# Global Attractor for the Generalized Dissipative KDV Equation with Nonlinearity

Authors: Zai-yun Zhang; Zhen-hai Liu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We discuss global attractor for the generalized dissipative KDV equation with nonlinearity under the initial conditionu(x,0)=u0(x). We prove existence of a global attractor in space H2(Ω), by using decomposition method with cut-off function and Kuratowskiα-measure in order to overcome the noncompactness of the classical Sobolev embedding.


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## Body

## 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 (1.1)ut+uxxx+γu=f-uux,x∈[0,L],
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 (1.2)ut+uxxx-ηuxx+γu=f-g(u)x,u(x,0)=u0.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:(1.3)ut-ϕ(u)x+uxxxx+βuxxx-αuxx+g(u)=f(x),u(x,0)=u0, 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 Kuratowskiα-measure to prove our result in order to overcome the noncompactness of the classical Sobolev embedding.This paper is organized as follows.In Section2, 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 Section3, we show the existence of a absorbing set in H2(Ω).In Section4, we prove the existence of global attractor.


## 2. Preliminaries and Main Result

We consider the generalized dissipative four-order KDV equation (1.3), where Ω⊂Rn is unbounded domain and the initial data u0∈H3(Ω), f∈H1(Ω), g(u) is nonlinearity.Throughout the paper, we use the notationH=L2(Ω), Hs=Hs,2(Ω) with the scalar product and norms given, respectively, by (·,·), |·|, and ((·,·)), ∥·∥. In the space H2, we consider the scalar product ((u,v))=(∫Ω∇u·∇vdx) and the norm ∥u∥=(∫Ω|∇u|2dx)1/2. While in the space H, we consider the scalar product (u,v)=(∫Ωu·vdx) and the norm |u|=(∫Ωu2dx)1/2.Notice
Ei,C,c,ci denote for different positive constants.First, we assume thatf∈H, and ϕ(u),g(u) satisfy the following conditions: (A1):
ϕ(u)∈C3, |ϕ(u)|≤A|u|5-σ, (A>0, 0<σ≤4),(A2):
|ϕ′′(u)|≤A|u|3-σ, |ϕ′′(u)|≤A|u|2-σ,(A3):
g(u)∈C2, g(u)=g1(u)+g2(u), where g1(u)=γu, g2(u)=K|u|5(K>0,γ>0),(A4):
g(0)=0, |g′(u)|≤C.Secondly, we can rewrite (1.3) as the following equation with the above assumption:(2.1)ut-ϕ(u)x+uxxxx+βuxxx-αuxx+γu+g2(u)=f(x),u(x,0)=u0, 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, u0∈H2,f∈H1, 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∈H2 in which sense the trajectories are attract to 𝒜, such that
(2.2)A=ω(B)=⋂s≥0⋃t≥sS(t)B¯,
where S(t) is semigroup operator generated by the problem (2.1).


## 3. Existence of Absorbing Set in SpaceH2(Ω)

In this section, we will  show the existence of an absorbing set in spaceH2(Ω) by obtaining uniformly in time estimates. In order to do this, we start with the following lemmas.Lemma 3.1.
Assume thatg(u) satisfied (A4), furthermore, u0∈H,f∈H, then for the solution u of the problem (2.1), one has the estimates
(3.1)‖u‖2≤‖u0‖2exp⁡(-ct)+‖f‖2C-exp⁡(-ct),(3.2)lim⁡t→∞‖u‖2≤‖f‖2C=E0,(3.3)lim⁡t→∞1t∫0t‖ux‖2dt≤‖f‖22αC.Proof.
Taking the inner  product of (2.1) with u, we have
(3.4)(ut-ϕ(u)x+uxxxx+βuxxx-αuxx+γu+g2(u),u)=(f,u),
where
(3.5)(u,ϕ(u)x)=0,(u,uxxx)=0,(u,g(u))≥C‖u‖2,|(f,u)̃|≤‖f‖2C+C‖u‖22,
here, we apply Young’s inequality and the condition (A4).
Thus, from (3.4), we get
(3.6)ddt‖u‖2+2α‖u‖+2C‖u‖2≤‖f‖2C.
