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Báo cáo hóa học: " Global attractor of the extended FisherKolmogorov equation in Hk spaces"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Global attractor of the extended FisherKolmogorov equation in Hk spaces | Luo Boundary Value Problems 2011 2011 39 http www.boundaryvalueproblems.eom content 2011 1 39 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Global attractor of the extended Fisher-Kolmogorov equation in Hk spaces Hong Luo1 2 Correspondence lhscnu@163.com 1College 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 0 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 77 dA2u Au u3 u in dt u 0 Au 0 in u x 0 p in X 0 to 3 X 0 to Q 1.1 where b 0 is given A is the Laplacian operator and o denotes an open bounded set of Rn n 1 2 3 with smooth boundary do. 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.