Skip to main content

Uniform decay rates of a coupled suspension bridges with temperature

Abstract

In this paper, we investigate the decay properties of the thermoelastic suspension bridges model. We prove that the energy is decaying exponentially. To our knowledge, our result is new and our method of proof is based on the energy method to build the appropriate Lyapunov functional.

Introduction

In recent years, increasing attention has been turned to the analysis of some classical engineering structures describing the dynamics of linear and nonlinear vibrations of suspension bridges. The stabilization of suspension bridges and their oscillations plays an important and crucial role nowadays.

In this paper, we consider the following coupled system of partial differential equations, given by:

$$\begin{aligned} \left\{ \begin{array}{ll} u_{tt}+u_{xxxx}+\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) u+2\left( \int _{0}^{l}uv\ \mathrm{d}x\right) v+\alpha \theta _{xx}=0, &{} in\;\left( 0,\,l\right) \times {\mathbb {R}} _{+}, \\ v_{tt}-v_{xx}+2\left( \int _{0}^{l}uv\ \mathrm{d}x\right) u+\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) v+\beta \varphi _{x}=0, &{} in\;\left( 0,\,l\right) \times {\mathbb {R}} _{+}, \\ \theta _{t}-\theta _{xx}-\alpha u_{txx}=0, &{} in\;\left( 0,\,l\right) \times {\mathbb {R}} _{+}, \\ \varphi _{t}-\varphi _{xx}+\beta v_{tx}=0, &{} in\;\left( 0,\,l\right) \times {\mathbb {R}} _{+}, \end{array} \right. \end{aligned}$$
(1.1)

with the following initial data:

$$\begin{aligned} \left( u,\,u_{t},v,\,v_{t},\,\theta ,\,\varphi \right) (x,\,0)=\left( u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0}\right) , \end{aligned}$$
(1.2)

and the following boundary conditions:

$$\begin{aligned} \begin{array}{l} u(0,\,t)=u(l,\,t)=u_{xx}(0,\,t)=u_{xx}(l,\,t)=v(0,\,t)=v(l,\,t)=0,\; \\ \theta (0,\,t)=\theta (l,\,t)=\varphi _{x}(0,\,t)=\varphi _{x}(l,\,t)=0, \end{array} \,\forall t\ge 0. \end{aligned}$$
(1.3)

The given system models a one-dimensional suspension bridge with thermal effects, where u is the vertical displacement,  v is the torsional angle, and the couple (\(\theta ,\) \(\varphi )\) is the two temperatures. We need to mention here that the modeling for the first two coupled equations in (1.1) was considered in [7] with intermediate piers, and the authors proved the well-posedness of the problem and investigated the stability of the solutions.

In recent years, the analysis and the stability of various nonlinear suspension bridge models have been attracting many researchers (see [1,2,3,4,5,6,7,8,9,10]) and the references therein.

In this paper, our interest is the stability properties of the coupling model (1.1). Therefore, from the mathematical point of view, it is important to study the stability properties of this system and determine whether the indirect dissipation coming via the two temperatures is enough to stabilize the whole system.

We prove that the coupling with the two temperatures is strong in a way that the solutions of System (1.1) decay exponentially. Our method of proof is based on multipliers’ techniques and the construction of the appropriate Lyapunov functional.

The paper is organized as follows. In Sect. 2, we state and prove the exponential stability result for System (1.1). We give some concluding remarks in Sect. 3.

Exponential stability

In this section, we state and prove the exponential stability result for (1.1) with the boundary and initial conditions given by (1.3) and (1.2).