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 Lemma3.1, one supposes that
(3.7)ϕ(u)∈C2,g(u)∈C1,|ϕ(u)|≤A|u|5-σ(A>0,0<σ≤4),
then one has the estimate
(3.8)‖ux‖2≤4exp⁡(-2c8t)φ(u0)+2c8(2‖f‖2α+c7)(1-exp⁡(-2c8t)),
where
(3.9)φ(u)=‖ux‖2-1β∫Ωϕ(u)dx.Proof.
Taking the inner product of (2.1) with uxx, we have
(3.10)(ut-ϕ(u)x+uxxxx+βuxxx-αuxx+γu+g2(u),uxx)=(f,uxx),
where
(3.11)(uxx,ϕ(u)x)=1β(ut-ϕ(u)x+uxxxx-αuxx+g(u)-f).
Noticing that(3.12)1β(ϕ(u),ut)=1β∫Ωϕ(u)utdx=1βddt∫Ωϕ(u)dx,(ϕ(u),ϕ(u)x)=0.
Using Nirberg's interpolation inequality and the Sobolev embedding theory (see [11]), we have
(3.13)|1β(ϕ(u),uxxxx)|≤α2‖uxx‖2+c2,|1β(ϕ(u),-αuxx)|≤α2‖uxx‖2+c3.
Due to Lemma3.1 and conditions of Lemma 3.2, we get that
(3.14)|1β(ϕ(u),g(u))|≤AKβ‖u‖5-σ≤c4,|1β(ϕ(u),f)|≤Aβ‖f‖‖u‖5-σ≤c5,|(g(u),uxx)|≤C‖uxx‖2,|(f,uxx)|≤α2‖uxx‖2+2α‖f‖2.
From (3.10) and above inequalities, we get
(3.15)ddt(‖ux‖2-2α∫Ωϕ(u)dx)+α2‖uxx‖2+(C-γ)‖ux‖2≤2α‖f‖2+c6=c7.
Setting C-γ=c8, then we can obtain that
(3.16)φ(u)=12∫Ω‖ux‖2dx-1β∫Ω∫0uϕ(s)dsdx≥14‖uxx‖2-c9.
Thus, by Gronwall’s inequality and (3.15), we get that
(3.17)‖ux‖2≤4exp⁡(-2c8t)φ(u0)+2c8(1-exp⁡(-2c8t))(2‖f‖2α+c7),
which implies
(3.18)lim⁡t→∞‖ux‖2≤E1,lim⁡t→∞1t∫0t‖uxx‖2dt≤C.
Therefore, we prove Lemma 3.2.Lemma 3.3.
Suppose thatϕ(u),g(u) satisfy (A2), (A3) and, moreover, the following conditions hold true:(1)
ϕ(u)∈C3,g(u)∈C2,(2)
u0∈H2,f∈H1,then for the solution u of the problem of (2.1), one has the following estimate
(3.19)‖uxx‖2≤‖uxx(0)‖2exp⁡(-2c13t)+c14c13(1-exp⁡(-2c13t));
furthermore,
(3.20)lim⁡t→∞‖uxx‖2≤E2,lim⁡t→∞1t∫0t‖uxxx‖2dt≤C.Proof.
Taking the inner product of (2.1) with uxxxx, we have
(3.21)(ut-ϕ(u)x+uxxxx+βuxxx-αuxx+γu+g2(u),uxxxx)=(f,uxxxx),
where
(3.22)(uxxxx,ϕ(u)x)=-(ϕ(u)xx,uxxx)=(ϕ′′(u)ux2+ϕ′(u)uxx,uxxxx)≤A‖u‖3-σ‖uxx‖2‖uxxx‖+A‖u‖4-σ‖uxx‖2‖uxxx‖.
By Young’s inequality and Lemmas 3.1 and 3.2, thus from (3.22), we have
(3.23)|(ϕ(u),uxxxx)|≤c10uxx2+α‖uxxx‖2+c11.
Due to Lemmas 3.1 and 3.2 and (A3), we obtain
(3.24)(g(u),uxxxx)=(γu,uxxxx)+(g2(u),uxxxx)=γ‖uxx‖2+(g2(u),uxxxx)=γ‖uxx‖2+(g2′′(u)uxx2+g2′(u)uxx,uxx)≤γ‖uxx‖2+K‖u‖2‖ux‖2‖uxx‖+K‖u‖3‖uxx‖2.
Using Young’s inequality, we have(3.25)|(g(u),uxxxx)|≤γ‖uxx‖2+α2∥uxx∥2+c12,|(f,uxxxx)|=|(-fx,uxxx)|≤α2‖uxxx‖2+2α‖fx‖2.
