Global Attractor for the Generalized Dissipative KDV Equation with Nonlinearity

Zhang, Zai-yun; Liu, Zhen-hai

doi:https://doi.org/10.1155/2011/725045

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2011 | Article ID 725045 | https://doi.org/10.1155/2011/725045

Global Attractor for the Generalized Dissipative KDV Equation with Nonlinearity

Zai-yun Zhang^1,2and Zhen-hai Liu³

Academic Editor: Marco Squassina

Received20 Dec 2010

Accepted23 Feb 2011

Published03 May 2011

Abstract

We discuss global attractor for the generalized dissipative KDV equation with nonlinearity under the initial condition . We prove existence of a global attractor in space , by using decomposition method with cut-off function and Kuratowski-measure in order to overcome the noncompactness of the classical Sobolev embedding.

1. Introduction

In order to study the longtime behavior of a dissipative evolutionary equation, we generally aim to show that the dynamics of the equation is finite dimensional for long time. In fact, one possible way to express this fact is to prove that dynamical systems describing the evolutional equation comprise the existence of the global attractor [1]. The KDV equation without dissipative and forcing was initially derived as a model for one direction water waves of small amplitude in shallow water, and it was later shown to model a number of other physical stems. In recent years, the KDV equations has been always being an important nonlinear model associated with the science of solids, liquids, and gases from different perspectives both mathematics and physics. As for dissipative KDV equation, existence of a global attractor is a significant feature. In [2], Ghidaglia proved that for the dissipative KDV equation with periodic boundary condition , there exists a weak global attractor of finite dimension. Later, there are many contributions to the global attractor of the dissipative KDV equation (see [3–10]). In [3], Guo and Wu proved the existence of global attractors for the generalized KDV equation

However, few efforts are devoted to the existence of global attractor for generalized dissipative four-order KDV equation with nonlinearity in unbounded domains. In this paper, we consider the existence of global attractor for generalized dissipative four-order KDV equation with nonlinearity as follows:, where , and is unbounded domain.

As we all know, the solutions to the dissipative equation can be described by a semigroup of solution operators. When the equation is defined in a bounded domain, if the semigroup is asymptotically compact, then the classical theory of semiflow yields the existence of a compact global attractor (see [11–13]). But, when the equation is defined in a unbounded domain, which causes more difficulties when we prove the existence of attractors. Because, in this case, the Sobolev embedding is not compact. Hence, we cannot obtain a compact global attractor using classical theory.

Fortunately, as far as we concerned, there are several methods which can be used to show the existence of attractors in the standard Sobolev spaces even the equations are defined in unbounded domains. One method is to show that the weak asymptotic compactness is equivalent to the strong asymptotic compactness by an energy method (see [9, 10, 14]). A second method is to decompose the solution operator into a compact part and asymptotically small part (see [15–17]). A third method is to prove that the solutions uniformly small for large space and time variables by a cut-off function (see [18, 19]) or by a weight function (see [20]).

Generally speaking, the energy method proposed by Ball depends on the weak continuity of relevant energy functions (see [21, 22]). However, for (1.3) in unbounded domains, it seems that the energy method is not easy to use. Consequently, in this paper, we will show the idea to obtain the existence of global attractor in unbounded domains by showing the solutions are uniformly small for large space by a cut-off function or weight function, and at the same time, we apply decomposition method and Kuratowski -measure to prove our result in order to overcome the noncompactness of the classical Sobolev embedding.

This paper is organized as follows.

In Section 2, firstly, we recall some basic notations; secondly, we make precise assumptions on the nonlinearity and ; finally, we state our main result of the global attractor for (1.3).

In Section 3, we show the existence of a absorbing set in .

In Section 4, we prove the existence of global attractor.

2. Preliminaries and Main Result

We consider the generalized dissipative four-order KDV equation (1.3), where is unbounded domain and the initial data , , is nonlinearity.

Throughout the paper, we use the notation , with the scalar product and norms given, respectively, by , , and , . In the space , we consider the scalar product and the norm . While in the space , we consider the scalar product and the norm .

Notice
denote for different positive constants.

First, we assume that , and satisfy the following conditions: (A1):, , (, ),(A2):, ,(A3):, , where , ,(A4):, .

Secondly, we can rewrite (1.3) as the following equation with the above assumption:, where .

Finally, we state our main result is the following theorem.

Theorem 2.1. Let the generalized dissipative of four-order KDV equation with nonlinearity given by (2.1). Assume that satisfy conditions (A1)–(A4) and, moreover, , then for , there exists a global attractor of the problem (2.1), that is, there is a bounded absorbing set in which sense the trajectories are attract to , such that where is semigroup operator generated by the problem (2.1).

