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