Throughout this paper, c is used to denote a generic positive constant. First, we introduce the following Hilbert spaces:

$$\begin{aligned} \begin{array}{l} {\widehat{H}}=\left\{ \varphi \in L^{2}(0,\,l):\int _{0}^{l}\varphi \left( x\right) \mathrm{d}x=0\right\} ,\ {\widehat{H}}_{1}=H^{1}(0,\,l)\cap {\widehat{H}},\ \\ H_{0}^{2}(0,\,l)=H^{2}(0,\,l)\cap H_{0}^{1}(0,\,l), \text { and } H_{4}\!=\!\left\{ u\in H^{4}(0,\,l):u(0,\,t)=u(l,\,t)=u_{xx}(0,\,t)=u_{xx}(l,\,t)\!=\!0\right\} , \\ {\mathcal {H}}\!=\!H_{0}^{2}(0,\,l)\times L^{2}(0,\,l)\times H_{0}^{1}(0,\,l)\times L^{2}(0,\,l)\times L^{2}(0,\,l)\times {\widehat{H}}. \end{array} \end{aligned}$$

The space \(L^{2}(0,l)\) is the pivot space and the related standard inner product and norm will be denoted along the paper by \(\left\langle u,\,v\right\rangle \) and \(\left\| .\right\| _{2}\).

As in [5], we introduce the biharmonic operator A : \( D\left( A\right) \subset L^{2}(0,\,l)\longrightarrow L^{2}(0,\,l)\) defined as

$$\begin{aligned} Au=u_{xxxx}\text { with }D\left( A\right) =H_{4}; \end{aligned}$$

we consider the compactly nested family of Hilbert spaces generated by the powers of A (r will be always omitted whenever zero)

$$\begin{aligned} H_{r}=D\left( A^{\frac{r}{4}}\right) ,\qquad \left\langle u,\,v\right\rangle _{r}=\left\langle A^{\frac{r}{4}}u,\,A^{\frac{r}{4}}v\right\rangle ,\qquad \left\| u\right\| _{r}=\left\| A^{\frac{r}{4}}u\right\| _{2},\qquad for\;r\in {\mathbb {R}} . \end{aligned}$$

The well-posedness of (1.1, 1.2) is stated in the following theorem.

Theorem 2.1

Assume that \((u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0})\in {\mathcal {H}}\). Then, Problem (1.1, 1.2) admits a unique weak solution \((u,\,u_{t},\,v,\,v_{t},\,\theta ,\,\varphi )\in C([0,\,\infty ),{\mathcal {H}}).\)

Proof

This theorem can be established using standard method such as Galerkin method following the same steps as in [3]. \(\square \)

Now, we introduce the first-order energy associated with (1.1, 1.2) given by

$$\begin{aligned} E\left( t\right) :=\frac{1}{2}\left( \int _{0}^{l}(u_{t}^{2}+v_{t}^{2}+u_{xx}^{2}+v_{x}^{2})\mathrm{d}x+\int _{0}^{l}(\theta ^{2}+\varphi ^{2})\mathrm{d}x+\frac{1}{2}\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \,\mathrm{d}x\right) ^{2}+2\left( \int _{0}^{l}uv\,\mathrm{d}x\right) ^{2}\right) . \nonumber \\ \end{aligned}$$
(2.1)

Our aim in this paper is to prove the uniform decay of the energy using the multipliers techniques.

Theorem 2.2

Let \((u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0})\in {\mathcal {H}}\) and \((u,\,v,\,\theta ,\,\varphi )\) be the solution of ( 1.1, 1.2). Then, there exist two positive constants \( C_{0}\) and \(C_{1}\), such that

$$\begin{aligned} E\left( t\right) \le C_{1}\,e^{-C_{0}\,t}\qquad \forall \ge 0, \end{aligned}$$
(2.2)

where \(C_{0}\) and \(C_{1}\) are two positive constants.

The proof of this theorem will be established through several lemmas.

Lemma 2.3

Let \((u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0})\in {\mathcal {H}}\) and \(\left( u,\,v,\,\theta ,\,\varphi \right) \) be the solution of (1.1, 1.2). Then, the energy functional E defined by (2.1) satisfies

$$\begin{aligned} \frac{d}{dt}E\left( t\right) =-\frac{1}{2}\left\| \theta _{x}\right\| _{2}^{2}-\frac{1}{2}\left\| \varphi _{x}\right\| _{2}^{2}. \end{aligned}$$
(2.3)

Proof

Multiplying the first equation of (1.1) by \(u_{t}\), the second equation by \(v_{t}\), the third equation by \(\theta \), the fourth equation by \(\varphi \), integrating by parts over \(\left( 0,\,l\right) \), and using the boundary conditions (1.3), then summing up, we obtain the desired result. \(\square \)

