On a spectral criterion for almost periodicity of solutions of periodic evolution equations

T. Naito, University of Electro-Communications, Tokyo, Japan
N. V. Minh, University of Electro-Communications, Tokyo, Japan
J. S. Shin, Korea University, Tokyo, Japan

E. J. Qualitative Theory of Diff. Equ., No. 1. (1999), pp. 1-28.

Abstract: This paper is concerned with equations of the form: $u'=A(t)u + f(t)$, where $A(t)$ is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if $u$ is a bounded uniformly continuous mild solution and $P$ is the monodromy operator, then their spectra satisfy $e^{i sp_{AP(u)}}\subset \sigma(P)\cap S^1$, where $S^1$ is the unit circle. This result is then applied to find almost periodic solutions to the above­mentioned equations. In particular, parabolic and functional differential equations are considered. Existence conditions for almost periodic and quasi­periodic solutions are discussed.

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