Global Attractor for the Generalized Dissipative KDV Equation with Nonlinearity

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


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 International Journal of Mathematics and Mathematical Sciences with periodic boundary condition u x, t u x L, t , 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: u x, 0 u 0 , where α, β > 0, x, t ∈ Ω × 0, T , and Ω 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 αmeasure 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 g u and φ u ; 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 H 2 Ω .In Section 4, we prove the existence of global attractor.

Preliminaries and Main Result
We consider the generalized dissipative four-order KDV equation 1.3 , where Ω ⊂ R n is unbounded domain and the initial data Secondly, we can rewrite 1.3 as the following equation with the above assumption: u x, 0 u 0 , where α, β > 0, x, t ∈ Ω × 0, T .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 φ u , g u satisfy conditions (A1)-(A4) and, moreover, u 0 ∈ H 2 , f ∈ H 1 , then for α, β, γ > 0, there exists a global attractor A of the problem 2.1 , that is, there is a bounded absorbing set B ∈ H 2 in which sense the trajectories are attract to A, such that where S t is semigroup operator generated by the problem 2.1 .

Existence of Absorbing Set in Space H 2 Ω
In this section, we will show the existence of an absorbing set in space H 2 Ω by obtaining uniformly in time estimates.In order to do this, we start with the following lemmas.
Lemma 3.1.Assume that g u satisfied (A4), furthermore, u 0 ∈ H, f ∈ H, then for the solution u of the problem 2.1 , one has the estimates Proof.Taking the inner product of 2.1 with u, we have where 3.5 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 International Journal of Mathematics and Mathematical Sciences 5 where Proof.Taking the inner product of 2.1 with u xx , we have where Noticing that

3.12
Using Nirberg's interpolation inequality and the Sobolev embedding theory see 11 , we have

3.13
Due to Lemma 3.1 and conditions of Lemma 3.2, we get that

6
International Journal of Mathematics and Mathematical Sciences From 3.10 and above inequalities, we get

3.15
Setting C − γ c 8 , then we can obtain that Thus, by Gronwall's inequality and 3.15 , we get that
Lemma 3.3.Suppose that φ u , g u satisfy (A2), (A3) and, moreover, the following conditions hold true: then for the solution u of the problem of 2.1 , one has the following estimate

3.24
Using Young's inequality, we have
Lemma 3.4.Suppose that φ u , g u satisfy (A2), (A3) and, moreover, the following conditions hold true: then for the solution u of the problem of 2.1 , we have the following estimates: where

3.33
Proof.Taking the inner product of 2. In a similar way as above, we can get the uniformly estimates of u xxxx , u t 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: 4 g u satisfies (A3), (A4) and g u is Lipschitz continuous, that is, then there exists a unique global solution u for the problem 2.1 such that u ∈ L ∞ 0, T; H m Ω , and furthermore,the semigroup operator S t associated with the problem of 2.1 is continuous and there exists an absorbing set B ⊂ H 2 Ω , where

3.46
Proof.Similar to the proof of Lemmas in H 2 Ω .But as for the continuity of semigroup S t , we can apply the following Lemmas 3.6, and 3.7 to prove the result.

International Journal of Mathematics and Mathematical Sciences
Now, we use the decomposition method to prove the continuity of S t for sake of overcoming the difficult of noncompactness. Set

3.57
Assume that u η is solution of the following equation:

3.60
Now, we prove the Lemma 3.6.
Proof.We take the scalar product in space H of 3.58 with u η , we get 1 2

3.61
Due to A3 and Young's inequality, we get

3.62
International Journal of Mathematics and Mathematical Sciences 13 From 3.61 , we obtain the following inequality:

3.63
By Gronwall's inequality, one has Hence, there exists C > 0, such that and implies

3.66
We take the scalar product in space H of 3.58 with u ηxx and similar to the proof of Lemma 3.2, we have

3.67
By application of Gronwall's inequality, we deduce that So, there exists C > 0, such that and implies u ηx 2 ≤ Cη, ∀t ≥ t 0 .

3.70
We take the scalar product in space H of 3.58 with u ηxxxx and similar to the proof of Lemma 3.3, we have 1 2

International Journal of Mathematics and Mathematical Sciences
It is easy to prove that We take the scalar product in space H of 3.58 with u ηxxxxxx and similar to the proof of Lemma 3.4, we have 1 2 that is,

3.75
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

3.77
where Proof.We take the scalar product in space H of 3.60 with x 2 w η and noticing that Let Ω {x||x| < A|}, then the imbedding H s Ω → H s 1 Ω is compact.Thus, . Indeed, for any u ∈ B, u u| Ω , then there exists a u k such that

4.5
This completes the lemma.Otherwise, then there exists a bounded subset B 0 of E such that dist S t B 0 , A does not tend to 0 as t → ∞.Thus, there exists a δ > 0 and a sequence t n → ∞ such that dist S t n B 0 , A ≥ δ > 0, ∀n ∈ N. 4

Remark
Proof.Taking the inner product of 2.1 with u xxxx , we haveu t − φ u x u xxxx βu xxx − αu xx γu g 2 u , u xxxx f, u xxxx , Next, we will prove the uniqueness of the global solution.Assume that u, v are two solutions of the problem 2.1 and w u − v, then 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.
In this section, we prove that the semigroup operator S t associated with the problem 2.1 possesses a global attractor in space H 2 Ω .In order to prove our result, we need the following results.Assume that s > s 1 , s, s 1 ∈ N , then the following embedding 1 |x| 2 dx be a bounded set.It suffices to prove that B has a finite ε-net for any ε > 0. First, since International Journal of Mathematics and Mathematical Sciences Definition 4.7 see 12, 27 .Kuratowski α-measure of set B is defined by the formula Thirdly, we prove Theorem 2.1.Proof.Using the result of 11 , we have S t is ω-limit compact and B is bounded, for any ε > 0, there exists t ≥ 0 such that , if ψ ∈ ω B , by 4.9 , we can find two sequences φ n ∈ B and t n → ∞ such that S t n φ n → φ.We need to prove that {S t n − t φ n } has a subsequence which converges in E. For any ε > 0, there exists a t ε such that Notice that α n≥N S t n − t φ n ≤ ε contains only a finite number of elements, where N 0 is fixed such that t n − t ≥ 0, as n ≥ N 0 .By properties 1 -4 in Remark 4.8, we haveThis implies that {S t n − t φ n } is relatively compact.So, there exists a subsequence t n j → ∞ and ψ ∈ E, such thatS t n j − t φ n j −→ ∞, as t n j −→ ∞.International Journal of Mathematics and Mathematical SciencesNext, by virtue of Lemma 4.2 and the result of 11, 12 , we prove that A ω B is an global attractor in E and attracts all bounded subsets of E.
.23 For each n, there exist b n ∈ B 0 , n 1, 2, . . .satisfying Whereas B is an absorbing set, S t n B 0 and S t n b n belong to B, for n sufficiently large.As in the discussion above, we obtain that S t n b n is relatively compact admits at least one cluster point γ, S t n i − t 1 S t 1 b n i , 4.25 where t 1 follows S t 1 B 0 ⊂ B. So, γ ∈ A ω B and this contradicts 4.24 .The proof is complete.