Research Article | Open Access
Strong Attractor of Beam Equation with Structural Damping and Nonlinear Damping
This paper is mainly concerned with the existence of a global strong attractor for the nonlinear extensible beam equation with structural damping and nonlinear external damping. This kind of problem arises from the model of an extensible vibration beam. By the asymptotic compactness of the related continuous semigroup, we prove the existence of a strong global attractor which is connected with phase space .
Global attractor is a basic concept in the study of long-time behavior of nonlinear dissipative evolution equations with various dissipation. There have been many methods to prove the existence of the global attractor. It can be proved by the theory of -contractions of the solution semigroup , such as [1–3] and the reference therein. It can also be proved by the decomposition of the solution semigroup (see Hale , Temam , etc.).
In this paper, we use the method of the asymptotically compact property of the solution semigroup which is different from the method of [1–5] to prove the existence of a strong global attractor for the Kirchhoff type equations with structural damping and nonlinear external damping which arises from the model of the nonlinear vibration beam where , and are all positive constants, is a bounded domain of with smooth boundary , , , , and are nonlinear functions specified later, and is an external force term. represents the vertical deflection of the beam, and is a real-valued function on .
In this context of problem (1), based on the vibrating beams equation which is proposed by Woinowsky-krieger ; Ma and Narciso  considered problem (1) without structural damping and posed a weak global attractor in weak phase space . Eden and Milani  considered the existence of exponential attractor for problem (1) with and a linear weak damping , being a nonlinear function and without structural damping. Ball  presented the existence and uniqueness of global solutions for problem (1) with are all linear functions.
On the other hand, the existence of the attractor for a related problem, with the boundary conditions of (2) replaced with , was considered by Ma and Narciso , Eden and Milani  with a linear damping or nonlinear damping without structural damping, respectively. Chueshov and Lasiecka  considered a kind of boundary condition which is but without structural damping.
Generally speaking, there have been many works on the long-time behavior for nonlinear beam equations [6–10]. But for the beam equation (1) with structural damping, in strong phase space , the global solutions and the strong global attractor have not still been proved until now.
The outline of this paper is arranged as follows: in Section 2 we give the existence and uniqueness of global solutions in space , in Section 3 we give the boundedness of solutions in phase space , and finally in Section 4, we give the proof of the existence of a strong global attractor in phase space .
2. Some Assumptions and Existence of Global Solution
In (1), we assume that damping term and the source term are in the form of with
We assume that the nonlinear functions are all class , and satisfying and
The functions are also class , with , , and for all , where , and are all constants. There also exists constants such that
In addition, nonlinear function also satisfies where , , and are all constants, .
Our analysis is based on the following Sobolev spaces: , , with the usual inner products and norms as follows, respectively:
Consider with the inner products and the norms .
Take and with the inner products and norms as follows, respectively:
Note that assumption (6) implies that , with or .
According by Theorem 1, for any , we may introduce the mapping It maps into itself, and it enjoys the usual semigroup properties as follows: And it is obvious that the map , for all , is continuous in space . In the following, we will introduce the existence of bounded absorbing set and global attractor in space for map .
3. The Existence of Bounded Absorbing Set in Space
Proof. Taking the inner products of with both sides of (1) and then making summation, we have
where , and is fixed at arbitrary time, and here the energy function is defined on by
Considering the assumption , we have
With , we have
With the assumptions , and and by using Mean Value Theorem and Mean Value inequality, we have
where among and . Set
So (15) is transformed into
Considering the assumptions , , , , and , and letting , we have
Substituting (23) into (22), we have
On the one hand, applying the Gronwall inequality to (24), we get
Note that and are bounded; then there exists a positive constant such that is bounded; so
On the other hand, considering that , fixing , and assuming that , then as , we have
Take the inner products by in both sides of (1); then make summation to get Considering the continuity of the functions and , we have where are all positive constants. Also In addition, with , there exists a constant such that ; so Also by using Schwarz and Mean Value inequalities and Mean Value Theorem, we have where among and . Set and write ; then (29) is transformed into Here the function is obtained by the energy function being changed slightly.
Let , we have , and so Then an application of the Gronwall inequality leads to If , there exists a positive constant such that .
Putting satisfing , then as , we get So The global estimate (40) shows the existence of an absorbing set of .
4. The Existence of Global Attractor in Space
The general theory  indicates that the continuous semigroup defined on a Banach space has a global attractor which is connected when the following conditions are satisfied.(i) There exists a bounded absorbing set such that for any bounded set , (ii) is asymptotically compact; that is, for any bounded sequence in and tending to , there exists a subsequence such that is convergent as .
Theorem 3. Under the assumptions of Theorem 1, the continuous semigroup has a global attractor which is connected to .
Proof. Let be two solutions of Problems (1)–(3) in space as shown above corresponding to the initial data and with , respectively. Then satisfies
Taking the inner products in both sides of (42) by , , and , respectively, we have
Equation (46)(45)(44) yields
where is among 0 and is among and , and
where is among 0 and , is among and .
Also considering for all , , for all , and , , by Hölder inequality we have Setting then substituting (48)–(56) into (47), by Schwarz inequality and Young inequality, and taking and , we have Again setting , and considering that , we have . On the one hand, from (58) we have Applying the Gronwall inequality to (59), we get On the other hand, with and setting , we get Hence
Now, let be a bound sequence in , and the corresponding solutions of problems (1)–(3) in . We assume . Let and . Then, applying estimate (62) to , we have By taking in the above, we have By Sobolev embedding Theorem, for any , we can extract a subsequence which is convergent in for any . For any , we first fix such that And, next, taking large , we have Then by (62) we have that We conclude that is asymptotically compact on . The theorem is now proved.
This work is partially supported by the Natural Science Foundation of China (11172194) and Shanxi Province (2011021002-2 and 2010011008). The authors also wish to give their thanks to the referees for their comments to improve the presentation of this paper.
- A. Eden and V. Kalantarov, “Finite-dimensional attractors for a class of semilinear wave equations,” Turkish Journal of Mathematics, vol. 20, no. 3, pp. 425–450, 1996.
- A. Eden and V. K. Kalantarov, “On the discrete squeezing property for semilinear wave equations,” Turkish Journal of Mathematics, vol. 22, no. 3, pp. 335–341, 1998.
- A. Eden, C. Foias, and V. Kalantarov, “A remark on two constructions of exponential attractors for -contractions,” Journal of Dynamics and Differential Equations, vol. 10, no. 1, pp. 37–45, 1998.
- J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25, American Mathematical Society, Providence, RI, USA, 1988.
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988.
- S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, NY, USA, 1959.
- T. F. Ma and V. Narciso, “Global attractor for a model of extensible beam with nonlinear damping and source terms,” Nonlinear Analysis, vol. 73, no. 10, pp. 3402–3412, 2010.
- A. Eden and A. J. Milani, “Exponential attractors for extensible beam equations,” Nonlinearity, vol. 6, no. 3, pp. 457–479, 1993.
- J. M. Ball, “Stability theory for an extensible beam,” Journal of Differential Equations, vol. 14, no. 3, pp. 399–418, 1973.
- I. Chueshov and I. Lasiecka, “Long-time behavior of second order evolution equations with nonlinear damping,” Memoirs of the American Mathematical Society, vol. 195, no. 912, pp. 12–27, 2008.
- Z. J. Yang, “Longtime behavior of the Kirchhoff type equation with strong damping on ,” Journal of Differential Equations, vol. 242, no. 2, pp. 269–286, 2007.
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