Open Access

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 ,
(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, X1 X 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 .
(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 .
     

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 U X of Σ such that for any u0 U,
inf v Σ S ( t ) u 0 - v X 0 , as t .

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 D X, we define the ω-limit set of D as follows:
ω ( D ) = s 0 t s S ( t ) D ¯ ,

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 B X, i.e., for any bounded set A X there exists a time t A ≥ 0 such that S(t)u0 B, u0 A and t > t A ;

     
  2. (2)

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

     

Then the ω-limit set A = ω ( B ) of B is a global attractor of (2.1), and A 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 and fractional order spaces X α for α R1, where L = - ( L - λ I ) . Without loss of generality, we assume that L generates the fractional power operators L α and fractional order spaces X α as follows:
L α = ( - L ) α : X α X , α R 1 ,

where X α = D ( L α ) is the domain of L α . 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(u0 X, 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 B X α such that for any u0 X α there exists t u 0 > 0 with
    u ( t , u 0 ) B , t > t u 0 ;
     
  2. (2)
    there is a β > α, for any bounded set U X β there are T > 0 and C > 0 such that
    u ( t , u 0 ) X β C , t > T , u 0 U .
     

Then, Equation (2.1) has a global attractor A X α 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 ,

     
  2. (2)

    < L u + G u , u > X - C 1 u X α 2 + C 2 ,

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

    Equation (2.1) has a global attractor A X 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 ) we have
  1. (1)

    T(t): XX α is bounded for all α R1 and t > 0,

     
  2. (2)

    T ( t ) L α x = L α T ( t ) x , x X α ,

     
  3. (3)
    for each t > 0, L α T ( t ) : X X is bounded, and
    L α T ( t ) C α t - α e - δ t ,
     
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 ,
    (2.2)
     
  2. (5)
    if L is symmetric, for any α, β R1 we have
    < L α u , v > X = < L α - β u , L β v > X .
     

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 } .
(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 ,
(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 . The space D(-L) = H1 is the same as (3.1), H 1 2 is given by H 1 2 = 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 ) .

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

We define functional I : H 1 2 R , as
I ( u ) = 1 2 Ω ( - β Δ u 2 - u 2 + u 2 - 1 2 u 4 ) d x ,
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 ,
(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 ,
< L u + G ( u ) , u > - C 1 u H 1 2 2 + C 2 ,
(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 in H α , and A 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 φ H. 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 .
(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 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 φ U 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 ,
(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 .
(3.7)
By (3.2) and (3.6), we get
d u d t = L u + G ( u ) = - D F ( u ) .
(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 ,
(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 ) .
(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 .
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 ,
which implies
u ( t , φ ) H 1 2 C . t 0 , φ U H 1 2 ,
(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 ) there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U , α < 1 .
(3.12)
By H 1 2 ( Ω ) L 6 ( Ω ) , 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 .

which implies that G : H 1 2 H 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 ,

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 ) there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U H α , α < 3 2 .
(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 ,
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 ) ,
which implies
G : H α H 1 2 is bounded for α 1 2 .
(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 .
(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 ,

where β = α - 1 2 ( 0 < β < 1 ) . Hence, (3.13) holds.

Step 4. We prove that for any bounded set U H α (α ≥ 0) there exists C > 0 such that
u ( t , φ ) H α C , t 0 , φ U H α , α 0 .
(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 .
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 )
which implies
G : H α H 1 is bounded for α 1 .
(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 .
(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) 

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 .

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 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 ,

     
  2. (2)

    the set S = { u H 1 2 D F ( u ) = 0 } 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 .
(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 .
Using Hölder inequality and the above inequality, we have
Ω ( Δ u 2 + u 2 + u 4 ) d x C ,

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 , i.e., for any bounded set U H 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 .
(3.20)
From the above discussion, we know that (3.20) holds as α = 1 2 . 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 .
(3.21)
Let B H 1 2 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 .
(3.22)
It is well known that
e t L C e - t λ 1 2 ,
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 .
Hence, for any given T > 0 and φ U H α ( α 1 2 ) . 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 .
(3.23)
From (3.21),(3.22) and Lemma 2.4, for any 1 2 α < 1 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 ,
(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 . By the iteration method, we have that (3.20) holds for all α 1 2 .

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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