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Boundary Value Problems

, 2011:39 | Cite as

Global attractor of the extended Fisher-Kolmogorov equation in H k spaces

  • Hong Luo
Open Access
Research

Abstract

The long-time behavior of solution to extended Fisher-Kolmogorov equation is considered in this article. Using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of global attractor, we prove that the extended Fisher-Kolmogorov equation possesses a global attractor in Sobolev space H k for all k > 0, which attracts any bounded subset of H k (Ω) in the H k -norm.

2000 Mathematics Subject Classification: 35B40; 35B41; 35K25; 35K30.

Keywords

semigroup of operator global attractor extended Fisher-Kolmogorov equation regularity 

1 Introduction

This article is concerned with the following initial-boundary problem of extended Fisher-Kolmogorov equation involving an unknown function u = u(x, t):
u t = - β Δ 2 u + Δ u - u 3 + u i n Ω × ( 0 , ) , u = 0 , Δ u = 0 , i n Ω × ( 0 , ) , u ( x , 0 ) = φ , i n Ω , Open image in new window
(1.1)

where β > 0 is given, Δ is the Laplacian operator, and Ω denotes an open bounded set of R n (n = 1, 2, 3) with smooth boundary ∂Ω.

The extended Fisher-Kolmogorov equation proposed by Dee and Saarloos [1, 2, 3] in 1987-1988, which serves as a model in studies of pattern formation in many physical, chemical, or biological systems, also arises in the theory of phase transitions near Lifshitz points. The extended Fisher-Kolmogorov equation (1.1) have extensively been studied during the last decades. In 1995-1998, Peletier and Troy [4, 5, 6, 7] studied spatial patterns, the existence of kinds and stationary solutions of the extended Fisher-Kolmogorov equation (1.1) in their articles. Van der Berg and Kwapisz [8, 9] proved uniqueness of solutions for the extended Fisher-Kolmogorov equation in 1998-2000. Tersian and Chaparova [10], Smets and Van den Berg [11], and Li [12] catch Periodic and homoclinic solution of Equation (1.1).

The global asymptotical behaviors of solutions and existence of global attractors are important for the study of the dynamical properties of general nonlinear dissipative dynamical systems. So, many authors are interested in the existence of global attractors such as Hale, Temam, among others [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

In this article, we shall use the regularity estimates for the linear semigroups, combining with the classical existence theorem of global attractors, to prove that the extended Fisher-Kolmogorov equation possesses, in any k th differentiable function spaces H k (Ω), a global attractor, which attracts any bounded set of H k (Ω) in H k -norm. The basic idea is an iteration procedure which is from recent books and articles [20, 21, 22, 23].

2 Preliminaries

Let X and X1 be two Banach spaces, X1X a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on X, given by
d u d t = L u + G ( u ) , u ( x , 0 ) = u 0 . Open image in new window
(2.1)

where u(t) is an unknown function, L: X1X a linear operator, and G: X1X a nonlinear operator.

A family of operators S(t): XX(t ≥ 0) is called a semigroup generated by (2.1) if it satisfies the following properties:
  1. (1)

    S(t): XX is a continuous map for any t ≥ 0,

     
  2. (2)

    S(0) = id: XX is the identity,

     
  3. (3)
    S(t + s) = S(t) · S(s), ∀t, s ≥ 0. Then, the solution of (2.1) can be expressed as
    u ( t , u 0 ) = S ( t ) u 0 . Open image in new window
     

Next, we introduce the concepts and definitions of invariant sets, global attractors, and ω-limit sets for the semigroup S(t).

Definition 2.1 Let S(t) be a semigroup defined on X. A set Σ ⊂ X is called an invariant set of S(t) if S(t)Σ = Σ, ∀t ≥ 0. An invariant set Σ is an attractor of S(t) if Σ is compact, and there exists a neighborhood UX of Σ such that for any u0U,
inf v Σ S ( t ) u 0 - v X 0 , as t . Open image in new window

In this case, we say that Σ attracts U. Especially, if Σ attracts any bounded set of X, Σ is called a global attractor of S(t) in X.

For a set DX, we define the ω-limit set of D as follows:
ω ( D ) = s 0 t s S ( t ) D ¯ , Open image in new window

where the closure is taken in the X-norm. Lemma 2.1 is the classical existence theorem of global attractor by Temam [17].

