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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. [ABSTRACT FROM AUTHOR]

The article focuses on a study which explores the Massera problem for the existence of a periodic mild solution of a class of non-densely non-autonomous semilinear differential equations with delay. The existence of a periodic solution is examined for non-autonomous inhomogeneous linear differential equations with delay. The use of a fixed point theorem regarding set-valued maps is mentioned.