Miklavčič, Milan A sharp condition for existence of an inertial manifold. (English) Zbl 0727.34048 J. Dyn. Differ. Equations 3, No. 3, 437-456 (1991). Author’s abstract: “It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applied also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semilinear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel’s theorem for Banach valued functions are used.” Reviewer: J.Andres (Olomouc) Cited in 1 ReviewCited in 35 Documents MSC: 34D45 Attractors of solutions to ordinary differential equations 34C30 Manifolds of solutions of ODE (MSC2000) 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35L10 Second-order hyperbolic equations Keywords:exponentially attracting; linear evolution equations; perturbation; nonlinear; inertial manifolds; nonlinear hyperbolic equations; Fourier transform; Plancherel’s theorem PDFBibTeX XMLCite \textit{M. Miklavčič}, J. Dyn. Differ. Equations 3, No. 3, 437--456 (1991; Zbl 0727.34048) Full Text: DOI References: [1] Babin, A. V., and Vishik, M. I. (1986). Unstable invariant sets of semigroups of nonlinear operators and their perturbations.Russ. Math. Surv. 41, 1–41. · Zbl 0624.47065 · doi:10.1070/RM1986v041n04ABEH003375 [2] Brunovský, P., and Tereščák, I. (1991). Regularity of invariant manifolds.J. Diff. Eq. 3, 313–337. · Zbl 0739.34037 · doi:10.1007/BF01049735 [3] Chow, S.-N., and Lu, K. (1988). Invariant manifolds for flows in Banach spaces.J. Diff. Eq. 74, 285–317. · Zbl 0691.58034 · doi:10.1016/0022-0396(88)90007-1 [4] Foias, C., Sell, G. R., and Titi, E. S. (1989). Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations.J. Dynam. Diff. Eq. 1, 199–244. · Zbl 0692.35053 · doi:10.1007/BF01047831 [5] Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Am. Math. Soc., Providence. · Zbl 0642.58013 [6] Henry, D. (1981).Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, New York. · Zbl 0456.35001 [7] Mallet-Paret, J., and Sell, G. R. (1988). Inertial manifolds for reaction diffusion equations in higher space dimensions.J. Am. Math. Soc. 1, 805–866. · Zbl 0674.35049 · doi:10.1090/S0894-0347-1988-0943276-7 [8] Matano, H. (1989). Personal communication. [9] Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York. · Zbl 0662.35001 [10] Vagi, S. (1969). A remark on Plancherel’s theorem for Banach space valued functions.Ann. Scuola Norm. Sup. Pisa 23, 305–315. · Zbl 0187.08102 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.