By (3.21), (3.23) and (3.25), we get
(3.26)12ddt‖uxx‖2+α2‖uxxx‖2+(c10-γ-α2)‖uxx‖2≤2α‖fx‖2+c11+c12,
that is,
(3.27)ddt‖uxx‖2+α‖uxxx‖2+c13‖uxx‖2≤c14,
where
(3.28)c13=2(c10-γ-α2),c14=2(2α‖fx‖2+c11+c12).
By virtue of Gronwall’s inequality, we have
(3.29)‖uxx‖2≤‖uxx(0)‖2exp⁡(-2c13t)+c14c13(1-exp⁡(-2c13t)),
and (3.27) implies
(3.30)lim⁡t→∞‖uxx‖2≤E2,lim⁡t→∞1t∫0t‖uxxx‖2dt≤C.
Therefore, we prove Lemma 3.3.Lemma 3.4.
Suppose thatϕ(u),g(u) satisfy (A2), (A3) and, moreover, the following conditions hold true:(1)
ϕ(u)∈C3, g(u)∈C2,(2)
u0∈H3, f∈H2,then for the solution u of the problem of (2.1), we have the following estimates:
(3.31)‖uxxx‖2≤‖uxxx(0)‖2exp⁡(-2γt)+c17γ(1-exp⁡(-2γt)),
where
(3.32)c17=2(c16+2α‖fxx‖2);
furthermore,
(3.33)lim⁡t→∞‖uxxx‖2≤E3,lim⁡t→∞1t∫0t‖uxxxx‖2dt≤C.Proof.
Taking the inner product of (2.1) with uxxxxxx, we have
(3.34)(ut-ϕ(u)x+uxxxx+βuxxx-αuxx+γu+g2(u),uxxxxxx)=(f,uxxxxxx),
where
(3.35)(ut,uxxxxxx)=-12ddt‖uxxx‖2,(uxxx,uxxxxxx)=0,(αuxx,uxxxxxx)=α‖uxxxx‖2,|(uxxxxxx,ϕ(u)x)|=(ϕ(u)xxx,uxxxx)=(ϕ′′′(u)ux3+3ϕ′′(u)uxuxx+ϕ′(u)uxxx,uxxxx)=(ϕ′′′(u)ux3)+3(ϕ′′(u)uxuxx,uxxxx).(3.36)|(uxxxxxx,ϕ(u)x)|=(ϕ(u)xxx,uxxxx)=(ϕ′′′(u)ux3+3ϕ′′(u)uxuxx+ϕ′(u)uxxx,uxxxx)=(ϕ′′′(u)ux3)+3(ϕ′′(u)uxuxx,uxxxx).
Using Nirberg’s interpolation inequality and Young’s inequality, from (3.36) and Lemmas 3.1–3.3, we have
(3.37)|(uxxxxxx,ϕ(u)x)|≤α‖uxxxx‖2+c15.
Due to the  condition (4.3), we get
(3.38)(g(u),uxxxxxx)=(γu,uxxxxxx)+(g2(u),uxxxxxx).
By direct calculations, it is easy to get that
(3.39)(γu,uxxxxxx)=-γ‖uxxx‖2,(g2(u),uxxxxxx)=(g2(u)xx,uxxxx)=(g2(u)x2+g2′(u)xx,uxxxx)≤α2‖uxxxx‖2+c16,|(f,uxxxxxx)|≤α2‖uxxxx‖2+2α‖fxx‖2.
Due to (3.34)–(3.39), we have
(3.40)12ddt‖uxxx‖2+α4‖uxxxx‖2+γ‖uxxx‖2≤2α‖fxx‖2+c16,
that is,
(3.41)ddt‖uxxx‖2+α2‖uxxxx‖2+2γ‖uxxx‖2≤c17,
where
(3.42)c17=2(2α‖fxx‖2+c16).