Lemma 2.4

Let \((u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0})\in {\mathcal {H}}\) and \((u,\,v,\,\theta ,\,\varphi )\) be the solution of (1.1, 1.2). Then, the functional

$$\begin{aligned} I_{1}\left( t\right) :=\int _{0}^{l}u\,u_{t}\,\mathrm{d}x+\int _{0}^{l}v\,v_{t}\,\mathrm{d}x, \end{aligned}$$

satisfies for \(\varepsilon _{1}>0\) the following estimate:

$$\begin{aligned} I_{1}^{\prime }\left( t\right)\le & {} \left\| u_{t}\,\right\| _{2}^{2}+\left\| v_{t}\right\| _{2}^{2}-\left( 1-\varepsilon _{1}\right) \left\| u_{xx}\,\right\| _{2}^{2}-\left( 1-\varepsilon _{1}\right) \left\| v_{x}\right\| _{2}^{2} \nonumber \\&-\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}-4\left( \int _{0}^{l}uv\right) ^{2}+c\left\| \theta _{x}\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}. \end{aligned}$$
(2.4)

Proof

Using the first and the second equations in (1.1), integrating by parts over \(\left( 0,\,l\right) \) and using the boundary conditions (1.3), we have

$$\begin{aligned} I_{1}^{\prime }\left( t\right)= & {} \left\| u_{t}\,\right\| _{2}^{2}+\left\| v_{t}\right\| _{2}^{2}-\left\| u_{xx}\,\right\| _{2}^{2}-\left\| v_{x}\right\| _{2}^{2} \nonumber \\&-\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}-4\left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}-\int _{0}^{l}\theta u_{xx}\ \mathrm{d}x+\int _{0}^{l}\varphi v_{x}\ \mathrm{d}x. \end{aligned}$$
(2.5)

Applying Young and Poincaré’s inequalities, we get, for any \(\varepsilon _{1}>0\)

$$\begin{aligned} \begin{array}{c} \left| \int _{0}^{l}\theta u_{xx}\ \mathrm{d}x\right| \le \varepsilon _{1}\left\| u_{xx}\,\right\| _{2}^{2}+c\left\| \theta \,\right\| _{2}^{2}\le \varepsilon _{1}\left\| u_{xx}\,\right\| _{2}^{2}+c\left\| \theta _{x}\,\right\| _{2}^{2}, \\ \left| \int _{0}^{l}\varphi v_{x}\ \mathrm{d}x\right| \le \varepsilon _{1}\left\| v_{x}\,\right\| _{2}^{2}+c\left\| \varphi \,\right\| _{2}^{2}\le \varepsilon _{1}\left\| v_{x}\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}; \end{array} \end{aligned}$$
(2.6)

by (2.5) and (2.6), we deduce the result. \(\square \)

Lemma 2.5

Let \((u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0})\in {\mathcal {H}}\) and \(\left( u,\,v,\,\theta ,\,\varphi \right) \) be the solution of (1.1, 1.2). Then, the functional

$$\begin{aligned} I_{2}(t)=\int _{0}^{l}v_{t}\,(\int _{0}^{x}\varphi \left( y,t\right) \mathrm{{d}}y)\ \mathrm{d}x, \end{aligned}$$

satisfies for \(\varepsilon _{2},\,\varepsilon _{3},\,\varepsilon _{4}>0\), the following estimate:

$$\begin{aligned} I_{2}^{\prime }\left( t\right)\le & {} -\left( 1-\varepsilon _{2}\right) \left\| v_{t}\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}+\varepsilon _{3}\left\| v_{x}\,\right\| _{2}^{2} \nonumber \\&+4\varepsilon _{4}\,E\left( 0\right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}+\varepsilon _{4}\,E\left( 0\right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}. \end{aligned}$$
(2.7)

