Global Attractor for Coupled Beam Equations with Nonhomogeneous Boundaries Conditions

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Introduction
In this paper, we are concerned with the longtime dynamics for the 1-dimensional nonlinear thermoelastic coupled beam equations with strong damping and past history thermal memory for every ∈ [0, ]. This problem arises from a model of the nonlinear thermoelastic coupled vibration beam with a nonlinear damping acting on the = end, which simultaneously considers the medium damping, the viscous effect, the nonlinear constitutive relation, and thermoelasticity based on a theory of non-Fourier heat flux.
Here the unknown functions ( , ) and ( , ) are the elevation of the surface of beam and vertical component of the temperature gradient, respectively, 0 ( ), 1 ( ) and 0 ( ) are the given initial value functions, the subscripts and denote the derivatives with respect to and , respectively, (⋅) is the nonlinearity of the material and continuous nonnegative nonlinear real function, ( ( )) and ( ) and ( ( )) are essentially | ( )| ( ) and | | and | ( )| ( ), respectively, with , , > 0, : + → is memory 2 Mathematical Problems in Engineering kernel and assumed to be a positive bounded convex function vanishing at infinity, and ℎ( ) is the lateral load distribution, ( ) is the external heat supply. The detailed assumptions on nonlinear functions (⋅), (⋅), (⋅), and (⋅) and the external force functions ℎ( ) and ( ) will be specified later.
It is well known that the infinite dimensional dynamical systems determined by the elastic beam are different because of the difference of boundary conditions. Berti et al. [1] and Fastovska T. [2] and Giorgi et al. [3] and Wu [4] all considered the longtime dynamics of one-dimensional thermoelastic coupled beam equations under null boundary conditions. Barbose et al. [5] studied the longtime behavior for a class of two-dimensional thermoelastic coupled beam equations subjected to the hinged conditions Dell'Oro et al. [6] dealt with the long-term properties of the thermoelastic nonlinear string-beam system related to the well-known Lazer-McKenna suspension bridge model We also refer the reader to [7][8][9][10][11] and the references therein for thermoelastic coupled beam equations with null boundary conditions. However, abovementioned longtime dynamics for the thermoelastic coupled beam equations are all subjected to null boundary conditions. Under nonlinear boundary conditions, we refer the reader to the following works. One of the first studies in this direction was done by Pazoto and Perla Menzala [12], where stabilization of a thermoelastic extensible beam was considered. Motivated by the result, Ma proved the existence of global solutions and the existence of a global attractor for the Kirchhoff-type beam equation in [13] and [14], respectively. In addition, we also found the works [15].
In this paper, we use the method of the asymptotically compact property of the solution semigroup ( ) to prove the existence of a global attractor for the system (1)-(6).
To prove the main result, we need the following Lemmas.

Basic Space and Global Solutions
First we proceed as in Barbose et al. [5], Giorgi [18], and Dafermos [19] and define a new variable = ( , ) by From the definition of , for all ≥ 0, we have ( , 0) = 0 in [0, ], ∈ + and 0 ( ) = 0 ( ) in [0, ], ∈ + , where 0 ( ) = ∫ 0 0 ( ) , ∈ + . Differentiate (21) with respect to and on both sides, respectively, and make the sum to get So Therefore thermal memory can be rewritten to be Then, from the assumption (19) of kernel ( ), problems (1)- (6) are transformed into the new system with the initial conditions and nonhomogeneous boundary conditions and with respect to the new variable , we define the weighted space which is a Hilbert space with innerproduct and the norm defined by 4

Mathematical Problems in Engineering
Motivated by the boundary condition (4), we assume, for regular solutions, that data 0 and 1 satisfy the compatibility condition Then for regular solutions we consider the phase space In the case of weak solutions we consider the phase space In H 0 we adopt the norm defined by Using the classical Galerkin method, we can establish the existence and uniqueness of regular solution and weak solution to problems (26)-(31) as in the work of Ma et al. [14] and Cavalcanti et al. [20].