Using Gronwall's inequality, we deduce that
(3.43)‖uxxx‖2≤‖uxxx(0)‖2exp⁡(-2γt)+c17γ(1-exp⁡(-2γt)),
moreover, (3.41) implies
(3.44)lim⁡t→∞‖uxxx‖2≤E3,lim⁡t→∞1t∫0t‖uxxxx‖2dt≤C;
Therefore, we prove Lemma 3.4.In a similar way as above, we can get the uniformly estimates of∥uxxxx∥,∥ut∥ 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)
u0∈Hm+1, f∈Hm,(2)
ϕ(u)∈Cm+1,∥ϕ(u)∥≤A|u|5-σ, (σ,A>0),(3)
g(u)∈Cm,
|g2(u)|≤K|u|5,(4)
g(u)  satisfies (A3), (A4) and g(u) is Lipschitz continuous, that is,(3.45)|g(u)-g(v)|≤C|u-v|,
then there exists a unique global solution u for the problem (2.1) such that u∈L∞(0,T;Hm(Ω)), and furthermore,the semigroup operator S(t) associated with the problem of (2.1) is continuous and there exists an absorbing set B⊂H2(Ω), where
(3.46)B∈{u∣u∈H2(Ω),‖u‖≤E0,‖ux‖≤E1,‖uxx‖≤E2}.Proof.
Similar to the proof of Lemmas3.1–3.4, we have
(3.47)‖u‖Hm≤C.
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 thatu,v are two solutions of the problem (2.1) and w=u-v, then we have
(3.48)wt-(ϕ′(u)-ϕ′(v))wx+wxxxx+βwxxx-αwxx+g(u)-g(v)=0.
Taking the inner product inH of (2.1) with w, we have
(3.49)12ddt‖w‖2+((ϕ′(u)-ϕ′(v))wx,w)+‖wxx‖2+β(wxxx,w)+α‖wx‖2+(g(u)-g(v),w)=0,
where
(3.50)(w,wx)=0,(wxxx,w)=0.
Due to the condition
(3.51)|g(u)-g(v)|≤C|u-v|,
and from (3.49), we obtain
(3.52)12ddt‖w‖2+‖wxx‖2+α‖wx‖2+C‖w‖2≤0,
that is,
(3.53)12ddt‖w‖2+C‖w‖2≤0.
By application of Gronwall's inequality, we get w=0.
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
(3.54)B∈{u∣u∈H2(Ω),‖u‖≤E0,‖ux‖≤E1,‖uxx‖≤E2}
in H2(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.Lemma 3.6.
Suppose thatu0∈H2,f∈H1, and ϕ(u),g(u) satisfy (A1)–(A4), there exists constant C>0, such that
(3.55)‖uη‖2,‖uηx‖2,‖uηxx‖2,‖uηxxx‖2,‖uηxxxx‖2≤C,∀η∈(0,1),t≥0,‖uη‖2,‖uηx‖2,‖uηxx‖2≤Cη,∀η∈(0,1),t≥t0>0.Now, we use the decomposition method to prove the continuity ofS(t) for sake of overcoming the difficult of noncompactness.Setf∈H1(Ω), λL(x)∈C0∞(Ω), 0≤λL≤1, satisfies
(3.56)λL(x)={1|x|≤L,0|x|≤L+1,then,forallη∈(0,1), there exists Lη>0, such that
(3.57)‖f-fη‖1,22≤η,fη=f×λL(η).Assume thatuη is solution of the following equation:
(3.58)uηt+ϕ(uη)x+uηxxxx+βuηxxx-αuηxx+g(uη)=f-fη,uη(x,0)=0.Setting(3.59)S1η(t)u0=uη,wη=S2η(t)u0=S(t)u0-S1η(t)u0
is a solution of the equation as follows:
(3.60)wηt+ϕ(wη)x+wηxxxx+βwηxxx-αwηxx+g(wη)=fη,wη(x,0)=0.Now, we prove the Lemma3.6.Proof.
We take the scalar product in spaceH of (3.58) with uη, we get
(3.61)12ddt‖uη‖2+λ‖uηxx‖2+α‖uηx‖2+γ‖uη‖2+(g2(uη),uη)≤‖f-fη‖‖uη‖.
Due to (A3) and  Young’s inequality, we get
(3.62)(g2(uη),uη)≤γ2‖uη‖2,‖f-fη‖‖uη‖≤γ2‖uη‖2+η2γ.
From (3.61), we obtain the following inequality:
(3.63)ddt‖uη‖2+2γ‖uη‖2≤η2γ.
By Gronwall’s inequality, one has
(3.64)‖uη‖2≤‖u0‖2exp⁡(-2γt)+η22γ2(1-exp⁡(-2γt)).
Hence, there exists C>0, such that
(3.65)‖uη‖2≤C,
and implies
(3.66)‖uη‖2≤Cη,∀t≥t0.
