An invariant set bifurcation theory for nonautonomous nonlinear evolution equations
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 →...
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| Dokumentumtípus: | Folyóirat |
| Megjelent: |
2020
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| Sorozat: | Electronic journal of qualitative theory of differential equations
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| Kulcsszavak: | Egyenletek - nemlineáris, Bifurkációelmélet |
| doi: | 10.14232/ejqtde.2020.1.57 |
| Online Access: | http://acta.bibl.u-szeged.hu/70941 |
| Tartalmi kivonat: | 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(·). |
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| ISSN: | 1417-3875 |