On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory

This paper deals with existence and uniqueness of permanent (i.e. defined for all time $t\in \mathbb{R}$) solutions of non-autonomous linear evolution equations governed by strongly stable (at $-\infty $) evolution families in Banach spaces and driven by permanent bounded forcing terms. In particular, we study the existence and uniqueness of (asymptotically) almost-periodic solutions driven by (asymptotically) almost-periodic forcing terms. Systematic applications to some non-autonomous linear kinetic equations in arbitrary geometries relying on their dispersive properties are given.