We take the scalar product in spaceH of (3.58) with uηxx and similar to the proof of Lemma 3.2, we have
(3.67)ddt‖uηx‖2+‖uηxxx‖2+2γ‖uηx‖2≤Cη.
By application of Gronwall's inequality, we deduce that
(3.68)‖uηx‖2≤‖ux(0)‖2exp⁡(-2γt)+Cη22γ(1-exp⁡(-2γt)).
So, there exists C>0, such that
(3.69)‖uηx‖2≤C,
and implies
(3.70)‖uηx‖2≤Cη,∀t≥t0.
We take the scalar product in spaceH of (3.58) with uηxxxx and similar to the proof of Lemma 3.3, we have
(3.71)12ddt‖uηxx‖2+12‖uηxxxx‖2+γ‖uηxx‖2≤Cη.
It is easy to prove that
(3.72)‖uηxx‖2≤C,‖uηxx‖2≤Cη,∀t≥t0.
We take the scalar product in spaceH of (3.58) with uηxxxxxx and similar to the proof of Lemma 3.4, we have
(3.73)12ddt‖uηxxx‖2+12‖uηxxxx‖2+γ‖uηxxx‖2≤Cη2‖f-fη‖2
that is,
(3.74)ddt‖uηxxx‖2+2γ‖uηxxx‖2≤Cη.
Hence, by Gronwall’s inequality, we get(3.75)‖uηxxx‖2≤C.
At the same time, we have
(3.76)‖uηxxxx‖2≤C
and we omit them here.Lemma 3.7.
Under the conditions of Lemma3.6, one has the following estimates
(3.77)‖xwη‖2≤C1(η),‖xwηx‖2≤C2(η),‖xwηxx‖2≤C3(η),
where Ci(η)>0, (i=1,2,3).Proof.
We take the scalar product in spaceH of (3.60) with x2wη and noticing that
(3.78)(wηxxxx,x2wη)=‖xwηxx‖2+4(wηxx,xwηx)-2‖wηx‖2,
it is easy to get that
(3.79)ddt‖xwη‖2+‖xwηxx‖2+γ‖xwη‖2≤C(η).
By Gronwall’s inequality, we have
(3.80)‖xwη‖2≤C1(η).
From (3.60), we obtain
(3.81)wηtxx+ϕ(wη)xxx+wηxxxxxxx+βwηxxxxxx-αwηxxxx+g(wη)xx=fηxx
We take the scalar product in spaceH of (3.61) with x2wηxx and noticing that
(3.82)(wηxxxxxx,x2wηxxx)=‖xwηxxxx‖2+4(wηxx,xwηxxx)-2‖wηxxx‖2,(wηxxxxx,x2wηxx)=-4(wηxx,xwηxxxx),(fηxx,x2wηxx)=-4(fη,x2wηxxxx),(ϕ(wη)xxx,x2wηxx)=(ϕ(wη)xx,(x2wηxx)xx)=(ϕ′(wη)wηx,2wηxx+4xwηxxx+x2wηxxxx)=4(ϕ′(wη)wηx,xwηxxx),(g(wη)xx,x2wηxx)=-(g(wη)xx)x,x2wηxx)x),=-(g(wη)xx)x,2xwηxx+x2wηxxx=(-g′(wη)wηx,x2wηxxx).
By Young’s inequality and  the Sobolev embedding theory (see [11]) and (3.81)-(3.82), we deduce that
(3.83)ddt‖xwηxx‖2+2γ‖xwηxx‖2≤C(η).
Using Gronwall’s inequality, we obtain
(3.84)‖xwηxx‖2≤C3(η),‖xwηx‖2=∫Ωx2wηx2dx-2∫Ωxwηxdx-∫Ωx2wηxxwηdx≤2‖xwη‖‖wηx‖+‖xwηxx‖‖xwη‖≤C2(η).
The proof of Lemma 3.7 is completed.Using Lemmas3.6 and 3.7, we can prove that S(t) is continuous.


## 4. Existence of Global Attractor in SpaceH2(Ω)

In this section, we prove that the semigroup operatorS(t) associated with the problem (2.1) possesses a global attractor in space H2(Ω).In order to prove our result, we need the following results.Lemma 4.1 (see [23]).
Assume thats>s1, (s,s1∈N), then the following embedding Hs(Rn)∩Hs1(Rn,(1+|x|2dx))intoHs1(Rn) is compact.Proof.