Lemma 2.1 Let S(t): XX be the semigroup generated by (2.1). Assume the following conditions hold:
  1. (1)

    S(t) has a bounded absorbing set BX, i.e., for any bounded set AX there exists a time t A ≥ 0 such that S(t)u0B, ∀u0A and t > t A ;

     
  2. (2)

    S(t) is uniformly compact, i.e., for any bounded set UX and some T > 0 sufficiently large, the set t T S ( t ) U ¯ Open image in new window is compact in X.

     

Then the ω-limit set A = ω ( B ) Open image in new window of B is a global attractor of (2.1), and A Open image in new window is connected providing B is connected.

Note that we used to assume that the linear operator L in (2.1) is a sectorial operator which generates an analytic semigroup e tL . It is known that there exists a constant λ ≥ 0 such that L - λI generates the fractional power operators L α Open image in new window and fractional order spaces X α for αR1, where L = - ( L - λ I ) Open image in new window. Without loss of generality, we assume that L generates the fractional power operators L α Open image in new window and fractional order spaces X α as follows:
L α = ( - L ) α : X α X , α R 1 , Open image in new window

where X α = D ( L α ) Open image in new window is the domain of L α Open image in new window. By the semigroup theory of linear operators [24], we know that X β X α is a compact inclusion for any β > α.

Thus, Lemma 2.1 can equivalently be expressed in Lemma 2.2 [20, 21, 22, 23].

Lemma 2.2 Let u(t, u0) = S(t)u0(u0X, t ≥ 0) be a solution of (2.1) and S(t) be the semigroup generated by (2.1). Let X α be the fractional order space generated by L. Assume:
  1. (1)
    for some α ≥ 0, there is a bounded set BX α such that for any u0X α there exists t u 0 > 0 Open image in new window with
    u ( t , u 0 ) B , t > t u 0 ; Open image in new window
     
  2. (2)
    there is a β > α, for any bounded set UX β there are T > 0 and C > 0 such that
    u ( t , u 0 ) X β C , t > T , u 0 U . Open image in new window
     

Then, Equation (2.1) has a global attractor A X α Open image in new window which attracts any bounded set of X α in the X α -norm.

For Equation (2.1) with variational characteristic, we have the following existence theorem of global attractor [20, 22].

Lemma 2.3 Let L: X1X be a sectorial operator, X α = D((-L) α ) and G: X α X(0 < α < 1) be a compact mapping. If
  1. (1)

    there is a functional F: X α R such that DF = L + G and F ( u ) - β 1 u X α 2 + β 2 Open image in new window,

     
  2. (2)

    < L u + G u , u > X - C 1 u X α 2 + C 2 Open image in new window,

     
then
  1. (1)
    Equation (2.1) has a global solution
    u C ( [ 0 , ) , X α ) H 1 ( [ 0 , ) , X ) C ( [ 0 , ) , X ) , Open image in new window
     
  2. (2)

    Equation (2.1) has a global attractor A X Open image in new window which attracts any bounded set of X, where DF is a derivative operator of F, and β1, β2, C1, C2 are positive constants.

     

For sectorial operators, we also have the following properties which can be found in [24].

Lemma 2.4 Let L: X1X be a sectorial operator which generates an analytic semigroup T(t) = e tL . If all eigenvalues λ of L satisfy Reλ < -λ0 for some real number λ0 > 0, then for L α ( L = - L ) Open image in new window we have
  1. (1)

    T(t): XX α is bounded for all αR1 and t > 0,

     
  2. (2)

    T ( t ) L α x = L α T ( t ) x , x X α Open image in new window,

     
  3. (3)
    for each t > 0, L α T ( t ) : X X Open image in new window is bounded, and
    L α T ( t ) C α t - α e - δ t , Open image in new window
     
where δ > 0 and C α > 0 are constants only depending on α,
  1. (4)
    the X α -norm can be defined by
    x X α = L α x X , Open image in new window
    (2.2)
     
  2. (5)
    if L Open image in new window is symmetric, for any α, βR1 we have
    < L α u , v > X = < L α - β u , L β v > X . Open image in new window
     