Proof

Using the first and the fourth equations in (1.1), integrating by parts over \(\left( 0,\,l\right) \) and using the boundary conditions (1.3), we have

$$\begin{aligned} I_{2}^{\prime }\left( t\right)= & {} \int _{0}^{l}v_{tt}(\int _{0}^{x}\varphi \left( y,t\right) \mathrm{{d}}y)\;\mathrm{d}x+\int _{0}^{l}v_{t}(\int _{0}^{x}\varphi _{t}\left( y,t\right) \mathrm{{d}}y)\;\mathrm{d}x \nonumber \\= & {} -\int _{0}^{l}v_{x}\varphi \;\mathrm{d}x-2\left( \int _{0}^{l}uv\;\mathrm{d}x\right) \int _{0}^{l}u(\int _{0}^{x}\varphi \left( y,t\right) \mathrm{{d}}y)\;\mathrm{d}x \nonumber \\&-\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) \int _{0}^{l}v(\int _{0}^{x}\varphi \left( y,t\right) \mathrm{{d}}y)\;\mathrm{d}x+\left\| \varphi \right\| _{2}^{2}-\left\| v_{t}\right\| _{2}^{2}+\int _{0}^{l}v_{t}\varphi _{x}\ \mathrm{d}x. \end{aligned}$$
(2.8)

Using Cauchy–Schwarz, Young and Poincaré’s inequalities, we get

$$\begin{aligned} \begin{aligned} \left| \int _{0}^{l}v_{x}\varphi \ \mathrm{d}x\right|&\le \varepsilon _{3}\left\| v_{x}\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}, \\ \left| \int _{0}^{l}v_{t}\varphi _{x}\ \mathrm{d}x\right|&\le \varepsilon _{2}\left\| v_{t}\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}, \\ \left| \left( \int _{0}^{l}uv\ \mathrm{d}x\right) \int _{0}^{l}u(\int _{0}^{x}\varphi \left( y,t\right) \mathrm{{d}}y)\ \mathrm{d}x\right|&\le \varepsilon _{4}\left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}\left\| u\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2} \\&\le 2\varepsilon _{4}\,E\left( 0\right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}, \\ \left| \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) \int _{0}^{l}v(\int _{0}^{x}\varphi \left( y,t\right) \mathrm{{d}}y)\ \mathrm{d}x\right|&\le \dfrac{\varepsilon _{4}}{2}\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}\left\| v\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2} \\&\le \varepsilon _{4}\,E\left( 0\right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}; \end{aligned} \end{aligned}$$
(2.9)

then from (2.8) and (2.9), we deduce that

$$\begin{aligned} I_{2}^{\prime }\left( t\right)\le & {} -\left( 1-\varepsilon _{2}\right) \left\| v_{t}\,\right\| _{2}^{2}+c\left\| \varphi _{x}\,\right\| _{2}^{2}+\varepsilon _{3}\left\| v_{x}\,\right\| _{2}^{2} \\&+4\varepsilon _{4}\,E\left( 0\right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}+\varepsilon _{4}\,E\left( 0\right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}. \end{aligned}$$

\(\square \)

Lemma 2.6

Let \((u_{0},\,u_{1},\,v_{0},\,v_{1},\,\theta _{0},\,\varphi _{0})\in {\mathcal {H}}\) and \(\left( u,\,v,\,\theta ,\,\varphi \right) \) be the solution of (1.1, 1.2). Then, the functional

$$\begin{aligned} I_{3}(t)=\left\langle u_{t},\,\theta \right\rangle _{-1}:=\int _{0}^{l}(\partial _{xxxx})^{\frac{-1}{4}}u_{t}\ (\partial _{xxxx})^{\frac{-1}{4}}\theta \ \mathrm{d}x, \end{aligned}$$

satisfies for \(\varepsilon _{5},\,\varepsilon _{6},\,\varepsilon _{7}>0\), the following estimate:

$$\begin{aligned} I_{3}^{\prime }\left( t\right)\le & {} -\left( 1-\varepsilon _{6}\right) \left\| u_{t}\,\right\| _{2}^{2}+\varepsilon _{5}\left\| u_{xx}\,\right\| _{2}^{2}+c\left\| \theta _{x}\right\| _{2}^{2} \nonumber \\&+\varepsilon _{7}E\left( 0\right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}+4\varepsilon _{7}E\left( 0\right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}. \end{aligned}$$
(2.10)

