Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions

Abstract

Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equation with the structural damping and the rotational inertia term. Little attention is paid to the longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.

1. Introduction

In this paper, we will consider a kind of more general Kirchhoff-type beam equation. The physical origin of the problem lies in the theory of vibrations of an extensible beam of length ; moreover, during vibration, the elements of a beam not only perform a translatory motion but also rotate.

A mathematical model for this problem is an initial boundary value problem for the nonlinear Kirchhoff-type beam equation We assume the nonlinear boundary conditions and the initial conditions Here the unknown function is the elevation of the surface of beam, and are the given initial value functions, and the subscript and denote derivative with respect to and , respectively. expresses the rotational inertia, and nonlinear terms and represent the extensibility effects and the structural damping, respectively. is a static load. Moreover the assumptions on nonlinear functions , , , and and the external force function will be specified later.

In (1), when the structural damping term and the rotational inertia term are absent, (1) is a model for vibrations of tensible beam. This was proposed by Woinowsky-Krieger [1] in the form One of the first mathematical analyses for equation was done by Ball [2] which was later extended to an abstract setting by defining a linear operator by Medeiros [3]. In [4], Patcheu obtained the decay of the energy for above equation when a nonlinear damping was effective in . In addition, the attractor on extensible beams with null boundary conditions was considered by several authors. We quote, for instance, [5–7], and so on. But the longtime behavior of the beam equation with nonlinear boundary conditions was paid little attention. We also refer the reader to a few works. One of the first studies in this direction was done by Pazoto and Menzala [8], 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 in [9] and [10], respectively, for the Kirchhoff-type beam equation with the absence of the structural damping and the rotational inertia, subjected to the nonlinear boundary conditions

In the following, we mentioned some results on longtime behavior of beam equation with the rotational inertia term. Under null boundary conditions, Geredeli and Lasiecka [11] considered the existence of a compact attractor of beam with a rotational inertia term. Under nonlinear boundary conditions Ji and Lasiecka [12] considered the semilinear Kirchhoff equation with rotational inertia, and they showed that the above problem is uniformly stabilized with uniform energy decay rates.

In addition, we also mentioned some results on longtime behavior of the equation with the structural damping term. Chueshov [13] studied the global attractor with a structural damping of the form with . Chueshov [14] and Yang et al. [15] considered the global attractor for the Kirchhoff-type equation with structural damping under null boundary conditions, respectively.

On (1) under the following other nonlinear boundary conditions we also can get the same on the existence of global solutions and the existence of global attractor.

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

Assumption 1. We assume that , are all nondecreasing and satisfy where . Moreover

Assumption 2. The function is of class and satisfies , and there exist constants and such that

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

Assumption 4. Consider .

Under the above assumptions, we prove the existence of global solutions and the existence of a global attractor of extensible beam equation system (1)–(4). And the paper is organized as follows. In Section 2, we introduce some Sobolev spaces. In Section 3, we discuss the existence of global strong and weak solutions. In Section 4, we establish the result of the existence of a global attractor in .

2. Preliminaries

Our analysis is based on the following Sobolev spaces. Let Motivated by the boundary condition (3) we assume, for regular solutions, that data satisfies the following compatibility condition: Then for regular solutions we consider the phase space In the case of weak solutions we consider the phase space which guarantees that, for regular data, the nonlinear condition (22) holds. 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 (1)–(4). We state it as follows.

Theorem 1. Assume Assumptions 1–4 and (22)-(23) hold, for any initial data ; then problem (1)–(4) has a unique regular solution with Moreover, where depends on the initial data and , but not on .

Proof. Let us consider the variational problem associated with (1)–(4): find such that for all . This is done with the Galerkin approximation methods 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, uniqueness, and . In the following we give the estimates of approximation solution, the proof of uniqueness of solution, and the proof of .

Estimate  1. In approximate equation of (28), putting and considering , , using Schwarz inequality, and then integrating from to , we see that Taking into account the assumptions , , and of , , , and , we see that there exists such that for all and for all .

Estimate  2. In approximate equation of (28), integrating by parts with and and considering the compatibility condition (22) and then using Schwarz inequality and the mean value inequality, we see that there exists such that for all and for all .

Estimate 3. Let us fix such that . Taking the difference of approximate equation of (28) with and and replacing by , we can find a constant , depending only on , such that

Uniqueness. Let be two solutions of (1)–(4) with the same initial data. Then writing and taking the difference (28) with and and replacing by and then using mean value theorem and the Young inequalities combined with Estimates 1 and 3, we deduce that, for some constant , Then from Gronwall’s lemma we see that .

The Proof of . Since , we get . Similarly, . The proof of Theorem 1 is completed.

Theorem 2. Assume the assumptions of Theorem 1 and (24) hold; if the initial data , then there exists a unique weak solution of problem (1)–(4) which depends continuously on initial data with respect to the norm of  .

Proof. Let us consider , and since is dense in , there exists , such that We observe that, for each , there exists , smooth solution of the initial boundary value problem (1)–(4) which satisfies Considering the arguments used in the estimate of the existence of solution, we obtain where is a positive constant independent of .
Defining , where and are regular solutions of (35), following the steps already used in the uniqueness of regular solution for (1)–(4) 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 2 is proved.

