  In this paper, using the subvariant functional method due to Favard
\cite{Favard}, we prove the existence of aunique compact almost automorphic
solution for a class of semilinear evolution equations in Banach spaces. More
specifically, we improve the assumptions in \cite{CieuEzz}, we show that the
almost automorphy of the coefficients in a weaker sense (Stepanov almost
automorphy of order $1\leq p <\infty$) is enough to obtain solutions that are
almost automorphic in a strong sense (Bochner almost automorphy). We
distinguish two cases, $ p=1 $ and $ p>1$. Moreover, we propose to study a
class of reaction-diffusion problems.
