Almost periodicity of inhomogeneous parabolic evolution equations

We show the (asymptotic) almost periodicity of the bounded solution to the parabolic evolution equation u′(t) = A(t)u(t) + f(t) on ℝ (on ℝ₊) assuming that the linear operators A(t) satisfy the ‘Acquistapace–Terreni’ conditions, that the evolution family generated by A(·) has an exponential dichotomy, and that R(ω, A(·)) and f are (asymptotically) almost periodic.