LetB⊂Hs(Rn)∩Hs1(Rn,(1+|x|2dx))be a bounded set. It suffices to prove that B has a finite ε-net   for any ε>0. First,  since
(4.1)∫Rn|x|2∑l>s1|Dlu|2dx≤C,foru∈B,
there exists an integer A>0, such that
(4.2)∫|x|<A∑l>s1|Dlu|2dx≤1A2∫|x|<A|x|2∑l>s1|Dlu|2dx≤CA2≤ε22.
Let Ω={x||x|<A|}, then the imbedding Hs(Ω)↪Hs1(Ω) is compact. Thus,
(4.3)B|Ω={u∣u=v|Ω,v∈B}⊂Hs(Ω)
is relatively compact in Hs1(Ω) and has a finite (ε/2)-net B(uk̃,(ε/2)), k=1,2,…,m with uk̃∈B|Ω,uk̃=uk|Ω¯ and uk∈B. We claim that {B(uk̃,(ε/2))} is an ε-net  of B in Hs1(Ω).
Indeed, for anyu∈B, ũ=u|Ω¯, then there exists a uk̃ such that
(4.4)‖uk̃-ũ‖Hs1(Ω)<ε2.
Hence,
(4.5)‖ũ-uk̃‖Hs1(Rn)2=‖ũ-uk̃‖Hs1Ω2+‖ũ-uk̃‖HRn∖Ω2≤ε22+∫|x|<A|x|2∑l≤s1|Dlu|2dx<ε2.
This completes the lemma.Lemma 4.2 (see [7, 11]).
LetE be Banach space and {S(t),t≥0} a set of semigroup operators, that is,S(t):E→E satisfy
(4.6)S(t)S(τ)=S(t+τ),S(0)=I,
where I is the identity operator and E is space H2(Ω). We also assume that(1)
S(t) is bounded, that is, for each R>0, there exists a constant C>0 such that ∥u∥E≤C implies ∥S(t)u∥E≤R, (t≥0),(2)
there is an bounded absorbing setB0⊂E, that is, for any bounded set B⊂E, there exists a constant T, such that S(t)B⊂B0, for t≥T,(3)
S(t) is a continuous operator for t>0, then S(t) has a compact global attractor(4.7)A=ω(B0)=⋂s≥0⋃t≥sS(t)B0¯
in the space E0, such that(1)
S(t)𝒜=𝒜, t≥0,(2)
dist(S(t)B,𝒜)E→0 as t→+∞, and dist(S(t)B,𝒜)E denotes the Hausdorff semidistance defined as(4.8)dist⁡(S(t)B,A)=sup⁡x∈S(t)Binf⁡y∈Ad(x,y),
for any bounded set B⊂H2(Ω) in which sense the trajectories are attracted to 𝒜 (see[9, 24]), using Kuratowski α-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{S(t)}t≥0 be a semigroup in complete metric space E. For any subset B⊂E, the set ω(B) defined by ω(B)=⋂s≥0⋃t≥sS(t)B¯ is called the ω-limit set of B.Remark 4.4.
(1) It is easy to see thatψ∈ω(B) if and only if there exists a sequence of element ψn∈B and a sequence tn→∞, such that
(4.9)S(tn)ψn→ψ,asn⟶∞.
(2) Ifω(B) is ω-limit compact set, then, for every bounded subset B of E and for any ε>0, there exists a t0>0, such that α(⋃t≥t0S(t)B)≤ε.Definition 4.5 (see [11, 26]).
Let{S(t)}t≥0 be a semigroup in complete metric space E. A subset B0 of E is called an absorbing set in E if, for any bounded subset B of E, there exists some t0≥0, such that S(t)B⊂B0, for all t≥t0.Definition 4.6 (see [11, 26]).
Let{S(t)}t≥0 be a semigroup in complete metric space E. A subset 𝒜 of E is called global attractor for the semigroup if 𝒜 is compact and enjoys the following properties:(1)
𝒜 is a invariant set, that is, S(t)𝒜=𝒜, for any t≥t0,(2)
𝒜 attract all bounded set of E, that is, for any bounded subset B of E, dist(S(t)B,𝒜)→0, as t→∞, where dist(B,A) is Hausdorff semidistance of two set B and A in space E: dist(B,A)=supx∈Binfy∈Ad(x,y).Definition 4.7 (see [12, 27]).