(39)
In addition, if the initial data ( 0 , 1 , 0 , 0 ) ∈ H 0 , there exists a unique weak solution ( , , ) of problems (26)-(31) such that which depends continuously on initial data with respect to the norm of H 0 . What is more, in both cases, is that where is a constant and denotes different constant in different expression in this paper.
The existence of weak solutions follows from density arguments as shown in [14,21]. Theorem

Global Attractor
The main result of this paper reads as follows.

Theorem 4. Assume the hypotheses of Theorem 3; then the corresponding semigroup ( ) of problem (26)-(31) has an
Proof. Now we show that semigroup ( ) has as absorbing set B in H 0 . Firstly, we can calculate the total energy functional Also, since (0) = (0) = 0, the following inequalities hold: and Let us fix an arbitrary bounded set ⊂ H 0 and consider the solutions of problems (26)-(31) given by Our analysis is based on the modified energy functioñ where 1 > 0 is the first eigenvalue of the operator in 2 (0, ). It is easy to see that̃( ) dominates ‖( ( ), ( ), ( ))‖ 2 H 0 and ‖ ( )‖ 2 ≤ 4̃( ). By multiplying (26) by and integrating over [0, ] and inserting it intõ( ), then integrating it from 1 to 2 , and considering (12) and (14), we obtain that . Now let us begin to estimate the right hand side of (47) to use the above Lemma 1 of Nakao.
The main result of a global attractor reads as follows.

Theorem 5. Assume the hypotheses of Theorem 3; then the corresponding semigroup ( ) of problems (26)-(31) is asymptotically smooth in space
Proof. We are going to apply Lemmas 1 and 2 to prove the asymptotic smoothness. Given initial data ( 0 , 1 , 0 , 0 ) and Let us define We can assume formally that is sufficiently regular by density. Then, multiplying the first equation in (78) by and integrating over [0, ], multiplying the second equation in (78) by and integrating over [0, ] and taking the inner with Π for the third equation in (78) in space 2 ( + , ), and then taking the sum, we get Let us estimate the right hand side of (81). Considering the continuity of (⋅) and the estimates (41) Applying the mean value theorem, then considering (44) and the estimates (41), we have where 1 is among ‖ ‖ 2 and ‖V ‖ 2 . Also considering (43) and (44), then using Young inequality, we have Also applying the mean value theorem combined with the estimates (41), we get Considering that the assumption (13) of (⋅) combined with the estimates (41) and (43), then Young inequality (( +1)/( + 2) + 1/( + 2) = 1) −Δ ( ) On the other hand, combining with the assumption (19) on ( ) Considering the assumption (15) on (⋅) Also by view of the assumption (16) on (⋅), we have Thus by inserting (82)-(89) into (81), with 1/2 ≤ < 1, we get that Then integrating from to + 1 and defining an auxiliary function 2 ( ), we get 10

Mathematical Problems in Engineering
Thus we can get Then by multiplying first equation in (78) by and integrating over [0, ], then integrating from 1 to 2 , we get that Now let us estimate the right hand side of (94). Firstly, from the first inequality of (92), we infer that Thus we combine with the first inequality of (92), and there exists 1 ∈ [ , + 1/4] and 2 ∈ [ + 3/4, + 1] such that then we can deduce that The assumption (13) on (⋅) combined with (43) and the estimates (41) implies that By the assumption (17) on (⋅) combined with (43) and the estimates (41), and using the Holder inequality and Young inequality, we have where 2 > 0 is the first eigenvalue of the operator in 1 (0, ). Applying the Mean value theorem and using Schwarz inequality and Young inequality, we have where 2 is among ‖ ‖ 2 and ‖V ‖ 2 . Also apply the Mean value theorem combined with the estimates (41) to get Mathematical Problems in Engineering 11 Using Young inequality,we have Finally, using twice Holder inequality with /( + 2) + 2/( + 2) = 1, combined with the fifth inequality of (92), we have thus by Schwarz inequality and Young inequality, we get By inserting (95)and (97)-(104) into (94), we obtain that Then from the definition of ( ) combined with the third inequality of (92), and (95) and (103) From (91), we see that  Proof. In view of Theorems 3-5, we directly get Theorem 6.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.