Asymptotically almost periodic solutions of some linear evolution equations

Asymptotically almost periodic (a.a.p.) solutions of the evolution equation
$$ \frac{du}{dt}+A(t)u=f(t) $$
in certain Hilbert spaces are studied.
Under the assumption of an a.a.p. operator $A(t)$ and function $f(t)$, it is proved that the solution $u(t)$ is a.a.p. in various Hilbert spaces, i.e., the solution can be represented in the form $u(t)=v(t)+\alpha(t)$, where $v(t)$ is an almost periodic function and $\alpha(t)\to0$ as $t\to\infty$ in the corresponding space.
The first boundary value problem for a second-order parabolic equation is considered as an example.
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