Global attractor of the extended Fisher-Kolmogorov equation in H k spaces
© Luo; licensee Springer. 2011
Received: 31 May 2011
Accepted: 25 October 2011
Published: 25 October 2011
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.
Keywordssemigroup of operator global attractor extended Fisher-Kolmogorov equation regularity
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 [1–3] 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 [4–7] 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 , Smets and Van den Berg , and Li  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 [13–23].
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 [20–23].
where u(t) is an unknown function, L: X1 → X a linear operator, and G: X1 → X a nonlinear operator.
S(t): X → X is a continuous map for any t ≥ 0,
S(0) = id: X → X is the identity,
- (3)S(t + s) = S(t) · S(s), ∀t, s ≥ 0. Then, the solution of (2.1) can be expressed as
Next, we introduce the concepts and definitions of invariant sets, global attractors, and ω-limit sets for the semigroup S(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.
where the closure is taken in the X-norm. Lemma 2.1 is the classical existence theorem of global attractor by Temam .
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 ;
S(t) is uniformly compact, i.e., for any bounded set U ⊂ X and some T > 0 sufficiently large, the set is compact in X.
Then the ω-limit set of B is a global attractor of (2.1), and is connected providing B is connected.
where is the domain of . By the semigroup theory of linear operators , we know that X β ⊂ X α is a compact inclusion for any β > α.
- (1)for some α ≥ 0, there is a bounded set B ⊂ X α such that for any u0 ∈ X α there exists with
- (2)there is a β > α, for any bounded set U ⊂ X β there are T > 0 and C > 0 such that
Then, Equation (2.1) has a global attractor which attracts any bounded set of X α in the X α -norm.
there is a functional F: X α → R such that DF = L + G and ,
- (1)Equation (2.1) has a global solution
Equation (2.1) has a global attractor 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 .
T(t): X → X α is bounded for all α ∈ R1 and t > 0,
- (3)for each t > 0, is bounded, and
- (4)the X α -norm can be defined by(2.2)
- (5)if is symmetric, for any α, β ∈ R1 we have
3 Main results
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: H1 → H given by (3.2) is a sectorial operator and . The space D(-L) = H1 is the same as (3.1), is given by = 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.
Proof. Clearly, L = -β Δ2 + Δ: H1 → H is a sectorial operator, and is a compact mapping.
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 in H α , and attracts any bounded set of H α in the H α -norm.
Next, according to Lemma 2.2, we prove Theorem 3.2 in the following five steps.
which implies that (3.6) is a Lyapunov function.
where C1, C2, and C are positive constants, and C only depends on φ.
which implies that is bounded.
where β = α(0 < β < 1). Hence, (3.12) holds.
where . Hence, (3.13) holds.
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 .
the set is bounded.
where C > 0 is a constant. Thus, property (2) is proved.
where C > 0 is a constant independent of φ.
Then, we infer from (3.23) and (3.24) that (3.20) holds for all . By the iteration method, we have that (3.20) holds for all .
Finally, this theorem follows from (3.16), (3.20) and Lemma 2.2. The proof is completed.
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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