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Stepanov-like almost automorphic functions and monotone evolution equations. (English) Zbl 1140.34399
Summary: We are concerned with a (new) class of (Stepanov-like) almost automorphic ($$S^p$$-a.a.) functions with values in a Banach space $$X$$. This class contains the space $$AA(X)$$ of all (Bochner) almost automorphic functions. We use the results obtained to prove the existence and uniqueness of a weak $$S^p$$-a.a. solution to the parabolic equation
$u^{\prime }(t)+A(t)u=f(t)$
in a reflexive Banach space, assuming some appropriate conditions of monotonicity, coercitivity of the operators $$A(t)$$ and $$S^{p^{\prime }}$$-almost automorphy of the forced term $$f(t)$$. This result extends a known result in the case of almost periodicity. An application is also given.

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