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Almost automorphy of semilinear parabolic evolution equations. (English) Zbl 1170.34344
Summary: This paper studies the existence and uniqueness of almost automorphic mild solutions to the semilinear parabolic evolution equation $u'(t)=A(t)u(t)+f(t, u(t)),$ assuming that the linear operators $$A(\cdot)$$ satisfy the Acquistapace-Terreni conditions, the evolution family generated by $$A(\cdot)$$ has an exponential dichotomy, and the resolvent $$R(\omega,A(\cdot))$$, and $$f$$ are almost automorphic.

##### MSC:
 34G10 Linear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 35K90 Abstract parabolic equations 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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