Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 2908964, 13 pages
https://doi.org/10.1155/2017/2908964

Research Article

Global Attractor of Thermoelastic Coupled Beam Equations with Structural Damping

Peirong Shi,¹ Danxia Wang,² and Weiyi Chen¹

¹Mechanics College, Taiyuan University of Technology, Taiyuan 030024, China
²Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Correspondence should be addressed to Danxia Wang; danxia.wang@163.com

Received 17 November 2016; Accepted 21 December 2016; Published 22 March 2017

Academic Editor: Rahmat Ellahi

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.

Abstract.

In this paper, we study the existence of a global attractor for a class of -dimension thermoelastic coupled beam equations with structural damping , and . Here is a bounded domain of , and and are both continuous nonnegative nonlinear real functions and is a static load. The source terms and and nonlinear external damping are essentially , , and respectively.

1. 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 initial-boundary 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, [4–8].

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

subjected to the conditions

In addition, we also refer the reader to [11–15] 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

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; and are the given initial value functions; the subscript denotes derivative with respect to and the assumptions on nonlinear functions , , , , , and the external force function will be specified later.

Our fundamental assumptions on , , , , , and are given as follows.

Assumption 1. We assume that satisfying

where . This condition is promptly satisfied if is nondecreasing with .

Assumption 2. We also assume that satisfying and is nondecreasing and

Assumption 3. The function is of class and satisfies , and there exist constants and such that

where .

Assumption 4. The function is of class and satisfies , and there exist constants and such that

Assumption 5. The function is of class and satisfies , and there exist constants and such that

Assumption 6. .

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.

2. 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 we adopt the norm defined by

3. 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 , for any initial data , then problem (8)–(11) has a unique regular solution with

Proof. Let us consider the variational problem associated with (8)–(11): find such that

for all 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 and , by using Schwarz inequality, and then integrating from to , we see that

independent of and , where is the same as of (47). Taking into account assumptions (13), (17), and (18) of , , and and Assumption 6, we see that there exists depending only on such that

for all and for all .
Estimate 2. In the first approximate equation and the second approximate equation of (25), respectively, putting and and making a computation of addition by using Schwarz inequality and Young inequality and considering the assumptions of , , and , we see that there exists depending only on such that

for all and for all .
Estimate 3. In the first approximate equation and the second approximate equation of (25), respectively integrating by parts with and with and using Schwarz inequality and Young inequality, we see that there exists depending only on such that

for all and for all .
Estimate 4. Let us fix such that . Respectively taking the difference of the first approximate equation and the second approximate equation of (25) with and and respectively replacing by and by , we can find constants , depending only on , such that

Estimate 5. Taking the scalar product in with for the second approximate equation of (25), after a computation we can find a constant , depending only on such that

With the estimates - and -, 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 .
Uniqueness. Let be two solutions of (8)–(11) with the same initial data. Then writing and taking the difference (25) with and and respectively replacing by and then making a computation of addition, we have

where and . Using Mean Value Theorem and the Young inequalities combined with the estimates - and -, we deduce that for some constant ,

Then from Gronwall’s Lemma we see that . The proof of Theorem 7 is completed.

Theorem 8. Under the assumptions of Theorem 7, if the initial data , there exists a unique weak solution of problem (8)–(11) which depends continuously on initial data with respect to the norm of .

Proof. By using density arguments, we can obtain the existence of a weak solution in .
Let us consider . Since is dense in , then there exists , such that

We observe that for each , there exists , smooth solution of the initial-boundary value problem (8)–(11) which satisfies

Respectively multiplying the first equation in (35) by and multiplying the second equation in (35) by and integrating over and taking the sum and then considering the arguments used in the estimate of the existence of solution, we obtain

where is a positive constant independent of .
Defining , 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.

Remark 9. In both cases

where is a constant depending on the initial data in different expression.

In addition, in this paper, denotes different constant in different expression.

Remark 10. Theorem 8 implies that problem (8)–(11) defines a nonlinear -semigroup on . Indeed, let us set , where is the unique solution corresponding to initial data . Moreover, the operator defined in meets the usual semigroup properties

To prove the main result, we need the following Lemma 11 of Nakao and Lemma 12

Lemma 11 (see [16]). Let be a nonnegative continuous function defined on , which satisfies

where are positive constants. Then we have

Lemma 12 (see [17]). Assume that for any bounded positive invariant set and for any , there exists such that

where satisfies for any sequence

Then is asymptotically smooth.

4. 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 in .

