Abstract: This paper is mainly concerned with almost automorphy for a class of finite delay differential equations $u'(t)=Au(t)+Lu_t+f(t,u_t),\ t\in \mathbb {R}$ on a Banach space $X$, where $A$ is a Hille-Yosida operator with the domain being not dense, $L$ is a bounded linear operator, and $f$ is a binary $S^p-$almost automorphic function. Compared with the previous research results, we do not require the semigroup generated by the Hille-Yosida operator to be compact, and only under weaker Lipschitz hypothesis of $f$ and $S^p-$almost automorphy hypothesis, which is weaker than almost automorphy, of $f$, the solution of the above delay differential equation is showed to be compact almost automorphic (stronger than almost automorphic). Moreover, the abstract results are applied to a class of partial differential equations arising in age-structured models.

Key words: Hille-Yosida operator, Almost automorphy, Almost periodicity, Abstract delay differential equation