Stepanov ergodic perturbations for nonautonomous evolution equations in Banach spaces

Abstract

In this work, we prove the existence and uniqueness of $\mu$-pseudo almost automorphic solutions for some 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 densely defined operators acting on a Banach space $X$ that generates a strongly continuous evolution family which has an exponential dichotomy on $\mathbb{R}$. The nonlinear term $f: \mathbb{R} \times X \longrightarrow X$ is just $\mu$-pseudo almost automorphic in Stepanov sense in $t$ and Lipshitzian with respect to the second variable. For illustration, an application is provided for a class of nonautonomous reaction diffusion equations on $\mathbb{R}$.