3. Existence of Absorbing Set in Space

In this section, we will show the existence of an absorbing set in space by obtaining uniformly in time estimates. In order to do this, we start with the following lemmas.

Lemma 3.1. Assume that satisfied (A4), furthermore, , then for the solution of the problem (2.1), one has the estimates

Proof. Taking the inner product of (2.1) with , we have where here, we apply Young’s inequality and the condition (A4).
Thus, from (3.4), we get By virtue of Gronwall's inequality and (3.6), one has (3.1) and which implies (3.2) and (3.3).

Lemma 3.2. In addition to the conditions of Lemma 3.1, one supposes that then one has the estimate where

Proof. Taking the inner product of (2.1) with , we have where
Noticing that
Using Nirberg's interpolation inequality and the Sobolev embedding theory (see [11]), we have
Due to Lemma 3.1 and conditions of Lemma 3.2, we get that
From (3.10) and above inequalities, we get Setting , then we can obtain that Thus, by Gronwall’s inequality and (3.15), we get that which implies Therefore, we prove Lemma 3.2.

Lemma 3.3. Suppose that satisfy (A2), (A3) and, moreover, the following conditions hold true:(1), (2), then for the solution of the problem of (2.1), one has the following estimate furthermore,

Proof. Taking the inner product of (2.1) with , we have where By Young’s inequality and Lemmas 3.1 and 3.2, thus from (3.22), we have Due to Lemmas 3.1 and 3.2 and (A3), we obtain
Using Young’s inequality, we have By (3.21), (3.23) and (3.25), we get that is, where By virtue of Gronwall’s inequality, we have and (3.27) implies Therefore, we prove Lemma 3.3.

Lemma 3.4. Suppose that satisfy (A2), (A3) and, moreover, the following conditions hold true:(1), , (2), ,then for the solution of the problem of (2.1), we have the following estimates: where furthermore,

Proof. Taking the inner product of (2.1) with , we have where
Using Nirberg’s interpolation inequality and Young’s inequality, from (3.36) and Lemmas 3.1–3.3, we have Due to the condition (4.3), we get By direct calculations, it is easy to get that Due to (3.34)–(3.39), we have that is, where Using Gronwall's inequality, we deduce that moreover, (3.41) implies Therefore, we prove Lemma 3.4.

In a similar way as above, we can get the uniformly estimates of and we omit them here.

Next, we will show the existence of global solution for the problem (2.1) as follows.

Lemma 3.5. Suppose that the following conditions hold true:(1), ,(2), ,(3), , (4) satisfies (A3), (A4) and is Lipschitz continuous, that is, then there exists a unique global solution for the problem (2.1) such that , and furthermore,the semigroup operator associated with the problem of (2.1) is continuous and there exists an absorbing set , where

Proof. Similar to the proof of Lemmas 3.1–3.4, we have At the same time, we use the Galerkin method (see [11]) and Lemmas 3.1–3.4 to prove the existence of global solution for the problem (2.1). So, we omit them here.
Next, we will prove the uniqueness of the global solution.
Assume that are two solutions of the problem (2.1) and , then we have
Taking the inner product in of (2.1) with , we have where Due to the condition and from (3.49), we obtain that is, By application of Gronwall's inequality, we get .
Finally, we recall some basic results in [11, 23] and by Lemmas 3.1–3.3, it is easy to prove that there exists an absorbing set in . But as for the continuity of semigroup , we can apply the following Lemmas 3.6, and 3.7 to prove the result.

Lemma 3.6. Suppose that , and satisfy (A1)–(A4), there exists constant , such that

Now, we use the decomposition method to prove the continuity of for sake of overcoming the difficult of noncompactness.

Set , , , satisfies

then, , there exists , such that

Assume that is solution of the following equation:

Setting is a solution of the equation as follows:

Now, we prove the Lemma 3.6.

Proof. We take the scalar product in space of (3.58) with , we get Due to (A3) and Young’s inequality, we get From (3.61), we obtain the following inequality: By Gronwall’s inequality, one has Hence, there exists , such that and implies
We take the scalar product in space of (3.58) with and similar to the proof of Lemma 3.2, we have By application of Gronwall's inequality, we deduce that So, there exists , such that and implies
We take the scalar product in space of (3.58) with and similar to the proof of Lemma 3.3, we have It is easy to prove that
We take the scalar product in space of (3.58) with and similar to the proof of Lemma 3.4, we have that is,
Hence, by Gronwall’s inequality, we get At the same time, we have and we omit them here.

Lemma 3.7. Under the conditions of Lemma 3.6, one has the following estimates where , .

