Global Attractor of Thermoelastic Coupled Beam Equations with Structural Damping

Copyright © 2017 Peirong Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we study the existence of a global attractor for a class of n-dimension thermoelastic coupled beam equations with structural damping utt +Δ2u+Δ2ut − [σ(∫Ω (∇u)2dx) + φ(∫Ω ∇u∇utdx)]Δu+f1(u) + g(ut) + ]Δθ = q(x), in Ω×R+, and θt −Δθ+ f2(θ) − ]Δut = 0. Here Ω is a bounded domain of RN, and σ(⋅) and φ(⋅) are both continuous nonnegative nonlinear real functions and q is a static load.The source terms f1(u) and f2(θ) and nonlinear external damping g(ut) are essentially |u|ρu, |θ|󰜚θ, and |ut|rut respectively.


Introduction
This problem is based on the equation which was proposed by Woinowsky-Krieger [1] as a model for vibrating beam with hinged ends.Without thermal effects, Ball [2] studied the initialboundary value problem of more general beam equation subjected to homogeneous boundary condition.Ma and Narciso [3] proved the existence of global solutions and the existence of a global attractor for the Kirchhoff-type beam equation without structural damping, subjected to the conditions In fact, the plate equations without thermal effects were studied by several authors; we quote, for instance, [4][5][6][7][8].
In the following we also make some comments about previous works for the long-time dynamics of thermoelastic coupled beam system with thermal effects.
Giorgi et al. [9] studied a class of one-dimensional thermoelastic coupled beam equations and gave the existence and uniqueness of global weak solution and the existence of global attractor under Dirichlet boundary conditions.Barbosa and Ma [10] studied the long-time behavior for a class of two-dimension thermoelastic coupled beam equation In addition, we also refer the reader to [11][12][13][14][15] and the references therein.
A mathematical problem is the nonlinear -dimension thermoelastic coupled beam equations with structural damping which arise from the model of the nonlinear vibration beam with Fourier thermal conduction law: with the initial conditions  (, 0) =  0 () ,   (, 0) =  1 () ,  (, 0) =  0 () (10) and the boundary conditions To the our best knowledge, the existence of global attractor for thermoelastic coupled beam equations was not considered in the presence of nonlinear structure damping.Here the unknown function (, ) is the elevation of the surface of beam;  0 () and  1 () are the given initial value functions; the subscript  denotes derivative with respect to  and the assumptions on nonlinear functions (⋅), (⋅),  1 (⋅),  2 (⋅), (⋅), and the external force function () will be specified later.
Under the above assumptions, we prove the existence of global solutions and the existence of a global attractor of extensible beam equation system ( 8)- (11).And the paper is organized as follows.In Section 2, we introduce some Sobolev spaces.In Section 3, we discuss the existence and uniqueness of global strong solution and weak solution.In Sections 4 and 5, we establish the result of the existence of a global attractor.

Basic Spaces
Our analysis is based on the following Sobolev spaces.Let Then for regular solutions we consider the phase space In the case of weak solutions we consider the phase space In  0 we adopt the norm defined by

The Existence of Global Solutions
Firstly, using the classical Galerkin method, we can establish the existence and uniqueness of regular solution to problem (8)- (11).We state it as follows.
Theorem 7.Under assumptions ( 1 )-( 6 ), for any initial data ( 0 ,  1 ,  0 ) ∈  1 , then problem ( 8)-( 11) has a unique regular solution (, ) with Proof.Let us consider the variational problem associated with ( 8)-( 11): find for all  ∈  2 0 (Ω) and  ∈ .This is done with the Galerkin approximation method which is standard.Here we denote the approximate solution by (  (),   ()).We can get the theorem by proving the existence of approximation solution, the estimate of approximation solution, convergence, and uniqueness.In the following we give the estimates of approximation solution and the proof of uniqueness of solution.
Estimate 1.In the first approximate equation and the second approximate equation of (25), respectively putting  =    () and  =   () and making a computation of addition and considering σ() = ∫  0 () and f1 () = ∫  0  1 (), by using Schwarz inequality, and then integrating from 0 to  <   , we see that With the estimates 1-2 and 4-5, we can get the necessary compactness in order to pass approximate equation of (25) to the limit.Then it is a matter of routine to conclude the existence of global solutions in [0, ].
Uniqueness.Let (, ), (V, θ) be two solutions of ( 8)-( 11) with the same initial data.Then writing  =  − V,  =  − θ and taking the difference (25) with  = ,  =  and  = V,  = θ and respectively replacing ,  by   ,  and then making a computation of addition, we have where Using Mean Value Theorem and the Young inequalities combined with the estimates 1-2 and 4-5, we deduce that for some constant Then from Gronwall's Lemma we see that  = V,  = θ.The proof of Theorem 7 is completed.
Theorem 8.Under the assumptions of Theorem 7, if the initial data ( 0 ,  1 ,  0 ) ∈  0 , there exists a unique weak solution of problem ( 8)- (11) which depends continuously on initial data with respect to the norm of  0 .
Proof.By using density arguments, we can obtain the existence of a weak solution in  0 .Let us consider { 0 ,  1 ,  0 } ∈  1 .Since  1 is dense in  0 , then there exists We observe that for each  ∈ , there exists (  ,   ), smooth solution of the initial-boundary value problem ( 8)-( 11) which satisfies where  0 is a positive constant independent of  ∈ .
Defining  , =   −   , Z, =   − θ : ,  ∈ , following the steps already used in the uniqueness of regular solution for ( 8)- (11), and considering the convergence given in (34), we deduce that there exists (, ) such that From the above convergence, we can pass to the limit using standard arguments in order to obtain Theorem 8 is proved.
where  is a constant depending on the initial data in different expression.
In addition, in this paper,  denotes different constant in different expression.

The Existence of Absorbing Set
The main result of an absorbing set reads as follows.
Theorem 13.Assume the hypotheses of Theorem 8; then the corresponding semigroup () of problem ( 8)- (11) has an absorbing set B in  0 .
then (70) can be rewritten as Using Nakao's Lemma 11, we conclude that As  → ∞, the first term of the right side of (74) goes to zero; thus, with Ẽ(), we conclude is an absorbing set for () in  0 .

The Existence of a Global Attractor
The main result of a global attractor reads as follows.
Proof.We are going to apply Lemmas 11 and 12 to prove the asymptotic smooth.Given initial data ( 0 ,  1 ,  0 ) and ) in a bounded invariant set  ⊂  0 , let (, ), (V, θ) be the corresponding weak solutions of problem ( 8)- (11).Then the differences  =  − V,  =  − θ are the weak solutions of where where Let us estimate the right hand side of (79).