Asymptotic behavior of a Balakrishnan-Taylor suspension bridge

1.   Introduction

This paper is concerned with well-posedness and exponential stability, in $\Omega \times (0, +\infty)$, for the following Balakrishnan-Taylor suspension bridge

with the boundary conditions

and initial conditions

where $\kappa$ and $\alpha$ are positive constants. The constant $\mu$ is the Poisson ratio and ranges in value from $0.1$ to $0.2$ for concrete and for metals it is about $0.3$. Accordingly, we will make the assumption that $0 < \mu < \frac{1}{2}$.

The constant $\sigma > 0$ is the Balakrishnan-Taylor damping coefficient, with the understanding that $\xi_1$ is positive for compressed plates and negative for stretched plates (^[1]HY__HY, Section 5]), $\xi_2 > 0$ relies on the material elasticity of the deck, and the term $\xi_2 \Vert z_x\Vert^2$ measures the geometric nonlinearity of the plate due to its stretching.

Here, the notation $(\cdot, \cdot)$ stands for the inner product in $L^2(\Omega)$ and its corresponding norm will be denoted by $\Vert \cdot \Vert$.

The function $\gamma\in L^{\infty}(\Omega)$ satisfies

where $\omega$ is an open subset of $\Omega$.

In this paper, we take into account the following conditions:

(H1): $f: \mathbb{R}\to \mathbb{R}$ is a nondecreasing $C^{1}$ function such that there exist positive constants $\varepsilon$, $c_1$, $c_2$ and a strictly increasing function $F\in C^{1}([0, +\infty))$, with $F(0) = 0$, and $F$ is a linear or strictly convex $C^{2}$ function on $(0, \varepsilon],$ such that

(H2): $h:\mathbb{R}\to\mathbb{R}$ is a Lipschitz function with $h(0) = 0$.

(H3): $H(t) = \int_0^t h(s)\; ds$ is positive such that

Remark 1.1. Hypothesis (H1) implies that $sf(s) > 0$ for all $s\neq0$.

Model (1.1) describes the vibrations of a thin and narrow rectangular plate in the presence of Balakrishnan-Taylor damping (the term $\sigma (z_x, z_{xt})$), strong damping (the term $\Delta^2z_t$), and nonlinear localized damping (the term $\gamma(x) f(z_t)$). Bridges are important, and they have long been a part of daily life for people. Bridges allow for uninterrupted travel over rivers and dangerous terrain, saving time and fuel while also minimizing traffic congestion, shortening travel distances, and reducing the number of accidents that may otherwise occur on the road. Nonetheless, difficulties with stability and collapse brought on by natural disasters like earthquakes and strong winds have been encountered during bridge building. The study of suspension bridges has been an interest for many researchers who have made efforts to discover the best designs and models feasible to tackle such challenges. Early results concerning suspension bridges go back to the works of McKenna and Walter ^[2] and McKenna et al. ^[3], where the authors gave a model describing the dynamics of a suspension bridge and proved the existence of nonlinear oscillations. The asymptotic dynamics and global attractors for coupled suspension bridge equations were investigated by Bochicchio et al. ^[4] and Ma and Zhong ^[5], respectively. Recently, a new model for a suspension bridge through a plate was given in Ferrero and Gazzola ^[6]. Further details on suspension bridge models can also be found in ^[7]. The bending and stretching energies of the model given in ^[6] were analyzed in ^[1]. Later, Berchio et al. ^[8] discussed the structural instability of nonlinear plates modelling suspension bridges. The finite time blow-up and uniform stability of a suspension bridge has recently been the subject of various works, and special cases of (1.1)–(1.3) have been investigated. We mention that all works that will be discussed hereafter treated similar problems to (1.1)–(1.3), but without strong damping. In ^[9], Wang considered problem (1.1) when $\kappa = \alpha = \xi_1 = \xi_2 = \sigma = 0$, $\gamma = 1$, $h(z) = az$ ($a = a(x, y, t)$ is a sign-changing and bounded measurable function), and $f(z) = z$ and with an external force. The author provided necessary and sufficient conditions for the uniqueness and existence of global solutions and finite time blow-up of these solutions. Next, Liu et al. ^[10] extended the work of Wang ^[9] by taking $f(z) = \vert z\vert^{m-2} z (m\geq 2)$. In ^[11], the authors considered system (1.1) in the case where $\kappa = \alpha = \xi_1 = \xi_2 = \sigma = 0$ and $\gamma(x) = \gamma$, and with linear damping, i.e., $f(z_t) = z_t$. Using the multiplier techniques, the authors showed a uniform decay of energy. Afterwards, Cavalcanti et al. ^[12] (resp., ^[13]) studied problem (1.1) when $\kappa = \alpha = \sigma = h = 0$ with a localized linear (resp., nonlinear) damping distributed around a neighborhood of the boundary, and showed the exponential decay of energy in both cases. Let us also mention works ^{[14,15,16,17,18,19]}, which dealt with suspension bridges and where other types of damping (structural and viscoelastic) are presented.

