Ju Xuewei and Qi Ailing: An invariant set bifurcation theory for nonautonomous nonlinear evolution equations. (2020)
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Abstract
In this paper we establish an invariant set bifurcation theory for the nonautonomous dynamical system (ϕλ, θ)X,H generated by the evolution equation ut + Au = λu + p(t, u), p ∈ H = H[ f(·, u)] (0.1) on a Hilbert space X, where A is a sectorial operator, λ is the bifurcation parameter, f(·, u) : R → X is translation compact, f(t, 0) ≡ 0 and H[ f ] is the hull of f(·, u). Denote by ϕλ := ϕλ(t, p)u the cocycle semiflow generated by the system. Under some other assumptions on f , we show that as the parameter λ crosses an eigenvalue λ0 ∈ R of A, the system bifurcates from 0 to a nonautonomous invariant set Bλ(·) on one-sided neighborhood of λ0. Moreover, lim λ→λ0 HXα (Bλ(p), 0) = 0, p ∈ P, where HXα (·, ·) denotes the Hausdorff semidistance in X (here X (α ≥ 0) defined below is the fractional power spaces associated with A). Our result is based on the pullback attractor bifurcation on the local central invariant manifolds Mλ loc(·).
| Item Type: | Journal |
|---|---|
| Publication full: | Electronic journal of qualitative theory of differential equations |
| Date: | 2020 |
| Number: | 57 |
| ISSN: | 1417-3875 |
| DOI: | 10.14232/ejqtde.2020.1.57 |
| Uncontrolled Keywords: | Egyenletek - nemlineáris, Bifurkációelmélet |
| Additional Information: | Bibliogr.: p. 21-24. ; összefoglalás angol nyelven |
| Date Deposited: | 2020. Nov. 30. 13:57 |
| Last Modified: | 2021. Oct. 20. 13:52 |
| URI: | http://acta.bibl.u-szeged.hu/id/eprint/70941 |
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