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.
Highlights
This article is concerned with the following initial-boundary problem of extendedFisher-Kolmogorov equation involving an unknown function u = u(x, t): ⎧ ⎨ ∂u ∂t = −β 2u + u − u3 + u in ⎩ u = 0, u(x, 0) u φ, 0, in in
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 global attractor, which attracts any bounded set of Hk(Ω) in Hk-norm
(1) Equation (2.1) has a global solution u ∈ C([0, ∞), Xα) ∩ H1([0, ∞), X) ∩ C([0, ∞), X), (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 b1, b2, C1, C2 are positive constants
Summary
In 1995-1998, Peletier and Troy [4,5,6,7] studied spatial patterns, the existence of kinds and stationary solutions of the extended Fisher-Kolmogorov equation (1.1) in their articles. 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 global attractor, which attracts any bounded set of Hk(Ω) in Hk-norm. If Σ attracts any bounded set of X, Σ is called a global attractor of S(t) in X.
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