Finally, we recall some recent works on the plate equation that are related to our problem. In ^[20], the authors considered the equation

in a bounded domain of $\mathbb{R}^2$, established the existence of the solutions, and proved explicit and general decay rate results. Next, Al-Mahdi ^[21] considered the same problem as in ^[20] but with infinite memory. He proved existence and general decay results with a wider class of relaxation functions. Later on, in ^[22], the authors proved similar outcomes to ^[21] by adding a nonlinear damping.

We also cite the recent works about the evolutive plate equation with partially hinged boundary conditions ^[23,24]. Motivated by all these works, our goal here is to prove the existence of global solutions as well as the exponential decay of energy of these solutions under the influence of a localized nonlinear damping distributed in a subset of $\Omega$, combined with Balakrishnan-Taylor and strong damping.

The results presented here are new and different from previous works due to the presence of $\Delta^2 z_t$ and an external force source $h(z)$. Note that the external force generally promotes the blow-up of the solution.

The organization of the paper proceeds as follows: Section 2 is devoted to fixing notations, recalling some previous lemmas, and establishing a technical inequality. In Section 3, the well-posedness of system (1.1)–(1.3) is proved. The exponential stability is shown in the last part.

2.   Preliminaries

We define the space

along with the scalar product

It is a known fact that $(W, \langle \cdot, \cdot\rangle)$ is a Hilbert space, with the norm $\Vert. \Vert^2_{W}$ being equivalent to the standard $H^2$ norm (see ^[6]).

Lemma 2.1. (^[6]). Assume $0 < \mu < \frac{1}{2}$ and consider $g\in L^2(\Omega)$. Therefore, there is a unique function $z\in W$ such that

for all $v\in W.$

The function $z$ belonging to $W$ and fulfilling (2.1) is referred to as the weak solution for the stationary problem

We also bring to mind the subsequent lemma:

Lemma 2.2. (^[9]). Let $z\in W$ and consider that $1\leq q < +\infty$. Hence, we have

for some positive constant $C_* = C_*(\Omega, q) > 0$, and, where $\Vert \cdot\Vert_q$ denotes the usual $L^q(\Omega)$-norm.

The energy of the solutions of (1.1)–(1.3) is determined by

and satisfies

Remark 2.3. An interesting observation is that, when $\xi_1 < 0$, the energy $E(t)$ remains nonnegative for all $t\geq 0$. This scenario, unlike real bridges, indicates a stretched plate instead of compressed one in terms of elasticity. However, in the more realistic case where $\xi_1 > 0$, typical of bridges, the energy is no longer guaranteed to be nonnegative. This aspect holds significant importance in the stabilization of distributed systems. To address this issue, we will draw upon concepts from [^[1], Section 3]. We now introduce the following:

When equipped with the Dirichlet norm below, $H_*^1(\Omega)$ forms a normed space

Remark 2.4. We introduce $H^1_*(\Omega)$ as the completion of $C_*^\infty(\Omega)$ with the norm $||\cdot||_{H^1_*(\Omega)}$. It is apparent that the inclusion $W\hookrightarrow H_*^1(\Omega)$ is compact and the optimal embedding constant can be expressed by

Proposition 2.5. (^[1]) Assume that $0 < \xi_1 < \Lambda$. Then, $E(t) > 0.$

Proof. Using Remark 2.4, we obtain the Poincaré-type inequality

Then, for all $z\in W$, and since

we have

and therefore

Thus, if $0 < \xi_1 < \Lambda$, it follows that $\frac12 ||z||_{W}^2 -\frac{\xi_1}{2} ||z_x||^2 > 0$, and consequently $E(t) > 0$.

