Asymptotic almost periodicity of bounded solutions for some partial functional differential equations with delay

We investigate the solution's behavior for abstract inhomogeneous partial functional differential equations (PFDEs) accordingly to an asymptotically almost periodic exogenous term. Here, the linear operator is made of a sectorial operator and a functional operator in the phase space with a domain. Firstly, when the semigroup solution is hyperbolic, we establish that bounded solutions on $ \mathbb{R}^+ $ are asymptotically almost periodic. Moreover, necessary and sufficient conditions are provided for the initial condition to yield an asymptotically almost periodic solution. Secondly, in the non-hyperbolic case, using certain ergodicity techniques, sufficient conditions are established to determine whether a bounded solution on $ \mathbb{R}^+ $ is asymptotically almost periodic. The obtained theoretical results are applied to a partial differential equation involving spatial derivatives in the delayed term.