3 Main results

Let H and H1 be the spaces defined as follows:
H = L 2 ( Ω ) , H 1 = { u H 4 ( Ω ) : u Ω = Δ u Ω = 0 } . Open image in new window
(3.1)
We define the operators L: H1H and G: H1H by
L u = - β Δ 2 u + Δ u G ( u ) = - u 3 + u , Open image in new window
(3.2)

Thus, the extended Fisher-Kolmogorov equation (1.1) can be written into the abstract form (2.1). It is well known that the linear operator L: H1H given by (3.2) is a sectorial operator and L = - L Open image in new window. The space D(-L) = H1 is the same as (3.1), H 1 2 Open image in new window is given by H 1 2 Open image in new window = closure of H1 in H2(Ω) and H k = H2k(Ω) ∩ H1 for k ≥ 1.

Before the main result in this article is given, we show the following theorem, which provides the existence of global attractors of the extended Fisher-Kolmogorov equation (1.1) in H.

Theorem 3.1 The extended Fisher-Kolmogorov equation (1.1) has a global attractor in H and a global solution
u C ( [ 0 , ) , H 1 2 ) H 1 ( [ 0 , ) , H ) . Open image in new window

Proof. Clearly, L = -β Δ2 + Δ: H1H is a sectorial operator, and G : H 1 2 H Open image in new window is a compact mapping.

We define functional I : H 1 2 R Open image in new window, as
I ( u ) = 1 2 Ω ( - β Δ u 2 - u 2 + u 2 - 1 2 u 4 ) d x , Open image in new window
which satisfies DI(u) = Lu + G(u).
I ( u ) = 1 2 Ω ( - β Δ u 2 - u 2 + u 2 - 1 2 u 4 ) d x 1 2 Ω ( - β Δ u 2 + u 2 - 1 2 u 4 ) d x 1 2 Ω ( - β Δ u 2 + 1 ) d x , I ( u ) - β 1 u H 1 2 2 + β 2 , Open image in new window
(3.3)
which implies condition (1) of Lemma 2.3.
< L u + G ( u ) , u > = Ω ( - β u Δ 2 u + u Δ u + u 2 - u 4 ) d x = Ω ( - β Δ u 2 - u 2 + u 2 - u 4 ) d x Ω ( - β Δ u 2 + u 2 - u 4 ) d x Ω ( - β Δ u 2 + 1 ) d x , Open image in new window
< L u + G ( u ) , u > - C 1 u H 1 2 2 + C 2 , Open image in new window
(3.4)

which implies condition (2) of Lemma 2.3.

This theorem follows from (3.3), (3.4), and Lemma 2.3.

The main result in this article is given by the following theorem, which provides the existence of global attractors of the extended Fisher-Kolmogorov equation (1.1) in any k th-order space H k .

Theorem 3.2 For any α ≥ 0 the extended Fisher-Kolmogorov equation (1.1) has a global attractor A Open image in new window in H α , and A Open image in new window attracts any bounded set of H α in the H α -norm.

Proof. From Theorem 3.1, we know that the solution of system (1.1) is a global weak solution for any φH. Hence, the solution u(t, φ) of system (1.1) can be written as
u ( t , φ ) = e t L φ + 0 t e ( t - τ ) L G ( u ) d τ . Open image in new window
(3.5)

Next, according to Lemma 2.2, we prove Theorem 3.2 in the following five steps.

Step 1. We prove that for any bounded set U H 1 2 Open image in new window there is a constant C > 0 such that the solution u(t, φ) of system (1.1) is uniformly bounded by the constant C for any φU and t ≥ 0. To do that, we firstly check that system (1.1) has a global Lyapunov function as follows:
F ( u ) = 1 2 Ω ( β Δ u 2 + u 2 - u 2 + 1 2 u 4 ) d x , Open image in new window
(3.6)
In fact, if u(t, ·) is a strong solution of system (1.1), we have
d d t F ( u ( t , φ ) ) = < D F ( u ) , d u d t > H . Open image in new window
(3.7)
By (3.2) and (3.6), we get
d u d t = L u + G ( u ) = - D F ( u ) . Open image in new window
(3.8)
Hence, it follows from (3.7) and (3.8) that
d F ( u ) d t = < D F ( u ) , - D F ( u ) > H = - D F ( u ) H 2 , Open image in new window
(3.9)

which implies that (3.6) is a Lyapunov function.