Proof

Using the first and the third equations in (1.1), integrating by parts over \(\left( 0,\,l\right) \) and taking into consideration the boundary conditions (1.3), we obtain

$$\begin{aligned} I_{3}^{\prime }\left( t\right)= & {} \left\langle u_{tt},\,\theta \right\rangle _{-1}+\left\langle u_{t},\,\theta _{t}\right\rangle _{-1} \nonumber \\= & {} -\left\langle u_{xxxx},\,\theta \right\rangle _{-1}-\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \ \mathrm{d}x\right) \left\langle u,\,\theta \right\rangle _{-1}-2\left( \int _{0}^{l}uv\ \mathrm{d}x\right) \left\langle v,\,\theta \right\rangle _{-1} \nonumber \\&-\left\langle \theta _{xx},\,\theta \right\rangle _{-1}+\left\langle u_{t},\,\theta _{xx}\right\rangle _{-1}+\left\langle u_{t},\,u_{txx}\right\rangle _{-1} \nonumber \\= & {} \int _{0}^{l}u_{xx}\theta \ \mathrm{d}x-\left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) \left\langle u,\,\theta \right\rangle _{-1}-2\left( \int _{0}^{l}uv\ \mathrm{d}x\right) \left\langle v,\,\theta \right\rangle _{-1} \nonumber \\&+\left\| \theta \right\| _{2}^{2}-\int _{0}^{l}u_{t}\theta \ \mathrm{d}x-\left\| u_{t}\,\right\| _{2}^{2}, \end{aligned}$$
(2.11)

by exploiting the following inequalities:

$$\begin{aligned} \begin{aligned} \left| \int _{0}^{l}u_{xx}\theta \ \mathrm{d}x\right|&\le \varepsilon _{5}\left\| u_{xx}\,\right\| _{2}^{2}+c\left\| \theta \right\| _{2}^{2}\le \varepsilon _{5}\left\| u_{xx}\,\right\| _{2}^{2}+c\left\| \theta _{x}\right\| _{2}^{2}, \\ \left| \int _{0}^{l}u_{t}\theta \ \mathrm{d}x\right|&\le \varepsilon _{6}\left\| u_{t}\,\right\| _{2}^{2}+c\left\| \theta _{x}\right\| _{2}^{2}, \\ \left| \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) \left\langle u,\theta \right\rangle _{-1}\right|&\le \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) \left\| u\,\right\| _{2}^{2}\left\| \theta \right\| _{2}^{2} \\&\le \varepsilon _{7}E\left( 0\right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}+c\left\| \theta _{x}\right\| _{2}^{2}, \\ \left| \left( \int _{0}^{l}uv\ \mathrm{d}x\right) \left\langle v,\theta \right\rangle _{-1}\right|&\le \left| \left( \int _{0}^{l}uv\ \mathrm{d}x\right) \right| \left\| v\,\right\| _{2}^{2}\left\| \theta \right\| _{2}^{2} \\&\le 2\varepsilon _{7}E\left( 0\right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}+c\left\| \theta _{x}\right\| _{2}^{2}, \end{aligned} \end{aligned}$$
(2.12)

we deduce from (2.11) and (2.12) that

$$\begin{aligned} I_{3}^{\prime }\left( t\right)\le & {} -\left( 1-\varepsilon _{6}\right) \left\| u_{t}\,\right\| _{2}^{2}+\varepsilon _{5}\left\| u_{xx}\,\right\| _{2}^{2}+c\left\| \theta _{x}\,\right\| _{2}^{2} \\&+\varepsilon _{7}E\left( 0\right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2}+4\varepsilon _{7}E\left( 0\right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2}. \end{aligned}$$

\(\square \)

To complete the proof of our main result, we introduce the following Lyapunov functional:

$$\begin{aligned} {\mathcal {L}}\left( t\right) :=N\,E\left( t\right) +N_{1}I_{1}\left( t\right) +N_{2}I_{2}\left( t\right) +N_{3}I_{3}\left( t\right) , \end{aligned}$$
(2.13)

where N, \(N_{1}\), \(N_{2}\), and \(N_{3}\) are positive constants to be fixed later.

