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

Boundary Value Problems20112011:39

DOI: 10.1186/1687-2770-2011-39

Received: 31 May 2011

Accepted: 25 October 2011

Published: 25 October 2011

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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ1_HTML.gif
(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 [13] 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 [47] 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 [1323].

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 [2023].

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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ2_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equa_HTML.gif
     

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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equb_HTML.gif

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 ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equc_HTML.gif

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 ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq1_HTML.gif is compact in X.

     

Then the ω-limit set A = ω ( B ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq2_HTML.gif of B is a global attractor of (2.1), and A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq3_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq4_HTML.gif and fractional order spaces X α for αR1, where L = - ( L - λ I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq5_HTML.gif. Without loss of generality, we assume that L generates the fractional power operators L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq4_HTML.gif and fractional order spaces X α as follows:
L α = ( - L ) α : X α X , α R 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equd_HTML.gif

where X α = D ( L α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq6_HTML.gif is the domain of L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq4_HTML.gif. 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 [2023].

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq7_HTML.gif with
    u ( t , u 0 ) B , t > t u 0 ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Eque_HTML.gif
     
  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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equf_HTML.gif
     

Then, Equation (2.1) has a global attractor A X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq8_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq9_HTML.gif,

     
  2. (2)

    < L u + G u , u > X - C 1 u X α 2 + C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq10_HTML.gif,

     
then
  1. (1)
    Equation (2.1) has a global solution
    u C ( [ 0 , ) , X α ) H 1 ( [ 0 , ) , X ) C ( [ 0 , ) , X ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equg_HTML.gif
     
  2. (2)

    Equation (2.1) has a global attractor A X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq11_HTML.gif 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq12_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq13_HTML.gif,

     
  3. (3)
    for each t > 0, L α T ( t ) : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq14_HTML.gif is bounded, and
    L α T ( t ) C α t - α e - δ t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equh_HTML.gif
     
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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ3_HTML.gif
    (2.2)
     
  2. (5)
    if L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq15_HTML.gif is symmetric, for any α, βR1 we have
    < L α u , v > X = < L α - β u , L β v > X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equi_HTML.gif
     

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 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ4_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ5_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq16_HTML.gif. The space D(-L) = H1 is the same as (3.1), H 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq17_HTML.gif is given by H 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq17_HTML.gif = 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equj_HTML.gif

Proof. Clearly, L = -β Δ2 + Δ: H1H is a sectorial operator, and G : H 1 2 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq18_HTML.gif is a compact mapping.

We define functional I : H 1 2 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq19_HTML.gif, as
I ( u ) = 1 2 Ω ( - β Δ u 2 - u 2 + u 2 - 1 2 u 4 ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equk_HTML.gif
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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ6_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equl_HTML.gif
< L u + G ( u ) , u > - C 1 u H 1 2 2 + C 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ7_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq3_HTML.gif in H α , and A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq3_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ8_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq20_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ9_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ10_HTML.gif
(3.7)
By (3.2) and (3.6), we get
d u d t = L u + G ( u ) = - D F ( u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ11_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ12_HTML.gif
(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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ13_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equm_HTML.gif
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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equn_HTML.gif
which implies
u ( t , φ ) H 1 2 C . t 0 , φ U H 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ14_HTML.gif
(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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq21_HTML.gif there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U , α < 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ15_HTML.gif
(3.12)
By H 1 2 ( Ω ) L 6 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq22_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equo_HTML.gif

which implies that G : H 1 2 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq23_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equp_HTML.gif

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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq24_HTML.gif there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U H α , α < 3 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ16_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equq_HTML.gif
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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equr_HTML.gif
which implies
G : H α H 1 2 is bounded for α 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ17_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ18_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equs_HTML.gif

where β = α - 1 2 ( 0 < β < 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq25_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ19_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equt_HTML.gif
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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equu_HTML.gif
which implies
G : H α H 1 is bounded for α 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ20_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ21_HTML.gif
(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)  http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equv_HTML.gif

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq26_HTML.gif.

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq27_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq28_HTML.gif,

     
  2. (2)

    the set S = { u H 1 2 D F ( u ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq29_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ22_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equw_HTML.gif
Using Hölder inequality and the above inequality, we have
Ω ( Δ u 2 + u 2 + u 4 ) d x C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equx_HTML.gif

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq30_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ23_HTML.gif
(3.20)
From the above discussion, we know that (3.20) holds as α = 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq26_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ24_HTML.gif
(3.21)
Let B H 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq31_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ25_HTML.gif
(3.22)
It is well known that
e t L C e - t λ 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equy_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equz_HTML.gif
Hence, for any given T > 0 and φ U H α ( α 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq32_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ26_HTML.gif
(3.23)
From (3.21),(3.22) and Lemma 2.4, for any 1 2 α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq33_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_Equ27_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq34_HTML.gif. By the iteration method, we have that (3.20) holds for all α 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-39/MediaObjects/13661_2011_Article_84_IEq30_HTML.gif.

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

Declarations

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).

Authors’ Affiliations

(1)
College of Mathematics, Sichuan University
(2)
College of Mathematics and Software Science, Sichuan Normal University

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