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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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