Stepanov ergodic perturbations for nonautonomous evolution equations in Banach spaces

TL;DR

This paper establishes the existence and uniqueness of pseudo almost automorphic solutions for certain nonautonomous evolution equations in Banach spaces, extending the understanding of their long-term behavior under specific conditions.

Contribution

It introduces new conditions for solutions to exist and be unique for semilinear nonautonomous evolution equations with Stepanov pseudo almost automorphic nonlinearities.

Findings

Proved existence and uniqueness of solutions under exponential dichotomy.

Extended the class of equations with known long-term behavior.

Provided an application to reaction diffusion equations.

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 $\mathbb{R}$.