Proof. We take the scalar product in space of (3.60) with and noticing that it is easy to get that By Gronwall’s inequality, we have From (3.60), we obtain
We take the scalar product in space of (3.61) with and noticing that
By Young’s inequality and the Sobolev embedding theory (see [11]) and (3.81)-(3.82), we deduce that Using Gronwall’s inequality, we obtain The proof of Lemma 3.7 is completed.

Using Lemmas 3.6 and 3.7, we can prove that is continuous.

4. Existence of Global Attractor in Space

In this section, we prove that the semigroup operator associated with the problem (2.1) possesses a global attractor in space .

In order to prove our result, we need the following results.

Lemma 4.1 (see [23]). Assume that , , then the following embedding into is compact.

Proof. Let be a bounded set. It suffices to prove that has a finite -net for any . First, since there exists an integer , such that Let , then the imbedding is compact. Thus, is relatively compact in and has a finite -net , with , and . We claim that is an -net of in .
Indeed, for any , , then there exists a such that Hence, This completes the lemma.

Lemma 4.2 (see [7, 11]). Let be Banach space and a set of semigroup operators, that is, satisfy where is the identity operator and is space . We also assume that(1) is bounded, that is, for each , there exists a constant such that implies , ,(2)there is an bounded absorbing set , that is, for any bounded set , there exists a constant , such that , for ,(3) is a continuous operator for , then has a compact global attractor in the space , such that(1), ,(2) as , and denotes the Hausdorff semidistance defined as for any bounded set in which sense the trajectories are attracted to (see[9, 24]), using Kuratowski -measure in order to overcome the non-compactness of the classical Sobolev embedding.

Firstly, we need the following definitions.

Definition 4.3 (see [11, 25]). Let be a semigroup in complete metric space . For any subset , the set defined by is called the set of .

Remark 4.4. (1) It is easy to see that if and only if there exists a sequence of element and a sequence , such that
(2) If is compact set, then, for every bounded subset of and for any , there exists a , such that .

Definition 4.5 (see [11, 26]). Let be a semigroup in complete metric space . A subset of is called an absorbing set in if, for any bounded subset of , there exists some , such that , for all .

Definition 4.6 (see [11, 26]). Let be a semigroup in complete metric space . A subset of is called global attractor for the semigroup if is compact and enjoys the following properties:(1) is a invariant set, that is, , for any ,(2) attract all bounded set of , that is, for any bounded subset of , , as , where is Hausdorff semidistance of two set and in space : .

Definition 4.7 (see [12, 27]). Kuratowski -measure of set is defined by the formula for every bounded set of a Banach space .

Secondly, due to Definition 4.6, it is easy to see that Kuratowski -measure of set has the following properties.

Remark 4.8. (1) If is compact set, then ;
(2) ,
(3) ,
(4) if , ,
(5) .
Thirdly, we prove Theorem 2.1.

Proof. Using the result of [11], we have is compact and is bounded, for any , there exists such that Taking , we can find a sequence , , such that By , we get First, we prove that is variant. As a matter of fact, if , then , for some . So, there exists a sequence and such that , that is, which implies that and .
Conversely, if , by (4.9), we can find two sequences and such that . We need to prove that has a subsequence which converges in . For any , there exists a such that which implies that Hence, there exists an integer , such that Then, it follows that Notice that contains only a finite number of elements, where is fixed such that , as .
By properties (1)–(4) in Remark 4.8, we have Let , then we get that This implies that is relatively compact. So, there exists a subsequence and , such that It is easy to see that and furthermore, .
Next, by virtue of Lemma 4.2 and the result of [11, 12], we prove that is an global attractor in and attracts all bounded subsets of .
Otherwise, then there exists a bounded subset of such that does not tend to 0 as . Thus, there exists a and a sequence such that For each , there exist satisfying Whereas is an absorbing set, and belong to , for sufficiently large. As in the discussion above, we obtain that is relatively compact admits at least one cluster point , where follows . So, and this contradicts (4.24). The proof is complete.

Acknowledgments

The authors are highly grateful for the anonymous referee’s careful reading and comments on this paper. The Z.-Y. Zhang would like to thank Professor Dr. Zhen-hai Liu, who is his Ph. D supervisor, for his valuable attention to our paper. This work was supported by the Graduate Degree Thesis Innovation Foundation of Central South University Grant no. CX2010B115 (China), the Doctoral Dissertation Innovation Project of Central South University Grant no. 2010ybfz016 (China), the NNSF of China Grant no. 10971019, and the Foundation (2010) of Guangxi Education Department.

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Copyright © 2011 Zai-yun Zhang and Zhen-hai Liu. 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.

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