Almost automorphic solutions for some nonautonomous evolution equations under the light of integrable dichotomy

In this work, we prove the existence and uniqueness of \(\mu\)-pseudo almost automorphic solutions for a class of semilinear nonautonomous evolution equations of the form: \(u'(t)=A(t)u(t)+f(t,u(t)),\; t\in\mathbb{R}\) where \((A(t))_{t\in \mathbb{R}}\) is a family of closed linear operators acting in a Banach space \( X \) that generates an evolution family having an integrable dichotomy on \( \mathbb{R} \) and \( f : \mathbb{R} \times X \to X \) is \( \mu \)-pseudo almost automorphic with respect to \(t\) and Lipshitzian in the second variable. Moreover we provide an application illustrating our results.