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Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations. (English) Zbl 0924.34038
The authors consider mild solutions of nonhomogeneous 1-periodic evolution equations on the line. They show that for each almost-periodic inhomogeneity \(f\) there exists a unique almost-periodic mild solution if and only if the spectrum of the monodromy operator \(P\) does not intersect the unit circle. Under an additional technical condition (which holds e.g. in the parabolic case), they obtain an almost-periodic solution if \(\sigma(P)\) does not intersect the closure of \(\exp(i sp(f))\), where \(sp(f)\) is the spectrum of \(f\). The approach is based on spectral properties of the associated evolution semigroup. Applications to semilinear equations and abstract parabolic problems are given.

MSC:
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
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