Integrating (3.9) from 0 to t gives
F ( u ( t , φ ) ) = - 0 t D F ( u ) H 2 d t + F ( φ ) . Open image in new window
(3.10)
Using (3.6), we have
F ( u ) = 1 2 Ω ( β Δ u 2 + u 2 - u 2 + 1 2 u 4 ) d x 1 2 Ω ( β Δ u 2 - u 2 + 1 2 u 4 ) d x 1 2 Ω ( β Δ u 2 - 1 ) d x C 1 Ω Δ u 2 d x - C 2 . Open image in new window
Combining with (3.10) yields
C 1 Ω Δ u 2 d x - C 2 - 0 t D F ( u ) H 2 d t + F ( φ ) , C 1 Ω Δ u 2 d x + 0 t D F ( u ) H 2 d t F ( φ ) + C 2 , Ω Δ u 2 d x C , t 0 , φ U , Open image in new window
which implies
u ( t , φ ) H 1 2 C . t 0 , φ U H 1 2 , Open image in new window
(3.11)

where C1, C2, and C are positive constants, and C only depends on φ.

Step 2. We prove that for any bounded set U H α ( 1 2 α < 1 ) Open image in new window there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U , α < 1 . Open image in new window
(3.12)
By H 1 2 ( Ω ) L 6 ( Ω ) Open image in new window, we have
G ( u ) H 2 = Ω G ( u ) 2 d x = Ω u - u 3 2 d x = Ω u 2 - 2 u 4 + u 6 d x Ω ( u 2 + 2 u 4 + u 6 ) d x C Ω u 6 d x + 1 C u H 1 2 6 + 1 . Open image in new window

which implies that G : H 1 2 H Open image in new window is bounded.

Hence, it follows from (2.2) and (3.5) that
u ( t , φ ) H α = e t L φ + 0 t e ( t - τ ) L g ( u ) d τ H α φ H α + 0 t ( - L ) α e ( t - τ ) L G ( u ) H d τ φ H α + 0 t ( - L ) α e ( t - τ ) L G ( u ) H d τ φ H α + C 0 t ( - L ) α e ( t - τ ) L ( u H 1 2 6 + 1 ) d τ φ H α + C 0 t τ β e - δ t d τ C , t 0 , φ U H α , Open image in new window

where β = α(0 < β < 1). Hence, (3.12) holds.

Step 3. We prove that for any bounded set U H α ( 1 α < 3 2 ) Open image in new window there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U H α , α < 3 2 . Open image in new window
(3.13)
In fact, by the embedding theorems of fractional order spaces [24]:
H 2 ( Ω ) W 1 , 4 ( Ω ) , H 2 ( Ω ) H 1 ( Ω ) , H α C 0 ( Ω ) H 2 ( Ω ) , α 1 2 , Open image in new window
we have
G ( u ) H 1 2 2 = Ω ( - L ) 1 2 G ( u ) 2 d x = < ( - L ) 1 2 G ( u ) , ( - L ) 1 2 G ( u ) > = < ( - L ) G ( u ) , G ( u ) > = Ω [ ( β Δ 2 G ( u ) - Δ G ( u ) ) G ( u ) ] d x C Ω ( Δ G ( u ) 2 + G ( u ) 2 ) d x = C Ω ( ( 1 - 3 u 2 ) u 2 + Δ u - 6 u ( u ) 2 - 3 u 2 Δ u 2 ) d x C Ω ( u 4 u 2 + u 2 + Δ u 2 + u 2 u 4 + u 4 Δ u 2 ) d x C Ω ( s u p x Ω u 4 u 2 + u 2 + Δ u 2 + s u p x Ω u 2 u 4 + s u p x Ω u 4 Δ u 2 ) d x C [ s u p x Ω u 4 Ω u 2 d x + Ω u 2 d x + Ω Δ u 2 d x + s u p x Ω u 2 Ω u 4 d x + s u p x Ω u 4 Ω Δ u 2 d x ] C ( u C 0 4 u H 1 2 + u H 1 2 + u H 2 2 + u C 0 2 u W 1 , 4 4 + u C 0 4 u H 2 2 ) C ( u H α 4 u H 1 2 + u H 1 2 + u H 2 2 + u H α 2 u W 1 , 4 4 + u H α 4 u H 2 2 ) C ( u H α 6 + u H α 2 ) , Open image in new window
which implies
G : H α H 1 2 is bounded for α 1 2 . Open image in new window
(3.14)
Therefore, it follows from (3.12) and (3.14) that
G ( u ) H 1 2 < C , t 0 , φ U H α , 1 2 α < 1 . Open image in new window
(3.15)
Then, using same method as that in Step 2, we get from (3.15) that
u ( t , φ ) H α = e t L φ + 0 t e ( t - τ ) L G ( u ) d τ H α φ H α + 0 t ( - L ) α e ( t - τ ) L G ( u ) H d τ φ H α + C 0 t ( - L ) α - 1 2 e ( t - τ ) L G ( u ) H 1 2 d τ φ H α + C 0 t τ β e - δ t d τ C , t 0 , φ U H α , Open image in new window

