  Consider the nonautonomous semilinear evolution equation of type: $(\star) \;
u'(t)=A(t)u(t)+f(t,u(t)), \; t \in \mathbb{R},$ where $ A(t), \ t\in \mathbb{R}
$ is a family of closed linear operators in a Banach space $X$, the nonlinear
term $f$, acting on some real interpolation spaces, is assumed to be almost
periodic just in a weak sense (i.e. in Stepanov sense) with respect to $t$ and
Lipschitzian in bounded sets with respect to the second variable. We prove the
existence and uniqueness of almost periodic solutions in the strong sense (Bohr
sense) for equation $ (\star) $ using the exponential dichotomy approach. Then,
we establish a new composition result of Stepanov almost periodic functions by
assuming just the continuity of $f$ in the second variable. Moreover, we
provide an application to a nonautonomous system of reaction-diffusion
equations describing a Lotka-Volterra predator-prey model with diffusion and
time-dependent parameters in a generalized almost periodic environment.
