Evolution Semigroups and Spectral Criteria for Almost Periodic Solutions of Periodic Evolution Equations

We investigate spectral criteria for the existence of (almost) periodic solutions to linear 1-periodic evolution equations of the formdx/dt=A(t) x+f(t) with (in general, unbounded)A(t) and (almost) periodicf. Using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set , then the above equation has an almost periodic (mild) solutionxfwhich is unique if one requiressp(xf) . We emphasize that our method allows us to treat the equations without assumption on the existence of Floquet representation. This improves recent results on the subject. In addition we discuss some particular cases, in which the spectrum of monodromy operator does not intersect the unit circle, and apply the obtained results to study the asymptotic behavior of solutions. Finally, an application to parabolic equations is considered.