where β = α - 1 2 ( 0 < β < 1 ) Open image in new window. Hence, (3.13) holds.

Step 4. We prove that for any bounded set UH α (α ≥ 0) there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U H α , α 0 . Open image in new window
(3.16)
In fact, by the embedding theorems of fractional order spaces [24]:
H 4 ( Ω ) H 3 ( Ω ) H 2 ( Ω ) , H 4 ( Ω ) W 2 , 4 ( Ω ) , H α C 1 ( Ω ) H 4 ( Ω ) , α 1 . Open image in new window
we have
G ( u ) H 1 2 = ( L ) G ( u ) 2 C Ω ( Δ 2 G ( u ) 2 + Δ G ( u ) 2 ) d x C Ω [ ( Δ 2 u + 30 u 2 Δ u + 12 u Δ u 2 + 18 u u Δ u + 3 u 2 Δ 2 u ) 2 + ( Δ u + 6 u u 2 + 3 u 2 Δ u ) 2 ] d x C Ω ( Δ 2 u 2 + u 4 Δ u 2 + u 2 Δ u 4 + u 2 u 2 Δ u 2 + u 4 Δ 2 u 2 + Δ u 2 + u 2 u 4 + u 4 Δ u 2 ) d x C Ω ( Δ 2 u 2 + s u p x Ω u 4 Δ u 2 + s u p x Ω u 2 Δ u 4 + s u p x Ω u 2 s u p x Ω u 2 Δ u 2 + s u p x Ω u 4 Δ 2 u 2 + Δ u 2 + s u p x Ω u 2 s u p x Ω u 4 + s u p x Ω u 4 Δ u 2 ) d x C [ Ω Δ 2 u 2 d x + s u p x Ω u 4 Ω Δ u 2 d x + s u p x Ω u 2 Ω Δ u 4 d x + s u p x Ω u 2 s u p x Ω u 2 Ω Δ u 2 d x + s u p x Ω u 4 Ω Δ 2 u 2 d x + Ω Δ u 2 d x + s u p x Ω u 2 s u p x Ω u 4 Ω d x + s u p x Ω u 4 Ω Δ u 2 d x ] C ( u H 4 2 + u C 1 4 u H 2 2 + u C 0 2 u W 2 , 4 4 + u C 0 2 u C 1 2 u H 3 2 + u C 0 4 u H 4 2 + u H 2 2 + u C 0 2 u C 1 4 + u C 0 4 u H 2 2 ) C ( u H 4 2 + u H α 4 u H 2 2 + u H α 2 u W 2 , 4 4 + u H α 4 u H 3 2 + u H α 4 u H 4 2 + u H 2 2 + u H α 6 + u H α 4 u H 2 2 ) C ( u H α 6 + u H α 2 ) Open image in new window
which implies
G : H α H 1 is bounded for α 1 . Open image in new window
(3.17)
Therefore, it follows from (3.13) and (3.17) that
G ( u ) H 1 < C , t 0 , φ U H α , 1 α < 3 2 . Open image in new window
(3.18)
Then, we get from (3.18) that
u ( t , φ ) H α = e t L φ + 0 t e ( t - τ ) L G ( u ) d τ H α φ H α + 0 t ( - L ) α e ( t - τ ) L G ( u ) H d τ (1) φ H α + 0 t ( - L ) α - 1 e ( t - τ ) L G ( u ) H 1 d τ (2) φ H α + C 0 t τ β e - δ t d τ C , t 0 , φ U H α , (3) (4)  Open image in new window

where β = α - 1(0 < β < 1). Hence, (3.16) holds.