Lemma 2.7

For N large enough, there exist two positive constants \(\alpha _{1}\) and \(\alpha _{2}\), such that

$$\begin{aligned} \alpha _{1}E\left( t\right) \le {\mathcal {L}}\left( t\right) \le \alpha _{2}E\left( t\right) . \end{aligned}$$
(2.14)

Proof

Let us define the following functional:

$$\begin{aligned} {\mathcal {L}}_{1}\left( t\right) =N_{1}I_{1}\left( t\right) +N_{2}I_{2}\left( t\right) +N_{3}I_{3}\left( t\right) . \end{aligned}$$

Using Cauchy–Schwarz, Young’s and Poincaré’s inequalities, we obtain

$$\begin{aligned} \left| {\mathcal {L}}_{1}\left( t\right) \right|\le & {} \frac{N_{1}}{2} \left\| u\,\right\| _{2}^{2}+\frac{1}{2}\left[ N_{1}+N_{3}\right] \left\| u_{t}\,\right\| _{2}^{2} \\&+\frac{N_{1}}{2}\left\| v\,\right\| _{2}^{2}+\frac{1}{2}\left[ N_{1}+N_{2}\right] \left\| v_{t}\,\right\| _{2}^{2} \\&+cN_{2}\left\| \varphi \,\right\| _{2}^{2}+cN_{3}\left\| \theta \right\| _{2}^{2}. \end{aligned}$$

Then, by (2.1), we get

$$\begin{aligned} \left| {\mathcal {L}}_{1}\left( t\right) \right| \le c\,E\left( t\right) . \end{aligned}$$

Consequently

$$\begin{aligned} \left| {\mathcal {L}}\left( t\right) -N\,E\left( t\right) \right| \le c\,E\left( t\right) , \end{aligned}$$

which implies that

$$\begin{aligned} \left( N-c\right) E\left( t\right) \le {\mathcal {L}}\left( t\right) \le \left( N+c\right) E\left( t\right) . \end{aligned}$$

Choosing N large enough, then we have (2.14). \(\square \)

Proof of Theorem 2.2:

Differentiating (2.13), and exploiting (2.4), (2.7) and (2.10), we obtain the following estimates:

$$\begin{aligned} {\mathcal {L}}^{\prime }\left( t\right)\le & {} \,-\left( \frac{N}{2}-\left( N_{1}+N_{3}\right) \,c\right) \left\| \theta _{x}\right\| _{2}^{2}-\left( \frac{N}{2}-\left( N_{1}+N_{2}\right) \,c\right) \left\| \varphi _{x}\right\| _{2}^{2} \nonumber \\&-\left( N_{1}-\left( \varepsilon _{4}\,N_{2}+\varepsilon _{7}N_{3}\right) E\left( 0\right) \right) \left( \int _{0}^{l}\left( u^{2}+v^{2}\right) \mathrm{d}x\right) ^{2} \nonumber \\&-4\left( N_{1}-\left( \varepsilon _{4}\,N_{2}+\varepsilon _{7}N_{3}\right) E\left( 0\right) \right) \left( \int _{0}^{l}uv\ \mathrm{d}x\right) ^{2} \nonumber \\&-\left( \left( 1-\varepsilon _{1}\right) N_{1}-\varepsilon _{5}N_{3}\right) \left\| u_{xx}\,\right\| _{2}^{2}-\left( \left( 1-\varepsilon _{1}\right) N_{1}-\varepsilon _{3}N_{2}\right) \left\| v_{x}\right\| _{2}^{2} \nonumber \\&-\left( \left( 1-\varepsilon _{2}\right) N_{2}-N_{1}\right) \left\| v_{t}\,\right\| _{2}^{2}-\left( \left( 1-\varepsilon _{6}\right) N_{3}-N_{1}\right) \left\| u_{t}\,\right\| _{2}^{2}. \end{aligned}$$
(2.15)