Remark 3. Theorem 2 implies that problem (1)–(4) defines a nonlinear -semigroup on . Indeed, let us set , where is the unique solution corresponding to initial data . Moreover, the operator defined in maps into itself and it enjoys the usual semigroup properties And it is obvious that the map is continuous in space .

4. The Existence of Global Attractor

In this section, we give the existence of a global attractor.

A global attractor for a -semigroup defined on a complete metric space is a bounded closed subset which is positive fully invariant, that is, , for  all , and uniformly attracting, that is, for any bounded set .

A bounded set is an absorbing set for if, for any bounded set , there exists such that which defines as a dissipative dynamical system.

Theorem 4. Assume the hypotheses of Theorem 2 and , is sufficiently small, and then the corresponding semigroup of problem (1)–(4) has an absorbing set in .

Proof. Now, we show that semigroup has an absorbing set in . Firstly, we can calculate the total energy functional Let us fix an arbitrary bounded set and consider the solutions of problem (1)–(4) given by with . Our analysis is based on the modified energy functional It is easy to see that .
Indeed, since , we have In a similar way, since , the following inequalities hold Now let us define By multiplying (1) by and integrating over , we have Then, multiplying (1) by and integrating over , we obtain Taking into account the boundary condition (3), we get Taking the sum of (47) with times (49) and using the Schwarz inequality, we have So, using (15), we obtain and using (16) and (44), we obtain Taking into account (18) and (44)-(45), we have Using the mean value inequality, we get Then inserting (51)–(54) into (50), we obtain Taking small enough, we get Since with sufficiently small we have , we get Using , , for all , we get So setting small enough, Considering that and , for all , we obtain Therefore with (56)–(60), (55) is transformed into Adding on both sides of inequality (61) and taking into account that , we obtain Now, let us set Then since from (45), also and , we can get Thus, dominates . Also, from (45), we have which implies that, for sufficiently small enough, Inserting (66) into (62), we get Applying Gronwall’s inequality, we obtain Since the given invariant set is bounded, is also bounded. Then there exists large enough such that Then from (64) we have This shows that is an absorbing set for in . The proof of Theorem 4 is ended.

A semigroup is asymptotically smooth in if for any bounded positive invariant set , there exists a compact set such that Then the following lemma is well known.

Lemma 5 (see [16], Theorem 2.3). Let be a dissipative -semigroup defined on a metric space ; then has a compact global attractor in if and only if it is asymptotically smooth in .
The asymptotic smoothness can be verified from a result by Khanmamedov [17] and Chueshov and Lasiecka [16]. Assume that is a Banach space.

Lemma 6 (see [16], Proposition 2.10). Assume that for any bounded positive invariant set and for any , there exists such that where satisfies for any sequence of . Then is asymptotically smooth in .

Theorem 7. Assume the hypotheses of Theorem 2 and ; then the corresponding semigroup of problem (1)–(4) is asymptotic compactness.

Proof. We are going to apply Lemma 6 to prove the asymptotic compactness. Given initial data and in a bounded invariant set , let , be the corresponding weak solutions of problem (1)–(4). Then the difference is a weak solution of where Let us assume and define As before, by density, we can assume formally that is sufficiently regular. Then, multiplying the first equation in (75) by and integrating over , we get Taking into account the third equation in (75) we see that By multiplying first equation in (75) by and integrating over , we obtain that Also taking into account the third equation in (75) we see that Then summing (80) with times (82) we obtain that In view of assumption (17) of the function , we obtain that Also since ,  , we see that In the following, let us estimate the right hand side of (83). We recall that , , and satisfy the estimate , then denoting by a generic positive constant which depends only on we can simplify several notations.
Firstly, since , , , and , we get , ; then we have From the mean value theorem and noting that and , we have From assumption (19) of the function and and inequalities (44), we get From assumption (18) of the function and and inequalities (44), we get Substituting (84)–(89) into (83) and using Schwarz inequality, we obtain that With sufficiently small enough, we have Defining and considering (44)-(45), we have which implies for sufficiently small enough that Let ; then inserting (93) into (91), we obtain that From Gronwall’s lemma, we get On the other hand, we have Therefore, combining (95) and (96), we can fix a constant , depending on the size of but not on , such that ; so Given , we choose large such that and define as Then from (97)–(99), we get for all .
Let be a given sequence of initial data in . Then the corresponding sequence of solutions of problem (1)–(4) 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 6 holds. Thus Theorem 7 is proved.

Remark 8. The novelty and difficulty of Theorems 4 lie in the appropriate definition on the functions , and the relationship between and the energy function and the relationship between and the function .

Lemma 9 ([18], Theorem 1.1 in Chapter I). One assumes that is a metric space and that the operators are given and satisfy the usual group properties and is continuous operator from into itself and the semigroup is asymptotically compact. One also assumes that there exist an open set and a bounded set of such that is absorbing in . Then is a compact attractor.

By view of Lemma 9, with Theorems 4 and 7, the main result of a global attractor reads as follows.

Theorem 10. The corresponding semigroup of problem (1)–(4) has a compact global attractor in the phase space .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant no. 11172194 and no. 11401420), the Natural Science Foundation of Shanxi Province, China (Grant no. 2010011008), and the Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant no. 2011021002-2).