This paper develops a modular framework for the study of time--dependent linear evolution processes via evolution families. We consider non--autonomous abstract Cauchy problems generated by families of operators depending on time and introduce a notion of $\varrho$--strong continuity compatible with the modular topology. Under suitable uniform $\varrho$--boundedness assumptions, we establish the existence of evolution families and derive modular growth estimates formulated in terms of the associated modular growth function. To address regularity issues, a Steklov--type averaging technique is employed, allowing differentiability and domain inclusion to be treated in the modular sense. Several examples, including time--varying multiplication processes in integral and Orlicz--type modular spaces, are presented to illustrate the scope and effectiveness of the proposed approach.
Abstract This paper develops a modular framework for the study of time--dependent linear evolution processes via evolution families. We consider non--autonomous abstract Cauchy problems [...]
Evolution Families and Time-Varying Processes in Modular Function Spaces 