Proof. Now we show that semigroup has as absorbing set in . Firstly, we can calculate the total energy functional

Let us fix an arbitrary bounded set and consider the solutions of problem (8)–(11) given by with . Our analysis is based on the modified energy function

where is the first eigenvalue of the operator in ; that is, satisfies

It is easy to see that dominates and . By multiplying (8) by and integrating over , we have

Inserting (48) into , we obtain

Considering (12) and (15) and integrating from to for (49), we obtain that

where .

Now let us begin to estimate the right hand side of (50) to use the above Lemma 11 of Nakao.

First, by multiplying (8) by and multiplying (9) by and integrating over and then taking the sum, we have

Then integrating from to , we get

Taking into account assumptions (13), (17), and (18) of , , and , we have

Then we define an auxiliary function by putting

Thus it is obvious that

Noting that and using twice Holder inequalities and considering assumption (18) of , we have

Using the Mean Value Theorem with and considering the estimate of (39) and then using Young inequality combined with (55), we have

where is among and .

Since , , from (56) we obtain

where is the first eigenvalue of the operator in .

Using Young inequality, we get

Since (57), in view of the Mean Value Theorem for integral, there exist number and number such that

Thus from Schwarz inequality combined with (47) and (61), we have

Considering assumption (19) of and using Young inequality and Holder inequality with , then from (47) and (57), we have

Also by Young inequality, we have

where is the first eigenvalue of the operator in ; that is, satisfies

Finally using Young inequality again, we get that

Inserting (57)–(60) and (62), (63), (64), and (66) into (50), we obtain

For the left hand side of (67), we use the Mean Value Theorem; then there exists number such that

So we conclude that

Inserting (67) into (69), we obtain that

Letting and noting that and are bounded with estimate (39), then from (70), we get

where is a constant which depends on .

Set

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 , we conclude

is an absorbing set for in .

5. The Existence of a Global Attractor

The main result of a global attractor reads as follows.

Theorem 14. Assume the hypotheses of Theorem 8; then the corresponding semigroup of problem (8)–(11) is asymptotic compactness.

Proof. We are going to apply Lemmas 11 and 12 to prove the asymptotic smooth. Given initial data and in a bounded invariant set , let be the corresponding weak solutions of problem (8)–(11). Then the differences are the weak solutions of

where

Let us define

As before, by density, we can assume formally that is sufficiently regular. Then, multiplying the first equation in (76) by and integrating over and multiplying the second equation in (76) by and integrating over and then taking the sum, we get

where

Let us estimate the right hand side of (79).
Considering the continuity of and estimate (39)

Applying the Mean Value Theorem combined with estimate (39), by Young inequality, we get

Also use the Mean Value Theorem combined with estimate (39) and Young inequality to get

where is among and , and is among and .
By the Holder inequality, Minkowski inequality combined with the estimate of (39), and Young inequality, we obtain

On the other hand, considering assumptions (17) and (18) of and ,

Thus by inserting (81)–(85) into (79), we get that

Then integrating from to and defining an auxiliary function , we get

It is obvious that

Then by multiplying first equation in (76) by and integrating over again, we obtain that

Integrating from to , we get

Now let us estimate the right hand side of (92). Firstly, from the first inequality of (90), by holder inequality we infer that

thus there exists and such that

then we can deduce that

Use Schwarz inequality combined with the estimate of (39) and Holder inequality to obtain

Apply the Mean Value Theorem combined with estimate (39) to get

Assumption (14) of and the estimate of (39) imply that

Also from assumption (19) of and the estimate of (39) combined with (94), we have

Using the Mean Value Theorem and considering the assumption of and the estimate of (39), we have

where is among and .
Finally, use Young inequality to get

By inserting (93) and (95)–(102) into (92), we obtain that

Considering (89) and (93), from (103), we have

Using Holder inequality with ,

Then from the definition of and (93), (104), and (105), we obtain that

For (106), by using Mean Value theorem, there exists such that

From (87), we see that

Let with ; then integrate (86) over and over to have

Inserting (107) into (109), we obtain

Therefore from the boundary of , we have

Therefore

From Nakao’s Lemma 11, there exists and such that

From the definition of , we have

Given , we choose large such that

and define as

Then from (114)–(116), we get

for all .
Let be a given sequence of initial data in . Then the corresponding sequence of solutions of the problem (8)–(11) is uniformly bounded in , because is bounded and positively invariant. So is bounded in . Since compactly, there exists a subsequence which converges strongly in . Therefore

So is asymptotically smooth in . That is, Lemma 12 holds. Thus Theorem 14 is proved.

In view of Theorems 13 and 14, we have the following.

Theorem 15. The corresponding semigroup of problem (8)–(11) has a compact global attractor in the phase space .

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant nos. 11172194 and 11401420), the Natural Science Foundation of Shanxi Province, China (Grant no. 2015011006 and Grant no. 2014011005-4).

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