Floquet bundles for scalar parabolic equations

Chow, Shui Nee; Lu, Kening; Mallet‐Paret, John

doi:10.1007/bf00383675

Cited by 37 publications

(40 citation statements)

References 16 publications

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“…The Floquet bundles obtained here are analogous to the ones obtained in [6] for time-dependent scalar parabolic equations. However, as the function σ is defined and continuous only on Λ and not on the whole R n \ {0} (while the zero-crossing number can be defined on the whole phase space X except for {0}), it is technically more difficult to construct the Floquet bundles.…”

Section: Introductionmentioning

confidence: 76%

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Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

Fang¹,

Gyllenberg²,

Wang³

2013

SIAM J. Math. Anal.

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Abstract. We consider a general time-dependent linear competitive-cooperative tridiagonal system of differential equations in the framework of skew-product flows and obtain canonical Floquet invariant bundles which are exponentially separated. Such Floquet bundles naturally reduce to the standard Floquet space when the system is assumed to be time-periodic. We apply the Floquet theory so obtained to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitivecooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.

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Section: Introductionmentioning

confidence: 76%

“…Our approach is motivated by the work of Chow, Lu, and Mallet-Paret [5,6] for time-dependent scalar parabolic equations and extends earlier work on linear autonomous tridiagonal equations in [33], linear time-periodic equations in [34], and linear asymptotically autonomous equations in [8].…”

Section: Introductionmentioning

confidence: 99%

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

Fang¹,

Gyllenberg²,

Wang³

2013

SIAM J. Math. Anal.

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“…Forget the Lyaponov theorem! Only in one spatial dimension in an impressive result of this kind for (9.2) achieved in beautiful works [64,65], where the inverse scattering method is used. Nothing comparable is available in higher dimensions, unless severe restrictions are imposed on the periodic term [244, Ch.…”

Section: Inhomogeneous Equationsmentioning

confidence: 99%

An overview of periodic elliptic operators

Kuchment

2016

Bull. Amer. Math. Soc.

189

186

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Abstract. The article surveys the main topics, techniques, and results of the theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which significantly influence most spectral features of such operators.The approaches described are applicable not only to the standard model example of Schrödinger operator with periodic electric potential −Δ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases.Important applications are mentioned. However, due to the size restrictions, they are not dealt with in detail.

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“…Suppose the coefficients of L are T -periodic in t for some T > 0. Consider the eigenvalue problem 6) where the boundary condition u w Ω = 0 on ∂Ω × R is defined in Section 4 (it reduces to u = 0 on ∂Ω × R if Ω is sufficiently regular). The principal eigenvalue λ 1 of this problem is the eigenvalue which is real and has a positive eigenfunction.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case

Húska

2008

Trans. Amer. Math. Soc.

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Abstract. We consider the Dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients. Under such minimal regularity assumptions, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. As a special case of our main theorem, assuming the coefficients are time-periodic, we obtain a new result on the existence of a principal eigenvalue of an associated (time-periodic) parabolic eigenvalue problem. We also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time-periodic uniformly parabolic operators. Finally, we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity.

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Floquet bundles for scalar parabolic equations

Cited by 37 publications

References 16 publications

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

Floquet Bundles for Tridiagonal Competitive-Cooperative Systems and the Dynamics of Time-Recurrent Systems

An overview of periodic elliptic operators

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case

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