Within the scope of this paper, $C$ represents a generic positive constant, and is not necessarily the same at different occurrences.

The proof of our main result relies heavily on the next proposition.

Proposition 2.6. The solution of (1.1) verifies

for some constants $A > 0$ and $0 < B < 2$.

Proof. Young's inequality and (2.3) lead to

for any $\beta > 0$.

The first term on the right hand side of (2.11) can be estimated as follows:

Using hypothesis (H1) and the fact that $f(z_t)z_t > 0$, it holds that

where $M = \max\{C, c_2\}$.

By taking $\beta < \frac{1}{C_{*}^2}$, (2.10) is satisfied with $A = \frac{M\Vert \gamma\Vert_{\infty}}{2\beta}$ and $B = 2\beta C_{*}^2$.

3.   Well-posedness

In this part, we shall use the Faedo-Galerkin approach to prove that system (1.1)–(1.3) is well-posed. We have the following result:

Theorem 3.1. Assume (H1)–(H3) and $0 < \xi_1 < \Lambda$. Let $(z_0, z_1)\in W\times W$. Then, problems (1.1)–(1.3) is well-posed, i.e., for any $T > 0$, there exists

satisfying

for a.e., $t\in [0, T]$.

Remark 3.2. The function $z$ satisfying (3.1) is called a weak solution of (1.1)–(1.3).

Proof. Let $\{w_j\}_{j = 1}^{\infty}$ be a basis of $W$ and $E_p = {\text span}\{w_1, w_2, ..., w_p\}, \; p\geq1$. Let us write

such that

We will seek approximate solutions

satisfying

Consequently, we obtain a system of ordinary differential equations for unknown functions $c_j (t)$ ^[20,25]. By a classical ODEs result, system (3.3) possesses a solution $z^p$ on $[0, t_p)$, $0 < t_p\leq T$, for each $p\geq 1$.

Now, we multiply (3.3) by $c^{\prime}_j(t)$ and sum over $j = 1, ..., p$ to get

where

Our choice of initial conditions implies that $E^p(0)$ is uniformly bounded. Let us integrate (3.4) over $(0, t)$, $0 < t < t_p$, which leads to

where $M_1$ represents a positive constant that does not depend on either $t$ or $p$.

Then, we can replace $t_p$ by $T$ and, in addition, we have

Next, we multiply (3.3) by $c^{\prime\prime}_j(t)$ and we sum over $j = 1, ..., p$ to get

With the help of The Cauchy-Schwarz inequality, Young's inequality, and (2.8), we find that

for any $\beta > 0$.

Additionally, Young's inequality leads to

and by (2.3) and the fact that $h$ is lipschitz with $h(0) = 0$, we find

where $C_h$ is the Lipschitz constant for the function $h$.

By proceeding as in the proof of Proposition 2.6, one gets that

Combining (3.8)-(3.13), we see that

By using (2.9), (3.6), and the definition of $E^p(t)$, we easily see that

which yields

and

Integrating (3.14) over $(0, T)$ and using the previous estimates, we derive that

Choosing $\beta$ small enough in (3.15), we infer that

implying that

From (3.7) and (3.17), we can find a subsequence of $(z^p)$, which we continue to label as $(z^p)$, satisfying

and

Analysis of the nonlinear terms:

Due to the compact embedding $W \subset L^2(\Omega)$, it follows that, up to a subsequence,

Since $h$ is Lipschitz continuous, it holds that

On the other hand, the fact that $h$ is Lipschitz and $(z^p)$ is bounded in $L^{\infty}(0, T; L^2(\Omega))$ implies that $h(z^p)$ is bounded in $L^{\infty}(0, T; L^2(\Omega))$. This latter fact combined with (3.20) leads to

By using (3.7), (3.17), and the Aubin-Lions theorem (see ^[19]), we have, up to a subsequence, that

and hence

From (2.3) and (3.6), one gets that

Consequently, by (3.21), (3.22), and the Lion's lemma ^[19,26], we see that

Concerning the term $\gamma(x) f(z^p_t)$, we can see from (3.6) that the sequence $\gamma(x) f(z_t^p)z_t^p$ is bounded in ${L}^2(0, T; {L}^2(\Omega))$. Then, by (2.12), $(f(z_t^p))_{p\geq1}$ is bounded in $L_{\gamma}^{2}(\Omega \times (0, T))$, where $L_{\gamma}^{2}$ is the weighted Lebesgue space. Hence, we get, up to a subsequence, that

