Uniform decay rates of a coupled suspension bridges with temperature

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: with the following initial data: (1.2) and the following boundary conditions: 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 (θ, ϕ) 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: 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 u, v and . 2 . As in [5], we introduce the biharmonic operator A : we consider the compactly nested family of Hilbert spaces generated by the powers of A (r will be always omitted whenever zero) The well-posedness of (1.1, 1.2) is stated in the following theorem.
Proof This theorem can be established using standard method such as Galerkin method following the same steps as in [3]. Now, we introduce the first-order energy associated with (1.1, 1.2) given by (2.1) Our aim in this paper is to prove the uniform decay of the energy using the multipliers techniques.
be the solution of ( 1.1, 1.2). Then, there exist two positive constants C 0 and C 1 , such that where C 0 and C 1 are two positive constants.
The proof of this theorem will be established through several lemmas.
Proof Multiplying the first equation of (1.1) by u t , the second equation by v t , the third equation by θ , the fourth equation by ϕ, integrating by parts over (0, l), and using the boundary conditions (1.3), then summing up, we obtain the desired result. 1, 1.2). Then, the functional satisfies for ε 1 > 0 the following estimate: Proof Using the first and the second equations in (1.1), integrating by parts over (0, l) and using the boundary conditions (1.3), we have Applying Young and Poincaré's inequalities, we get, for any ε 1 > 0 (2.6) by (2.5) and (2.6), we deduce the result.

Lemma 2.7
For N large enough, there exist two positive constants α 1 and α 2 , such that (2.14) Proof Let us define the following functional: Using Cauchy-Schwarz, Young's and Poincaré's inequalities, we obtain Then, by (2.1), we get Consequently which implies that Choosing N large enough, then we have (2.14).
Proof of Theorem 2.2: Differentiating (2.13), and exploiting (2.4), (2.7) and (2.10), we obtain the following estimates: Here, we choose carefully the constants as follows: Now, we choose N large enough and satisfying Consequently, using inequality (2.15) and (2.1), we get where k 0 is a positive constant. A combination of (2.14) and (2.16) gives A simple integration over (0, t) yields 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.
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