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

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Global attractor of the extended FisherKolmogorov equation in Hk spaces Hong Luo1,2 Correspondence: [email protected] 1 College of Mathematics, Sichuan University, Chengdu, Sichuan 610041, PR China Full list of author information is available at the end of the article

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 Hk for all k > 0, which attracts any bounded subset of Hk(Ω) in the Hk-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 = −β2 u + u − u3 + u in u = 0, u = 0, in ⎩ u(x, 0) = ϕ, in

 × (0, ∞), ∂ × (0, ∞), ,

(1:1)

where b > 0 is given, Δ is the Laplacian operator, and Ω denotes an open bounded set of Rn(n = 1, 2, 3) with smooth boundary ∂Ω. The extended Fisher-Kolmogorov equation proposed by Dee and Saarloos [1-3] in 19871988, 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 [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 [13-23]. © 2011 Luo; licensee Springer. 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.

Luo Boundary Value Problems 2011, 2011:39 http://www.boundaryvalueproblems.com/content/2011/1/39

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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 kth differentiable function spaces Hk(Ω), a glo