Kuratowskiα-measure of set B is defined by the formula
(4.10)α(B)=inf⁡{δ∣Bhasafinitecoverofdiameter<δ},
for every bounded set B of a Banach space X.Secondly, due to Definition4.6, it is easy to see that Kuratowskiα-measure of set B has the following properties.Remark 4.8.
(1) If𝒜 is compact set, then α(𝒜)=0;
(2)α(𝒜+ℬ)≤α(𝒜)+α(ℬ),
(3)α(𝒜∪ℬ)≤Max{α(𝒜),α(ℬ)},
(4) if𝒜≤ℬ, α(𝒜)≤α(ℬ),
(5)α(ℬ¯)≤α(ℬ).
Thirdly, we prove Theorem2.1.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
(4.11)α(⋃t≥t0S(t)B)≤ε.
Taking ε=1/n,(n=1,2,…), we can find a sequence {tn}, t1<t2<⋯tn<⋯, such that
(4.12)α(⋃t≥tnS(t)B)≤1n.
By α(ℬ¯)≤α(ℬ), we get
(4.13)α(⋃t≥tnS(t)B¯)≤1n,A=ω(B)=⋂s≥0⋃t≥sS(t)B¯=⋂n=1∞⋃t≥tnS(t)B¯.
First, we prove that 𝒜=ω(B) is variant. As a matter of fact, if ψ∈S(t)ω(B), then ψ=S(t)ϕ, for some ϕ∈ω(B). So, there exists a sequence ϕn∈B and tn→∞ such that S(t)ϕn→ϕ, that is,
(4.14)S(t)S(tn)ϕn=S(t+tn)ϕn⟶S(t)ϕ=ψ,
which implies that ψ∈ω(B) and S(t)ω(B)⊂ω(B).
Conversely, ifψ∈ω(B), by (4.9), we can find two sequences ϕn∈B and tn→∞ such that S(tn)ϕn→ϕ. We need to prove that {S(tn-t)ϕn} has a subsequence which converges in E. For any ε>0, there exists a tε such that
(4.15)α(⋃t′≥tεS(t′)B)≤ε,
which implies that
(4.16)α(⋃t′≥t+tεS(t′-t)B)≤ε.
Hence, there exists an integer N, such that
(4.17)⋃n≥NS(tn-t)ϕn⊂⋃t′≥t+tεS(t′-t)B.
Then, it follows that
(4.18)α(⋃n≥NS(tn-t)ϕn)≤ε.
Notice that α(⋃n≥NS(tn-t)ϕn)≤ε contains only a finite number  of elements, where N0 is fixed such that tn-t≥0, as n≥N0.
By properties (1)–(4) in Remark4.8, we have
(4.19)α(⋃n≥N0S(tn-t)ϕn)=α(⋃n≥NS(tn-t)ϕn)≤ε.
Let ε→0, then we get that
(4.20)α(⋃n≥N0S(tn-t)ϕn)=0.
This implies that {S(tn-t)ϕn} is relatively compact. So, there exists a subsequence tnj→∞ and ψ∈E, such that
(4.21)S(tnj-t)ϕnj⟶∞,astnj⟶∞.
It is easy to see that ψ∈ω(B) and
(4.22)ϕ=lim⁡j→∞S(tnj-t)ϕnj=lim⁡j→∞S(t)S(tnj-t)ϕnj=S(t)ψ
furthermore, ϕ∈S(t)ω(B).
Next, by virtue of Lemma4.2 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.
Otherwise, then there exists a bounded subsetB0 of E such that dist(S(t)B0,𝒜) does not tend to 0 as t→∞. Thus, there exists a δ>0 and a sequence tn→∞ such that
(4.23)dist⁡(S(tn)B0,A)≥δ>0,∀n∈N.
For each n, there exist bn∈B0,(n=1,2,…) satisfying
(4.24)dist⁡(S(tn)bn,A)≥δ2>0.
Whereas B is an absorbing set, S(tn)B0 and S(tn)bn belong to B, for n sufficiently large. As in the discussion above, we obtain that S(tn)bn is relatively compact admits at least one cluster point γ,
(4.25)γ=lim⁡ni→∞S(tni)bni=lim⁡ni→∞S(tni-t1)S(t1)bni,
where t1 follows S(t1)B0⊂B. So, γ∈𝒜=ω(B) and this contradicts (4.24). The proof is complete.



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*Source: 725045-2011-05-03.xml*