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.


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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