The integration of (3.3) over the interval $(0, t)$ leads to

By letting $p\to \infty$ and using ^[14], we get that

Consequently, we conclude that (3.25) holds for all $w\in W$. Moreover, within (3.25), the expressions on the righthand side are absolutely continuous, and hence we can deduce that (3.25) is differentiable for almost every $t\geq 0$. Thus, we derive that

Next, by using the same ideas as in ^[18] (where it was showed that $\chi = \vert u_t\vert^m u_t$ in the proof of Theorem 3.2 of ^[18]), we prove that $\chi = f(z_t)$, and then (3.26) turns to

In terms of the initial conditions, we may also apply (3.18) and (3.19) in a standard way to confirm that

Moreover, we can see that

and

which means that the solution $z(t)$ of (1.1)–(1.3) is bounded and global.

4.   Exponential stability

We start this section by defining the functional

We hen have the following:

Proposition 4.1. Suppose that $0 < \xi_1 < \Lambda$. Then, we have

for some constant $\lambda > 0$.

Proof. Thanks to the Cauchy-Schwarz inequality, Young's inequality, and (2.3), we obtain

with $\lambda = \max\Big\{\left(1+\alpha+\frac{C_{*}^2}{\kappa+1}\right)\frac{1}{1-\xi_1 \Lambda^{-1}}, \alpha+\frac{C_{*}^{2(\kappa+1)}}{\kappa+1} \left(\frac{2E(0)}{\alpha}\right)^{\kappa}\Big\}.$

Let $\eta$ be a positive constant satisfying, for the moment,

From (4.2) and (4.3), the functional $\mathcal{L}$ defined by

satisfies $\mathcal{L}(t)\sim E(t)$, and more precisely we have

Lemma 4.2. Assume (H1)–(H3) and $0 < \xi_1 < \Lambda$. Therefore, there exists a constant $\delta > 0$ such that

Proof. By combining $(1.1)$ and (2.5), we infer that

By (2.3), we can easily check that

Therefore, it holds that

where $M_2 = C_{*}^{\kappa+2} \left(\frac{2E(0)}{\alpha}\right)^{\frac{\kappa}{2}}\Big(\frac{1}{\kappa+2}+ \frac{1}{\kappa+1}\Big).$

By assumption (H3), we see that

Moreover, we have

and then we obtain

Finally, using Proposition 2.6, we have

Using the estimates above, we find that

At this point, we pick up $\eta$ satisfying (4.3) and

and since $B < 2$, then it holds that $1-\eta(1+M_2) > 0, \; 1-\frac{B}{2} > 0, \; \xi_2-\frac{\sigma\eta}{4} > 0$ and $1-\frac{\eta A}{2} > 0$. By taking

we get the desired inequality (4.5).

The following theorem establishes the uniform stability of system (1.1)–(1.3).

Theorem 4.3. Assume (H1)–(H3) and $0 < \xi_1 < \Lambda$. Then, the energy of the solutions of (1.1)–(1.3) decays exponentially, i.e., there exist positive constants $b$ and $\nu$ such that

Proof. By using (4.4) and (4.5), one finds that

which implies that

Integrating the last inequality over $(0, t)$, one has

Consequently,

Once more, relying on (4.4), we arrive at

Hence, (4.8) holds true with $b = \frac{1+\lambda\eta}{1-\lambda\eta}$ and $\nu = \frac{\delta}{1+\lambda\eta}.$

5.   Conclusions

In this paper, we focus on the study of the asymptotic behavior of the energy associated with a nonlinear problem in a rectangular domain, subject to Balakrishnan-Taylor, strong, and localized nonlinear damping and with the presence of a source term. This equation describes the deformation of the deck of either a footbridge or a suspension bridge, which is hinged along its short edges and has free vibrations on the remaining portion of the boundary. As a future work, we can study this problem with the presence of fractional damping ^[27,28,29].

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by Researchers Supporting Project number (RSPD2024R736), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The author declares there is no conflict of interest.