Here, we choose carefully the constants as follows:

$$\begin{aligned} \begin{array}{l} \varepsilon _{4}=\dfrac{N_{1}}{4N_{2}E\left( 0\right) },\;\varepsilon _{7}= \dfrac{N_{1}}{4N_{3}E\left( 0\right) },\varepsilon _{1}=\dfrac{1}{4} ,\;\varepsilon _{3}=\dfrac{N_{1}}{4N_{2}},\varepsilon _{5}=\dfrac{ N_{1}}{4N_{3}}, \\ N_{1}=1,\,N_{2}=2N_{1},\ \;\varepsilon _{2}=\dfrac{1}{4}{, } N_{3}=2N_{1},\text { and }\;\varepsilon _{6}=\dfrac{1}{4}\text {.} \end{array} \; \end{aligned}$$

Now, we choose N large enough and satisfying

$$\begin{aligned} \frac{N}{2}>3N_{1}c. \end{aligned}$$

Consequently, using inequality (2.15) and (2.1), we get

$$\begin{aligned} {\mathcal {L}}^{\prime }\left( t\right) \le -k_{0}E\left( t\right) ,\ \ \ \forall t>0, \end{aligned}$$
(2.16)

where \(k_{0}\) is a positive constant.

A combination of (2.14) and (2.16) gives

$$\begin{aligned} {\mathcal {L}}^{\prime }\left( t\right) \le -C_{0}{\mathcal {L}}\left( t\right) ;\qquad \forall t>0. \end{aligned}$$

A simple integration over \((0,\,t)\) yields

$$\begin{aligned} {\mathcal {L}}\left( t\right) \le {\mathcal {L}}\left( 0\right) \,e^{-C_{0}\,t};\qquad \forall t>0. \end{aligned}$$
(2.17)

Thus, using (2.14) and (2.17), the conclusion of our main result follows.

Concluding Remark

In this work, we proved that the coupling of the suspension bridges model with two temperatures is strong enough to stabilize exponentially the solutions of the model. It will be interesting to study the case where the suspension bridges are coupled with only one heat equation.

References

  1. 1.

    Ahmed, N.U.; Harbi, H.: Mathematical analysis of dynamic models of suspension bridges. SIAM J. Appl. Math. 109, 853–874 (1998)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Arioli, G.; Gazzola, F.: Torsional instability in suspension bridges: the Tacoma Narrows Bridge case. Commun. Nonlinear Sci. Numer. Simul. 42, 342–357 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Berchio, E.; Gazzola, F.: A qualitative explanation of the origin of torsional instability in suspension bridges. Nonlinear Anal. TMA 121, 54–72 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bochicchio, I.; Giorgi, C.; Vuk, E.: Long-term damped dynamics of the extensible suspension bridge. Int. J. Differ. Equ. 383420, 19 (2010)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Dell’Oro, F.; Giorgi, C.: Global attractors for the coupled suspension bridge system with temperature. Math. Methods Appl. Sci. 39, 864–875 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dell’Oro, F.; Giorgi, C.; Pata, V.: Asymptotic behavior of coupled linear systems modeling suspension bridges. Z. Angew. Math. Phys. 66, 1095–1108 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Garrione, M.; Gazzola, F.: Nonlinear Equations for Beams and Degenerate Plates with Piers. Springer, Berlin (2019)

    Book  Google Scholar 

  8. 8.

    Gazzola, F., Soufyane, A.: Long-time behavior of partially damped systems modeling degenerate plates with piers (Accepted in Nonlinearity, 2021)

  9. 9.

    Holubova, G.; Janoušek, J.: One-dimensional model of a suspension bridge: revision of uniqueness results. Appl. Math. Lett. 71, 6–13 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Giorgi, C.; Vuk, E.: Steady-state solutions for a suspension bridge with intermediate supports. Bound. Value Probl. 2013, 204 (2013)

    MathSciNet  Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Abdelaziz Soufyane.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Afilal, M., Alahyane, M. & Soufyane, A. Uniform decay rates of a coupled suspension bridges with temperature. Arab. J. Math. 10, 505–511 (2021). https://doi.org/10.1007/s40065-021-00342-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40065-021-00342-0

Mathematics Subject Classification

  • 35B40
  • 74D99
  • 93D15
  • 93D20