By doing the same procedures as Steps 1-4, we can prove that (3.16) holds for all α ≥ 0.

Step 5. We show that for any α ≥ 0, system (1.1) has a bounded absorbing set in H α . We first consider the case of α = 1 2 Open image in new window.

From Theorem 3.1 we have known that the extended Fisher-Kolmogorov equation possesses a global attractor in H space, and the global attractor of this equation consists of equilibria with their stable and unstable manifolds. Thus, each trajectory has to converge to a critical point. From (3.9) and (3.16), we deduce that for any φ H 1 2 Open image in new window the solution u(t, φ) of system (1.1) converges to a critical point of F. Hence, we only need to prove the following two properties:
  1. (1)

    F ( u ) u H 1 2 Open image in new window,

     
  2. (2)

    the set S = { u H 1 2 D F ( u ) = 0 } Open image in new window is bounded.

     
Property (1) is obviously true, we now prove (2) in the following. It is easy to check if DF(u) = 0, u is a solution of the following equation
β Δ 2 u - Δ u - u + u 3 = 0 , u Ω = 0 , Δ u Ω = 0 . Open image in new window
(3.19)
Taking the scalar product of (3.19) with u, then we derive that
Ω ( β Δ u 2 + u 2 - u 2 + u 4 ) d x = 0 . Open image in new window
Using Hölder inequality and the above inequality, we have
Ω ( Δ u 2 + u 2 + u 4 ) d x C , Open image in new window

where C > 0 is a constant. Thus, property (2) is proved.

Now, we show that system (1.1) has a bounded absorbing set in H α for any α 1 2 Open image in new window, i.e., for any bounded set UH α there are T > 0 and a constant C > 0 independent of φ such that
u ( t , φ ) H α C , t T , φ U . Open image in new window
(3.20)
From the above discussion, we know that (3.20) holds as α = 1 2 Open image in new window. By (3.5) we have
u ( t , φ ) = e ( t - T ) L u ( T , φ ) + 0 t e ( t - τ ) L G ( u ) d τ . Open image in new window
(3.21)
Let B H 1 2 Open image in new window be the bounded absorbing set of system (1.1), and T0 > 0 such that
u ( t , φ ) B , t T 0 , φ U H α α 1 2 . Open image in new window
(3.22)
It is well known that
e t L C e - t λ 1 2 , Open image in new window
where λ1 > 0 is the first eigenvalue of the equation
β Δ 2 u - Δ u = λ u , u Ω = 0 , Δ u Ω = 0 . Open image in new window
Hence, for any given T > 0 and φ U H α ( α 1 2 ) Open image in new window. We have
e ( t - τ ) L u ( t , φ ) H α = ( - L ) α e ( t - τ ) L u ( t , φ ) H 0 , a s t . Open image in new window
(3.23)
From (3.21),(3.22) and Lemma 2.4, for any 1 2 α < 1 Open image in new window we get that
u ( t , φ ) H α e ( t - T 0 ) L u ( T 0 , φ ) H α + T 0 t ( - L ) α e ( t - τ ) L G ( u ) d τ e ( t - T 0 ) L u ( T 0 , φ ) H α + C 0 t - T 0 τ - α e - λ 1 τ d τ , Open image in new window
(3.24)

where C > 0 is a constant independent of φ.

Then, we infer from (3.23) and (3.24) that (3.20) holds for all 1 2 α < 1 Open image in new window. By the iteration method, we have that (3.20) holds for all α 1 2 Open image in new window.

Finally, this theorem follows from (3.16), (3.20) and Lemma 2.2. The proof is completed.

Notes

Acknowledgements

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. Foundation item: the National Natural Science Foundation of China (No. 11071177).

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© Luo; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.College of MathematicsSichuan UniversityChengduPR China
  2. 2.College of Mathematics and Software ScienceSichuan Normal UniversityChengduPR China

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