Certain classes of solutions to non-autonomous evolution equations in interpolation spaces and applications

In this paper, we study the existence, uniqueness, and stability of various classes of solutions, including almost periodic, almost automorphic, Stepanov-like, and weighted Stepanov-like pseudo almost automorphic solutions for a broad class of non-autonomous evolution equations in interpolation spaces. Our analytical framework relies on two principal approaches: first, a blend of differential inequalities with interpolation functor techniques applied to the linearized version of the equations; and second, fixed-point arguments tailored to handle the nonlinear (semi-linear) case. As applications of the general theory developed, we examine its implications for the non-autonomous Oseen–Navier–Stokes equations (ONSE) in an exterior domain and the non-autonomous Ornstein–Uhlenbeck equations.