − I 5 in (4.10).

(4.11) I 1 = − ∫ s T γ 0 ′ d d t ∫ 0 ℓ E ( t ) ( u u t + v v t ) d x d t = − γ 0 ′ ∫ 0 ℓ E ( t ) ( u u t + v v t ) d x s T + ∫ s T γ 0 ″ ∫ 0 ℓ E ( t ) ( u u t + v v t ) d x d t .

Recalling (3.3) and embedding theory, we have

(4.12) 1 2 ( ‖ u ( t ) ‖ L 2 ( 0 , ℓ ) 2 + ‖ u t ( t ) ‖ L 2 ( 0 , ℓ ) 2 + ‖ v ( t ) ‖ L 2 ( 0 , ℓ ) 2 + ‖ v t ( t ) ‖ L 2 ( 0 , ℓ ) 2 ) ≤ 1 2 ( C e ‖ u x x ‖ L 2 ( 0 , ℓ ) 2 + ‖ u t ( t ) ‖ L 2 ( 0 , ℓ ) 2 + c p ‖ v x ‖ L 2 ( 0 , ℓ ) 2 + ‖ v t ( t ) ‖ L 2 ( 0 , ℓ ) 2 ) ≤ c E ( t ) ,

where C e and c p are the embedding and Poincaré constants, respectively. Using Young’s inequality, we have

(4.13) − γ 0 ′ ∫ 0 ℓ E ( t ) ( u u t + v v t ) d x s T ≤ ∫ 0 ℓ γ 0 ′ ( t ) E ( s ) ( u ( x , s ) u t ( x , s ) + v ( x , s ) v t ( x , s ) ) d x + ∫ 0 ℓ γ 0 ′ ( t ) E ( T ) ( u ( x , T ) u t ( x , T ) + v ( x , T ) v t ( x , T ) ) d x ≤ 1 2 ∫ 0 ℓ γ 0 ′ ( t ) E ( s ) ( u 2 ( x , s ) + u t 2 ( x , s ) + v 2 ( x , s ) + v t 2 ( x , s ) ) d x + 1 2 ∫ 0 ℓ γ 0 ′ ( t ) E ( T ) ( u 2 ( x , T ) + u t 2 ( x , T ) + v 2 ( x , T ) + v t 2 ( x , T ) ) d x .

Now, by using Young’s inequality and (4.12), we have

(4.14) ∫ s T γ 0 ″ ∫ 0 ℓ E ( t ) ( u u t + v v t ) d x d t ≤ c ∫ s T γ 0 ″ E 2 d t ≤ c E 2 ( s ) ∫ s T γ 0 ″ d t ≤ c γ 0 ′ ( s ) E 2 ( s ) ≤ c E 2 ( s ) .

Thus, combining (4.11)–(4.14), we obtain

(4.15) ∣ I 1 ∣ ≤ c E 2 ( s ) .

For I 2 , we have

(4.16) ∣ I 2 ∣ ≤ − ∫ s T γ 0 ′ ∫ 0 ℓ E ′ ( t ) ∣ u u t + v v t ∣ d x d t ≤ − 1 2 ∫ s T γ 0 ′ ∫ 0 ℓ E ′ ( t ) ( u 2 + u t 2 + v 2 + v t 2 ) d x d t ≤ − c ∫ s T γ 0 ′ ∫ 0 ℓ E ′ ( t ) ( u x x 2 + u t 2 + v x 2 + v t 2 ) d x d t ≤ − c ∫ s T γ 0 ′ E ( t ) E ′ ( t ) d t = − C ∫ s T ( E 2 ( t ) ) ′ d t ≤ c E 2 ( s ) .

Take t 1 ≥ t 0 such that γ 0 ′ ( t 1 ) < m , and consider the following partition:

(4.17) Ω 1 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ > m } Ω 2 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and ∣ u t ∣ ≤ Q 1 − 1 ( γ 0 ′ ) } Ω 3 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and ∣ u t ∣ > Q 1 − 1 ( γ 0 ′ ) } .

Consequently, using ( A ) , we have, for T > s > t 1 ,

γ 0 ′ ∫ Ω 1 u t 2 d x ≤ 1 c γ 0 ′ ∫ 0 ℓ u t q 1 ( u t ) d x ≤ − c E ′ ( t ) γ 0 ′ ∫ Ω 2 u t 2 d x ≤ γ 0 ′ ( Q 1 − 1 ( γ 0 ′ ) ) 2 ≤ γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ) ) 2 γ 0 ′ ∫ Ω 2 u t 2 d x ≤ m ∫ Ω 3 Q 1 ( ∣ u t ∣ ) ∣ u t ∣ d x ≤ m ∫ 0 ℓ u t q 1 ( u t ) d x ≤ − c E ′ ( t ) ,

which gives

(4.18) ∫ s T γ 0 ′ E ∫ 0 ℓ u t 2 d x ≤ c E 2 ( s ) + c E ( s ) ∫ s T γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ) ) 2 d t .

Similarly, we obtain

(4.19) ∫ s T γ 0 ′ E ∫ 0 ℓ v t 2 d x ≤ c E 2 ( s ) + c E ( s ) ∫ s T γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ) ) 2 d t .

By combining (4.18) and (4.19), we have

(4.20) I 3 ≤ c E 2 ( s ) + c E ( s ) ∫ s T γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ) ) 2 d t .

For I 4 , we consider the following partition of ( 0 , ℓ ) :

(4.21) Ω 1 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ > m } Ω 2 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and ∣ u t ∣ ≤ γ 0 ′ } Ω 3 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and ∣ u t ∣ > γ 0 ′ } .

Then, by using Hölder’s, Young’s and Poincare’s inequalities, ( A ) , and the embedding theory, we have

(4.22) γ 0 ′ ∫ Ω 1 u q 1 ( u t ) d x ≤ γ 0 ′ ∫ Ω 1 ∣ u ∣ p + 1 d x 1 p + 1 ∫ Ω 1 ∣ q 1 ( u t ) ∣ 1 + 1 p d x p p + 1 ≤ c γ 0 ′ ∫ 0 ℓ u x 2 d x 1 2 ∫ Ω 1 u t q 1 ( u t ) d x p p + 1 ≤ c γ 0 ′ E 1 2 ( t ) ( − E ′ ( t ) ) p p + 1 ≤ c γ 0 ′ ε E p p + 1 ( t ) − C ε E ′ ( t ) ≤ c γ 0 ′ E ( t ) − C ε E ′ ( t ) .

(4.23) γ 0 ′ ∫ Ω 2 u q 1 ( u t ) d x ≤ ε γ 0 ′ ∫ Ω 2 ∣ u ∣ 2 d x + C ε γ 0 ′ ∫ Ω 2 q 1 2 ( u t ) d x ≤ c ε γ 0 ′ ( t ) E ( t ) + C ε γ 0 ′ ( t ) ( Q 0 − 1 ( γ 0 ′ ) ) 2 ≤ c ε γ 0 ′ ( t ) E ( t ) + C ε γ 0 ′ ( t ) ( Q 0 − 1 ( γ 0 ′ ) ) 2 .

(4.24) γ 0 ′ ∫ Ω 3 u q 1 ( u t ) d x ≤ ε γ 0 ′ ∫ Ω 3 ∣ u ∣ 2 d x + C ε γ 0 ′ ∫ Ω 3 q 1 2 ( u t ) d x ≤ c ε γ 0 ′ ( t ) E ( t ) + C ε Q 1 − 1 ( m ) ∫ 0 ℓ u t q 1 ( u t ) d x ≤ c ε γ 0 ′ ( t ) E ( t ) − C ε E ′ ( t ) .

A combination of (4.22)–(4.24) leads to

(4.25) I 4 ≤ c ε ∫ s T γ 0 ′ E 2 d t + C ε E 2 ( s ) + C ε E ( s ) ∫ s T γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ) ) 2 d t .

Similarly, we can obtain

(4.26) I 5 ≤ c ε ∫ s T γ 0 ′ E 2 d t + C ε E 2 ( s ) + C ε E ( s ) ∫ s T γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ) ) 2 d t .

Combining (4.10), (4.15), (4.16), (4.20), (4.25), and (4.26) and taking ε small enough lead to

(4.27) ∫ s ∞ γ 0 ′ ( t ) E 2 ( t ) d t ≤ c E 2 ( s ) + c E ( S ) ∫ s ∞ γ 0 ′ ( Q 0 − 1 ( γ 0 ′ ( t ) ) ) 2 d t = c E 2 ( s ) + c E ( S ) ∫ γ 0 ( s ) ∞ Q 0 − 1 Q 0 1 s 2 d s = c E 2 ( s ) + c E ( s ) γ 0 ( s ) .

By using (4.27) and Lemma 4.1 with α = β = 1 and σ = γ 0 , we obtain

(4.28) E ( t ) ≤ c γ 0 2 ( t ) , ∀ t ≥ t 1 .

To prove (3.4), let us define Q ˜ ( s ) = s Q 0 ( s ) and take s 0 > t 0 such that Q 0 1 s 0 ≤ 1 .

Since Q 0 is increasing, then we have Q 0 1 s ≤ Q 0 1 s 0 ≤ 1 , ∀ s ≥ s 0 . Therefore, we obtain

(4.29) γ 0 − 1 ( s ) ≤ 1 + ( s − 1 ) 1 Q 0 1 s = Q 0 1 s + s − 1 Q 0 1 s ≤ s Q 0 1 s = 1 Q ˜ 1 s , ∀ s ≥ s 0 .

So, with letting t = 1 Q ˜ 1 s , we can see that

(4.30) K 1 s = 1 t ⇒ 1 s = Q ˜ − 1 1 t ⇒ s = 1 Q ˜ − 1 1 t .

Then, by using (4.29) and (4.30), we see that

(4.31) γ 0 − 1 ( s ) ≤ 1 Q ˜ Q ˜ − 1 1 t = 1 1 ∕ t = t .

Hence,

(4.32) s = 1 Q ˜ − 1 ( 1 t ) ≤ γ 0 ( t ) .

Now, it is easy to see that

(4.33) 1 γ 0 ( t ) ≤ Q ˜ − 1 1 t , ∀ t ≥ t 0 .

Therefore, combining (4.28) and (4.33), estimate (3.4) is established.

To prove (3.5), we assume, without loss of generality, that r = m . In fact, if r < m and r ≤ ∣ s ∣ ≤ m , then, using ( A ) , we have, for i = 1 , 2 ,

∣ q i ( s ) ∣ ≤ Q i − 1 ( ∣ s ∣ ) ∣ s ∣ p ∣ s ∣ p ≤ Q i − 1 ( m ) ∣ r ∣ p ∣ s ∣ p

and

∣ q i ( s ) ∣ ≥ Q i − 1 ( ∣ s ∣ ) ∣ s ∣ ∣ s ∣ ≤ Q i ( m ) m ∣ s ∣ .

This implies that

Q i ( ∣ s ∣ ) ≤ ∣ q i ( s ) ∣ ≤ Q i − 1 ( ∣ s ∣ ) , for all ∣ s ∣ ≤ r , i = 1 , 2 , c 1 ∣ s ∣ ≤ ∣ q i ( s ) ∣ ≤ c 2 ∣ s ∣ p , for all ∣ s ∣ ≥ r , i = 1 , 2 ,

which justifies our assumption ( r = m ) . Since Q 1 ( 0 ) = Q 2 ( 0 ) = Q 1 ′ ( 0 ) = Q 1 ′ ( 0 ) = 0 and, for s > 0 ,

0 < J 0 ( s ) = Q 0 ( s ) s = ( Q 1 − 1 + Q 2 − 1 ) − 1 ( s ) s ≤ Q i ( s ) s , i = 1 , 2 ,

then Q 0 ′ ( 0 ) = J 0 ( 0 ) = 0 . Also, one can easily conclude that Q 0 is strictly convex on ( 0 , m ) . Then, by using the mean value theorem and the strict convexity of Q i , i = 0 , 1 , 2 , on ( 0 , m ) , we deduce that

J i ( s ) = Q i ( s ) s , i = 0 , 1 , 2 ,

are strictly increasing on ( 0 , m ) . Now, we take γ 0 = J ˜ − 1 , where

J ˜ ( t ) ≔ 1 + ∫ 1 t 1 J 0 1 s d s , t ≥ t 0 .

In this case, we replace (4.17) and (4.21) by

(4.34) Ω 1 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ > m } Ω 2 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and ∣ u t ∣ ≤ J 1 − 1 ( γ 0 ′ ) } Ω 3 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and ∣ u t ∣ > J 1 − 1 ( γ 0 ′ ) }

and

(4.35) Ω 1 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ > m } Ω 2 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and Q 1 − 1 ( ∣ u t ∣ ) ≤ J 1 − 1 ( γ 0 ′ ) } Ω 3 = { x ∈ ( 0 , ℓ ) : ∣ u t ∣ ≤ m and Q 1 − 1 ( ∣ u t ∣ ) > J 1 − 1 ( γ 0 ′ ) } .

Consequently, we obtain

(4.36) γ 0 ′ ∫ Ω 3 u t 2 d x ≤ ∫ Ω 3 J 1 ( ∣ u t ∣ ) u t 2 d x = ∫ Ω 3 Q 1 ( ∣ u t ∣ ) ∣ u t ∣ d x ≤ ∫ 0 ℓ u t q 1 ( u t ) ≤ − E ′ ( t ) .

(4.37) γ 0 ′ ∫ Ω 3 q 1 2 ( u t ) d x ≤ ∫ Ω 3 J 1 ( Q 1 − 1 ( ∣ u t ∣ ) ) Q 1 − 1 ( ∣ u t ∣ ) ∣ q 1 ( u t ) ∣ d x = ∫ 0 ℓ u t q 1 ( u t ) ≤ − E ′ ( t ) .

The other cases can be dealt with similarly. Then, the same reasoning leads to (3.5).□

5 Conclusions

In this work, we have established explicit and generalized decay rate results for the energy for a coupled suspension bridge problem, without imposing any restrictive growth assumption near the origin on the damping terms. The results of this article allow a larger class of functions q 1 and q 2 , from which the energy decay rates are not necessarily of exponential or polynomial types. This work generalizes and improves several works in the literature.

Acknowledgement

The author would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project (No. INCB2404), is also greatly acknowledged. The author thanks an anonymous referee for his/her careful reading and valuable suggestions.

  1. Funding information: This work was funded by KFUPM, Grant No. INCB2404.

  2. Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation. The author read and approved the final version of the manuscript.

  3. Conflict of interest: The author declares that there is no conflict of interest.

  4. Data availability statement: No data were used to support this study.

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Received: 2023-11-08
Revised: 2024-03-09
Accepted: 2024-07-23
Published Online: 2024-08-14

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
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  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
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  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
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  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
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  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
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  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
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  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0042
Keywords for this article
suspension bridge; coupled; nonlinear damping; general decay
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